## Go Math Grade K Chapter 6 Answer Key Subtraction

### Subtraction Show What You Know

DIRECTIONS 1–2. Count and tell how many. Draw a set with one fewer counter. Write how many in each set. 3. Write the number of cubes in each set. Circle the number that is less than the other number.
Question 1.

Answer:

The number of counters in the first set are two and the number of counters in the second set are one.

Explanation:
In the first image, we can see that the number of counters is two. As there are two counters in the first image, given that to draw fewer counters than the first set. So we will take only one counter in the second set.

Question 2.

Answer:
The number of counters in the first set is 4 and the number of counters in the second set is 3.

Explanation:
In the first image we can see that the number of counters is 4. As there are four counters in the first image, given that to draw fewer counters than the first set. So we will take only three counters in the second image.

Compare Numbers to 10
Question 3.

Answer:
The number of cubes in first set is 7 and the number of cubes in second set is 4.

Explanation:
In the above image, we can see two types of sets. The first set contains seven cubes which are in blue color and the second set contains four cubes which are in red color. So we have to circle it for less number which is four.

### Subtraction Vocabulary Builder

DIRECTIONS Add the set of bees and the set of butterflies. Write how many insects altogether.

Answer:
The sum of the set of bees and the set of butterflies is 10.

Explanation:
The set of bees are five and the set of butterflies are five. Here we performed addition to know the insects altogether and the sum can be defined as the resulting of two or more numbers by adding. So here the sum of the set of bees and the set of butterflies is
5 + 5 is 10.

### Subtraction Game Spin for More

DIRECTIONS Play with a partner. Decide who goes first. Take turns spinning to get a number from each spinner. Use cubes to model a cube train with the number from the first spin. Say the number. Add the cubes from the second spin. Compare your number with your partner’s. Mark an X on the table for the player who has the greater number. The first player to have five Xs wins the game.

MATERIALS two paper clips, connecting cubes

### Subtraction Vocabulary Game

DIRECTIONS Players take turns. A player chooses a secret word from the Word Box and then sets the timer. The player draws pictures to give hints about the secret word. If the other player guesses the secret word before time runs out, he or she puts a counter in the chart. The first player who has counters in all his or her boxes is the winner.

MATERIALS timer, drawing paper, two-color counters for each player

The Write Way
DIRECTIONS Draw to show how to solve a subtraction problem. Write a subtraction sentence. Reflect Be ready to tell about your drawing.

### Lesson 6.1 Subtraction: Take From

Essential Question How can you show subtraction as taking from?

Share and Show

DIRECTIONS 1. Listen to the subtraction word problem. Trace the number that shows how many children in all. Write the number that shows how many children are leaving. Write the number that shows how many children are left.
Question 1.

Answer:

Explanation:
In the above image, we can see that there are four children and two children are leaving. So here we will subtraction
The number that shows how many children in all are four by observing the above image.  The number that shows how many children are leaving are two. The number that shows how many children left are two.

DIRECTIONS 2. Listen to the subtraction word problem. Write the number that shows how many children in all. Write the number that shows how many children are leaving. Write the number that shows how many children are left.
Question 2.

Answer:

Explanation:
The total number of children is five by observing the above image.  The number that shows how many children are leaving is one. The number that shows how many children left are four.

Problem Solving • Applications

DIRECTIONS 3. Blair has two marbles. His friend takes one marble from him. Draw to show the subtraction. Write the numbers. 4. Write the number that shows how many marbles Blair has now.
Question 3.

Answer:

Explanation:
Blair has two marbles by observing the above image. His friend takes one marble from him. Subtract one marble from total number of marbles. So number of marbles left over are one.

Question 4.
_________
_ _ _ _ _ _ _
_________
Answer:

Explanation:
Blair has one marble as his friend has one marble from the two marbles he has.

HOME ACTIVITY • Show your child a set of four small objects. Have him or her tell how many objects there are. Take one of the objects from the set. Have him or her tell you how many objects there are now.

### Subtraction: Take From Homework & Practice 6.1

DIRECTIONS 1. Tell a subtraction word problem about the children. Write the number that shows how many children in all. Write the number that shows how many children are leaving. Write the number that shows how many children are left.
Question 1.

Answer:

Explanation:
The number that shows how many children in all are three by observing the above image.  The number that shows how many children are leaving is one. The number that shows how many children left are two.

DIRECTIONS 1. Tell a subtraction word problem about the frogs. Write the number that shows how many frogs are left. 2. Tell an addition word problem about the birds. Write and trace to complete the addition sentence. 3. How many more counters would you place to model a way to make 8? Draw the counters.
Lesson Check
Question 1.

Answer:

Explanation:
The total number of frogs are three and one is taken away. The number that shows how many frogs left are two by observing the above image.  So leftover frogs are two.

Spiral Review
Question 2.

Answer:

Explanation:
In the above image, we can see two boxes. The first box contains five birds and the second box contains three birds. By adding birds in these two boxes results in the total number of birds. The total number of birds is eight.

Question 3.

Answer:

Explanation:
In the given image, we can see five counters are present.
In order to model a way to make counters eight, three more counters need to be added. So, we have to place three more counters to model a way to make eight counters.

### Lesson 6.2 Subtraction: Take Apart

Essential Question How can you show subtraction as taking apart?

Listen and Draw

DIRECTIONS Listen to the subtraction word problem. Place seven counters in the ten frame as shown. Trace the counters. Trace the number that shows how many in all. Trace the number that shows how many are red. Write the number that shows how many are yellow.

Answer:

Explanation:
We have to place seven counters in the ten frames as shown in the above image. The number that shows the total number of counters in all is seven. The number that shows the number of counters in red is two. The number of yellow counters is obtained by subtracting red counters from the total number of counters. Thus, we have five yellow counters.

Share and Show

DIRECTIONS 1. Listen to the subtraction word problem. Place eight counters in the ten frame. Draw and color the counters. Trace the number that shows how many in all. Write the number that shows how many are yellow. Write the number that shows how many are red.
Question 1.

Answer:

Explanation:
Place eight counters in the ten frames as shown in the above figure. The number of red counters is one. By subtracting the number of red counters from the total number of counters, we will get the number of yellow counters as seven.

DIRECTIONS 2. Listen to the subtraction word problem. Place ten counters in the ten frame. Draw and color the counters. Write the number that shows how many in all. Write the number that shows how many are red. Write the number that shows how many are yellow.
Question 2.

Answer:

Explanation:
Place ten counters in the ten frames as shown in the above figure. The total number of yellow counters is four. By subtracting the number of yellow counters from the total number of counters will give us the number of red counters as six.

Problem Solving • Applications

DIRECTIONS 3. Juanita has nine apples. One apple is red. The rest of the apples are yellow. Draw the apples. Write the numbers and trace the symbol. 4. Write the number that shows how many apples are yellow.
Question 3.

Answer:

Explanation:
Juanita has nine apples. One apple is red. If we subtract one red apple from the total number of apples, then we will get the rest of the apples which are yellow in color are 8.

Question 4.
__________
_ _ _ _ _ _ _
__________
Answer:

Explanation:
The total number of yellow apples are eight.

HOME ACTIVITY • Show your child a set of seven small objects. Now take away four objects. Have him or her tell a subtraction word problem about the objects.

### Subtraction: Take Apart Homework & Practice 6.2

DIRECTIONS 1. Listen to the subtraction word problem. Jane has nine counters. Three of her counters are red. The rest of her counters are yellow. How many are yellow? Place nine counters in the ten frame. Draw and color the counters. Write the number that shows how many in all. Write the number that shows how many are red. Write the number that shows how many are yellow.
Question 1.

Answer:

Explanation:
Jane has nine counters as shown in the above image. Three of her counters are red. The rest of her counters are yellow. We placed nine counters in the ten frames and colored the counters. The number that shows how many in all are nine. The number that shows how many are red is three. Subtract three red counters from the total number of counters to get the number of yellow counters. The number of yellow counters is six.

DIRECTIONS 1. Clyde has eight counters. Two of his counters are yellow. The rest of his counters are red. How many are red? Draw and color the counters. Write the number that shows how many are red. 2. Count the number of leaves in each set. Circle the set that has the greater number of leaves. 3. Compare the cube trains. Write how many. Circle the number that is greater.
Lesson Check
Question 1.

Answer:

Explanation:
Clyde has eight counters by observing the above image. Two of his counters are yellow. The rest of his counters are red. In order to get the number of red counters,  we have to subtract yellow counters from the total number of counters. So, the number of red counters is six.

Spiral Review
Question 2.

Answer:

Explanation:
In the above image, we can see the number of leaves in the first set is four. The number of leaves in the second set is five. So circle the set that has the greater number of leaves. The second set contains a greater number of leaves. So, circle for the second set.

Question 3.

Answer:

Explanation:
In the above image, we can see two cube trains. The total number of cubes in the first train is six and the total number of cubes in the second train is four. By Comparing these two cube trains, circle for the number that is greater. So we have to circle for the first train which has a greater number of cubes in it.

### Lesson 6.3 Problem Solving • Act Out Subtraction Problems

Essential Question How can you solve problems using the strategy act it out?

Try Another Problem

DIRECTIONS 1. Listen to and act out the subtraction word problem. Trace the numbers and the symbols. Write the number that shows how many children are left.
Question 1.

Answer:

Explanation :
Total two children are there in the above image. One child left the place and another child remains there in the same place.  Subtract total number of child left from total number of child. So, only one child leftover at that place.

Share and Show

DIRECTIONS 2. Listen to and act out the subtraction word problem. Trace the numbers and the symbols. Write the number that shows how many children are left.
Question 2.

Answer:

Explanation:
The total number of children in the above image is four. Two children left the place and two children remain in the same place. So, the number of children leftover is two.

On Your Own

DIRECTIONS 3. Tell a subtraction word problem about the kittens. Trace the numbers and the symbols. Write the number that shows how many kittens are left. 4. Draw to show what you know about the subtraction sentence. Write how many are left. Tell a friend a subtraction word problem to match.
Question 3.

Answer:

Explanation:
In the above image, we can see four kittens. One kitten left the place and some kittens remain in the same place. Subtract number of kitten left from the total number of kittens .So, the number of kittens leftover is three.

Question 4.

Answer:

Explanation:
The total number of flowers in the above image is four. Three flowers are taken away from the place and one flower remains in the same place. In order to get the left over flowers we have subtract total number of flowers taken away from the total number of flowers. So the number of flowers leftover is one.

HOME ACTIVITY • Tell your child a short subtraction word problem. Have him or her use objects to act out the word problem.

### Problem Solving • Act Out Subtraction Problems Homework & Practice 6.3

DIRECTIONS 1. Tell a subtraction word problem about the beavers. Trace the numbers and the symbols. Write the number that shows how many beavers are left. 2. Draw to tell a story about the subtraction sentence. Write how many are left. Tell a friend about your drawing.
Question 1.

Answer:

Explanation:
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. In the above image we can see the total number of beavers are three. One beaver left the place and two beavers remains in the same place. So, the number of beavers leftover are two.

Question 2.

Answer:

Explanation:
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. The result of a subtraction is called a difference. In the above image we can see four penguins. Three penguins left the place and one penguin remains in the same place.

DIRECTIONS 1. Tell a subtraction word problem about the birds. Trace the numbers and the symbols. Write the number that shows how many birds are left. 2. Count and tell how many bees. Write the number. 3. How many more counters would you place to model a way to make 7? Draw the counters.
Lesson Check
Question 1.

Answer:

Explanation:
The process of taking one number or amount away from another number is called as subtraction. In the above image we can observe the total number of birds are five. Four birds left the place and one bird remains in the same place.

Spiral Review
Question 2.

Answer:

Explanation:
In the above image, we can see a group of bees. The addition is taking two or more numbers and adding them together. The total number of bees are nine.

Question 3.

Answer:

Explanation:
Four more counters are needed to place the model in a way to make seven. The addition is taking two or more numbers and adding them together. In the above image we can observe Four counters are added.

### Lesson 6.4 Algebra • Model and Draw Subtraction Problems

Essential Question How can you use objects and drawings to solve subtraction word problems?

Share and Show

DIRECTIONS 1. Model a four-cube train. One cube is blue and the rest are green. Take apart the train to show how many cubes are green. Draw and color the cube trains. Trace and write to complete the subtraction sentence. 2. Model a three-cube train. Two cubes are orange and the rest are blue. Take apart the train to show how many cubes are blue. Draw and color the cube trains. Trace and write to complete the
Question 1.

Answer:

Explanation:
In the above image we can see a four-cube train. One cube is blue and the rest are green. The number of green cubes are three. Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.  Here the collections has four cubes and we are removing one object(blue cube). So the remaining green cubes are three.

Question 2.

Answer:

Explanation:
In the above image, we can see a three-cube train. Two cubes are orange and the rest are blue.  In order to get the number of blue cubes we have to subtract two orange cubes from total number of cubes. So we got the total number of blue cubes are one.

DIRECTIONS 3. Model a four-cube train. Three cubes are red and the rest are blue. Take apart the train to show how many cubes are blue. Draw and color the cube trains. Trace and write to complete the subtraction sentence. 4. Model a five-cube train. Three cubes are yellow and the rest are green. Take apart the train to show how many cubes are green. Draw and color the cube trains. Trace and write to complete the subtraction sentence.
Question 3.

Answer:

Explanation:
In the above image, we can see a four-cube train. Three cubes are red and the rest are blue. In order to get the number of blue cubes we have to subtract three red cubes from total number of cubes. So we got the total number of blue cubes are one.

Question 4.

Answer:

Explanation:
In the above image, we can see a five-cube train. Three cubes are yellow and the rest are green. In order to get the number of green cubes we have to subtract three yellow cubes from total number of cubes. So we got the total number of green cubes are two.

HOME ACTIVITY • Show your child two small objects. Take apart the set of objects. Have him or her tell a word problem to match the subtraction.

### Algebra • Model and Draw Subtraction Problems Homework & Practice 6.4

DIRECTIONS 1. Model a three-cube train. Two cubes are red and the rest are blue. Take apart the cube train to show how many cubes are blue. Draw and color the cube trains. Trace and write to complete the subtraction sentence. 2. Model a five-cube train. One cube is yellow and the rest are green. Take apart the train to show how many cubes are green. Draw and color the cube trains. Trace and write to complete the subtraction sentence.
Question 1.

Answer:

Explanation:
In the above image, we can observe a three-cube train. Two cubes are red and the rest are blue. In order to get the number of blue cubes we have to subtract two red cubes from total number of cubes. So we got the total number of blue cubes are one.

Question 2.

Answer:

Explanation:
In the above image, we can observe a five-cube train. One cube is yellow and the rest are green. In order to get the number of green cubes we have to subtract one yellow cube from total number of cubes. So we got the total number of green cubes are four.

DIRECTIONS 1. Ellie makes the cube train shown. She takes the cube train apart to show how many cubes are gray. Trace and write to show the subtraction sentence for Ellie’s cube train. 2. Count the dots in the ten frames. Begin with 5. Write the numbers in order as you count forward. 3. Complete the addition sentence to show the numbers that match the cube train.
Lesson Check
Question 1.

Answer:

Explanation:
There are total five cubes in the train shown. In those some are colored gray. We need to separate total cube train into two parts to know how many cubes are gray. Substract the number of non-gray colored cubes from the total number of cubes to get the number of gray colored cubes. So, if we substract three from total five cubes, then the total number of gray colored cubes are two.

• Spiral Review
Question 2.

Answer:

Explanation:
• In the first frame, we can see that there are total ten number of counters and five are not colored grey. If we substract the number of non-colored counters from total number of counters, then we will get the total number of grey colored counters. So, substracting five from the total ten counters gives us five grey colored counters.
• Similar to the first frame, if we see in the second frame, there are total ten number of counters and four are not colored grey. If we substract the number of non-colored counters from total number of counters, then we will get the total number of grey colored counters. So, substracting four from the total ten counters gives us six grey colored counters.
• For the third frame, the same logic will apply here as well an substracting three from ten gives us seven grey colored counters.
• For the fourth frame, substracting two from the total ten counters will give us eight grey colored counters.
• For the fifth frame, substracting one from the total ten counters will give us nine grey colored counters.

Question 3.

Answer:

Explanation:
In the above image we can see five grey cubes and three white cubes .Perform addition to get total number of cubes. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD five black cubes with three white cubes then we get total number of cubes. The total number of cubes are eight.

### Subtraction Mid-Chapter Checkpoint

Concepts and Skills

DIRECTIONS 1. Listen to the subtraction word problem. Draw and color the six circles in the ten frame. Write the number that shows how many in all. Write the number that shows how many are yellow. (K.OA.A.1) 2. Model a five-cube train. Four cubes are blue and the rest are orange. Take apart the cube train to show how many are orange. Draw and color the cube trains. Trace and write to complete the subtraction sentence. (K.OA.A.5) 3. Choose Yes or No. Does the subtraction sentence match the model? (K.OA.A.5)
Question 1.

Answer:

Explanation:

Question 2.

Answer:

Explanation:
In the above image, we can observe a five-cube train. In that four cubes are blue and the rest are orange. The process of taking one number or amount away from another number is called as subtraction. We have subtract total number of blue cubes from a five cube train .Then we get total number of orange cubes. The total number of orange cube is one.
Question 3.
THINK SMARTER+

Answer:

Explanation:
In the above image, we can see a four cube train. In that, one is red cube and three are blue cubes. We have to choose the correct answer for number of red cubes by subtracting the number of blue cubes from total number of cubes. If the answer matches with the number of red cubes, then the answer is ‘Yes’ or else ‘No’.

• In the first number sentence, subtracting two blue cubes from the total four cubes will not give the correct number of red cubes. The answer doesn’t match with the number of red cubes and so the answer is ‘No’.
• In the second number sentence, substracting three blue cubes from the total four cubes equals to one red cube. So, the answer is ‘Yes’.
• In the third number sentence, substracting one blue cube from three cubes will not give correct number of red cubes. So the answer is ‘No’.

### Lesson 6.5 Algebra • Write Subtraction Sentences

Essential Question How can you solve subtraction word problems and complete the equation?

Share and Show

DIRECTIONS 1. Listen to the subtraction word problem. Trace the circle and X to show how many are being taken from the set. Trace to complete the subtraction sentence. 2–3. Listen to the subtraction word problem. How many are being taken from the set? Circle and mark an X to show how many are being taken from the set. Trace and write to complete the subtraction sentence.
Question 1.

Answer:
Explanation:
In the above image we can see five scorpions. In that two scorpions are taken away from the set. Circle the scorpions that are taken away from the set and marked with X. In order to get the left over scorpions we have to subtract two scorpions from four scorpions. Then we will get the number of scorpions left over. The left over scorpion is one.

Question 2.

Answer:

Explanation:
In the above image we can see four tortoises. In that one tortoise is taken away from the set. Circle the tortoises that are taken away from the set and marked with X. In order to get the left over tortoises we have to subtract one tortoise from four tortoises. Then we will get the number of tortoises left over. The left over tortoises are three.

Question 3.

Answer:

Explanation:
In the above image we can see four scorpions. In that three scorpions are taken away from the set. Circle the scorpions that are taken away from the set and marked with X. In order to get the left over scorpions we have to subtract three scorpions from four scorpions. Then we will get the number of scorpions left over. The left over scorpion is one.

DIRECTIONS 4–6. Listen to the subtraction word problem. How many are being taken from the set? Circle and mark an X to show how many are being taken from the set. Trace and write to complete the subtraction sentence.
Question 4.

Answer:

Explanation:
In the above image we can see five fish. In that three fish are taken away from the set. Circle the fish that are taken away from the set and marked with X. In order to get the left over fish we have to subtract three fish from five fish that we have, and we will get the number of fish left over. The left over fish are two as shown in the above image.
Question 5.

Answer:

Explanation:
In the above image we can see three fish. In that two fish are taken away from the set. Circle the fish that are taken away from the set  and marked with X. In order to get the left over fish we have to subtract two fish from three fish that we have, and we will get the number of fish left over. The left over fish is one.

Question 6.

Answer:

Explanation:
In the above image we can see five fish. In that four fish are taken away from the set.  Circle the fish that are taken away from the set and marked with X. In order to get the left over fish we have to subtract four fish from five fish that we have, and we will get the number of fish left over. The left over fish are one as shown in the above image.

Problem Solving • Applications

DIRECTIONS 7. Kristen has four flowers. She gives her friend some flowers. Now Kristen has two flowers. How many did Kristen give her friend? Draw to solve the problem. Complete the subtraction sentence. 8. Tell a different subtraction word problem about the flowers. Draw to solve the problem. Tell a friend about your drawing. Complete the subtraction sentence.
Question 7.

Answer:

Explanation:
Kristen has four flowers as shown in the above image. She gave some flowers to her friend. Subtract two flowers from total number of flowers which are four. Now Kristen has two flowers. Kristen gave two flowers to her friend.
Question 8.

Answer:

Explanation:
The total number of flowers in the above image are four. Three flowers are taken away from the place and one flower remains in the same place. In order to get the left over flowers we have subtract total number of flowers taken away from the total number of flowers. So the number of flowers leftover is one.

HOME ACTIVITY • Have your child draw a set of five or fewer balloons. Have him or her circle and mark an X on some balloons to show that they have popped. Then have your child tell a word problem to match the subtraction.

### Algebra • Write Subtraction Sentences Homework & Practice 6.5

DIRECTIONS 1–3. Listen to the subtraction word problem about the animals. There are _____ _____. _____ are taken from the set. Now there are ____. How many are taken from the set? Circle and mark an X to show how many are being taken from the set. Trace and write to complete the subtraction sentence.
Question 1.

Answer:

Explanation:
In the above image we can see four animals. In that three animals are taken away from the set. Circle the animals that are taken away from the set and marked with X. In order to get the left over animals we have to subtract three animals from total number of animals. Then we will get the number of animals left over. The left over animal is one.

Question 2.

Answer:

Explanation:
In the above image we can see three animals. In that one animal is taken away from the set. Circle the animals that are taken away from the set and marked with X. In order to get the left over animals we have to subtract one animal from total number of animals. Then we will get the number of animals left over. The left over animals are two.
Question 3.

Answer:

Explanation:
In the above image we can see five animals. In that four animals are taken away from the set. Circle the animals that are taken away from the set and marked with X. In order to get the left over animals we have to subtract four animals from total number of animals. Then we will get the number of animals left over. The left over animal is one.

DIRECTIONS 1. Trace and write to show the subtraction sentence for the set. 2. Count the number of counters in each set. Circle the set that has the greater number of counters. 3. How many more counters would you place to model a way to make 9? Draw the counters.
Lesson Check
Question 1.

Answer:

Explanation:
In the above image we can observe some tortoise. Three tortoise are taken away from the set .The left over tortoise is one. Total number of tortoise are four. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of tortoise taken away from the set from total number of tortoise. Then we get left over tortoise. So subtract three tortoise from four tortoise. The total number of tortoise left over is one.

Spiral Review
Question 2.

Answer:

Explanation:
In the above image, we can see the number of counters in the first set are three. The number of counters in the second set are four. So circle the set that has the greater number of counters. The second set contains a greater number of counters. So, circle for the second set.
Question 3.

Answer:

Explanation:
Three more counters are needed to place the model in a way to make nine. The addition is taking two or more numbers and adding them together. In the above image we can observe three counters are added.

### Lesson 6.6 Algebra • Write More Subtraction Sentences

Essential Question How can you solve subtraction word problems and complete the equation?

Share and Show

DIRECTIONS 1–3. Listen to the subtraction word problem. How many birds are taken from the set? Trace the circle and X. How many birds are left? How many birds were there in all to begin with? Write and trace to complete the subtraction sentence.
Question 1.

Answer:
Subtract six from eight in order to get the difference. The difference is two.
Explanation:
In the above image we can observe some birds. Six birds are taken away from the set .The left over birds are two. Total number of birds are eight. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract six birds from eight birds. The total number of birds left over are two.
Question 2.

Answer:

Explanation:
In the above image we can observe some birds. Three birds are taken away from the set .The left over birds are three. Total number of birds are six. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract three birds from six birds. The total number of birds left over are three.
Question 3.

Answer:

Explanation:
In the above image we can observe some birds. Four birds are taken away from the set .The left over birds are five. Total number of birds are nine. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract four birds from nine birds. The total number of birds left over are five.

DIRECTIONS 4–6. Listen to the subtraction word problem. How many birds are taken from the set? Trace the circle and X. How many birds are left? How many birds were there in all to begin with? Write and trace to complete the subtraction sentence.
Question 4.

Answer:

Explanation:
In the above image we can observe some birds. Five birds are taken away from the set .The left over bird is one. Total number of birds are six. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract five birds from six birds. The total number of birds left over are five.

Question 5.

Answer:

Explanation:
In the above image we can observe some birds. six birds are taken away from the set .Now there are three birds. Total number of birds are nine. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract six birds from nine birds. The total number of birds left over are three.

Question 6.

Answer:

Explanation:
In the above image we can observe some birds. Three birds are taken away from the set .Now there are five birds. Total number of birds are eight. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract three birds from eight birds. The total number of birds left over are five.

Problem Solving • Applications

DIRECTIONS 7. Complete the subtraction sentence. Draw a picture of real objects to show what you know about this subtraction sentence. Tell a friend about your drawing.
Question 7.

Answer:

Explanation:
In the above image we can see eight oranges. In that six oranges are taken away from the set. Circle the animals that are taken away from the set and marked with X. In order to get the left over oranges we have to subtract six oranges from total number of oranges. Then we will get the number of oranges left over. The left over oranges are two.

HOME ACTIVITY • Tell your child you have some small objects in your hand. Tell him or her that you are taking two objects from the set and now there are five objects left. Ask him or her to tell you how many objects were in the set to start with.

### Algebra • Write More Subtraction Sentences Homework & Practice 6.6

DIRECTIONS 1–3. Listen to a subtraction word problem about the birds. There are some birds. _____ birds are taken from the set. Now there are _____ birds. How many birds in all did you start with? Write the number to complete the subtraction sentence.
Question 1.

Answer:

Explanation:
In the above image we can observe some birds. Four birds are taken from the set .Now there are three birds. Total number of birds are seven. We have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. The process of taking one number or amount away from another number is called as subtraction. So subtract four birds from seven birds. The total number of birds left over are three.

Question 2.

Answer:

Explanation:
In the above image we can observe some birds. Three birds are taken away from the set .Now there are six birds. Total number of birds are nine. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract three birds from nine birds. The total number of birds left over are six.
Question 3.

Answer:

Explanation:
In the above image we can observe some birds. One bird is taken away from the set .Now there are five birds. Total number of birds are six. The process of taking one number or amount away from another number is called as subtraction. So we have to subtract number of birds taken away from the set from total number of birds. Then we get left over birds. So subtract one bird from six birds. The total number of birds left over are five.

DIRECTIONS 1. Trace and write to show the subtraction sentence for the buses. 2. How many lunch boxes are there? Write the number. 3. Count the dots in the ten frames. Begin with 1. Write the numbers in order as you count forward.
Lesson Check
Question 1.

Answer:

Explanation:
In the above image we can see the total number of buses are seven. In that four buses are taken away .We have to find how many buses left over there.  Number of buses taken away are four and the total number of buses are six. So we have to subtract total number of buses taken away from total number of buses . Subtract four buses from six buses.  So we got the left over buses. The left over buses are two.

Spiral Review
Question 2.

Answer:

Explanation:
In the above image we can observe some lunch boxes. We have to perform addition operation to calculate the total number of lunch boxes. An addition sentence is a mathematical expression that shows two or more values added together and their sum. Total number of lunch boxes are seven

Question 3.

Answer:

Explanation:

• In the first frame, we can see that there are total ten number of counters and nine are not colored grey. If we substract the number of non-colored counters from total number of counters, then we will get the total number of grey colored counters. So, substracting nine from the total ten counters gives us one grey colored counters.
• Similar to the first frame, if we see in the second frame, there are total ten number of counters and eight are not colored grey. If we substract the number of non-colored counters from total number of counters, then we will get the total number of grey colored counters. So, substracting eight from the total ten counters gives us two grey colored counters.
• For the third frame, the same logic will apply here as well an substracting seven from ten gives us three grey colored counters.
• For the fourth frame, substracting four from the total ten counters will give us four grey colored counters.
• For the fifth frame, substracting five from the total ten counters will give us five grey colored counters.

### Lesson 6.7 Algebra • Addition and Subtraction

Essential Question How can you solve word problems using addition and subtraction?

Share and Show

DIRECTIONS Tell addition and subtraction word problems. Use cubes to add and to subtract. 1. Trace the number sentences. 2. Complete the number sentences.
Question 1.

Answer:
The first cube train contains five red cubes and two blue cubes. ADD five red cubes with two blue cubes .The total number of cubes are seven.
Total number of cubes in the second cube train are seven. Subtract two blue cubes from total number of cubes are seven. Total number of red cubes are five.
Explanation:
In the first image we can see five red cubes and two blue cubes. We have to find total number of cubes. In order to get the total number of cubes we have to perform addition operation. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD five red cubes with two blue cubes then we get total number of cubes. So the total number of cubes are seven.
In the second image we can see the total number of cubes are seven. In that two blue cubes are taken away .We have to find how many red cubes are left over there.  Number of cubes taken away are two and the total number of cubes are seven. So we have to subtract total number of  blue cubes taken away from total number of cubes . Subtract two blue cubes from seven cubes.  So we got the left over red cubes. The left over red cubes are five.

Question 2.

Answer:

Explanation:
In the first image we can see six red cubes and four blue cubes. We have to find total number of cubes. In order to get the total number of cubes we have to perform addition operation. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD six red cubes with four blue cubes then we got total number of cubes. So the total number of cubes are ten.
In the second image we can see the total number of cubes are ten. In that four blue cubes are taken away .We have to find how many red cubes are left over there.  Number of cubes taken away are four and the total number of cubes are ten. So we have to subtract total number of  blue cubes taken away from total number of cubes . Subtract four blue cubes from ten cubes.  So we got the left over red cubes. The left over red cubes are six.

DIRECTIONS 3–4. Tell addition and subtraction word problems. Use cubes to add and subtract. Complete the number sentences.
Question 3.

Answer:

Explanation:
In the first image we can see six red cubes and two blue cubes. We have to find total number of cubes. In order to get the total number of cubes we have to perform addition operation. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD six red cubes with two blue cubes then we get total number of cubes. So the total number of cubes are eight.
In the second image we can see the total number of cubes are eight. In that two blue cubes are taken away .We have to find how many red cubes are left over there.  Number of cubes taken away are two and the total number of cubes are eight. So we have to subtract total number of  blue cubes taken away from total number of cubes . Subtract two blue cubes from eight cubes.  So we got the left over red cubes. The left over red cubes are six.

Question 4.

Answer:

Explanation:
In the first image we can see eight red cubes and one blue cube. We have to find total number of cubes. In order to get the total number of cubes we have to perform addition operation. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD eight red cubes with one blue cube then we get total number of cubes. So the total number of cubes are nine.
In the second image we can see the total number of cubes are nine . In that one blue cube is taken away .We have to find how many red cubes are left over there.  Number of cubes taken away are one and total number of cubes are nine .So we have to subtract total number of  blue cubes taken away from total number of cubes . Subtract one blue cube from nine cubes. The left over red cubes are eight .

Problem Solving • Applications

DIRECTIONS Look at the addition sentence at the top of the page. 5–6. Tell a related subtraction word problem. Complete the subtraction sentence.
Question 5.

Answer:

Explanation:
The process of taking one number or amount away from another number is called as subtraction. The result of a subtraction is called a difference. Subtraction is signified by the minus sign. For example, there are nine birds on a tree. If three birds flew away, then the number of birds left on the tree will be calculated by substracting the number of birds flew away from the total number of birds on the tree. If we substract three from six, then we get the answer as three.

Question 6.

Answer:

Explanation:
The process of taking one number or amount away from another number is called as subtraction. The result of a subtraction is called a difference. Subtraction is signified by the minus sign. For example, there are nine birds on a tree. If three birds flew away, then the number of birds left on the tree will be calculated by substracting the number of birds flew away from the total number of birds on the tree. If we substract three from nine, then we get the answer as six.

HOME ACTIVITY • Ask your child to use objects to model a simple addition problem. Then have him or her explain how to make it into a subtraction problem.

### Algebra • Addition and Subtraction Homework & Practice 6.7

DIRECTIONS 1–2. Tell an addition or subtraction word problem. Use cubes to add or subtract. Complete the number sentence.
Question 1.

Answer:

Explanation:
In the above image we can see five grey cubes and three white cubes .Perform addition to get total number of cubes. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD five black cubes with three white cubes then we get total number of cubes. The total number of cubes are eight.
Question 2.

Answer:

Explanation:
In the above image we can see the total number of cubes are eight .Three cubes are taken away .We have to find how many cubes are left over there.  Number of cubes taken away are three and total number of cubes are eight .So we have to substract total number of cubes taken away from total number of cubes . The left over cubes are five .

DIRECTIONS 1. Tell a subtraction word problem. Use cubes to subtract. Complete the number sentence. 2. Complete the addition sentence to show the numbers that match the cube train. 3. Compare the numbers. Circle the number that is greater.
Lesson Check
Question 1.

Answer:

Explanation:
In the above image we can see the total number of cubes are eight .Three cubes are taken away .We have to find how many cubes are left over there.  Number of cubes taken away are three and total number of cubes are eight .So we have to substract total number of cubes taken away from total number of cubes . The left over cubes are five .

Spiral Review
Question 2.

Answer:

Explanation:
In the above image the total number of cubes are ten. In that image we can see four black cubes and six white cubes .Perform addition to get total number of cubes. An addition sentence is a mathematical expression that shows two or more values added together and their sum. ADD four black cubes with six white cubes then we get total number of cubes are ten.

Question 3.

Answer:

Explanation:
In the above image we can see two numbers. First one is eight and second one is nine. By comparing these two numbers nine is greater than eight .So circle it for nine.

### Subtraction Review/Test

DIRECTIONS 1. Write how many owls are flying away. Write how many owls are left. 2. Which answers show how many counters are red? Choose Yes or No. 3. Model a five-cube train. Two cubes are yellow and the rest are blue. Take apart the cube train to show how many are blue. Draw the cube trains. Trace and write to complete the subtraction sentence.
Question 1.

Answer:

Explanation:
In the above image total number of owls on the tree are four. One owl is flying away. The number of owls left on the tree are calculated by the subtraction method. Subtract number of owls flying away from the total number of owls present on the tree which means subtract one owl from four owls. Owls left on a tree are three.

Question 2.

Answer:

Explanation:
In the above image, we can see there are total nine counters. In that, four are red counters and five are yellow counters. We have to choose the correct answer for number of red counters by subtracting the number of yellow counters from total umber of counters. If the answer matches with the number of red counters, then the answer is ‘Yes’ or else ‘No’.

• In the first number sentence, subtracting one yellow counter from the total nine counters will not give the correct number of red counters. The answer doesn’t match with the number of red counters and so the answer is ‘No’.
• In the second number sentence, substracting five yellow counters from the total nine counters equals four red counters. So, the answer is ‘Yes’.
• In the third number sentence, substracting three yellow counters from eight counters will not give correct number of red counters. So the answer is ‘No’.

Question 3.
THINK SMARTER+

Answer:

Explanation:
In the above image we can see a five-cube train. Two cubes are yellow and the rest are blue. In order to find how many cubes are blue, we have to substract the yellow cubes from total number of cubes.

• Total number of cubes are five.
• Number of Yellow color cubes are two.
• Substracting two yellow cubes from the total five cubes will give us the total number of blue cubes which are three.

DIRECTIONS 4. There are 4 penguins. Two penguins are taken from the set. How many penguins are left? Trace and write to complete the subtraction sentence. 5. There are some birds. Three birds are taken from the set. How many birds are left? How many birds were there in all to start? Write and trace to complete the subtraction sentence. 6. Does the number sentence match the picture? Circle Yes or No. 7. Mark under all the number sentences that match the cubes.
Question 4.

Answer:

Explanation:
In the above image there are four penguins. Two penguins are taken from the set. Substract two penguins from the total four penguins and so we get left over penguins. The left over penguins are two.

Question 5.

Answer:

Explanation:
In the above image, the total number of birds are seven. In that, three birds are taken from the set. So, substract three birds which are taken away from the set from the total seven birds  and we will get the number of birds left over in the set and they are four.

Question 6.

Answer:

Explanation:
In the above picture, we can see a five cube train. In that, three are red cubes and two are blue cubes . The picture says that all the blue cubes are taken apart from the train and only the red cubes are remaining there.

• The first number sentence doesn’t match with the five cube train. Because in the above picture two blue cubes are taken away , So substracting four from five is not relevant here. So answer is ‘No’ for first number sentence.
• In the second sentence we can observe addition process takes place , which is irrelevant to the cube train as we are taking apart the blue cubes from the train actually. .So answer is ‘No’ for second number sentence also.
• In third number sentence, we can see total number of cubes are 5 and taken away blue cubes are two by substracting these we can get three red cubes . So answer is ‘Yes’ for third sentence.

Question 7.

Answer:

Explanation:
In the above image we can see a ten cube train and three number sentences. In that image we have to match the cube train with number sentence. A ten cube train contains three red cubes and seven blue cubes. Add three red cubes with seven blue cubes to get ten cube train which matches with the third number sentence. Mark for third number sentence.

DIRECTIONS 8. Model a four-cube train. Three cubes are red and the rest are blue. Take apart the train to show how many cubes are blue. Draw the cube trains. Complete the subtraction sentence. 9–10. Complete the subtraction sentence to match the picture.
Question 8.

Answer:

Explanation:
In the above image we can see a four-cube train. In that three cubes are red and rest are blue. In order to get how many cubes are blue ,subtract three red cubes from the total four cubes so as to get one blue cube .
Question 9.

Answer:

Explanation:
In the above image we can see an eight cube train . In that, one cube is blue and remaining are red. Blue cube is taken apart from the train, so substract one blue cube from the total eight cube train and then we will get the number of red cubes present in the train .The number of red cubes in the above image are seven.

Question 10.

Answer:

Explanation:
In the above image we can see four fish are there. In that two fish are taken away which represents to subtract two fish from four fish that we have and we will get the number of fish left over. The left over fish are two in the image.

DIRECTIONS 11. There were some apples on a tree. Some were taken away. Now there are zero apples left. Draw to show how many apples there could have been to start. Cross out apples to show how many were taken away. Complete the subtraction sentence. 12. There are some birds. Two birds are taken from the set. How many birds are left? How many birds were there in all to begin with? Write the number to complete the subtraction sentence. 13. Erica has 6 balloons. She gives some of her balloons to a friend. Now Erica has 4 balloons. How many did Erica give to her friend? Draw to solve the problem. Complete the subtraction sentence.
Question 11.
THINK SMARTER+

Answer:

Explanation:
In the first image, we can see there are five apples on a tree and in the second image five apples are taken away. Now the apples on the tree are zero as we have taken away all the five apples.

Question 12.

Answer:

Explanation:
In the above image we can see five birds are there in total and two birds were taken away from the set. So, substract two birds from the set of five birds and then the remaining birds are three.

Question 13.

Answer:

Explanation:
In the above image we can see Erica has six balloons and She gave two of her balloons to her friend.  Subtract two balloons from total number of balloons are two. Now Erica has four balloons with her.

## Go Math Grade 1 Chapter 12 Answer Key Two-Dimensional Geometry

Go Math Grade 1 Chapter 12 Answer Key Two-Dimensional Geometry is here to provide you the help to have your kids a fun way of learning. It helps children to improve their practical skills by recognizing the real life geometrical shapes and here we concentrated on the two dimensional shapes with colorful methods and answers will be an extra point to the children to gain the knowledge easily with playful learning.

## Go Math Grade 1 Chapter 12 Answer Key Two-Dimensional Geometry

Now parents have an easy way to teach their kids about shapes. Go Math Grade 1 Chapter 12 Answer Key Two-Dimensional Geometry has all the answers for the kids who are learning the concepts of two dimensional shapes. In this chapter each and every question is answered in a colorful and fun way to help your kids to have a happy learning. The difficulty level of this chapter is very easy as other kids also seem to be enjoying to learn together . This chapter highlights the concepts of shapes , sizes, equal shares, by visualizing them practically.

Two-Dimensional Geometry Show What You Know

Two-Dimensional Geometry Vocabulary Builder

Two-Dimensional Geometry Game Rocket Shapes

Two-Dimensional Geometry Vocabulary Game

Lesson 1 Sort Two-Dimensional Shapes

Lesson 2 Describe Two-Dimensional Shapes

Lesson 3 Combine Two-Dimensional Shapes

Lesson 4 Combine More Shapes

Lesson 5 Problem Solving • Make New Two-Dimensional Shapes

Lesson 6 Find Shapes in Shapes

Lesson 7 Take Apart Two-Dimensional Shapes

Lesson 8 Equal or Unequal Parts

Lesson 9 Halves

Lesson 10 Fourths

Two-Dimensional Geometry Review/Test

### Two-Dimensional Geometry Show What You Know

Sort by Shape
Circle the shape that belongs in each group.
Question 1.

Answer:

Explanation:
Given, in the question above,
All the figures in the picture are triangles,
The odd one in the second picture is triangle
So, triangle shape is circled.

Question 2.

Answer:

Explanation:
Given, in the question above,
All the figures in the picture are rectangle ,
The odd one in the second picture is rectangle,
So, rectangle shape is circled.

Sort Shapes
Circle the shapes with 4 sides.
Question 3.

Answer:

Explanation:
The shapes having four sides are square and rectangle,
and shapes having three sides are triangles.

Identify Two-Dimensional Shapes
Color each square blue. Color each rectangle yellow.
Color each circle red.
Question 4.

Answer:

Explanation:
There are totally 8 circles in the picture,
All of them are colored in red.

### Two-Dimensional Geometry Vocabulary Builder

Visualize It
Complete the chart.
Mark each row with a ✓.

Understand Vocabulary
Write the number of each shape.

Question 1.
_____ circles
Answer: 2

Explanation:
There are 9 shapes in the picture,
And there are 2 circles in the picture.

Question 2.
_____ squares
Answer: 4

Explanation:
There are 9 shapes in the picture,
And there are 4 squares in the picture.

Question 3.
_____ triangles
Answer: 3

Explanation:
There are 9 shapes in the picture,
And there are 3 triangles in the picture.

### Two-Dimensional Geometry Game Rocket Shapes

Materials

Play with a partner.
Take turns.

1. Spin the .
2. Name the shape you spin.
3. Place that shape on the rocket if you can.
4. If you cannot place the shape, your turn is over.
5. The first player to cover a whole rocket wins.

Answer:

Explanation:
who ever wins the points between the player1 and player2,
The picture with the colors is the output.

### Two-Dimensional Geometry Vocabulary Game

Guess the Word

Materials
timer
How to Play
Play with a partner.

1. Choose a math word from the Word Box. Do not tell your partner.
2. Set the timer.
3. Give a one-word clue.
4. Your partner tries to guess the secret word.
5. Repeat with a new one-word clue until your partner guesses correctly or time runs out.
6. Take turns.
7. The first player to correctly guess 5 words wins.

Answer:  Circle, Triangle, Rectangle, Square, Trapezoid.

Explanation:
From the picture we have the 5 words,
That is Circle, Triangle, Rectangle, Square, Trapezoid.
circles and trapezoid are in the cycle  from the picture,
Rectangle is from fencing of the play ground in the picture,
Square is from the sandbox in the picture,
Triangle is from the support of the swings in the frame.

The Write Way
Reflect
Choose one idea. Draw and write about it.

• Tell about two of your favorite shapes.
• Explain how you can combine shapes to make a new shape.

Answer: we can use circle and triangle to form a Cone.

Explanation:
From the shapes Circle and Triangle, combined together,
Thus forming a Cone.

### Lesson 12.1 Sort Two-Dimensional Shapes

Essential Question How can you use attributes to classify and sort two-dimensional shapes?

Listen and Draw

Draw to sort the shapes.
Write the sorting rule.

Answer: Sorting rule is defined as the method to identify the patterns by sort of size, color and shapes for two-dimensional objects and also helpful to recognizing the shapes which are similar and different.

The different ways to sort out the two-dimensional shapes are given below

Explanation:
shapes with 3 vertices are:

shapes with 4 vertices are:

shapes with open figures are:

MATHEMATICAL PRACTICES
Explain Are there shapes that did not go in your groups?
Answer: No, There are no shapes that did not go in your group , because here we are dealing with two dimensional shapes and we have triangle, square, circle, rectangle and so on. from the above questions so all  of them are grouped.

Model and Draw

Here are some ways to sort two-dimensional shapes.

Answer: By sorting rule we have , circles, triangles, rectangles

Explanation:
From the question, we know that
curved and closed shapes here are circles,
closed shapes with 3 sides here are triangles,
closed shapes with 4 sides here are rectangles.

Share and Show

Read the sorting rule. Circle the shapes that follow the rule.
Question 1.
4 vertices (corners)

Answer: There are 3 figures following the given sorting rule.

Explanation:

Here we have rectangle and 2 squares.

Question 2.
not curved

Answer: There are 4 figures following the given sorting rule.

Explanation:

Here we have shapes with out curved end are triangle, square and rectangle.

Question 3.
only 3 sides

Answer: There are 2 figures following the given sorting rule.

Explanation:

Here we have shapes with 3 sides are triangle

Question 4.
more than 3 sides

Answer: There are 3 figures following the given sorting rule.

Explanation:

Here we have shapes with more than 3 sides are 2 squares and  a rectangle.

On Your Own

MATHEMATICAL PRACTICE Use Math Vocabulary
Circle the shapes that follow the rule.
Question 5.
curved

Answer: shape with round and curved surface

Explanation:
The shape that follow the given rule is shape with curved and round surface is

Question 6.
only 3 vertices (corners)

Answer:  shape with only 3 equal sides

Explanation:
The shape that follow the given rule is shape with only 3 equal sides is

Question 7.
4 sides

Answer: shape with only 4 sides

Explanation:
The shape that follow the given rule is shape with only 4 sides is

Question 8.
4 sides are the same length

Answer: shape with 4 sides are the same length

Explanation:
The shape that follow the given rule is shape with 4 sides are the same length is

THINK SMARTER Draw 2 different two-dimensional shapes that follow both parts of the sorting rule.

Question 9.
3 sides and 3 vertices (corners)
Answer: Triangle

Explanation:

By sorting rule given in the question,
A Triangle has 3 sides and 3 vertices (corners)

Question 10.
2 sides are long and 2 sides are short
Answer: Rectangle

Explanation:

By sorting rule given in the question,
A Rectangle has 2 sides are long and 2 sides are short

Problem Solving • Applications

Ted sorted these shapes three different ways. Write sorting rules to tell how Ted sorted.

Question 11.

Answer: He sorted out by using 3 sides and 3 vertices

Explanation:
Because only triangle has 3 sides and 3 vertices

Question 12.

Answer: He sorted out by using 4 vertices and 4 sides

Explanation:
Because rectangles and squares have 4 vertices and 4 sides

Question 13.
THINK SMARTER

Answer: He sorted out by using shapes having 3 or more sides and 3 vertices

Explanation:
The shapes with having 3 or more sides and 3 vertices  are triangle, square, rectangle. are given in the picture.

Question 14.
THINK SMARTER
Which shapes have more than 3 sides? Choose all that apply.

Answer:  The shapes have more than 3 sides are square, rectangle, parallelogram and pentagon.

Explanation:

The shapes have more than 3 sides are square has 4 sides,
rectangle has 4 sides,
parallelogram has 4 sides,
pentagon has 5 sides.

TAKE HOME ACTIVITY • Gather some household objects such as photos, buttons, and napkins. Ask your child to sort them by shape.

### Sort Two-Dimensional Shapes Homework & Practice 12.1

Read the sorting rule. Circle the shapes that follow the rule.
Question 1.
not curved

Answer: The shapes having are 4 figures following the given sorting rule.

Explanation:

The shapes having are 4 figures following the given sorting rule,
They are triangle, square.

Question 2.
4 vertices

Answer: The shapes having are 3 figures following the given sorting rule.

Explanation:

The shapes having are 3 figures following the given sorting rule,
They are rectangle, square.

Problem Solving
Question 3.
Katie sorted these shapes. Write a sorting rule to tell how Katie sorted.

Answer: shapes having 3 vertices

Explanation:
Katie sorted these shapes by choosing 3 vertices ,
Because all triangles have 3 vertices irrespective of their length of the sides

Question 4.
WRITE
Explain how you would name a sorting rule for 1 square, 1 rectangle, and 1 triangle.
____________________
____________________
Answer: shapes having 3 or more sides

Explanation:

Here we have 1 square, 1 rectangle, and 1 triangle,
shapes having 3 or more sides.

Lesson Check
Question 1.
Circle the shape that would not be sorted into this group.

Answer: triangle and circle.

Explanation:

From the first group , we have all of them with 4 sides except triangle,
So, triangle does not belong to the group.
From the second group, we have sides and corners except circle
So, circle does not belong to the group.

Question 2.
Circle the shape that has fewer than 4 sides.

Answer: Triangle

Explanation:

From the group we have all of them having 4 sides Except triangle,
So, triangle does not belong to the group.

Spiral Review
Solve. Draw or write to explain.
Question 3.
Clue 1: A black line is shorter than a white line.
Clue 2: The white line is shorter than a gray line. Is the black line longer or shorter than the gray line?

Answer: The black line is shorter than the gray line.

Explanation:

Given, A black line is shorter than a white line.
The white line is shorter than a gray line,
So, from the picture we have, The black line is shorter than the gray line.

### Lesson 12.2 Describe Two-Dimensional Shapes

Essential Question What attributes can you use to describe two-dimensional shapes?

Listen and Draw

Use two-dimensional shapes. Sort them into two groups. Draw to show your work.

MATHEMATICAL PRACTICES
Look for Structure How did you sort the shapes into two groups? Name the shapes in each group.
Answer: shapes with 3 vertices and shapes with 4 vertices

Explanation:

Table shows , shapes with 3 vertices and shapes with 4 vertices

Share and Show

Use two-dimensional shapes.
Draw and write to complete the chart.

Answer:

On Your Own

Use to trace each straight side.
Use to circle each vertex (corner).
Write the number of sides and vertices (corners).
Question 6.

_____ sides
_____ vertices
Answer: 4 sides and 4 vertices.

Explanation:

There are 4 sides and 4 vertices in the given figure.

Question 7.

_____ sides
_____ vertices
Answer: 4 sides and 4 vertices

Explanation:

There are 4 sides and 4 vertices in the given figure.

Question 8.

_____ sides
_____ vertices
Answer: 3 sides and 3 vertices

Explanation:

There are 3 sides and 3 vertices in the given figure.

Question 9.

_____ sides
_____ vertices
Answer:  3 sides and 3 vertices

Explanation:

There are 3 sides and 3 vertices in the given figure.

Question 10.

_____ sides
_____ vertices
Answer: 6 sides and 6 vertices

Explanation:

There are 6 sides and 6 vertices in the given figure.

Question 11.

_____ sides
_____ vertices
Answer: 4 sides and 4 vertices

Explanation:

There are 4 sides and 4 vertices in the given figure.

THINK SMARTER Draw a picture to solve.

Question 12.
I am a shape with 3 straight sides and 3 vertices.
Answer: Triangle

Explanation:

A Triangle has 3 straight sides and 3 vertices.

Question 13.
I am a shape with 4 straight sides that are the same length and 4 vertices.
Answer: Square

Explanation:

A Square has 4 straight sides that are the same length and 4 vertices.

Problem Solving • Applications

MATHEMATICAL PRACTICE Use Math Vocabulary
Draw shapes to match the clues.
Question 14.
Jake draws a shape that has fewer than 5 sides. It has 3 vertices.
Answer: Triangle

Explanation:

A Triangle has 3 straight sides and 3 vertices.

Question 15.
Meg draws a shape with 4 sides. She labels it as a trapezoid.
Answer: yes, Trapezoid

Explanation:

A Trapezoid has 4 sides

Question 16.
GO DEEPER
Ben draws two different shapes. They each have only 4 vertices.
Answer: Rectangle and Square.

Explanation:
and
Both of them have 4 vertices

Question 17.
THINK SMARTER
Circle the number that makes the sentence true.

Answer: A triangle has 3 vertices.

Explanation:

TAKE HOME ACTIVITY • Have your child draw a square, a trapezoid, and a triangle. For each shape, have him or her show you the sides and vertices and tell how many of each.

### Describe Two-Dimensional Shapes Homework & Practice 12.2

Use to trace each straight side.
Use to circle each vertex. Write the number of sides and vertices.
Question 1.

______ sides
______ vertices
Answer: 6 sides and 6 vertices

Explanation:

There are 6 sides and 6 vertices in the given figure.

Question 2.

______ sides
______ vertices
Answer: 4 sides and 4 vertices

Explanation:

There are 4 sides and 4 vertices in the given figure.

Question 3.

______ sides
______ vertices
Answer: 4 sides and 4 vertices

Explanation:

There are 4 sides and 4 vertices in the given figure.

Question 4.

______ sides
______ vertices
Answer: 3 sides and 3 vertices

Explanation:

There are 3 sides and 3 vertices in the given figure.

Problem Solving
Draw a shape to match the clues.
Question 5.
Ying draws a shape with 4 sides. She labels it as a rectangle.
Answer: Yes, rectangle

Explanation:

Rectangle has 4 sides.

Question 6.
WRITE
Use pictures and words to show the attributes of a hexagon.
Answer:

Explanation:
Characteristics of hexagon are,
All the sides are equal in length.
All the interior angles measure 120°.
The sum of all the interior angles of a regular hexagon is 720°.

Lesson Check
Question 1.
How many vertices does a triangle have?

______ vertices
Answer:  A Triangle has 3 vertices

Explanation:

A Triangle has 3 vertices

Question 2.
How many vertices does a ☐ have?
______ vertices
Answer: A Square has 4 vertices

Explanation:

A Square has 4 vertices

Spiral Review
Question 3.
Circle the greater addend.
Count on to find the sum.

Answer: 9 + 2 = 11

Explanation:

9 + 2 = 11

Question 4.
Corey measures a crayon box with his paper clip ruler. About how long is the box?

Answer: The box is about 5 inches

Explanation:
He used  4 paper clips in the figure,
So, crayon box is about 5 inches in length.

### Lesson 12.3 Combine Two-Dimensional Shapes

Essential Question How can you put two-dimensional shapes together to make new two-dimensional shapes?

Listen and Draw

Use pattern blocks. Draw to show your work.
Answer: we can use 2squares to form a rectangle.

Explanation:

we need 2 squares to form 1 rectangle,
Each square has 4 sides ,
And combined to form a Rectangle having four sides

MATHEMATICAL PRACTICES
Use Tools Describe the new shape Karen made.
Answer: we can use 2 triangles to form a rhombus

Explanation:

we need 2 triangles to make 1 rhombus
Each triangle has 3 vertices,
And combined to form a Rhombus which has 4 sides.

Share and Show

Use pattern blocks. Draw to show the blocks. Write how many blocks you used.
Question 1.

Answer: we need 3 Rhombus to form a Hexagon.

Explanation:

We need 3 Rhombus to form a Hexagon.
Each rhombus have 4 sides,
and combined forming a hexagon having 6 sides

Question 2.

Answer: we need 3 triangles to form a trapezoid.

Explanation:

we need 3 triangles to form a trapezoid.
Each triangle has 3 sides
And combined to form a Trapezoid having 4 sides.

On Your Own

MATHEMATICAL PRACTICE Use a Concrete Model
Use pattern blocks. Draw to show the blocks.
Write how many blocks you used.
Question 3.

Answer: we need 6 triangles to make a Hexagon.

Explanation:

we need 6 triangles to make a Hexagon.
Each triangle has 3 sides,
And combined to form a Hexagon having 6 sides.

Question 4.

Answer: we need 2 triangles to make 1 rhombus

Explanation:

we need 2 triangles to make 1 rhombus
Each triangle has 3 vertices,
And combined to form a Rhombus which has 4 sides.

Question 5.
THINK SMARTER
Use me two times to make this shape. Which block am I? Circle a block to show your answer.

Answer: we need 2 Rhombus to get the given shape.

Explanation:

we need 2 Rhombus to get the given shape.
Each rhombus has 4 sides,
And combined to form the given shape which also has 4 sides.

Question 6.
GO DEEPER
Use these pattern blocks to make the shape. Write how many times you used each block.

Answer: we used 6 triangles to make a hexagon,
we used 3 rhombus to make a hexagon,

Explanation:

We need 3 Rhombus to form a Hexagon.
Each rhombus have 4 sides,
and combined forming a hexagon having 6 sides

we need 6 triangles to make a Hexagon.
Each triangle has 3 sides,
And combined to form a Hexagon having 6 sides.

we need 2 trapezoids to make a Hexagon.
Each trapezoid has 4 sides,
And combined to form a Hexagon having 6 sides.

Problem Solving • Applications

GO DEEPER Use pattern blocks. Draw to show your answer.
Question 7.

Answer: we need 2 triangles to make 1 rhombus, for 3 rhombus we need 6 triangles .

Explanation:

we need 2 triangles to make 1 rhombus
So, for 3 rhombus we need 6 triangles .

Question 8.
THINK SMARTER
How many make a ? Use pattern blocks. Draw to show the blocks you used.
Answer: we need 3 triangles to form a trapezoid.

Explanation:

we need 3 triangles to form a trapezoid.
Each triangle has 3 sides
And combined to form a Trapezoid having 4 sides.

TAKE HOME ACTIVITY • Have your child explain how he or she solved Exercise 7.

### Combine Two-Dimensional Shapes Homework & Practice 12.3

Use pattern blocks. Draw to show the blocks. Write how many blocks you used.
Question 1.

Answer: we need 3 triangles to form a trapezoid.

Explanation:

we need 3 triangles to form a trapezoid.
Each triangle has 3 sides
And combined to form a Trapezoid having 4 sides.

Question 2.

Answer: we can use 2 triangles to form a rhombus

Explanation:

we need 2 triangles to make 1 rhombus
Each triangle has 3 vertices,
And combined to form a Rhombus which has 4 sides.

Problem Solving
Use pattern blocks. Draw to show your answer.
Question 3.

Answer:  we need 6 trapezoids for 4 hexagons

Explanation:

we need 2 trapezoids to make a Hexagon.
Each trapezoid has 4 sides,
And combined to form a Hexagon having 6 sides.
So, we need 6 trapezoids for 4 hexagons

Question 4.
WRITE
Draw the shapes you could put together to make a rectangle.
Answer: we can use 2squares to form a rectangle.

Explanation:

we need 2 squares to form 1 rectangle,
Each square has 4 sides ,
And combined to form a Rectangle having four sides

Lesson Check
Question 1.

Answer: we need 6 triangles to make a Hexagon.

Explanation:

we need 6 triangles to make a Hexagon.
Each triangle has 3 sides,
And combined to form a Hexagon having 6 sides.

Question 2.

Answer: we need 3 Rhombus to form a Hexagon.

Explanation:

We need 3 Rhombus to form a Hexagon.
Each rhombus have 4 sides,
and combined forming a hexagon having 6 sides

Spiral Review
Question 3.
Use . Which string is about 5 long?
Circle the string that is about 5 long.

Answer: First string has minimum length of 6 inches that is 5 paper clips.

Explanation:

So, First string has minimum length of 6 inches that is 5 paper clips.

Question 4.
Look at the hour hand. Write the time.

_________
Answer: 5 O’clock

Explanation:
Because in the clock ,
short hand is the hour hand and long hand is the minutes hand .
So, it is 5 O’clock , in the watch.

### Lesson 12.4 Combine More Shapes

Essential Question How can you combine two-dimensional shapes to make new shapes?

Listen and Draw

Use shapes to fill each outline.
Draw to show your work.

Answer: we can use right angle triangle and square for the the first shape,
we can use triangle and rectangle for the second shape.

Explanation:

we can use right angle triangle and square for the the first shape,
we can use triangle and rectangle for the second shape.

MATHEMATICAL PRACTICES
Represent Use the outline on the left to describe how two shapes can make another shape.
Answer:  we are representing the above figures by outline to make another shape

Explanation:

we are representing the above figures by outline to make another shape
By, using two shapes to fill the outline on the left, and drawing a line to show the two shapes. Then using three shapes to fill the outline on the right,

Share and Show

Circle two shapes that can combine to make the shape on the left.
Question 1.

Answer:  two semi circles give one full circle

Explanation:

So, Two semi circles give one full circle.

Question 2.

Answer: Two squares will form a new rectangle

Explanation:

So, Two squares will form a new rectangle

Question 3.

Answer: Two Quarter circles form a new shape

Explanation:

So, Two Quarter circles form a new shape

On Your Own

MATHEMATICAL PRACTICE Use Diagrams Circle two shapes that can combine to make the shape on the left.
Question 4.

Answer: One rectangle and one quarter circle form a new shape

Explanation:

Thus , One rectangle and one quarter circle form a new shape

Question 5.

Answer: one triangle and one square form a new shape

Explanation:

Thus, one triangle and one square form a new shape

THINK SMARTER Draw lines to show two different ways to combine the shapes on the left to make new shapes on the right.
Question 6.

Answer: we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Explanation:

Here, we have Two squares and one Rectangle

Question 7.

Answer: we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Explanation:

Here, we have Two Squares and Two Triangles

Problem Solving • Applications

THINK SMARTER Draw lines to show how the shapes on the left combine to make the new shape.
Question 8.

Answer: we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Explanation:

Here, we have one Rectangle and one Triangle.

Question 9.

Answer: we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Explanation:

Here, we have  a semi circle and a quarter circle.

Question 10.

Answer: we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Explanation:

Here, we have Two squares and a Rectangle .

Question 11.
THINK SMARTER
Circle the two shapes that can combine to make this new shape.

Answer: A Triangle and a Hexagon forma a new shape.

Explanation:

Thus, A Triangle and a Hexagon forma a new shape.

TAKE HOME ACTIVITY • Ask your child to draw a new shape he or she can make by combining two triangles.

### Combine More Shapes Homework & Practice 12.4

Circle two shapes that can combine to make the shape on the left.
Question 1.

Answer: A Rectangle and a Quarter circle forms a new shape,

Explanation:

Hence, A Rectangle and a Quarter circle forms a new shape.

Question 2.

Answer: A Triangle and a Square forms a new shape.

Explanation:

Hence, A Triangle and a Square forms a new shape.

Problem Solving
Question 3.
Draw lines to show how the shapes on the left combine to make the new shape.

Answer: we are representing the above figures by outline to make another shape

Explanation:

we are using the shapes given  on left to identify the shapes on the right to by drawing lines

Question 4.
WRITE
Draw two shapes. Then draw what they would look like if you put them together to make a new shape.
Answer:  we have one Rectangle and one Triangle.

Explanation:

Hence, we have one Rectangle and one Triangle.

Lesson Check
Question 1.
Circle the shapes that can combine to make this new shape.

Answer: Two circles can form a full  circle.

Explanation:

Thus, Two circles can form a full  circle.

Spiral Review
Use the picture graph to answer each question.

Question 2.
How many more children chose than ?
______ more children
Answer: 3 more children.

Explanation:
From given , data we have,
3 more children in drawing  than swimming.

Question 3.
How many children chose and ?
______ children
Answer: 6 children chose both of them.

Explanation:
From given , data we have,
Totally 6 children chose Dancing and Swimming.

### Lesson 12.5 Problem Solving • Make New Two-Dimensional Shapes

Essential Question How can acting it out help you make new shapes from combined shapes?

Cora wants to combine shapes to make a circle. She has . How can Cora make a circle?

Unlock the Problem

Show how to solve the problem.
Step 1 Use shapes. Combine to make a new shape.

Step 2 Then use the new shape.

Answer: we can make a new circle by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

HOME CONNECTION • Recognizing how shapes can be put together and taken apart provides a foundation for future work with fractions.

Try Another Problem

Use shapes to solve.
Draw to show your work.
Question 1.
Use ☐ to make a larger ☐.
Step 1 Combine shapes to make a new shape.

Step 2 Then use the new shape.

Answer: we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

Question 2.

Answer:  we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

MATHEMATICAL PRACTICES
Model How did you make the rectangle in Exercise 2?
Answer: we used the new blocks of  squares from the set of triangles,
And the two squares are formed as rectangle.

Explanation:

Share and Show

MATHEMATICAL PRACTICE Analyze Relationships Use shapes to solve. Draw to show your work.
Question 3.

Answer: we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

Question 4.
THINK SMARTER

Answer: we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

TAKE HOME ACTIVITY • Have your child explain how he or she solved Exercise 3.

### Problem Solving • Make New Two-Dimensional Shapes Homework & Practice 12.5

Use shapes to solve.
Draw to show your work.
Question 1.

Answer: we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

Question 2.
WRITE
Use pictures to show how you can make a new shape using a combined shape made from two trapezoids.
Answer: we can make a new shape using a combined shape made from two trapezoids.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

Lesson Check
Follow the steps.
Question 1.
Which new shape could you make? Circle your answer.

Answer: we can make a new shape by using given shape.

Explanation:
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

Spiral Review
Question 2.
Circle the shape that has no flat surfaces.

Answer:  A circle has no flat surface

Explanation:

Because, A circle has no flat surface

Question 3.
Which flat surface does a cylinder have? Circle your answer.

Answer: The flat surface which a cylinder have is rectangle.

Explanation:

The flat surface which a cylinder have is rectangle.

### Two-Dimensional Geometry Mid-Chapter Checkpoint

Concepts and Skills

Write the number of sides and vertices (corners).
Question 1.

_____ sides
______ vertices
Answer: the shape has 4 corners and 4 sides.

Explanation:

So, the shape has 4 corners and 4 sides.

Question 2.

_____ sides
______ vertices
Answer: the shape has 6 corners and 6 sides

Explanation:

So, the shape has 6 corners and 6 sides

Circle the shapes that can combine to make the new shape.
Question 3.

Answer: we have a rectangle and a quarter circle

Explanation:

To form a new shape, we have a rectangle and a quarter circle

Question 4.
THINK SMARTER
Which new shape can you make?

Answer:

Explanation:
we can make a new shape by using given shape.
step 1 : use the given shape, combine to make a new shape

step2 : Then use the new shape.

### Lesson 12.6 Find Shapes in Shapes

Essential Question How can you find shapes in other shapes?

Listen and Draw

Use pattern blocks. What shape can you make with 1 and 2 ?
Draw to show your shape.
Answer: we can make a new shape by using given shape.

Explanation:

We can forma new shape from a hexagon and a triangle.

MATHEMATICAL PRACTICES
Use Tools Can you use the same pattern blocks to make a different shape?
Answer: yes, we can use the same pattern blocks to make a different shape

Explanation:
or
Here we used same pattern blocks to make a different shape

Share and Show

Use two pattern blocks to make the shape.
Draw a line to show your model.
Circle the blocks you use.
Question 1.

Answer:

Explanation:

We can form a new shape from a hexagon and a triangle.

Question 2.

Answer:

Explanation:

We can form a new shape from a hexagon and a rhombus.

On Your Own

MATHEMATICAL PRACTICE Use a Concrete Model Use two pattern blocks to make the shape. Draw a line to show your model. Circle the blocks you use.
Question 3.

Answer:

Explanation:

we can form a new shape with a rhombus and a triangle.

Question 4.

Answer:

Explanation:

we can form a new shape with a rhombus and a triangle.

Question 5.

Answer:

Explanation:

Just rotate the give triangle

Question 6.

Answer:

Explanation:

we can form a new shape with a rhombus and a triangle.

Question 7.
THINK SMARTER
Use three pattern blocks to make the shape. Draw lines to show your model. Circle the blocks you use.

Answer: we use three figures to form a new shape.

Explanation:

Here, we are using trapezoid, rhombus and triangle.

Problem Solving • Applications

THINK SMARTER
Make this shape. Use the number of pattern blocks listed in the exercise. Write how many of each block you use.

Question 8.
Use 3 blocks.

Answer:

Explanation:
=
Here, we used one hexagon and two triangles

Question 9.
Use 5 blocks.

Answer:

Explanation:

Here, we used 3 triangles, 1 rhombus, 1 trapezoid to form a new shape.

Question 10.
Use 7 blocks.

Answer:

Explanation:

Here, we used 6 triangles, 1 rhombus, to form a new shape.

Question 11.
Use 8 blocks.

Answer:

Explanation:

Here, we used 8 triangles to form a new shape.

Question 12.
THINK SMARTER
Use 4 pattern blocks to fill the shape. Draw to show the blocks you used.

Answer:

Explanation:

Here, we used 1 triangle, 1 rhombus, 2 trapezoids to form a new shape.

TAKE HOME ACTIVITY • Have your child use this page to explain how to find shapes within the given shape.

### Find Shapes in Shapes Homework & Practice 12.6

Use two pattern blocks to make the shape. Draw a line to show your model. Circle the blocks you use.
Question 1.

Answer:

Explanation:

Here, we used two pattern blocks to form a new shape.

Question 2.

Answer:

Explanation:

Here, we used two pattern blocks to form a new shape.

Problem Solving
Make the shape to the right. Use the number of pattern blocks listed in the exercise. Write how many of each block you use.

Question 3.
Use 3 blocks.

Answer:

Explanation:

Here, we used two pattern blocks to form a new shape.

Question 4.
WRITE
Use pictures or words to explain what shapes can be put together to make a hexagon shape.
Answer: I put together 2 trapezoids to make a hexagon. It has 6 sides and 6 vertices.
It has 2 equal parts. My new shape has 2 trapezoids and 4 triangles.

Lesson Check
Question 1.
Circle the pair of pattern blocks that can make this shape.

Answer:

Explanation:

Here, we used two pattern blocks to form a new shape.

Spiral Review
Question 2.
Write the time.

_______
Answer: It is 2 hours and 30 minutes in the clock.

Explanation:
Because, the hours hand is in between 2 and 3,
And minutes hand is on 6 , that implies an half an hour or 30 minutes.
So, It is 2 hours and 30 minutes in the clock.

Question 3.
Write tally marks to show the number 8.
_________
Answer:

Explanation:
It is a form of numeral used for counting. The general way of writing tally marks is as a group or set of five lines. The first four lines are drawn vertically and each of the fifth line runs diagonally over the previous four vertical lines, i.e. from the top of the first line to the bottom of the fourth line.

Question 4.
How many vertices does a ☐ have?
_____ vertices
Answer: 4 vertices.

Explanation:

A square has 4 vertices and 4 sides.

### Lesson 12.7 Take Apart Two-Dimensional Shapes

Essential Question How can you take apart two dimensional shapes?

Listen and Draw

Color rectangles orange.
Color triangles purple.

Answer:
In the picture there are 4 triangles and are colored in purple,
And 2 rectangles and are colored in orange

MATHEMATICAL PRACTICES
Look for Structure What shapes did Angelina make?
Answer:

Explanation:
Angelina put some triangles and rectangles together.
She drew pictures to show what she made.
We have Colored them to show how Angelina put the shapes together.

Share and Show

Draw a line to show the parts.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

On Your Own

MATHEMATICAL PRACTICE Identify Relationships Draw a line to show the parts.
Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

THINK SMARTER
Draw two lines to show the parts.

Question 9.

Answer:

Question 10.

Answer:

Problem Solving • Applications

Question 11.
THINK SMARTER
How many squares are there?

________ squares
Answer:
There 4 squares in the given figure.

Question 12.
THINK SMARTER
Draw a line to show the parts.

Answer:

TAKE HOME ACTIVITY • Ask your child to explain how he or she solved Exercise 11.

### Take Apart Two-Dimensional Shapes Homework & Practice 12.7

Draw a line to show the parts.
Question 1.

Answer:

Question 2.

Answer:

Problem Solving
Question 3.
How many triangles are there?

_______ triangles
Answer:
There are 4 triangles in the given picture.

Question 4.
WRITE
Draw a shape. Then draw one or two lines to show parts of the shape.
Answer:
We used a triangle and a rectangle to have a new shape.

Lesson Check
Question 1.
Look at the picture.
Circle the pair that shows the parts.

Answer:

Explanation:

Spiral Review
Question 2.
Use the graph.
How many children chose

______ children
Answer: 2 children chose soccer.

Question 3.
Which new shape can you make?
Circle your answer.

Answer:

### Lesson 12.8 Equal or Unequal Parts

Essential Question How can you identify equal and unequal parts in two-dimensional shapes?

Listen and Draw

Draw to show the parts.

Answer:  and

MATHEMATICAL PRACTICES
Explain how the triangles shown in each square compare.
Answer: For the first square we have 2 equal triangles,
For the second square we have 3 unequal triangles.

Share and Show

Circle the shape that shows equal parts.
Question 1.

Answer:
Here we have 2 equal rectangles.

Question 2.

Answer:
Here we have 4 equal triangles

Question 3.

Answer:
Here we have 2 equal semi circles.

Circle the shape that shows unequal parts.
Question 4.

Answer:
Here we have 4 equal quarters of a circle.

Question 5.

Answer:
Here we have 2 equal parts of rectangles.

Question 6.

Answer:
Here we have 4 equal parts of the given shape.

On Your Own

MATHEMATICAL PRACTICE Use Math Vocabulary Color the shapes that show unequal shares.
Question 7.

Answer:

Question 8.

Answer:

Color the shapes that show equal shares.
Question 9.

Answer:

Question 10.

Answer:

THINK SMARTER Write the number of equal shares.
Question 11.

______ equal shares
Answer: 2 equal shares

Explanation:

Question 12.

_______ equal shares
Answer: 4 equal shares.

Explanation:

Problem Solving • Applications

THINK SMARTER Draw lines to show the parts.
Question 13.
2 equal parts

Answer:

Question 14.
2 unequal parts

Answer:

Question 15.
4 equal shares

Answer:

Question 16.
4 unequal shares

Answer:

Question 17.
THINK SMARTER
Does the shape show equal shares? Choose Yes or No.

Answer:

TAKE HOME ACTIVITY• Draw a circle on a piece of paper. Ask your child to draw a line so the circle shows 2 equal shares

### Equal or Unequal Parts Homework & Practice 12.8

Color the shapes that show unequal shares.
Question 1.

Answer:

Color the shapes that show equal shares.
Question 2.

Answer:

Problem Solving
Draw lines to show the parts.
Question 3.
4 equal shares

Answer:

Question 4.
WRITE
Draw two rectangles. Draw lines on one rectangle to show equal parts. Draw lines on the other rectangle to show unequal parts.
Answer: First rectangle is
second rectangle is

Lesson Check
Question 1.
Color the shape that shows unequal shares.

Answer:

Spiral Review
Question 2.
Which food did the most children choose?
Circle your answer.

Answer: Most of the children chose Pancakes, Because it has the highest number of 6 children

Question 3.
Use the graph. How many children chose

__________ children
Answer: The number of children whos chose stuffed animals are 4 members .

### Lesson 12.9 Halves

Essential Question How can a shape be separated into two equal shares?

Listen and Draw

Draw to solve

Answer: First equal share =
second equal share =

MATHEMATICAL PRACTICES
Analyze Will all four friends get the same amount of sandwich?
Answer: Yes all of them get the same amount of sandwich.

Explanation:
Two friends share the sandwich on the left, they cut the sandwich so each gets an equal share.

Two other friends share the sandwich on the right, this sandwich be cut a different way so each friend gets an equal share.

Share and Show

Draw a line to show halves.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

On Your Own

MATHEMATICAL PRACTICE Analyze Relationships Circle the shapes that show halves.
Question 5.

Answer: This shape has  equal shares

Question 6.

Answer: This shape has  unequal shares

Question 7.

Answer: This shape has  equal shares

Question 8.

Answer: This shape has  equal shares

Question 9.

Answer: This shape has  equal shares

Question 10.

Answer: This shape has  equal shares

Question 11.

Answer: This shape has  equal shares

Question 12.

Answer: This shape has  equal shares

Question 13.

Answer: This shape has  unequal shares

Question 14.
THINK SMARTER
Use the picture.
Write numbers to solve

The picture shows ______ halves.
The _____ equal shares make _____ whole.
Answer:  The picture shows 2 halves.
The 2 equal shares make 1 whole.

Problem Solving • Applications

Draw or write to solve.
Question 15.
Color half of each shape.

Answer:

Question 16.
Linus cut a circle into equal shares. He traced one of the parts. Write half of or halves to name the part.

Answer:

Question 17.
GO DEEPER
Draw three different ways to show halves.

Answer:

Question 18.
THINK SMARTER
Circle the shapes that show halves.

Answer:

TAKE HOME ACTIVITY • Draw a rectangle on a piece of paper. Ask your child to draw a line to show halves.

### Halves Homework & Practice 12.9

Circle the shapes that show halves.
Question 1.

Answer: This shape has  equal shares

Question 2.

Answer: This shape has  equal shares

Question 3.

Answer: This shape has  equal shares

Question 4.

Answer: This shape has  unequal shares

Question 5.

Answer: This shape has  equal shares

Question 6.

Answer: This shape has  unequal shares

Problem Solving
Draw or write to solve.
Question 7.
Kate cut a square into equal shares. She traced one of the parts. Write half of or halves to name the part.

Answer:

Question 8.
WRITE
Draw a circle and separate it into halves. Color each half a different color.
Answer:

Lesson Check
Question 1.
Circle the shape that shows halves.

Answer:

Spiral Review
Question 2.
Circle the new shape you can make.

Answer:

Question 3.
Circle the shape that has both flat and curved surfaces.

Answer: A Cylinder shape has both flat and curved surfaces.

Question 4.
How many ∆ do you use to make a ?
Draw to show your answer.

Answer:
3 triangles make a 1 trapezoid.

6 triangles make 1 Hexagon.

### Lesson 12.10 Fourths

Essential Question How can a shape be separated into four equal shares?

Listen and Draw

Use what you know about halves.
Draw to solve. Write how many.

There are _____ equal shares.
Answer: There are 2 equal share.

MATHEMATICAL PRACTICES
Explain How did you decide how to cut the pizza?
Answer: There are 4 equal shares.

Explanation:
Two friends will share a pizza, Then two more friends come. Now four friends will share the pizza.
The pizza can be cut into 4 equal pieces so each friend gets an equal share.

Share and Show

Color a fourth of the shape.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Color a quarter of the shape.
Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

On Your Own

MATHEMATICAL PRACTICE Use Diagrams Circle the shapes that show fourths.
Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer: There are no fourths in this shape because, it is divided in to two equal shares.

Question 10.

Answer: There are no fourths in this shape because, it is divided in to two equal shares.

Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Answer: There are no fourths in this shape because, it is divided in to unequal shares.

Question 15.

Answer: There are no fourths in this shape because, it is divided in to unequal shares.

Question 16.
GO DEEPER
Draw three different ways to show fourths.

Answer:

Problem Solving • Applications

Solve.
Question 17.
Write halves, fourths, or quarters to name the equal shares.

Answer:

Question 18.
THINK SMARTER
Circle the shape that shows quarters.

Answer:

Explanation:

Question 19.
THINK SMARTER+
Alano has a small pizza.
He wants to share the pizza with friends.
He cuts the pizza into fourths.
Draw lines to show how he cuts the pizza.

How many equal shares did you draw? ______
How many halves can you show in a circle? _______
Tell how you can solve this problem in a different way.
Answer: He has 4 equal shares
and we can show 2 halves in a circle.

Explanation:
He has 4 equal shares

and we can show 2 halves in a circle.

TAKE HOME ACTIVITY • Draw a circle on a piece of paper. Ask your child to draw lines to show fourths.

### Fourths Homework & Practice 12.10

Circle the shapes that show fourths.
Question 1.

Answer: There are no fourths in this shape because, it is divided in to unequal shares.

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer: There are no fourths in this shape because, it is divided in to unequal shares.

Question 6.

Answer:

Problem Solving
Solve.
Question 7.
Chad drew a picture to show a quarter of a circle. Which shape did Chad draw? Circle it.

Answer:

Explanation:

Question 8.
WRITE
Draw two squares. Draw lines to show fourths. Color a fourth of the first square. Color the second square to show a whole.
Answer:

Lesson Check
Question 1.
Circle the shape that shows fourths.

Answer:

Spiral Review
Question 2.
What shapes did Leila use to build the wall? Circle the shapes she used.

Answer:

Question 3.
Use the graph to answer the question. How many fewer children answered yes than no?

________ fewer children
Answer: 2 fewer children answered yes than no

### Two-Dimensional Geometry Review/Test

Question 1.
Which shapes have only 3 sides?
Choose all that apply.

Answer:

Question 2.
Circle the number that makes the sentence true.

Answer:

Question 3.

Use pattern blocks. Draw to show the blocks you used.
Answer:
Here we used 2 trapezoids to form a Hexagon.

Question 4.
Circle two shapes that can combine to make this new shape.

Answer:

Question 5.
THINK SMARTER+

Answer: No.

Explanation:

We need four quarters to form a whole circle.

Question 6.
Use 4 pattern blocks to fill the shape.
Draw to show the blocks you used.

Answer:

Question 7.
Draw a line to show the parts.

Answer:

Question 8.
Does the shape show equal shares? Choose Yes or No.

Answer:

Question 9.
Circle the shapes that show halves.

Answer:

Question 10.
GO DEEPER
Draw lines to show fourths.

How many equal shares did you draw?

How many halves can you show in a rectangle?

Tell how you can solve this problem in a different way.
Answer: we have 4 equal shares.

we can have 2 halves in a rectangle.

## Go Math Grade 2 Answer Key Chapter 5 2-Digit Subtraction

Go Math Grade 2 Answer Key Chapter 5 2-Digit Subtraction answer key is useful for students who are preparing for their examinations and can download this pdf for free of cost. In this chapter, each and every question was explained in detail which helps students to understand easily. Go Math Grade 2 Answer Key Chapter 5 2-Digit Subtraction explains different types of questions on 2 Digit Subtraction.

## Go Math Grade 2 Answer Key Chapter 5 2-Digit Subtraction

In this chapter, we can see different topics on Break Apart Ones to Subtract, Break Apart Numbers to Subtract, Model Regrouping for Subtraction, Model and Record 2-Digit Subtraction, 2-Digit Subtraction, etc. Those topics were being set up by the mathematical professionals as indicated by the most recent release. Look down this page to get the answers to all the inquiries. Click on the links to look at the subjects shrouded in this chapter 2-Digit Subtraction.

Chapter: 9-  2-Digit Subtraction

Lesson 1: Algebra • Break Apart Ones to Subtract

Lesson 2: Algebra • Break Apart Numbers to Subtract

Lesson 3: Model Regrouping for Subtraction

Lesson 4: Model and Record 2-Digit Subtraction

Lesson 5: 2-Digit Subtraction

Lesson 5.5 2-Digit Subtraction
2-Digit Subtraction Homework & practice 5.5

Lesson 6: Practice 2-Digit Subtraction

Mid-Chapter Checkpoint

2-Digit Subtraction Mid-Chapter Checkpoint

Lesson 7: Rewrite 2-Digit Subtraction

Lesson 8: Add to Find Differences

Lesson 9: Problem Solving • Subtraction

Lesson 10: Algebra • Write Equations to Represent Subtraction

Lesson 11: Solve Multistep problems

Chapter 5 Review/Test

2-Digit Subtraction Chapter 5 Review Test

Curious about Math

There are hundreds of different kinds of dragonflies. If 52 dragonflies are in a garden and 10 fly away, how many dragonflies are left? How many are left if 10 more fly away?

Answer:
The total number of files left is 32 files.

Explanation:
As there are hundreds of different kinds of dragonflies and if 52 dragonflies are in a garden and 10 flies away, so the number of files left is 52 – 10= 42 files and again there are 10 more flies away. So there will be 42 – 10= 32 files left.

### 2-Digit Subtraction Show What You Know

Subtraction Patterns
Subtract 2. Complete each subtraction sentence.

Question 1.

Question 2.
6 – __ = ___

Answer:
On subtracting 6 – 2 we will get the result as 4.

Explanation:
By subtracting 2 with 6 we will get the result as 6 – 2= 4.

Question 3.
5 – __ = __

Answer:
On subtracting 5 – 2 we will get the result as 3.

Explanation:
By subtracting 2 with 5 we will get the result as 5 – 2= 3.

Question 4.
4 – ___ = __

Answer:
On subtracting 4 – 2 we will get the result as 2.

Explanation:
By subtracting 2 with 4 we will get the result as 4 – 2= 2.

Question 5.
3 – __ = __

Answer:
On subtracting 3 – 2 we will get the result as 1.

Explanation:
By subtracting 2 with 3 we will get the result as 3 – 2= 1.

Question 6.
2 – _ = __

Answer:
On subtracting 2 – 2 we will get the result as 0.

Explanation:
By subtracting 2 with 2 we will get the result as 2 – 2= 0.

Subtraction Facts
Write the difference.

Question 7.

Answer:
On subtracting 8 – 5 we will get the result as 3.

Explanation:
By subtracting 5 with 8 we will get the result as 8 – 5= 3.

Question 8.

Answer:
On subtracting 14 – 6 we will get the result as 8.

Explanation:
By subtracting 6 with 14 we will get the result as 14 – 6= 8.

Question 9.

Answer:
On subtracting 9 – 6 we will get the result as 3.

Explanation:
By subtracting 6 with 9 we will get the result as 9 – 6= 3.

Question 10.

Answer:
On subtracting 16 – 7 we will get the result as 9.

Explanation:
By subtracting 7 with 16 we will get the result as 16 – 7= 9.

Question 11.

Answer:
On subtracting 12 – 6 we will get the result as 6.

Explanation:
By subtracting 6 with 12 we will get the result as 12 – 6= 6.

Question 12.

Answer:
On subtracting 10 – 8 we will get the result as 2.

Explanation:
By subtracting 8 with 10 we will get the result as 10 – 8= 2.

Tens and Ones
Write how many tens and ones are in each model.

Question 13.
54

__ tens __ ones

Answer:
The number of tens is 5 and the number of ones is 4.

Explanation:
In the above image, we can see 10 blocks with 5 rows. Which are 10 × 5= 50 blocks, so there are 5 tens and we can see 4 blocks which are 4 ones. So the total number of blocks is 50 + 4= 54 blocks.

Question 14.
45

__ tens __ ones

Answer:
The number of tens is 4 and the number of ones is 5.

Explanation:
In the above image, we can see 10 blocks with 4 rows. Which are 10 × 4= 40 blocks, so there are 4 tens and we can see 5 blocks which are 5 ones. So the total number of blocks is 40 + 5= 45 blocks.

### 2-Digit Subtraction Vocabulary Builder

Visualize It
Fill in the boxes of the graphic organizer.

Understand Vocabulary
Draw a line to complete the sentence.

Answer:
1. A digit can be 0,1,2,3,4,5,6,7,8, or 9.
2. You can regroup to trade 10 ones for 1 ten.
3. 20 ones are the same as 2 tens.

Explanation:

1.  A digit is a single symbol that is used to make numerals, so the digits can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 are the ten digits which we will use every day.
2. Here, regrouping is defined as the process of making and then carrying out the operation like addition with two-digit numbers or larger than the two-digit numbers. And we use regrouping in addition when the sum of two digits in the place value column is greater than nine. And Regrouping in subtraction is a method of interchanging one ten into ten ones. This regrouping of subtraction is used to work out the different subtraction problems. We can regroup to trade 10 ones for 1 ten. And we can regroup to trade 10 ones for 1 ten.
3. 20 ones are the same as 2 tens. As two tens mean 2 × 10= 20 and 20 × 1= 20, so 20 ones are the same as 2 tens.

### 2-Digit Subtraction Game: Subtraction Search

Materials Search
• 3 sets of number cards 4 – 9 • 18
Play with a partner.
1. Shuffle all the cards. Place them face down in one stack.
2. Take one card. Find a square with a subtraction problem with this number as the difference. Your partner checks your answer.
3. If you are correct, place a on that square. If there is no match, skip your turn.
4. Take turns. The first player to have on all the squares wins.

### 2-Digit Subtraction Vocabulary Game

Going Places with GOMATH! Words

Bingo

For 3 to 6 players

Materials

• 1 set of word cards
• 1 Bingo board for each player
• game markers

How to Play

1. The caller chooses a word card and reads the word. Then the caller puts the word card in a second pile.
2. Players put a marker on the word each time they find it on their Bingo boards.
3. Repeat steps 1 and 2 until a player marks 5 boxes in a line going down, across, or on a slant and calls “Bingo.”
4. Check the answers. Have the player who said “Bingo” read the words aloud while the caller checks the word cards in the second pile.

The Write Way

Reflect
Choose one idea. Write about it in the space below.

• Explain how drawing quick pictures helps you add 2-digit numbers.
• Tell about all the different ways you can add 2-digit numbers.
• Write about a time that you helped explain something to a classmate. What was your classmate having trouble with? How did you help him or her?

### Lesson 5.1 Algebra • Break Apart Ones to Subtract

Essential Question
How does breaking apart a number make subtracting easier?

Listen and Draw

Write two addends for each sum.

Answer:
The two addends for the first image are 3 and 4.
The two addends for the second image are 4 and 5.
The two addends for the third image are 2 and 3.
The two addends for the fourth image are 3 and 3.
The two addends for the fifth image are 2 and 2.
The two addends for the sixth image are 4 and 4.

Explanation:

As addend can be defined as the numbers are added together to get the sum. Here the given sum is 7, so to get the sum 7 we will take the two addends as 3 + 4 so we can get the sum as 7. The given sum is 9, so to get the sum 9 we will take the two addends 4 + 5 so we can get the sum as 9. The given sum is 5, so to get the sum 5 we will take the two addends 2 + 3 so we can get the sum as 5. The given sum is 6, so to get the sum 6 we will take the two addends 3 + 3 so we can get the sum as 6. the given sum is 4, so to get the sum 4 we will take the two addends 2 + 2 so we can get the sum as 4. The given sum is 8, so to get the sum 8 we will take the two addends
4 + 4 so we can get the sum as 8.

Math Talk
MATHEMATICAL PRACTICES

Describe how you chose addends for each sum.

Answer:
The addends are chosen by the given sum. As addend can be defined as the numbers are added to together to get the sum. So we have chosen the two addends by the given sum.

Model and Draw

Break apart ones. Subtract in two steps.

So, 63 − 7 = ___.

Answer:
The subtraction of 63 – 7 is 56.

Explanation:
Here, we have started at 63 and subtracted 3, to subtract 3 on the number line jump makes a jump from 63 to 60 and the size of the jump is 3 and we will get 60. Then we have to subtract 4, so we will start from 60 on the number line jump makes a jump from 60 to 56 and the size of the jump is 4 and we will get 56. So the subtraction of 63 – 7 is 56.

Share and Show MATH BOARD

Break apart ones to subtract. Write the difference.

Question 1.

Answer:
The subtraction of 55 – 8 is 47.

Explanation:
Here, we have started at 55 and subtracted 5, to subtract 5 on the number line jump makes a jump from 55 to 50 and the size of the jump is 5 and we will get 50. Then we have to subtract 3, so we will start from 50 on the number line jump makes a jump from 50 to 47 and the size of the jump is 3 and we will get 47. So the subtraction of 55 – 8 is 47.

Question 2.

Answer:
The subtraction of 42 – 5 is 37.

Explanation:
Here, we have started at 42 and subtracted 2, to subtract 2 on the number line jump makes a jump from 42 to 40 and the size of the jump is 2 and we will get 40. Then we have to subtract 3, so we will start from 40 on the number line jump makes a jump from 40 to 37 and the size of the jump is 3 and we will get 37. So the subtraction of 42 – 5 is 37.

Question 3.
41 – 9 = __

Answer:
The subtraction of 41 – 9 is 32.

Explanation:
Here, we have started at 41 and subtracted 4, to subtract 4 on the number line jump makes a jump from 41 to 37 and the size of the jump is 4 and we will get 37. Then we have to subtract 5, so we will start from 37 on the number line jump makes a jump from 37 to 32 and the size of the jump is 5 and we will get 32. So the subtraction of 41 – 9 is 32.

Question 4.
53 – 6 = __

Answer:
The subtraction of 53 – 6 is 47.

Explanation:
Here, we have started at 53 and subtracted 3, to subtract 3 on the number line jump makes a jump from 53 to 50 and the size of the jump is 3 and we will get 50. Then we have to subtract 3, so we will start from 50 on the number line jump makes a jump from 50 to 47 and the size of the jump is 3 and we will get 47. So the subtraction of 53 – 6 is 47.

Question 5.
44 – 7 = __

Answer:
The subtraction of 44 – 7 is 37.

Explanation:
Here, we have started at 44 and subtracted 3, to subtract 3 on the number line jump makes a jump from 44 to 41 and the size of the jump is 3 and we will get 41. Then we have to subtract 4, so we will start from 41 on the number line jump makes a jump from 41 to 37 and the size of the jump is 4 and we will get 37. So the subtraction of 44 – 7 is 37.

Question 6.
52 – 8 = __

Answer:
The subtraction of 52 – 8 is 44.

Explanation:
Here, we have started at 52 and subtracted 4, to subtract 4 on the number line jump makes a jump from 52 to 48 and the size of the jump is 4 and we will get 48. Then we have to subtract 4, so we will start from 48 on the number line jump makes a jump from 48 to 44 and the size of the jump is 4 and we will get 44. So the subtraction of 52 – 8 is 44.

On Your Own

Break apart ones to subtract. Write the difference.

Question 7.

Answer:
The subtraction of 75 – 7 is 68.

Explanation:
Here, we have started at 75 and subtracted 3, to subtract 3 on the number line jump makes a jump from 75 to 72 and the size of the jump is 3 and we will get 72. Then we have to subtract 4, so we will start from 72 on the number line jump makes a jump from 72 to 68 and the size of the jump is 4 and we will get 68. So the subtraction of 75 – 7 is 68.

Question 8.

Answer:
The subtraction of 86 – 8 is 78.

Explanation:
Here, we have started at 86 and subtracted 4, to subtract 4 on the number line jump makes a jump from 86 to 82 and the size of the jump is 4 and we will get 82. Then we have to subtract 4, so we will start from 82 on the number line jump makes a jump from 82 to 78 and the size of the jump is 4 and we will get 78. So the subtraction of 86 – 8 is 78.

Question 9.
82 – 5 = __

Answer:
The subtraction of 82 – 5 is 77.

Explanation:
Here, we have started at 82 and subtracted 3, to subtract 3 on the number line jump makes a jump from 82 to 79 and the size of the jump is 3 and we will get 79. Then we have to subtract 2, so we will start from 79 on the number line jump makes a jump from 79 to 77 and the size of the jump is 2 and we will get 77. So the subtraction of 82 – 5 is 77.

Question 10.
83 – 7 = __

Answer:
The subtraction of 83 – 7 is 76.

Explanation:
Here, we have started at 83 and subtracted 3, to subtract 3 on the number line jump makes a jump from 83 to 80 and the size of the jump is 3 and we will get 80. Then we have to subtract 4, so we will start from 80 on the number line jump makes a jump from 80 to 76 and the size of the jump is 4 and we will get 76. So the subtraction of 83 – 7 is 76.

Question 11.
72 – 7 = __

Answer:
The subtraction of 72 – 7 is 65.

Explanation:
Here, we have started at 72 and subtracted 3, to subtract 3 on the number line jump makes a jump from 72 to 69 and the size of the jump is 3 and we will get 69. Then we have to subtract 4, so we will start from 69 on the number line jump makes a jump from 69 to 65 and the size of the jump is 4 and we will get 65. So the subtraction of 72 – 7 is 65.

Question 12.
76 – 9 = __

Answer:
The subtraction of 76 – 9 is 67.

Explanation:
Here, we have started at 76 and subtracted 5, to subtract 5 on the number line jump makes a jump from 76 to 71 and the size of the jump is 5 and we will get 71. Then we have to subtract 4, so we will start from 71 on the number line jump makes a jump from 71 to 67 and the size of the jump is 4 and we will get 67. So the subtraction of 76 – 9 is 67.

Question 13.
85 – 8 = __

Answer:
The subtraction of 85 – 8 is 77.

Explanation:
Here, we have started at 85 and subtracted 4, to subtract 4 on the number line jump makes a jump from 85 to 81 and the size of the jump is 4 and we will get 81. Then we have to subtract 4, so we will start from 81 on the number line jump makes a jump from 81 to 77 and the size of the jump is 4 and we will get 77. So the subtraction of 85 – 8 is 77.

Question 14.
71 – 6 = __

Answer:
The subtraction of 71 – 6 is 65.

Explanation:
Here, we have started at 71 and subtracted 3, to subtract 3 on the number line jump makes a jump from 71 to 68 and the size of the jump is 3 and we will get 68. Then we have to subtract 3, so we will start from 68 on the number line jump makes a jump from 68 to 65 and the size of the jump is 3 and we will get 65. So the subtraction of 71 – 6 is 65.

Question 15.
THINK SMARTER
Cheryl brought 27 bagels for the bake sale. Mike brought 24 bagels. They sold all but 9 of them. How many bagels did they sell?

__ bagels

Answer:
The total number of bagels sold is 42 bagels.

Explanation:
Cheryl brought 27 bagels for the bake sale and Mike brought 24 bagels, so the total number of bagels did Mike and Cheryl bought is 27 + 24= 51 bagels. And they sold all but 9 bagels are left with them, so the number of bagels sold is 51 – 9= 42 bagels. So the total number of bagels sold is 42 bagels.

Question 16.
MATHEMATICAL PRACTICE
Analyze Lexi has 8 fewer crayons than Ken. Ken has 45 crayons. How many crayons does Lexi have?
__ crayons

Answer:
The number of crayons Lexi had is 37 crayons.

Explanation:
As Lexi has 8 fewer crayons than Ken and Ken has 45 crayons, so Lexi has 45 – 8= 37 crayons.

Problem Solving • Applications

Write or draw to explain.

Question 17.
Cheryl built a toy train with 27 train cars. Then she added 18 more train cars. How many train cars are on the toy train now?

__ train cars

Answer:
The number of train cars is on the toy train is 9 train cars.

Explanation:
Cheryl built a toy train with 27 train cars and then she added 18 more train cars, so the number of train cars are on the toy train is 27 – 18= 9 train cars.

Question 18.
MATHEMATICAL PrACTICE
Analyze
Samuel had 46 marbles. He gave some marbles to a friend and has 9 marbles left. How many marbles did Samuel give to his friend?
__ marbles

Answer:
The number of marbles did Samuel gave to his friend is 37 marbles.

Explanation:
Samuel had 46 marbles and he gave some marbles to a friend and has 9 marbles left, so the number of marbles did Samuel gave to his friend is 46 – 9= 37 marbles. So the number of marbles did Samuel gave to his friend is 37 marbles.

Question 19.
THINK SMARTER
Matthew had 73 blocks. He gave 8 blocks to his sister. How many blocks does Matthew have now? Draw or write to show how to solve the problem.

Matthew has __ blocks now.

Answer:
The total number of blocks Matthew had is 65 blocks.

Explanation:
Matthew had 73 blocks and he gave 8 blocks to his sister, so the number of blocks did Matthew had is 73 – 8= 65 blocks. So after giving 8 blocks to his sister Matthew had 65 blocks. By subtracting the number of blocks did Matthew had and the number of blocks did he gave to his sister we had solved the problem.

TAKE HOME ACTIVITY
• Ask your child to describe how to find 34 − 6.

Answer:
The subtraction of 6 from 34 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 6 from 34, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 6 from 34 is 34 – 6= 28 and there are two tens and eight ones.

### Algebra • Break Apart Ones to Subtract Homework & Practice 5.1

Break apart ones to subtract. Write the difference.

Question 1.
36 – 7 = __

Answer:
The subtraction of 36 – 7 is 29.

Explanation:
Here, we have started at 36 and subtracted 3, to subtract 3 on the number line jump makes a jump from 36 to 33 and the size of the jump is 3 and we will get 33. Then we have to subtract 4, so we will start from 33 on the number line jump makes a jump from 33 to 29 and the size of the jump is 4 and we will get 29. So the subtraction of 36 – 7 is 56.

Question 2.
35 – 8 = __

Answer:
The subtraction of 35 – 8 is 27.

Explanation:
Here, we have started at 35 and subtracted 4, to subtract 4 on the number line jump makes a jump from 35 to 31 and the size of the jump is 4 and we will get 31. Then we have to subtract 4, so we will start from 31 on the number line jump makes a jump from 31 to 27 and the size of the jump is 4 and we will get 27. So the subtraction of 35 – 8 is 27.

Question 3.
37 – 9 = __

Answer:
The subtraction of 37 – 9 is 28.

Explanation:
Here, we have started at 37 and subtracted 4, to subtract 4 on the number line jump makes a jump from 37 to 33 and the size of the jump is 4 and we will get 33. Then we have to subtract 5, so we will start from 33 on the number line jump makes a jump from 33 to 28 and the size of the jump is 5 and we will get 28. So the subtraction of 37 – 9 is 28.

Question 4.
41 – 6 = __

Answer:
The subtraction of 41 – 6 is 35.

Explanation:
Here, we have started at 41 and subtracted 3, to subtract 3 on the number line jump makes a jump from 41 to 38 and the size of the jump is 3 and we will get 38. Then we have to subtract 3, so we will start from 35 on the number line jump makes a jump from 38 to 35 and the size of the jump is 3 and we will get 35. So the subtraction of 41 – 6 is 35.

Question 5.
44 – 5 = __

Answer:
The subtraction of 44 – 5 is 39.

Explanation:
Here, we have started at 44 and subtracted 3, to subtract 3 on the number line jump makes a jump from 44 to 41 and the size of the jump is 3 and we will get 41. Then we have to subtract 2, so we will start from 41 on the number line jump makes a jump from 41 to 39 and the size of the jump is 2 and we will get 39. So the subtraction of 44 – 5 is 39.

Question 6.
33 – 7 = __

Answer:
The subtraction of 33 – 7 is 26.

Explanation:
Here, we have started at 33 and subtracted 3, to subtract 3 on the number line jump makes a jump from 33 to 30 and the size of the jump is 3 and we will get 30. Then we have to subtract 4, so we will start from 30 on the number line jump makes a jump from 30 to 26 and the size of the jump is 4 and we will get 26. So the subtraction of 33 – 7 is 26.

Question 7.
32 – 4 = __

Answer:
The subtraction of 32 – 4 is 28.

Explanation:
Here, we have started at 32 and subtracted 2, to subtract 2 on the number line jump makes a jump from 32 to 30 and the size of the jump is 2 and we will get 30. Then we have to subtract 2, so we will start from 30 on the number line jump makes a jump from 30 to 28 and the size of the jump is 2 and we will get 28. So the subtraction of 32 – 4 is 28.

Question 8.
31 – 6 = __

Answer:
The subtraction of 31 – 6 is 25.

Explanation:
Here, we have started at 31 and subtracted 3, to subtract 3 on the number line jump makes a jump from 31 to 28 and the size of the jump is 3 and we will get 28. Then we have to subtract 3, so we will start from 28 on the number line jump makes a jump from 28 to 25 and the size of the jump is 3 and we will get 25. So the subtraction of 31 – 6 is 25.

Problem Solving

Choose a way to solve. Write or draw to explain.

Question 9.
Beth had 44 marbles. She gave 9 marbles to her brother. How many marbles does Beth have now?
__ marbles.

Answer:
The total number of marbles does Beth had is 35 marbles.

Explanation:
Beth had 44 marbles and she gave 9 marbles to her brother and the remaining marbles do Beth had are 44 – 9= 35 marbles.

Question 10.
WRITE Math
Draw a number line and show how to find the difference for 24 – 6 using the break apart method in this lesson.

Lesson Check

Question 1.
What is the difference?

58 – 9 = __

Answer:
The difference of 58 – 9 is 49.

Explanation:
Here, we have started at 58 and subtracted 4, to subtract 4 on the number line jump makes a jump from 58 to 54 and the size of the jump is 4 and we will get 54. Then we have to subtract 5, so we will start from 54 on the number line jump makes a jump from 54 to 49 and the size of the jump is 5 and we will get 49. So the subtraction of 58 – 9 is 49.

Spiral Review

Question 2.
What is the difference?
14 – 6 = __

Answer:
The difference of 14 – 6 is 8.

Explanation:
Here, we have started at 14 and subtracted 3, to subtract 3 on the number line jump makes a jump from 14 to 11 and the size of the jump is 3 and we will get 11. Then we have to subtract 3, so we will start from 11 on the number line jump makes a jump from 11 to 3 and the size of the jump is 3 and we will get 8. So the subtraction of 14 – 6 is 8.

Question 3.
What is the sum?
3 + 6 + 2 = ___

Answer:
The sum of the three numbers is 11.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the three numbers
3 + 6 + 2 is 11.

Question 4.
What is the sum?
64 + 7 = __

Answer:
The sum of the two numbers is 71.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 64 + 7 is 71.

Question 5.
What is the sum?
56 + 18 = __

Answer:
The sum of the two numbers is 74.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 56 + 18  is 74.

### Lesson 5.2 Algebra • Break Apart Numbers to Subtract

Essential Question
How does breaking apart a number make subtracting easier?

Listen and Draw

Draw jumps on the number line to show how to break apart the number to subtract.

Math Talk
MATHEMATICAL PRACTICES
Describe a Method
For one of the problems, describe what you did.

Model and Draw

Break apart the number you are subtracting into tens and ones.
Subtract 10.
Next, subtract 2 to get to 60.
Then subtract 5 more.

So, 72 – 12 = ___

Share and Show MATH BOARD

Break apart the number you are subtracting.
Write the difference.

Question 1.

Answer:
The subtraction of 43 – 18 is 25.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 18 as 10 and 8 and we will subtract 10 and we will get the result as 33. Then we will start from 33 and subtract 3 to get to 30 and then subtract 5 more. So we will get the result as 25, and we will place it on the number line. So the subtraction of 43 – 18 is 25.

Question 2.

Answer:
The subtraction of 45 – 14 is 31.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 14 as 10 and 4 and we will subtract 10 and we will get the result as 35. Then we will start from 35 and subtract 2 to get to 33 and then subtract 2 more. So we will get the result as 31, and we will place it on the number line. So the subtraction of 45 – 14 is 31.

Question 3.
46 – 17 = __

Answer:
The subtraction of 46 – 17 is 29.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 17 as 10 and 7 and we will subtract 10 and we will get the result as 36. Then we will start from 33 and subtract 3 to get to 30 and then subtract 5 more. So we will get the result as 25, and we will place it on the number line. So the subtraction of 46 – 17 is 29.

Question 4.
44 – 16 = __

Answer:
The subtraction of 44 – 16 is 28.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 18 as 10 and 6 and we will subtract 10 and we will get the result as 34. Then we will start from 34 and subtract 4 to get to 30 and then subtract 2 more. So we will get the result as 28, and we will place it on the number line. So the subtraction of 44 – 16 is 28.

On Your Own

Break apart the number you are subtracting. Write the difference.

Question 5.

Answer:
The subtraction of 57 – 15 is 42.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 15 as 10 and 5 and we will subtract 10 and we will get the result as 47. Then we will start from 47 and subtract 3 to get to 44 and then subtract 2 more. So we will get the result as 42, and we will place it on the number line. So the subtraction of 57 – 15 is 42.

Question 6.

Answer:
The subtraction of 63 – 17 is 46.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 17 as 10 and 7 and we will subtract 10 and we will get the result as 53. Then we will start from 53 and subtract 3 to get to 50 and then subtract 4 more. So we will get the result as 46, and we will place it on the number line. So the subtraction of 63 – 17 is 46.

Question 7.
68 – 19 = __

Answer:
The subtraction of 68 – 19 is 49.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 19 as 10 and 9 and we will subtract 10 and we will get the result as 58. Then we will start from 58 and subtract 8 to get to 50 and then subtract 1 more. So we will get the result as 49, and we will place it on the number line. So the subtraction of 68 – 19 is 49.

Question 8.
61 – 18 = __

Answer:
The subtraction of 61 – 18 is 43.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 18 as 10 and 8 and we will subtract 10 and we will get the result as 51. Then we will start from 51 and subtract 1 to get to 50 and then subtract 7 more. So we will get the result as 43, and we will place it on the number line. So the subtraction of 61 – 18 is 43.

Question 9.
THINK SMARTER
Jane has 53 toys in a box. She takes some toys out. Now there are 36 toys in the box. How many toys did Jane take out of the box?

Answer:
The number of toys did Jane take out of the box is 17 toys.

Explanation:
As Jane has 53 toys in a box and she takes some toys out, so Now there are 36 toys in the box. So to find how many toys did Jane take out of the box we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So The subtraction of 36 from 53 is 53 – 36= 17. So the number of toys did Jane take out of the box is 17 toys.

Question 10.
GO DEEPER
Look at Tom’s steps to solve a problem. Solve this problem in the same way.
42 – 15 = ?

Answer:
The subtraction of 42 – 15 is 27.

Explanation:
As we can see in the above image that the subtraction is done by the break apart number process, so the given problem is 42 – 15, so here we will break the 15 into 10 and 5, and then we will break 5 apart into 3 and 2. So first we will subtract 10 then the result will be 42 – 10 is 32 and then we will 2 then the result will be 32 – 2= 30. Now we will subtract 3 then the result will be 30 – 3= 27. So the subtraction of 42 – 15 is 27.

Problem Solving • Applications

Question 11.
38 people are in the library. Then 33 more people go into the library. How many people are in the library now?
__ people

Answer:
The total number of people in the library is 71 people.

Explanation:
As there are 38 people in the library and then 33 more people go into the library, so to find the number of people in the library we will perform addition. So we will add 38 + 33 is 71. The total number of people in the library is 71 people.

Question 12.
MATHEMATICAL PRACTICE
Analyze Alex has 24 toys in a chest. He takes some toys out of the chest. Then there are 16 toys in the chest. How many toys did he take out of the chest?

__ toys

Answer:
There are 8 toys did Alex take out of the chest.

Explanation:
As Alex has 24 toys in a chest and he takes some toys out of the chest. Then there are 16 toys in the chest, so to find the number of toys did Alex has to take out of the chest we will perform subtraction. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 16 from 24 is 24 – 16= 8 and there are 8 toys did Alex take out of the chest.

Question 13.
THINK SMARTER
Gail has two piles of newspapers. There are 32 papers in the first pile. There are 19 papers in the second pile. How many more papers are in the first pile than in the second pile?
__ more papers
Write or draw to explain how you solved the problem.

Answer:
The number of papers more in the first pile than the second pile is 13 more papers.

Explanation:
As Gail has two piles of newspapers and there are 32 papers in the first pile and there are 19 papers in the second pile. So to know how many papers are in the first pile than the second pile we will perform subtraction which is 32 – 19= 13 more papers.

TAKE HOME ACTIVITY
• Ask your child to write a subtraction story that uses 2-digit numbers.

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

### Algebra • Break Apart Numbers to Subtract Homework & practice 5.2

Break apart the number you are subtracting. Write the difference.

Question 1.
81 – 14 = __

Answer:
The subtraction of 81 – 14 is 67.

Explanation:
Here, we have started at 81 and subtracted 7, to subtract 7 on the number line jump makes a jump from 81 to 74 and the size of the jump is 7 and we will get 74. Then we have to subtract 7, so we will start from 74 on the number line jump makes a jump from 74 to 67 and the size of the jump is 7 and we will get 67. So the subtraction of 81 – 14 is 67.

Question 2.
84 – 16 = __

Answer:
The subtraction of 84 – 16 is 68.

Explanation:
Here, we have started at 84 and subtracted 8, to subtract 8 on the number line jump makes a jump from 84 to 76 and the size of the jump is 8 and we will get 76. Then we have to subtract 8, so we will start from 76 on the number line jump makes a jump from 76 to 68 and the size of the jump is 8 and we will get 68. So the subtraction of 84 – 16 is 68.

Question 3.
77 – 14 = __

Answer:
The subtraction of 77 – 14 is 63.

Explanation:
Here, we have started at 77 and subtracted 7, to subtract 7 on the number line jump makes a jump from 77 to 70 and the size of the jump is 7 and we will get 70. Then we have to subtract 7, so we will start from 70 on the number line jump makes a jump from 70 to 63 and the size of the jump is 7 and we will get 63. So the subtraction of 77 – 14 is 63.

Question 4.
83 – 19 = __

Answer:
The subtraction of 83 – 19 is 64.

Explanation:
Here, we have started at 83 and subtracted 10, to subtract 10 on the number line jump makes a jump from 83 to 73 and the size of the jump is 10 and we will get 73. Then we have to subtract 9, so we will start from 73 on the number line jump makes a jump from 73 to 64 and the size of the jump is 9 and we will get 64. So the subtraction of 83 – 19 is 64.

Question 5.
81 – 17 = __

Answer:
The subtraction of 81 – 17 is 64.

Explanation:
Here, we have started at 81 and subtracted 10, to subtract 10 on the number line jump makes a jump from 81 to 71 and the size of the jump is 10 and we will get 71. Then we have to subtract 7, so we will start from 71 on the number line jump makes a jump from 71 to 64 and the size of the jump is 7 and we will get 64. So the subtraction of 81 – 17 is 64.

Question 6.
88 – 13 = __

Answer:
The subtraction of 88 – 13 is 75.

Explanation:
Here, we have started at 88 and subtracted 3, to subtract 3 on the number line jump makes a jump from 83 to 80 and the size of the jump is 3 and we will get 80. Then we have to subtract 10, so we will start from 80 on the number line jump makes a jump from 80 to 75 and the size of the jump is 10 and we will get 75. So the subtraction of 88 – 13 is 75.

Question 7.
84 – 19 = __

Answer:
The subtraction of 84 – 19 is 65.

Explanation:
Here, we have started at 84 and subtracted 10, to subtract 10 on the number line jump makes a jump from 84 to 74 and the size of the jump is 10 and we will get 74. Then we have to subtract 9, so we will start from 74 on the number line jump makes a jump from 74 to 65 and the size of the jump is 9 and we will get 65. So the subtraction of 84 – 19 is 65.

Question 8.
86 – 13 = __

Answer:
The subtraction of 86 – 13 is 73.

Explanation:
Here, we have started at 86 and subtracted 7, to subtract 7 on the number line jump makes a jump from 86 to 79 and the size of the jump is 7 and we will get 79. Then we have to subtract 6, so we will start from 79 on the number line jump makes a jump from 79 to 73 and the size of the jump is 6 and we will get 73. So the subtraction of 86 – 13 is 73.

Question 7.
84 – 19 = __

Answer:
The subtraction of 84 – 19 is 65.

Explanation:
Here, we have started at 84 and subtracted 10, to subtract 10 on the number line jump makes a jump from 84 to 74 and the size of the jump is 10 and we will get 74. Then we have to subtract 9, so we will start from 74 on the number line jump makes a jump from 74 to 65 and the size of the jump is 9 and we will get 65. So the subtraction of 84 – 19 is 65.

Question 8.
86 – 18 = __

Answer:
The subtraction of 86 – 18 is 68.

Explanation:
Here, we have started at 84 and subtracted 9, to subtract 9 on the number line jump makes a jump from 86 to 77 and the size of the jump is 9 and we will get 77. Then we have to subtract 9, so we will start from 77 on the number line jump makes a jump from 77 to 68 and the size of the jump is 9 and we will get 68. So the subtraction of 86 – 18 is 68.

Problem Solving

Solve. Write or draw to explain.

Question 9.
Mr. Pearce bought 43 plants. He gave 14 plants to his sister. How many plants does Mr. Pearce have now?
__ plants

Answer:
The number of plants does Mr. Pearce had is 29 plants.

Explanation:
Mr. Pearce bought 43 plants and he gave 14 plants to his sister. So the number of plants does Mr. Pearce has now is 43 – 14. Here, we have started at 43 and subtracted 7, to subtract 7 on the number line jump makes a jump from 43 to 36 and the size of the jump is 7 and we will get 36. Then we have to subtract 7, so we will start from 36 on the number line jump makes a jump from 7 to 29 and the size of the jump is 7 and we will get 29. So the subtraction of 43 – 14 is 29. So the number of plants does Mr. Pearce had is 29 plants.

Question 10.
WRITE Math
Draw a number line and show how to find the difference for 36 – 17 using the break apart method in this lesson.

Lesson Check

Question 1.
What is the difference?

63 – 19 = __

Answer:
The subtraction of 63 – 19 is 44.

Explanation:
Here, we have started at 63 and subtracted 10, to subtract 10 on the number line jump makes a jump from 63 to 53 and the size of the jump is 10 and we will get 53. Then we have to subtract 9, so we will start from 53 on the number line jump makes a jump from 53 to 44 and the size of the jump is 9 and we will get 44. So the subtraction of 63 – 19 is 44.

Spiral Review

Question 2.
What is the sum?

Answer:
The sum of the two numbers is 37.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 14 + 23 is 37.

Question 3.
What is the sum?
8 + 7 = _

Answer:
The sum of the two numbers is 15.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 8 + 7 is 15.

Question 4.
Write a related subtraction fact for 6 + 8 = 14.
____

Question 5.
John has 7 kites. Annie has 4 kites. How many kites do they have altogether?
__ kites

Answer:
The number of kites does they together have is 7 + 4= 11 kites.

Explanation:
John has 7 kites and Annie has 4 kites, so the number of kites do they together have is 7 + 4= 11 kites.

### Lesson 5.3 Model Regrouping for Subtraction

Essential Question
When do you regroup in subtraction?

Answer:
We will regroup in subtraction when the minuend digit is less than the subtrahend digit. Then we will take carry forward to the lesser digit and then we will perform subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend.

Listen and Draw

Use to model the problem. Draw quick pictures to show your model.

Math Talk
MATHEMATICAL PRACTICES

Describe why you traded a tens block for 10 ones blocks.

Model and Draw

How do you subtract 26 from 53?

Answer:
The subtraction of 53 and 26 is 27.

Explanation:
To perform subtraction for 26 from 53 we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So The subtraction of 26 from 53 is 27.

Step 1
Show 53. Are there enough ones to subtract 26?

Step 2
If there are not enough ones, regroup 1 ten as 10 ones.

Step 3
Subtract 6 ones from 13 ones.

Step 4
Subtract the tens. Write the tens and ones. Write the difference.

Share and Show MATH BOARD

Draw to show the regrouping. Write the difference two ways. Write the tens and ones. Write the number.

Question 1.
Subtract 13 from 41.

__ tens __ ones __

Answer:
The subtraction of 13 from 41 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are four tens blocks and one one’s block. So here we need to subtract 13 from 41, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 41 is 41 – 13= 28 and there are two tens and eight ones.

Question 2.
Subtract 9 from 48.

__ tens __ ones __

Answer:
The subtraction of 9 from 48 is 39. And there are three tens and nine ones.

Explanation:
In the above image, we can see that there is four tens blocks and eight one’s block. So here we need to subtract 9 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 48 is 48 – 9= 39 and there are three tens and nine ones.

Question 3.
Subtract 28 from 52.

__ tens __ ones __

Answer:
The subtraction of 28 from 52 is 24. And there are two tens and four ones.

Explanation:
In the above image, we can see that there is five tens blocks and two one’s block. So here we need to subtract 28 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 52 is 52 – 28= 24 and there are two tens and four ones.

On Your Own

Draw to show the regrouping. Write the difference two ways. Write the tens and ones. Write the number.

Question 4.
Subtract 8 from 23

__ tens __ ones __

Answer:
The subtraction of 8 from 23 is 15. And there are one ten and five ones.

Explanation:
In the above image, we can see that there is one tens block and five ones block. So here we need to subtract 8 from 23, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 8 from 23 is 23 – 8= 15 and there are one ten and five ones.

Question 5.
Subtract 36 from 45.

__ tens __ ones __

Answer:
The subtraction of 36 from 45 is 9. And there are no tens and nine ones.

Explanation:
In the above image, we can see that there is five tens blocks and two one’s block. So here we need to subtract 36 from 45, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 36 from 45 is 45 – 36= 9 and there are no tens and nine ones.

Question 6.
Subtract 6 from 43.

__ tens __ ones __

Answer:
The subtraction of 6 from 43 is 37. And there are three tens and seven ones.

Explanation:
In the above image, we can see that there is four tens blocks and three one’s block. So here we need to subtract 6 from 43, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 6 from 43 is 43 – 6= 37 and there are three tens and seven ones.

Question 7.
Subtract 39 from 67

__ tens __ ones __

Answer:
The subtraction of 39 from 67 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there is six tens blocks and seven ones block. So here we need to subtract 28 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 39 from 67 is 67 – 39= 28 and there are two tens and eight ones.

Question 8.
Subtract 21 from 50.

__ tens __ ones __

Answer:
The subtraction of 21 from 50 is 29. And there are two tens and nine ones.

Explanation:
In the above image, we can see that there is five tens blocks and zero ones in ones block. So here we need to subtract 21 from 50, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 21 from 50 is 50 – 21= 29 and there are two tens and nine ones.

Question 9.
Subtract 29 from 56

__ tens __ ones __

Answer:
The subtraction of 29 from 56 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are five tens blocks and two one’s blocks. So here we need to subtract 28 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 56 is 56 – 29= 27 and there are two tens and seven ones.

Question 10.
GO DEEPER
Draw to find what number was subtracted from 53.

Subtract __ from 53.
3 tens 4 ones
34

Problem Solving • Applications

Write or draw to explain.

Question 11.
THINK SMARTER
Billy has 18 fewer marbles than Sara. Sara has 34 marbles. How many marbles does Billy have?

__ marbles

Answer:
The total number of marbles does Billy had is 16 marbles.

Explanation:
As Billy has 18 fewer marbles than Sara and Sara has 34 marbles, so here fewer marbles means we will do subtraction. So the number of marbles do Billy had is 34 – 18= 16 marbles. Here we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the total number of marbles does Billy had is 16 marbles.

Question 12.
THINK SMARTER
There are 67 toy animals in the store. Then the clerk sells 19 toy animals. How many toy animals are in the store now?
Draw to show how to find the answer.

__ toy animals

Answer:
The number of toys in the store is 48 toys.

Explanation:
As there are 67 toy animals in the store and then the clerk sells 19 toy animals, so to find how many toys in the store is we will perform subtraction. Here we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of toys in the store is 67 – 19= 48 toys.

Describe how you solved the problem.
__________________________
__________________________
__________________________

Answer:
The problem is solved by using regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend.

TAKE HOME ACTIVITY
• Ask your child to write a subtraction story and then explain how to solve it.

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

### Model Regrouping for Subtraction Homework & Practice 5.3

Draw to show the regrouping. Write the difference two ways. Write the tens and ones. Write the number.

Question 1.
Subtract 9 from 35

__ tens __ ones __

Answer:
The subtraction of 9 from 35 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are three tens blocks and five one’s block. So here we need to subtract 9 from 35, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 35 is 35 – 9= 26 and there are two tens and six ones.

Question 2.
Subtract 14 from 52.

__ tens __ ones __

Answer:
The subtraction of 9 from 35 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are three tens blocks and five one’s block. So here we need to subtract 9 from 35, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 35 is 35 – 9= 26 and there are two tens and six ones.

Problem Solving

Choose a way to solve. Write or draw to explain.

Question 3.
Mr. Ortega made 51 cookies. He gave 14 cookies away. How many cookies does he have now?
__ cookies

Answer:
The cookies that Mr. Ortega had now are 37 cookies.

Explanation:
As Mr. Ortega made 51 cookies and he gave 14 cookies away. So to find the number of cookies does Ortega had we will do regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 51 – 14 is 37 cookies. The cookies that
Mr. Orteg had now is 37 cookies.

Question 4.
WRITE Math
Draw a quick picture for 37. Draw to show how you would subtract 19 from 37. Write to explain what you did.

__________________________
__________________________
__________________________

Answer:
The subtraction of 37 – 19 is 18. The subtraction is done by the regrouping of subtraction.

Explanation:

The subtraction of 37 – 19 is 18. The subtraction is done by the regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 37 – 19 is 18.

Lesson Check

Question 1.
Subtract 9 from 36. What is the difference?

Answer:
The subtraction of 9 from 36 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are three tens blocks and six one’s blocks. So here we need to subtract 9 from 36, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 36 is 36 – 9= 27 and there are two tens and seven ones.

Question 2.
Subtract 28 from 45. What is the difference?

Answer:
The subtraction of 28 from 45 is 17. And there are one ten and seven ones.

Explanation:
In the above image, we can see that there are four tens blocks and five one’s blocks. So here we need to subtract 28 from 45, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 45 is 45 – 28= 17 and there are one ten and seven ones.

Spiral Review

Question 3.
What is the difference?
51 – 8 = __

Answer:
The subtraction of 8 from 51 is 43. And there are four tens and three ones.

Explanation:
In the above image, we can see that there are three tens blocks and six one’s blocks. So here we need to subtract 8 from 51, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 8 from 51 is 51 – 8= 43 and there are four tens and three ones.

Question 4.
What is the sum?
38 + 35 = __

Answer:
The sum of the two numbers is 73.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 38 + 35 is 73.

Question 5.
What is the sum?

Answer:
The sum of the three numbers is 90.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the three numbers 63 + 18 + 9 is 90.

### Lesson 5.4 Model and Record 2-Digit Subtraction

Essential Question
How do you record 2-digit subtraction?

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

Listen and Draw
Use to model the problem. Draw quick pictures to show your model.

Math Talk
MATHEMATICAL PRACTICE

Explain a Method
Did you trade blocks in your model? Explain why or why not.

Model and Draw

Trace over the quick pictures in the steps. Subtract.

Share and Show MATH BOARD

Draw a quick picture to solve. Write the difference.

Question 1.

Answer:
The subtraction of 15 from 47 is 32. And there are three ten and two ones.

Explanation:
In the above image, we can see that there are four tens and seven ones. So here we need to subtract 15 from 47, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 15 from 47 is 47 – 15= 32 and there are three ten and two ones.

Question 2.

Answer:
The subtraction of 18 from 32 is 14. And there are one ten and four ones.

Explanation:
In the above image, we can see that there are three tens and five ones. So here we need to subtract 18 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 32 is 32 – 18= 14 and there are one ten and four ones.

On Your Own

Draw a quick picture to solve. Write the difference.

Question 3.

Answer:
The subtraction of 29 from 35 is 6. And there are zero tens and six ones.

Explanation:
In the above image, we can see that there are three tens and five ones. So here we need to subtract 29 from 35, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 35 is 35 – 29= 6 and there are zero tens and six ones.

Question 4.

Answer:
The subtraction of 5 from 28 is 23. And there are two tens and three ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 5 from 28, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 5 from 28 is 28 – 5= 23 and there are two ten and three ones.

Question 5.

Answer:
The subtraction of 26 from 53 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are five tens and three ones. So here we need to subtract 26 from 53, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 26 from 53 is 53 – 26= 27 and there are two tens and seven ones.

Question 6.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 7.
GO DEEPER
There are 16 robins in the trees. 24 more fly in. Then 28 robins fly away. How many robins are still in the trees?

__ robbins

Answer:
The total number of robins are still on the trees is 12 robins.

Explanation:
As there are 16 robins on the tree and 24 more fly-ins. So the number of robins is 24 + 16= 40. Now 28 robins fly away, so the total number of robins are still on the trees is 40 – 28= 12 robins.

Problem Solving • Applications

Question 8.
THINK SMARTER
Claire’s puzzle has 85 pieces. She has used 46 pieces so far. How many puzzle pieces have not been used yet?

__ puzzle pieces

Answer:
The number of puzzle pieces that have not been used is 39 puzzle pieces.

Explanation:
As Claire’s puzzle has 85 pieces and she has used 46 pieces so far. So the number of puzzle pieces that have not been used is
85 – 46= 39 puzzle pieces.

Question 9.
MATHEMATICAL PRACTICE
Analyze There were some people at the park. 24 people went home. Then there were 19 people at the park. How many people were at the park before?
__ people

Answer:
The total number of people at the park before is 43 people.

Explanation:
As there were some people at the park and 24 people went home. Then there were 19 people at the park, so the total number of people at the park before is 24 + 19= 43 people.

Question 10.
THINK SMARTER
Mr. Sims has a box of 44 erasers. He gives 28 erasers to his students. How many erasers does Mr. Sims have now? Show how you solved the problem.

__ erasers

Answer:
The number of erasers does Mr. Sims has now is 16 erasers.

Explanation:
As Mr. Sims has a box of 44 erasers and he gives 28 erasers to his students. So the number of erasers does Mr. Sims has now is
44 – 28= 16 erasers. Here we have solved by using regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 44 is 44 – 28= 16 erasers. The number of erasers does Mr. Sims has now is 16 erasers.

TAKE HOME ACTIVITY
• Write 73 − 28 on a sheet of paper. Ask your child if he or she would regroup to find the difference.

Answer:
The subtraction of 73 – 28 is 45.

Explanation:
The subtraction of 73 – 28 is 45. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 73 – 28 is 45.

### Model and Record 2-Digit Subtraction Homework & Practice 5.4

Draw a quick picture to solve. Write the difference.

Question 1.

Answer:
The subtraction of 17 from 43 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are four tens and three ones. So here we need to subtract 17 from 43, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 17 from 43 is 43 – 17= 26 and there are two tens and six ones.

Question 2.

Answer:
The subtraction of 29 from 38 is 9. And there are zero tens and nine ones.

Explanation:
In the above image, we can see that there are three tens and eight ones. So here we need to subtract 29 from 38, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 38 is 38 – 29= 9 and there are zero tens and nine ones.

Problem Solving

Solve. Write or draw to explain.

Question 3.
Kendall has 63 stickers. Her sister has 57 stickers. How many more stickers does Kendall have than her sister?
__ more stickers.

Answer:
The number of stickers does Kendall has than her sister is 6 more stickers.

Explanation:
As Kendall has 63 stickers and her sister has 57 stickers, so the number of stickers does Kendall has than her sister is 63 – 57= 6 more stickers. Here we will perform regrouping of subtraction. Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 57 from 63 is 63 – 57= 9 stickers.

Question 4.
WRITE Math
Draw a quick picture to show the number 24. Then draw a quick picture to show 24 after you have regrouped 1 ten as 1-ones. Explain how both pictures show the same number, 24.
__________________________
__________________________
__________________________

Lesson Check

Question 1.
What is the difference?

Answer:
The subtraction of 18 from 47 is 29. And there are two tens and nine ones.

Explanation:
In the above image, we can see that there are four tens and seven ones. So here we need to subtract 18 from 47, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 47 is 47 – 18= 29 and there are two tens and nine ones.

Question 2.
What is the difference?

Answer:
The subtraction of 29 from 33 is 4. And there are zero tens and four ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 29 from 33, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 33 is 33 – 29= 4 and there are zero tens and four ones.

Spiral Review

Question 3.
What is the difference?
10 – 6 = __

Answer:
The difference is 4.

Explanation:
The difference between 10 and 6 is 4.

Question 4.
What is the sum?
16 + 49 = __

Answer:
The sum of the two numbers is 65.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 16 + 49  is 65.

Question 5.
What is the sum?
28 + 8 = __

Answer:
The sum of the two numbers is 36.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 28 + 8  is 36.

Question 6.
What is the difference?
52 – 6 = __

Answer:
The difference is 46.

Explanation:
The difference between 52 and 6 is 46.

### Lesson 5.5 2-Digit Subtraction

Essential Question
How do you record the steps when subtracting 2-digit numbers?

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

Listen and Draw

Draw a quick picture to model each problem.

Math Talk
MATHEMATICAL PRACTICES
Use Reasoning Explain how you know when to regroup.

Answer:
We will perform regrouping when the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. And Regrouping in subtraction is a process of exchanging one ten into ten ones.

Model and Draw

Share and Show MATH BOARD

Regroup if you need to. Write the difference.

Question 1.

Answer:
The subtraction of 14 from 31 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and one in ones place. So here we need to subtract 14 from 31, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 14 from 31 is 31 – 14= 19 and there are one ten and nine ones.

Question 2.

Answer:
The subtraction of 21 from 56 is 35. And there are three tens and five ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 21 from 56, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 21 from 56 is 56 – 21= 35 and there are three tens and five ones.

Question 3.

Answer:
The subtraction of 35 from 72 is 37. And there are three tens and seven ones.

Explanation:
In the above image, we can see that there are seven tens and two ones. So here we need to subtract 35 from 72, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 35 from 72 is 72 – 35= 37 and there are three tens and seven ones.

On Your Own
Regroup if you need to. Write the difference.

Question 4.

Answer:
The subtraction of 14 from 23 is 9. And there are zero tens and nine ones.

Explanation:
In the above image, we can see that there are two tens and three one’s. So here we need to subtract 14 from 23, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 14 from 23 is 23 – 14= 9 and there are zero tens and nine ones.

Question 5.

Answer:
The subtraction of 57 from 87 is 30. And there are three ten and zero ones.

Explanation:
In the above image, we can see that there are eight tens and seven one’s. So here we need to subtract 57 from 87, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 57 from 87 is 87 – 57= 30 and there are three ten and zero ones.

Question 6.

Answer:
The subtraction of 18 from 34 is 16. And there are one ten and six ones.

Explanation:
In the above image, we can see that there are three tens and four one’s. So here we need to subtract 18 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 34 is 34 – 18= 16 and there are one ten and six ones.

Question 7.

Answer:
The subtraction of 13 from 61 is 48. And there are four ten and eight ones.

Explanation:
In the above image, we can see that there are six tens and one ones. So here we need to subtract 13 from 61, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 61 is 61 – 13= 48 and there are four ten and eight ones.

Question 8.

Answer:
The subtraction of 18 from 45 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 45 is 45 – 18= 27 and there are two ten and nine ones.

Question 9.

Answer:
The subtraction of 36 from 52 is 16. And there are one ten and six ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 36 from 52, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 36 from 52 is 52 – 36= 16 and there are one ten and six ones.

Question 10.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 11.

Answer:
The subtraction of 43 from 75 is 32. And there are three tens and two ones.

Explanation:
In the above image, we can see that there are seven tens and five ones. So here we need to subtract 43 from 75, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 43 from 75 is 75 – 43= 32 and there are three tens and two ones.

Question 12.

Answer:
The subtraction of 27 from 56 is 29. And there are two tens and nine ones.

Explanation:
In the above image, we can see that there are five tens and six ones. So here we need to subtract 27 from 56, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 27 from 56 is 56 – 27= 29 and there are two tens and nine ones.

Question 13.

Answer:
The subtraction of 29 from 94 is 65. And there are six tens and five ones.

Explanation:
In the above image, we can see that there are nine tens and four one’s. So here we need to subtract 29 from 94, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 94 is 94 – 29= 65 and there are six tens and five ones.

Question 14.

Answer:
The subtraction of 39 from 87 is 48. And there are four tens and eight ones.

Explanation:
In the above image, we can see that there are eight tens and seven one’s. So here we need to subtract 39 from 87, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 39 from 87 is 87 – 39= 48 and there are four tens and eight ones.

Question 15.

Answer:
The subtraction of 46 from 83 is 37. And there are three tens and seven ones.

Explanation:
In the above image, we can see that there are eight tens and three one’s. So here we need to subtract 46 from 83, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 46 from 83 is 83 – 46= 37 and there are three tens and seven ones.

Question 16.
THINK SMARTER
Spencer wrote 5 fewer stories than Katie. Spencer wrote 18 stories. How many stories did Katie write?

__ stories

Answer:
The number of stories written by Katie is 13 stories.

Explanation:
As Spencer wrote 5 fewer stories than Katie and Spencer wrote 18 stories. As Spencer wrote 5 fewer stories than Katie so we will perform subtraction. So the number of stories written by Katie is 18 – 5= 13 stories.

Problem Solving • Applications

Question 17.
MATHEMATICAL PRACTICE
Explain a Method Circle the problems below that you could use mental math to solve.
54 – 10 = __
63 – 27 = __
93 – 20 = __
39 – 2 = __
41 – 18 = __
82 – 26 = __
Explain your choices
__________________________
__________________________
__________________________

Answer:

Explanation:

Question 18.
THINK SMARTER
There are 34 chickens in the barn. If 16 chickens go outside into the yard, how many chickens will still be in the barn?
Circle the number from the box to make the sentence true.

Answer:
The number of chickens will be in the barn will be 18 chickens.

Explanation:
As there are 34 chickens in the barn and if 16 chickens go outside into the yard then the number of chickens will be in the barn will be 34 – 16= 18 chickens. Here we have performed regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of chickens will be in the barn will be 34 – 16= 18 chickens.

TAKE HOME ACTIVITY
• Ask your child to write a 2-digit subtraction problem with no regrouping needed. Have your child explain why he or she chose those numbers.

Answer:
The subtraction of 56 – 23 is 33.

Explanation:
Given that we need to perform the subtraction without regrouping, so we need to take the minuend number greater than the subtrahend. Then we need not perform the regrouping, so we will take the numbers to be subtracted be 56 – 23= 33.

### 2-Digit Subtraction Homework & practice 5.5

Regroup if you need to. Write the difference.

Question 1.

Answer:
The subtraction of 28 from 47 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 28 from 47, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 47 is 47 – 28= 19 and there are one ten and nine ones.

Question 2.

Answer:
The subtraction of 18 from 33 is 15. And there are one ten and five ones.

Explanation:
In the above image, we can see that there are three tens and three ones. So here we need to subtract 18 from 33, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 33 is 33 – 18= 15 and there are one ten and five ones.

Question 3.

Answer:
The subtraction of 14 from 28 is 19. And there are one ten and four ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 14 from 28, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 14 from 28 is 28 – 14= 14 and there are one ten and four ones.

Question 4.

Answer:
The subtraction of 19 from 66 is 47. And there are four ten and seven ones.

Explanation:
In the above image, we can see that there are six tens and six ones. So here we need to subtract 19 from 66, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 66 is 66 – 19= 47 and there are four ten and seven ones.

Question 5.

Answer:
The subtraction of 26 from 77 is 51. And there are five tens and one in one’s place.

Explanation:
In the above image, we can see that there are seven tens and seven ones. So here we need to subtract 26 from 77, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 26 from 77 is 77 – 26= 51 and there are five tens and nine one in one’s place.

Question 6.

Answer:
The subtraction of 34 from 58 is 24. And there are two tens and four ones.

Explanation:
In the above image, we can see that there are five tens and eight ones. So here we need to subtract 34 from 58, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 34 from 58 is 58 – 34= 24 and there are two tens and four ones.

Question 7.

Answer:
The subtraction of 25 from 52 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are five tens and five ones. So here we need to subtract 25 from 52, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 25 from 52 is 52 – 25= 27 and there are two tens and seven ones.

Question 8.

Answer:
The subtraction of 49 from 87 is 38. And there are three tens and eight ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 49 from 87, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 49 from 87 is 87 – 49= 38 and there are three tens and eight ones.

Problem Solving

Solve. Write or draw to explain.

Question 9.
Mrs. Paul bought 32 erasers. She gave 19 erasers to students. How many erasers does she still have?

Answer:
The number of erasers Mrs. Paul had is 13 erasers.

Explanation:
As Mrs. Paul bought 32 erasers and she gave 19 erasers to students. So to find how many erasers does she still has is we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of erasers Mrs. Paul had is
32 – 9= 13 erasers.

Question 10.
WRITE Math
Write a few sentences about different ways to show subtraction for a problem like 32 – 15.
__________________________
__________________________

Lesson Check

Question 1.
What is the difference?

Answer:
The subtraction of 39 from 48 is 9. And there are zero tens and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 39 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 39 from 48 is 48 – 39= 9 and there are zero tens and nine ones.

Question 2.
What is the difference?

Answer:
The subtraction of 66 from 84 is 18. And there are one ten and eight ones.

Explanation:
In the above image, we can see that there are eight tens and four ones. So here we need to subtract 66 from 84, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 66 from 84 is 84 – 66= 18 and there are one ten and eight ones.

Spiral Review

Question 3.
What is the difference?

Answer:
The subtraction of 19 from 32 is 13. And there are one ten and three ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 32 is 32 – 19= 13 and there are one ten and three ones.

Question 4.
Write an addition fact that will give the same sum as 8 + 7.
10 + __

Answer:
The number to get the sum as 15 is we should add 5 to the number 10.

Explanation:
The sum of 8 + 7 is 15. So to get the 15 as a result of the given number, we will add 5 to the number 10. So we will get the result as 15.

Question 5.
27 boys and 23 girls go on a field trip to the museum. How many children go to the museum?
__ children

Answer:
The number of children who went to the museum is 50 children.

Explanation:
As there are 27 boys 23 girls on the field trip to the museum, so the number of children who went to the museum is 27 + 23= 50 children

Question 6.
There were 17 berries in the basket. Then 9 berries are eaten. How many berries are there now?
__ berries

Answer:
The number of berries will be 8 berries.

Explanation:
As there are 17 berries in the basket and 9 berries are eaten, then the number of berries will be 17 – 9= 8 berries. Here we have performed regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of berries will be 17 – 9= 8 berries

### Lesson 5.6 Practice 2-Digit Subtraction

Essential Question
How do you record the steps when subtracting 2-digit numbers?

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

Listen and Draw

Choose one way to solve the problem.
Draw or write to show what you did.

Math Talk
MATHEMATICAL PRACTICES
Describe a different way that you could have solved the problem.

Model and Draw

Carmen had 50 game cards. Then she gave 16 game cards to Theo. How many game cards does Carmen have now?

Answer:
The number of game cards will be 50 – 16= 34 game cards.

Explanation:
As the Carmen had 50 game cards and she gave 16 game cards to Theo, then the number of game cards will be 50 – 16= 34 game cards. Here we have performed regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of game cards will be 50 – 16= 34 game cards.
Step 1
Look at the ones. There are not enough ones to subtract 6 from 0. So, regroup

Step 2
Subtract the ones.
10 – 6 = 4

Step 3
Subtract the tens.
4 – 1 = 3

Share and Show MATH BOARD

Write the difference.

Question 1.

Answer:
The subtraction of 19 from 38 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and eight ones. So here we need to subtract 19 from 38, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 38 is 38 – 19= 19 and there are one ten and nine ones.

Question 2.

Answer:
The subtraction of 32 from 65 is 33. And there are three tens and three ones.

Explanation:
In the above image, we can see that there are six tens and five ones. So here we need to subtract 32 from 65, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 32 from 65 is 65 – 32= 33 and there are three tens and three ones.

Question 3.

Answer:
The subtraction of 12 from 50 is 38. And there are three tens and eight ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 12 from 50, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 12 from 50 is 50 – 12= 38 and there are three tens and eight ones.

Question 4.

Answer:
The subtraction of 4 from 23 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are two tens and three ones. So here we need to subtract 4 from 23, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 4 from 23 is 23 – 4= 19 and there are one ten and nine ones.

Question 5.

Answer:
The subtraction of 38 from 70 is 32. And there are three tens and two ones.

Explanation:
In the above image, we can see that there are seven tens and zero one. So here we need to subtract 38 from 70, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 38 from 70 is 70 – 38= 32 and there are three tens and two ones.

Question 6.

Answer:
The subtraction of 17 from 52 is 35. And there are three tens and five ones.

Explanation:
In the above image, we can see that there are five tens and two one. So here we need to subtract 17 from 52, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 17 from 52 is 52 – 17= 35 and there are three tens and five ones.

On Your Own

Write the difference.

Question 7.

Answer:
The subtraction of 24 from 41 is 17. And there are one ten and seven ones.

Explanation:
In the above image, we can see that there are four tens and one in ones place. So here we need to subtract 24 from 41, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 24 from 41 is 41 – 24= 17 and there are one ten and nine ones.

Question 8.

Answer:
The subtraction of 16 from 58 is 42. And there are four tens and two ones.

Explanation:
In the above image, we can see that there are five tens and eight ones. So here we need to subtract 16 from 58, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 16 from 58 is 58 – 16= 42 and there are four tens and two ones.

Question 9.

Answer:
The subtraction of 13 from 60 is 47. And there are four tens and seven ones.

Explanation:
In the above image, we can see that there are six tens and zero ones. So here we need to subtract 13 from 60, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 60 is 60 – 13= 47 and there are four tens and seven ones.

Question 10.

Answer:
The subtraction of 47 from 52 is 5. And there are zero tens and five ones.

Explanation:
In the above image, we can see that there are five tens and two ones. So here we need to subtract 47 from 52, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 47 from 52 is 52 – 47= 5 and there are zero tens and five ones.

Question 11.

Answer:
The subtraction of 46 from 72 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are seven tens and two ones So here we need to subtract 46 from 72, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 46 from 72 is 72 – 46= 26 and there are two tens and six ones.

Question 12.

Answer:
The subtraction of 6 from 37 is 31. And there are three tens and one in ones place.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 6 from 37, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 6 from 37 is 37 – 6= 31 and there are three tens and one in ones place.

Question 13.

Answer:
The subtraction of 46 from 74 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 46 from 74, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 46 from 74 is 74 – 46= 28 and there are two tens and eight ones.

Question 14.

Answer:
The subtraction of 18 from 90 is 72. And there are seven tens and two ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 18 from 90, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 90 is 90 – 18= 72 and there are seven tens and two ones.

Question 15.
GO DEEPER
Write the missing numbers in the subtraction problems. The regrouping for each problem is shown.

Answer:
The missing values is 75 and 28 and 83 and 58.

Explanation:

Here, to find the missing values we will find with the help of the carry forward numbers. As the carry forward are 6 and 15 so the minuend is 75 as the result is 47, so the other number will be 75 – 47= 28. As the carry forward are 7 and 13 so the minuend is 83 as the result is 25, so the other number will be 83 – 58= 25.

Question 16.
THINK SMARTER
Adam takes 38 rocks out of a box. There are 23 rocks left in the box. How many rocks were in the box to start?

__ rocks

Answer:
The number of rocks were in the box to start is 61 rocks.

Explanation:
Adam takes 38 rocks out of a box and there are 23 rocks left in the box. So to find the number of rocks were in the box we will add the rocks that adam had took and the number of rocks left in the box. So the number of rocks were in the box to start is
38 + 28= 61 rocks.

TAKE HOME ACTIVITY
• Ask your child to show you one way to find 80 − 34.

Answer:
The subtraction of 34 from 80 is 46. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are eight tens and zero ones. So here we need to subtract 34 from 80, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 34 from 80 is 80 – 34= 46 and there are four tens and six ones.

### Practice 2-Digit Subtraction Homework & Practice 5.6

Write the difference.

Question 1.

Answer:
The subtraction of 18 from 50 is 32. And there are three tens and two ones.

Explanation:
In the above image, we can see that there are five tens and zero ones. So here we need to subtract 18 from 50, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 18 from 50 is 50 – 18= 32 and there are three tens and two ones.

Question 2.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 3.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 4.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 5.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 6.

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Problem Solving

Solve. Write or draw to explain.

Question 7.
Julie has 42 sheets of paper. She gives 17 sheets to Kari. How many sheets of paper does Julie have now?
__ sheets of paper

Question 8.
WRITE Math
Draw and write to explain how these two problems are different: 35 – 15 = ________ and 43 – 26 = _______
__________________________

Lesson Check

Question 1.
What is the difference?

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Question 2.
What is the difference?

Answer:
The subtraction of 13 from 32 is 19. And there are one ten and nine ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 32 is 32 – 13= 19 and there are one ten and nine ones.

Spiral Review

Question 3.
What is the sum?
9 + 9 = __

Answer:
The sum of the two numbers is 18.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 9 + 9 is 18.

Question 4.
What is the difference?
14 – 7 = __

Answer:
The subtraction of 7 from 14 is 7. And there are zero tens and seven ones.

Explanation:
In the above image, we can see that there are one ten and four ones. So here we need to subtract 7 from 14, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 7 from 14 is 14 – 7= 7 and there are zero tens and seven ones.

Question 5.
What is the sum?
36 + 25 = __

Answer:
The sum of the two numbers is 61.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 36 + 25 is 61.

Question 6.
What is the sum?
7 + 2 + 3 = __

Answer:
The sum of the three numbers is 12.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the three numbers 7 + 2 + 3 is 12.

### 2-Digit Subtraction Mid-Chapter Checkpoint

Concepts and Skills

Break apart the number you are subtracting. Use the number line to help. Write the difference.

Question 1.
34 – 8 = __

Answer:
The subtraction of 8 from 34 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are three tens and four ones. So here we need to subtract 8 from 34, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 8 from 34 is 34 – 8= 26 and there are two tens and six ones.

Question 2.
45 – 17 = __

Answer:
The subtraction of 17 from 45 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are four tens and five ones. So here we need to subtract 17 from 45, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 17 from 45 is 45 – 17= 28 and there are two tens and eight ones.

Draw a quick picture to solve. Write the difference.

Question 3.

Answer:
The subtraction of 29 from 42 is 13. And there are one ten and three ones.

Explanation:
In the above image, we can see that there are four tens and two ones. So here we need to subtract 29 from 42, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 42 is 42 – 29= 13 and there are one ten and three ones.

Question 4.

Answer:
The subtraction of 23 from 54 is 31. And there are three tens and one in ones place.

Explanation:
In the above image, we can see that there are five tens and four ones. So here we need to subtract 23 from 54, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 23 from 54 is 54 – 23= 31 and there are three tens and one in ones place

Write the difference.

Question 5.

Answer:
The subtraction of 43 from 78 is 35. And there are three tens and five ones.

Explanation:
In the above image, we can see that there are seven tens and eight ones. So here we need to subtract 43 from 78, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 43 from 78 is 78 – 43= 35 and there are three tens and five ones.

Question 6.

Answer:
The subtraction of 26 from 60 is 34. And there are three tens and four ones.

Explanation:
In the above image, we can see that there are six tens and zero ones. So here we need to subtract 26 from 60, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 26 from 60 is 60 – 26= 34 and there are three tens and four ones.

Question 7.

Answer:
The subtraction of 37 from 85 is 48. And there are four tens and eight ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 37 from 85, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 37 from 85 is 85 – 37= 48 and there are four tens and eight ones.

Question 8.
THINK SMARTER
Marissa had 51 toy dinosaurs. She gave 14 toy dinosaurs to her brother. How many toy dinosaurs does she have now?

__ toy dinosaurs

Answer:
The number of dinosaurs does Marissa has is 37 dinosaurs.

Explanation:
Marissa had 51 toy dinosaurs and she gave 14 toy dinosaurs to her brother. So to how many toy dinosaurs does she have we will perform regrouping of subtraction. Here we have solved by using regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of dinosaurs does Marissa has is 51 – 14= 37 dinosaurs.

### Lesson 5.7 Rewrite 2-Digit Subtraction

Essential Question
What are two different ways to write subtraction problems?

Math Talk
MATHEMATICAL PRACTICES
Explain why it is important to line up the digits of the numbers in columns.

Model and Draw

What is 81 – 36?
Rewrite the subtraction problem.
Then find the difference.

Step 1
For 81, write the tens digit in the tens column. Write the ones digit in the ones column.

Step 2
Look at the ones. Regroup if you need to.

Share and Show MATH BOARD

Rewrite the subtraction problem. Then find the difference.

Question 1.

Answer:
The subtraction of 37 from 4 is 33. And there are three tens and three ones.

Explanation:
In the above image, we can see that there are three tens and four ones. So here we need to subtract 4 from 37, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 4 from 37 is 37 – 4= 33 and there are three tens and three ones.

Question 2.

Answer:
The subtraction of 24 from 48 is 24. And there are two tens and four ones.

Explanation:
In the above image, we can see that there are four tens and eight ones. So here we need to subtract 24 from 48, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 24 from 48 is 48 – 24= 24 and there are two tens and four ones.

Question 3.

Answer:
The subtraction of 37 from 85 is 48. And there are four tens and eight ones.

Explanation:
In the above image, we can see that there are eight tens and five ones. So here we need to subtract 37 from 85, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 37 from 85 is 85 – 37= 48 and there are four tens and eight ones.

Question 4.

Answer:
The subtraction of 19 from 63 is 44. And there are four tens and four ones.

Explanation:
In the above image, we can see that there are six tens and three ones. So here we need to subtract 19 from 63, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 63 is 63 – 19= 44 and there are four tens and four ones.

Question 5.

Answer:
The subtraction of 37 from 62 is 25. And there are two tens and five ones.

Explanation:
In the above image, we can see that there are six tens and two ones. So here we need to subtract 37 from 62, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 37 from 62 is 62 – 37= 25 and there are two tens and five ones.

Question 6.

Answer:
The subtraction of 27 from 51 is 24. And there are two tens and four ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 27 from 51, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 27 from 51 is 51 – 27= 24 and there are two tens and four ones.

Question 7.

Answer:
The subtraction of 3 from 76 is 73. And there are seven tens and three ones.

Explanation:
In the above image, we can see that there are seven tens and three ones. So here we need to subtract 3 from 76, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 3 from 76 is 76 – 3= 73 and there are seven tens and three ones.

Question 8.

Answer:
The subtraction of 48 from 95 is 47. And there are four tens and seven ones.

Explanation:
In the above image, we can see that there are nine tens and five ones. So here we need to subtract 48 from 95, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 48 from 95 is 95 – 48= 47 and there are four tens and seven ones.

On Your Own

Rewrite the subtraction problem. Then find the difference.

Question 9.

Answer:
The subtraction of 8 from 49 is 41. And there are four tens and one in one’s place.

Explanation:
In the above image, we can see that there are four tens and nine ones. So here we need to subtract 8 from 49, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 8 from 49 is 49 – 8= 41 and there are four tens and one in one’s place.

Question 10.

Answer:
The subtraction of 47 from 85 is 38. And there are three tens and eight ones.

Explanation:
In the above image, we can see that there are eight tens and five ones. So here we need to subtract 47 from 85, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 47 from 85 is 85 – 47= 38 and there are three tens and eight ones.

Question 11.

Answer:
The subtraction of 23 from 63 is 40. And there are four tens and zero ones.

Explanation:
In the above image, we can see that there are six tens and three ones. So here we need to subtract 23 from 63, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 23 from 63 is 63 – 23= 40 and there are four tens and zero ones.

Question 12.

Answer:
The subtraction of 23 from 51 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are five tens and one in one’s place. So here we need to subtract 23 from 51, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 23 from 51 is 51 – 23= 28 and there are two tens and eight ones.

Question 13.

Answer:
The subtraction of 15 from 60 is 45. And there are four tens and five ones.

Explanation:
In the above image, we can see that there are six tens and zero ones. So here we need to subtract 15 from 60, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 15 from 60 is 60 – 15= 45 and there are four tens and four ones.

Question 14.

Answer:
The subtraction of 58 from 94 is 36. And there are three tens and six ones.

Explanation:
In the above image, we can see that there are nine tens and four ones. So here we need to subtract 58 from 94, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 58 from 94 is 94 – 58= 36 and there are three tens and six ones.

Question 15.

Answer:
The subtraction of 20 from 47 is 27. And there are two tens and seven ones.

Explanation:
In the above image, we can see that there are four tens and seven ones. So here we need to subtract 20 from 47, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 20 from 47 is 47 – 20= 27 and there are two tens and seven ones.

Question 16.

Answer:
The subtraction of 9 from 35 is 26. And there are two tens and six ones.

Explanation:
In the above image, we can see that there are three tens and five ones. So here we need to subtract 9 from 35, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 35 is 35 – 9= 26 and there are two tens and six ones.

Question 17.

Answer:
The subtraction of 10 from 78 is 68. And there are six tens and eight ones.

Explanation:
In the above image, we can see that there are seven tens and eight ones. So here we need to subtract 10 from 78, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 10 from 78 is 78 – 10= 68 and there are six tens and eight ones.

Question 18.

Answer:
The subtraction of 38 from 54 is 16. And there are one ten and six ones.

Explanation:
In the above image, we can see that there are five tens and four one. So here we need to subtract 38 from 54, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 38 from 54 is 54 – 38= 16 and there are one ten and six ones.

Question 19.

Answer:
The subtraction of 39 from 92 is 53. And there are five tens and three ones.

Explanation:
In the above image, we can see that there are nine tens and two ones. So here we need to subtract 39 from 92, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 39 from 92 is 92 – 39= 53 and there are five tens and three ones.

Question 20.

Answer:
The subtraction of 28 from 87 is 59. And there are five tens and nine ones.

Explanation:
In the above image, we can see that there are eight tens and seven ones. So here we need to subtract 28 from 87, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 87 is 87 – 28= 59 and there are eight tens and nine ones.

Question 21.
THINK SMARTER
For which of the problems above could you find the difference without rewriting it? Explain.

___________________________
___________________________
___________________________

Problem Solving • Applications

Read about the class trip. Then answer the questions.

Question 22.
How many more paintings were done by adults than by children?
__ more paintings

Answer:
The number of paints which were more by adults than the children is 53 – 26= 27 paintings.

Explanation:
As Pablo’s class went to the art museum, and they saw 26 paintings done by the children. After that, they saw 53 paintings done by the adults. So the number of paints which were more by adults than the children is 53 – 26= 27 paintings.

Question 23.
GO DEEPER
How many more paintings than sculptures did they see?
__ more paintings

Answer:
The number of paints that were more than the sculptures is 61 more paintings.

Explanation:
As Pablo’s class went to the art museum, and they saw 26 paintings done by the children. After that they saw 53 paintings done by the adults. So the total number of paintings is 53 + 26 = 79 paintings. And they saw 18 sculptures, so the number of paints which were more than the sculptures is 79 – 18= 61 more paintings.

Question 24.
THINK SMARTER
Tom drew 23 pictures last year. Beth drew 14 pictures. How many more pictures did Tom draw than Beth?
Fill in the bubble next to all the ways to show the problem.

Answer:
The number of more pictures did Tom draw than Beth is 23 – 14= 9 pictures.

Explanation:
As Tom draw 23 pictures last year and Beth draw 14 pictures, so the number of more pictures did Tom draw than Beth is 23 – 14= 9 pictures.

TAKE HOME ACTIVITY
• Ask your child to write and solve a subtraction problem about a family trip.

### Rewrite 2-Digit Subtraction Homework & Practice 5.7

Rewrite the subtraction problem. Then find the difference.

Question 1.

Answer:
The subtraction of 19 from 35 is 16. And there are one ten and six ones.

Explanation:
In the above image, we can see that there are three tens and five ones. So here we need to subtract 19 from 35, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 35 is 35 – 19= 16 and there are one ten and six ones.

Question 2.

Answer:
The subtraction of 23 from 47 is 24. And there are two tens and four ones.

Explanation:
In the above image, we can see that there are four tens and seven ones. So here we need to subtract 23 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 23 from 47 is 47 – 23= 24 and there are two tens and four ones.

Question 3.

Answer:
The subtraction of 28 from 58 is 30. And there are three tens and zero ones.

Explanation:
In the above image, we can see that there are three tens and five one’s blocks. So here we need to subtract 28 from 58, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 58 is 58 – 28= 30 and there are three tens and zero ones.

Problem Solving

Solve. Write or draw to explain.

Question 4.
Jimmy went to the toy store. He saw 23 wooden trains and 41 plastic trains. How many more plastic trains than wooden trains did he see?
___ more plastic trains

Answer:
There are 18 more plastic trains than wooden trains.

Explanation:
Jimmy went to the toy store and he saw 23 wooden trains and 41 plastic trains, so to find how many more plastic trains than wooden trains did he see is we will perform subtraction. So there will be 41 – 23= 18 more plastic trains than wooden trains.

Question 5.
WRITE Math
Is it easier to subtract when the numbers are written above and below each other? Explain your answer
__________________________
__________________________
__________________________
__________________________

Answer:
Yes, it is easier to subtract the numbers that are written above and below each other. As we can easily borrow from the place in front of the number and we can rewrite without a problem. It also helps to keep the math problems more organized so by writing the numbers above and below we will be confused.

Lesson Check

Question 1.
What is the difference for 43 − 17?

Answer:
The subtraction of 17 from 43 is 26. And there are one ten and nine ones.

Explanation:
In the above, we can see that there are four tens and three ones. So here we need to subtract 13 from 32, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 17 from 43 is 43 – 16= 26 and there are two tens and six ones.

Question 2.
What is the difference for 50 − 16?

Answer:
The subtraction of 16 from 50 is 34. And there are three tens and four ones.

Explanation:
In the above, we can see that there are five tens and zero ones. So here we need to subtract 16 from 50, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 16 from 50 is 50 – 16= 34 and there are three tens and four ones.

Spiral Review

Question 3.
What is the sum?

Answer:
The sum of the given numbers is 74.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the given numbers
29 + 4 + 25 + 16  is 74.

Question 4.
What is the sum of 41 + 19?
_______

Answer:
The sum of the given numbers is 60.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the given numbers
41 + 19 is 60.

Question 5.
Write an addition fact that will give the same sum as 5 + 9?
10 + ___

Answer:
The sum of the two numbers is 14.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 5 + 9 is 14. So to get the same sum we need to add 4, so the sum of the 10 + 4= 14, which is same as 5 + 9 sum.

Question 6.
What is the difference?
45 – 13 = ___

Answer:
The subtraction of 13 from 45 is 32. And there are three tens and two ones.

Explanation:
In the above, we can see that there are four tens and three ones. So here we need to subtract 13 from 45, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 13 from 45 is 45 – 13= 32 and there are three tens and two ones.

### Lesson 5.8 Add to Find Differences

Essential Question
How can you use addition to solve subtraction problems?
_______ ____ markers

Now draw pictures to show the next part of the problem. Write a number sentence for your drawing.
_______ ____ markers

Math Talk
MATHEMATICAL PRACTICES

Describe what happens when you add back the number that you had subtracted.

Model and Draw

Count up from the number you are subtracting to find the difference.

Start at 38. Count up to 40.

Then count up 5 more to 45.

So, 45 − 38 = __.

Share and Show MATH BOARD

Use the number line. Count up to find the difference.

Question 1.
36 – 27 = ___

Answer:
The subtraction of 27 from 36 is 9. And there are zero tens and nine ones.

Explanation:
Here we need to subtract 27 from 36, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 27 from 36 is 36 – 27= 9 and there are zero tens and nine ones.

Question 2.
56 – 49 = __

Answer:
The subtraction of 49 from 56 is 7. And there are zero tens and seven ones.

Explanation:
Here we need to subtract 49 from 56, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 49 from 56 is 56 – 49= 7 and there are zero tens and seven ones.

Question 3.
64 – 58 = __

Answer:
The subtraction of 58 from 64 is 6. And there are zero tens and six ones.

Explanation:
Here we need to subtract 58 from 64, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 58 from 64 is 64 – 58= 6 and there are zero tens and six ones.

On Your Own

Use the number line. Count up to find the difference.

Question 4.
33 – 28 = __

Answer:
The subtraction of 28 from 33 is 5. And there are zero tens and five ones.

Explanation:
In the above image, we can see that there are three tens and three ones. So here we need to subtract 28 from 33, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 33 is 33 – 28= 5 and there are zero tens and five ones.

Question 5.
45 – 37 = __

Answer:
The subtraction of 37 from 45 is 8. And there are zero tens and eight ones.

Explanation:
In the above image, we can see that there are four tens and five ones. So here we need to subtract 37 from 45, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 37 from 45 is 45 – 37= 8 and there are zero tens and eight ones.

Question 6.
58 – 49 = __

Answer:
The subtraction of 49 from 58 is 9. And there are zero tens and nine ones.

Explanation:
In the above image, we can see that there are five tens and eight ones. So here we need to subtract 49 from 58, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 49 from 58 is 58 – 49= 49 and there are zero tens and nine ones.

Question 7.
THINK SMARTER
There were 55 books on the table. Sandra picked up some of the books. Now there are 49 books on the table. How many books did Sandra pick up?

__ books

Answer:
The number of books picked up by the Sandra is 6 books.

Explanation:
As there were 55 books on the table and Sandra picked up some of the books. And Now there are 49 books on the table, so to find how many books did Sandra pick up is we will perform subtraction. So the number of books picked up by the Sandra is 55 – 49= 6 books.

Problem Solving • Applications

Solve. You may wish to use the number line to help.

Question 8.
There are 46 game pieces in a box. Adam takes 38 game pieces out of the box. How many game pieces are still in the box?

__ game pieces

Answer:
The number of game pieces is still in the box is 8 game pieces.

Explanation:
As there are 46 game pieces in a box and Adam takes 38 game pieces out of the box. So there will be 46 – 38= 8 game pieces.

Question 9.
THINK SMARTER
Rachel had 27 craft sticks. Then she gave 19 craft sticks to Theo. How many craft sticks does Rachel have now?
Circle the number from the box to make the sentence true.
Rachel has craft sticks now.
Explain how you can use addition to solve the problem.
__________________________
__________________________

Answer:
The number of craft sticks Rachel has now is 27 – 19 = 8 craft sticks.

Explanation:
As Rachel had 27 craft sticks and then she gave 19 craft sticks to Theo. So the number of craft sticks Rachel has now is 27 – 19 = 8 craft sticks.

TAKE HOME ACTIVITY
• Have your child describe how he or she used a number line to solve one problem in this lesson.

Answer:
The number line is used by performing we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. Then we will make the number of jumps by the difference what we had got and we will make jump from minuend number to the subtrahend and will count the number of jumps.

### Add to Find Differences Homework & practice 5.8

Use the number line. Count up to find the difference.

Question 1.
36 – 29 = __

Answer:
The subtraction of 29 from 36 is 7. And there are zero tens and seven ones.

Explanation:
In the above image, we can see that there are three tens and six ones. So here we need to subtract 29 from 36, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 29 from 36 is 36 – 29= 7 and there are zero tens and seven ones.

Question 2.
43 – 38 = __

Answer:
The subtraction of 38 from 43 is 5. And there are zero tens and five ones.

Explanation:
In the above image, we can see that there are three tens blocks and six one’s blocks. So here we need to subtract 8 from 51, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 38 from 43 is 43 – 38= 5 and there are zero tens and five ones.

Problem Solving

Solve. You may wish to use the number line.

Question 3.
Jill has 63 index cards. She uses 57 of them for a project. How many index cards does Jill have now?
__ index cards

Answer:
The number of index cards does Jill has now is 6 index cards.

Explanation:
As Jill has 63 index cards and she uses 57 of them for a project, so the number of index cards does Jill have now is 63 – 57 = 6 index cards.

Question 4.
WRITE Math
Explain how a number line can be used to find the difference for 34 – 28.
__________________________
__________________________

Answer:

Lesson Check

Use the number line. Count up to find the difference.

Question 1.
82 − 75 = __

Answer:
The subtraction of 75 from 82 is 7. And there are zero tens and seven ones.

Explanation:
In the above image, we can see that there are eight tens and two ones. So here we need to subtract 75 from 82, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 75 from 82 is 82 – 75= 7 and there are zero tens and seven ones.

Question 2.
90 − 82 = __

Answer:
The subtraction of 82 from 90 is 8. And there are zero tens and eight ones.

Explanation:
In the above image, we can see that there are nine tens and zero ones. So here we need to subtract 82 from 90, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 82 from 90 is 90 – 82= 8and there are zero tens and eight ones.

Spiral Review

Question 3.
Jordan has 41 toy cars at home. He brings 24 cars to school. How many cars are at home?
__ cars

Answer:
The number of cars is at home is 17 cars.

Explanation:
As Jordan has 41 toy cars at home and he brings 24 cars to school, so the number of cars are at home is 41 – 24= 17 cars.

Question 4.
Pam has 15 fish. 9 are goldfish and the rest are guppies. How many fish are guppies?
__ guppies

Answer:
The number of fishes is guppies is 6 guppies.

Explanation:
As Pam has 15 fishes and 9 are gold fishes and the rest are guppies, so the number of fishes are guppies is 15 – 9= 6 guppies.

Question 5.
What is the sum?

Answer:
The sum of the two numbers is 54.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 35 + 19 is 54.

Question 6.
Each table has 5 pencils. There are 3 tables. How many pencils are there altogether?
__ pencils.

Answer:
The total number of pencils will be 15 pencils are there altogether.

Explanation:
As each table has 5 pencils and there are 3 tables, so the total number of pencils will be 5 × 3= 15 pencils are there altogether.

### Lesson 5.9 Problem Solving • Subtraction

Essential Question
How can drawing a diagram help when solving subtraction problems?

Jane and her mom made 33 puppets for the craft fair. They sold 14 puppets. How many puppets do they still have?

Answer:
The total number of puppets do they left is 19 puppets.

Explanation:
As Jane and her mom made 33 puppets for the craft fair and they sold 14 puppets, so they still have 33 – 14= 19 puppets. Here we need to subtract 14 from 33, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 14 from 33 is 33 – 14= 19 and there are one ten and nine ones. So they will have 19 puppets left.

Unlock the Problem

What do I need to find?
they still have

What information do I need to use?

They made __ puppets.
They sold __ puppets.

Show how to solve the problem.

HOME CONNECTION
• Your child used a bar model and a number sentence to represent the problem. Using a bar model helps show what is known and what is needed to solve the problem.

Try Another Problem

Label the bar model. Write a number sentence with a for the missing number. Solve.

Question 1.
Carlette had a box of 46 craft sticks. She used 28 craft sticks to make a sailboat. How many craft sticks were not used?

Answer:
They will have 18 craft sticks left.

Explanation:
As Carlette had a box of 46 craft sticks and she used 28 craft sticks to make a sailboat. So the number of craft sticks that are not used is 46 – 28= 18 craft sticks. Here we need to subtract 28 from 46, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 28 from 46 is 46 – 28= 18 and there are one ten and eight ones. So they will have 18 craft sticks left.

Question 2.
Rob’s class made 31 clay bowls. Sarah’s class made 15 clay bowls. How many more clay bowls did Rob’s class make than Sarah’s class?

______ ___ more clay bowls

Answer:
The number of more clay bowls did Rob’s class make than Sarah’s class is 31 – 15= 16 clay bowls.

Explanation:
As Rob’s class made 31 clay bowls and Sarah’s class made 15 clay bowls. So the number of more clay bowls did Rob’s class make than Sarah’s class is 31 – 15= 16 clay bowls. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of more clay bowls did Rob’s class make than Sarah’s class is 16 clay bowls.

Share and Show MATH BOARD

Label the bar model. Write a number sentence with a for the missing number. Solve.

Question 3.
Mr. Hayes makes 32 wooden frames. He gives away 15 frames as gifts. How many frames does he still have?

Answer:
The number of many frames does he still have now is 32 – 15= 17 frames.

Explanation:
As Mr. Hayes makes 32 wooden frames and he gives away 15 frames as gifts. So the number of many frames does he still have now is 32 – 15= 17 frames. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of many frames does he still have now is 17 frames.

Question 4.
Wesley has 21 ribbons in a box. He has 15 ribbons on the wall. How many more ribbons does he have in the box than on the wall?

__ more ribbons

_____

Answer:
The number of many more ribbons does he have in the box than on the wall is 6 more ribbons.

Explanation:
As Wesley has 21 ribbons in a box and he has 15 ribbons on the wall. So the number of many more ribbons does he have in the box than on the wall is 21 – 15= 6 more ribbons. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of many more ribbons does he have in the box than on the wall is 6 more ribbons.

Question 5.
THINK SMARTER
Jennifer wrote 9 poems at school and 11 poems at home. She wrote 5 more poems than Nell. How many poems did Nell write?

__ poems

Answer:

Explanation:

On Your Own

Question 6.
GO DEEPER
There are 70 children. 28 children are hiking and 16 are at a picnic. The rest of the children are playing soccer. How many children are playing soccer?
Draw a model with bars for the problem. Describe how your drawing shows the problem. Then solve the problem.
__________________________
__________________________
__________________________

Question 7.
THINK SMARTER
There are 48 crackers in a bag. The children eat 25 crackers. How many crackers are still in the bag? Circle the bar model that can be used to solve the problem.

Write a number sentence with a for the missing number. Solve.
__________________________
___ crackers

TAKE HOME ACTIVITY
• Ask your child to explain how he or she solved one of the problems on this page.

Answer:
We will perform subtraction by breaking the two-digit number into tens place digit and ones place digit. Then the other number will be placed below the number. If the minuend number is less than the subtrahend number then we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So by this, we will perform the subtraction.

### Problem Solving • Subtraction Homework & Practice 5.9

Label the bar model. Write a number sentence with a for the missing number. Solve.

Question 1.
Megan picked 34 flowers. Some of the flowers are yellow and 18 flowers are pink. How many of the flowers are yellow?
___ yellow flowers

___

Question 2.
Alex had 45 toy cars. He put 26 toy cars in a box. How many toy cars are not in the box?
___ toy cars

Question 3.
WRITE Math
Explain how bar models show a problem in a different way.
__________________________
__________________________
__________________________
__________________________

Lesson Check

Question 1.
There were 39 pumpkins at the store. Then 17 of the pumpkins were sold. How many pumpkins are still at the store?
__ pumpkins

Question 2.
There were 48 ants on a hill. Then 13 of the ants marched away. How many ants are still on the hill?
__ ants

Spiral Review

Question 3.
Ashley had 26 markers. Her friend gave her 17 more markers. How many markers does Ashley have now?
__ markers

Question 4.
What is the sum?

Question 5.
Write a subtraction fact that will give the same difference as 15 − 7.
10 – __

Question 6.
What is the sum?
34 + 5 = __

### Lesson 5.10 Algebra • Write Equations to Represent Subtraction

Essential Question
How do you write a number sentence to represent a problem?

Listen and Draw

Draw to show the problem. Write a number sentence. Then solve.
_______________

Math Talk
MATHEMATICAL PRACTICES
Describe how your drawing shows the problem.

Model and Draw

You can write a number sentence to
show a problem.
Liza has 65 postcards. She gives 24 postcards to Wesley. How many postcards does Liza have now?

Liza has __ postcards now.

Share and Show MATH BOARD

Write a number sentence for the problem. Use a for the missing number. Then solve.

Question 1.
There were 32 birds in the trees. Then 18 birds flew away. How many birds are in the trees now?

__ birds

Question 2.
Carla read 43 pages in her book. Joe read 32 pages in his book. How many more pages did Carla read than Joe?
________
__ more pages

On Your Own

Write a number sentence for the problem. Use a for the missing number. Then solve.

Question 3.
There were 40 ants on a rock. Some ants moved to the grass. Now there are 26 ants on the rock. How many ants moved to the grass?
___
__ ants

Question 4.
THINK SMARTER
Keisha had a bag of ribbons. She took 29 ribbons out of the bag. Then there were 17 ribbons still in the bag. How many ribbons were in the bag to start?

_______
__ ribbons

Question 5.
GO DEEPER
There are 50 bees in a hive. Some bees fly out. If fewer than 20 bees are still in the hive, how many bees could have flown out?
__ bees

Problem Solving • Applications

Question 6.
MATHEMATICAL PRACTICE
Make Connections Brendan made this number line to find a difference. What was he subtracting from 100? Explain your answer.

__________________________
__________________________
__________________________
__________________________

Question 7.
THINK SMARTER
There are 52 pictures on the wall. 37 are wild cats and the rest are birds. How many of the pictures are birds? Use the numbers and symbols on the tiles to complete the number sentence for the problem.

_______________
___ birds

TAKE HOME ACTIVITY
• Have your child explain how he or she solved one problem in this lesson.

### Algebra • Write Equations to Represent Subtraction Homework & Practice 5.10

Write a number sentence for the problem. Use a for the missing number. Then solve.

Question 1.
29 children rode their bikes to school. After some of the children rode home, there were 8 children with bikes still at school. How many children rode their bikes home?
________
__ children

Answer:
The number of students who rode their bikes home is 29 – 8= 21 children rode their bike home.

Explanation:
As 29 children rode their bikes to school and after that some of the children rode home, there were 8 children with bikes still at school. So the number of students who rode their bikes home is 29 – 8= 21 children rode their bike home. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of students who rode their bikes home is 21 children rode their bike home.

Problem Solving

Solve. Write or draw to explain.

Question 2.
There were 21 children in the library. After 7 children left the library, how many children were still in the library?
__ children

Question 3.
WRITE Math
Describe different ways that you can show a story problem. Use one of the problems in this lesson as your example.
__________________________
__________________________
__________________________
__________________________

Lesson Check

Question 1.
Cindy had 42 beads. She used some beads for a bracelet. She has 14 beads left. How many beads did she use for the bracelet?
__ beads

Question 2.
Jake had 36 baseball cards. He gave 17 cards to his sister. How many baseball cards does Jake have now?
__ cards

Spiral Review

Question 3.
What is the sum?
6 + 7 = __

Question 4.
What is the difference?
16 – 9 = __

Answer:
The subtraction of 9 from 16 is 7. And there are zero tens and seven ones.

Explanation:
In the above image, we can see that there are nine tens and zero ones. So here we need to subtract 9 from 16, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 9 from 16 is 16 – 9= 7 and there are zero tens and seven ones.

Question 5.
What is the difference?

Answer:
The subtraction of 39 from 46 is 7. And there are zero tens and seven ones.

Explanation:
In the above image, we can see that there are four tens and six ones. So here we need to subtract 39 from 46, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 39 from 46 is 46 – 39= 7 and there are zero tens and seven ones.

Question 6.
Write an addition fact that will give the same sum as 6 + 8.
10 + ___

Answer:
The sum of the two numbers is 14.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 6 + 8 is 14. So to get the same sum we need to add 4, so the sum of the 10 + 4= 14, which is same as 6 + 8 sum.

### Lesson 5.11 Solve Multistep Problems

Essential Question
How do you decide what steps to do to solve a problem?

Listen and Draw

Label the bar model to show each problem. Then solve.

Math Talk
MATHEMATICAL PRACTICES

Describe how the two bar models are different.

Model and Draw

Bar models help you know what to do to solve a problem.
Ali has 27 stamps. Matt has 38 stamps. How many more stamps are needed so they will have 91 stamps?

They need __ more stamps

Share and Show MATH BOARD

Complete the bar models for the steps you do to solve the problem.

Question 1.
Jen has 93 beads. Ana has 46 red beads and 29 blue beads. How many more beads does Ana need to have 93 beads also?

__ more beads

Answer:
Ana needs 18 beads to have 93 beads.

Explanation:
As Jen has 93 beads and Ana has 46 red beads and 29 blue beads, so the total number of beads does Ana had is 46 + 29= 75 beads. So the number of many more beads does Ana need to have 93 beads is 93 – 75= 18 beads. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So Ana needs 18 beads to have 93 beads.

On Your Own

Complete the bar models for the steps you do to solve the problem.

Question 2.
Max has 35 trading cards. He buys 22 more cards. Then he gives 14 cards to Rudy. How many cards does Max have now?

Answer:
The number of many cards does Max have now is 43 trading cards.

Explanation:
As Max has 35 trading cards and he buys 22 more cards, so the total number of cards will be 35 + 22= 57 trading cards. Then he gives 14 cards to Rudy. So the number of trading cards will be 57 – 14= 43 trading cards. So the number of many cards does Max have now is 43 trading cards.

Question 3.
Drew has 32 toy cars. He trades 7 of those cars for 11 other toy cars. How many toy cars does Drew have now?

Question 4.
Marta and Debbie each have 17 ribbons. They buy 1 package with 8 ribbons in it. How many ribbons do they have now?

Answer:
The total number of ribbons do they had now is 42 ribbons.

Explanation:
As Marta and Debbie each have 17 ribbons, so the total number of ribbons did Marta and Debbie had together is 17 + 17= 34 ribbons. And they bought 1 package with 8 ribbons in it. So the total number of ribbons do they had now is 34 + 8= 42 ribbons.

Problem Solving • Applications WRITE Math

Question 5.
Shelby had 32 rocks. She finds 33 more rocks at the park and gives 28 rocks to George. How many rocks does she have now?

__ rocks

Answer:
So the number of many rocks does she have now is 37 rocks.

Explanation:
As Shelby had 32 rocks and she finds 33 more rocks at the park, so the total number of rocks did Shelby had now is 32 + 33= 65 rocks. And Shelby gives 28 rocks to George. So the number of many rocks does she have now is 65 – 28= 37 rocks. so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of many rocks does she have now is 37 rocks.

Question 6.
Benjamin finds 31 pinecones at the park. Together, Jenna and Ellen find the same number of pinecones as Benjamin. How many pinecones could each girl have found?

Jenna: __ pinecones
Ellen: __ pinecones

Answer:

Explanation:
As Benjamin finds 31 pinecones at the park and Jenna and Ellen together finds the same number of pinecones as Benjamin. How many pinecones could each girl have found

Question 7.
THINK SMARTER
Tanya finds 22 leaves. Maurice finds 5 more leaves than Tanya finds. How many leaves do the children find? Draw to show how you solve the problem.

__ leaves

Answer:
The number of many leaves do the children finds together is 22 + 5=  27 leaves.

Explanation:
As Tanya finds 22 leaves and Maurice finds 5 more leaves than Tanya finds. So the number of many leaves do the children finds together is 22 + 5=  27 leaves.

TAKE HOME ACTIVITY
• Have your child explain how he or she would solve Exercise 6 if the number 31 was changed to 42.

### Solve Multistep Problems Homework & Practice 5.11

Complete the bar models for the steps you do to solve the problem

Question 1.
Greg has 60 building blocks. His sister gives him 17 more blocks. He uses 38 blocks to make a tower. How many blocks are not used in the tower?

__ blocks

Answer:
The number of many blocks are not used in the tower is 5 blocks.

Explanation:
As Greg has 60 building blocks and his sister gives him 17 more blocks, so the number of blocks did Greg had now is 60 – 17= 43 blocks and he uses 38 blocks to make a tower so the number of many blocks are not used in the tower is 43 – 38= 5 blocks. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of many blocks are not used in the tower is 5 blocks.

Problem Solving

Solve. Write or draw to explain.

Question 2.
Ava has 25 books. She gives away 7 books. Then Tom gives her 12 books. How many books does Ava have now?
__ books

Answer:
The number of many books does Ava have now is 6 books.

Explanation:
As Ava has 25 books and she gives away 7 books, then Ava will have 25 – 7= 18 books and then Tom gives her 12 books, so the number of books did Ava have now is 18 – 12= 6 books. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the number of many books does Ava have now is 6 books.

Question 3.
WRITE Math
Choose one of the problems on this page. Describe how you decided what steps were needed to solve the problem.
___________________________
___________________________
___________________________
___________________________

Lesson Check

Question 1.
Sara has 18 crayons. Max has 19 crayons. How many more crayons do they need to have 50 crayons altogether?
__ crayons

Answer:
They need 13 crayons more do they need to have 50 crayons altogether.

Explanation:
As Sara has 18 crayons and Max has 19 crayons, so the total number of crayons do they have together is 18 + 19= 37 crayons. So to get 50 crayons altogether we need to perform subtraction. So we need 50 – 37= 13 crayons more do they need to have 50 crayons altogether. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So They need 13 crayons more do they need to have 50 crayons altogether.

Question 2.
Jon has 12 pennies. Lucy has 17 pennies. How many more pennies do they need to have 75 pennies altogether?
__ pennies

Answer:
They need 46 pennies more to have 75 pennies altogether.

Explanation:
As Jon has 12 pennies and Lucy has 17 pennies, so the total number of pennies together they have is 29 pennies. So to 75 pennies together we need to perform subtraction 75 – 29= 46 pennies. So the number of many frames does he still have now is 32 – 15= 17 frames. So we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the total number of pennies together they have is 29 pennies. So to 75 pennies together we need to perform subtraction 75 – 29= 46 pennies.

Spiral Review

Question 3.
What is the difference?
58 – 13 = __

Answer:
The subtraction of 8 from 51 is 43. And there are four tens and three ones.

Explanation:
In the above image, we can see that there are three tens blocks and six one’s blocks. So here we need to subtract 8 from 51, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 8 from 51 is 51 – 8= 43 and there are four tens and three ones.

Question 4.
What is the sum?

Answer:
The sum of the two numbers is 62.

Explanation:
The sum can be defined as the resulting of two or more numbers by adding. So here the sum of the two numbers 47 + 15 is 62.

Question 5.
There are 26 cards in a box. Bryan takes 12 cards. How many cards are still in the box?
__ cards

Answer:
The number of cards still in the box is 14 cards.

Explanation:
As there are 26 cards in a box and Bryan takes 12 cards, so the number of cards still in the box is 26 – 12= 14 cards. we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 12 from 26 is 26 – 12= 14 and there are zero tens and four ones.

### 2-Digit Subtraction Chapter 5 Review Test

Question 1.
Do you need to regroup to subtract? Choose Yes or No.

Question 2.
Use the number line. Count up to find the difference.
52 – 48 = __

Answer:
The subtraction of 48 from 52 is 4. And there are zero tens and four ones.

Explanation:
Here we need to subtract 48 from 52, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 48 from 52 is 52 – 48= 4 and there are zero tens and four ones.

Question 3.
Ed has 28 blocks. Sue has 34 blocks. Who has more blocks? How many more? Label the bar model. Solve.

Circle the word and number from each box to make the sentence true.

Break apart the number you are subtracting. Write the difference?

Answer:
Sue has more number of blocks tha Ed. And had 6 more blocks than Ed.

Explanation:
Ed has 28 blocks and Sue has 34 blocks. As 34 is greater than 28, so Sue has more number of blocks and has 34 – 28= 6 more blocks than the Ed. So to perform Break apart the number subratction for 34 – 28 first we will break into tens and ones. So here we will break 28 as 20 and 8 and we will subtract 20 and we will get the result as 14. Then we will start from 14 and subtract 8 to get to 6. So we will get the result as 6. The subtraction of 34 – 28 is 6.

Question 4.

Answer:
The subtraction of 42 – 8 is 34.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 8 as 2 and 6 and we will subtract 2 and we will get the result as 42. Then we will start from 42 and subtract 6 to get to 34 and then. So we will get the result as 34. The subtraction of 42 – 8 is 34.

Question 5.

Answer:
The subtraction of 53 – 16 is 37.

Explanation:
Here, we will break apart the number that we are subtracting. So first we will break into tens and ones. So here we will break 10 as 6 and we will subtract 10 and we will get the result as 43. Then we will start from 43 and subtract 6 to get to 37 and then. So we will get the result as 37. The subtraction of 53 – 16 is 37.

Question 6.
What is 33 − 19? Use the numbers on the tiles to rewrite the subtraction problem. Then find the difference.

Answer:
The subtraction of 19 from 33 is 14. And there are one ten and four ones.

Explanation:
In the above image, we can see that there are three tens and three ones. So here we need to subtract 19 from 33, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 19 from 33 is 33 – 19= 14 and there are one ten and four ones.

Question 7.
GO DEEPER
Jacob’s puzzle has 84 pieces. Jacob puts together 27 pieces in the morning. He puts together 38 more pieces in the afternoon. How many pieces does Jacob need to put together to finish the puzzle?

Complete the bar models for the steps you do to solve the problem.
__ more pieces

Answer:
Jacob needs 19 pieces to complete the puzzle.

Explanation:
Jacob’s puzzle has 84 pieces and Jacob puts together 27 pieces in the morning and he puts together 38 more pieces in the afternoon. So the total number of puzzles kept by Jacob together in the morning and the afternoon is 27 + 38= 65 pieces. So Jacob needs 84 – 65= 19 pieces to complete the puzzle.

Regroup if you need to. Write the difference.

Question 8.

Answer:
The subtraction of 5 from 28 is 23. And there are two tens and three ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 5 from 28, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 5 from 28 is 28 – 5= 23 and there are two ten and three ones.

Question 9.

Answer:
The subtraction of 12 from 32 is 23. And there are two tens and three ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 5 from 28, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 5 from 28 is 28 – 5= 23 and there are two ten and three ones.

Question 10.
Find the difference.

Answer:
The subtraction of 62 from 90 is 28. And there are two tens and eight ones.

Explanation:
In the above image, we can see that there are nine tens and zero ones. So here we need to subtract 62 from 90, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 62 from 90 is 90 – 62= 28 and there are two tens and eight ones.

Fill in the bubble next to one number from each column to show the difference.

Question 11.
There are 22 children at the park. 5 children are on the swings. The rest of the children are playing ball. How many children are playing ball?

Answer:
The number of children is playing ball is 17 children.

Explanation:
As there are 22 children at the park and 5 children are on swings and the rest of the children are playing ball. So the number of children are playing ball is 22 – 15= 17 children.

Question 12.
THINK SMARTER
Subtract 27 from 43. Draw to show the regrouping.
Fill in the bubble next to all the ways to write the difference.

Answer:
The subtraction of 27 from 43 is 16. And there are one ten and six ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 27 from 43, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 27 from 43 is 43 – 27= 43 and there are one ten and six ones.

Question 13.
Jill collects stamps. Her stamp book has space for 64 stamps. She needs 18 more stamps to fill the book. How many stamps does Jill have now?
Write a number sentence for the problem.
Use a for the missing number. Then solve.
________
Jill has __ stamps.

Answer:
The number of does Jill have now is 46 stamps.

Explanation:
Jill collects stamps and her stamp book has space for 64 stamps then she needs 18 more stamps to fill the book. So the number of does Jill have now is 64 – 18= 46 stamps.

Question 14.
Draw a quick picture to solve. Write the difference.

Explain what you did to find the difference.
_______________
_______________

Answer:
The subtraction of 25 from 62 is 37. And there are three tens and seven ones.

Explanation:
In the above image, we can see that there are two tens and eight ones. So here we need to subtract 25 from 62, so we will perform regrouping of subtraction. Here Regrouping in subtraction is a process of exchanging one ten into ten ones. We use regrouping in subtraction when the minuend is smaller than the subtrahend. So the subtraction of 25 from 62 is 62 – 25= 37 and there are three tens and seven ones.

Conclusion:

The information which was discussed in the above section which is Go Math Grade 2 Answer Key Chapter 5 2-Digit Subtraction is useful for you. Exchange these pdf links with your beloved ones and help them to acquire knowledge in maths. Stay connected with us to get the recent updates regarding the Go Math Grade 2 Answer Key for all the chapters.

## Big Ideas Math Answers Grade 8 Chapter 7 Functions

Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions is here to provide you the help to get into the race of the examinations as well as the knowledge on the daily life examples. This chapter has conceptualized lessons on functions, relations, linear and non linear equations for better use in practical skills. The chapter functions is provided with all the possible methods of explanation of each question in a detailed way. So sort out all of your doubts and get clarified with the Big Ideas Math Answers Grade 8 Chapter 7 Functions.

## Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions

Having difficulty on finding the solutions for your math problems? Then you have come to the right place, Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions have all the solutions that you are looking for. The quick way of solving problems will help the students to save time. This chapter gives the most accurate answers for each and every problem in Functions. Hence, This chapter is providing all answers to the problems.

STEAM Video/Performance Task

Getting Ready for Chapter 7

Lesson 1 Relations and Functions

Lesson 2 Representations of Functions

Lesson 3 Linear Functions

Lesson 4 Comparing Linear and Non Linear Functions

Lesson 5 Analyzing and Sketching Graphs

Functions Connecting Concepts

### Functions STEAM Video/Performance Task

STEAM Video

Apparent Temperature
Sometimes it feels hotter or colder outside than the actual apparent temperature. How hot or cold it feels is called the temperature. What weather factors might contribute to the apparent temperature?
Watch the STEAM Video “Apparent Temperature.” Then answer the following questions.

1. Robert says that the Wet-Bulb Globe Temperature (WBGT)index is used as a measure of apparent temperature.

In the formula, TW is the natural wet-bulb temperature, TG is the black-globe temperature, TD and is the dry-bulb temperature. Find WBGT when TW = 75ºF, TG = 100ºF, and TD = 84ºF.
2. Different categories of Wet-Bulb Globe Temperatures are shown in the chart. Each category can be represented by a different-colored ﬂag. Which flag color is displayed when WGBT = 87.5ºF?

Performance Task

Heat Index
After completing this chapter, you will be able to use the STEAM concepts you learned to answer the questions in the Video Performance Task. You will be given information about heat index.

You will be asked to create a graph of the temperatures and heat indices. Why is it useful to know the heat index?

### Functions Getting Ready for Chapter 7

Chapter Exploration

Work with a partner. Copy and complete the diagram.

1.  Answer: ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Area A of a rectangle we have , A = w × l
for x = 1 , A = 2 × 1 = 2 ,
for x = 2 , A = 2 × 2 = 4 ,
for x = 3 , A = 2 × 3 = 6 ,
for x = 4 , A = 2 × 4 = 8 ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output A = 2 , 4 , 6 , 8 , respectively .
That is ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

2. Answer: ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ).

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Perimeter of a rectangle we have , P = 2( l + w )
for x = 1 ,P = 2( 1 + 2 ) = 2 × 3 = 6 ,
for x = 2 , P = 2( 2 + 2 ) = 2 × 4 = 8  ,
for x = 3 , P = 2( 3 + 2 ) = 2 × 5 = 10  ,
for x = 4 , P = 2( 4 + 2 ) = 2 × 6 = 12  ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output P = 6 , 8 , 10 , 12 , respectively .
That is ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ) .

3. Answer : ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

Explanation:
Given , Radius of a circle , where as  r = 1 , 2 , 3 , 4
To find the circumference of a circle , we have C = 2Òr , Ò = 3.14 , or we can write it as 3 .
for r = 1 , C = 2 × 3 × 1 = 6 ,
for r = 2 , C = 2 × 3 × 2 = 12 ,
for r = 3 , C = 2 × 3 × 3 = 18 ,
for r = 4 , C = 2 × 3 × 4 = 24 ,
So, for every value of Input r = 1 , 2 , 3 , 4 we have Output C = 6 , 12 , 18 , 24 , respectively .
That is ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

4. Answer: ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 )

Explanation:
Given , Two Edges of a cube = 3 , h = 1 , 2 , 3 , 4
To find the Volume of the cube we have , V = a³
for h = 1 , V = 3 × 3 × 1 = 9 ,
for h = 2 , V = 3 × 3 × 2 = 12 ,
for h = 3 , V = 3 × 3 × 3 = 27 ,
for h = 4 , V = 3 × 3 × 4 = 36 ,
So, for every value of Input h = 1 , 2 , 3 , 4 we have Output V = 9 , 18 , 27 , 36 , respectively .
That is ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 ) .

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
input
mapping diagram
nonlinear function
output
linear function

Answer : Input : The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

mapping diagram : A relation pairs inputs with outputs , A relation can be represented by ordered pairs or a mapping diagram .

nonlinear function : nonlinear functions are functions which are not linear. Quadratic functions are one type of nonlinear function. It is a relation between two variables , function that does not form a line when graphed.

output ; The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

linear function : A linear function is a relation between two variables that produces a straight line when graphed. And it has one dependent variable and one independent variable .

### Lesson 7.1 Relations and Functions

EXPLORATION 1

Interpreting Diagrams
Work with a partner. Describe the relationship between the inputs and outputs in each diagram. Then complete each diagram. Is there more than one possible answer? Explain your reasoning.

Answer: a. The relation between the inputs and outputs is outputs are the result of twice as many times the inputs.
b. The relation between the inputs and outputs is outputs are the result of colors of inputs . In this case we can notice that , for any one input we can have more than one output .

Explanation:
a. As shown in the diagrams , The relation between the inputs and outputs is outputs are the result of twice as many times the inputs , so for input 1 = 1 × 1 = 1 as output ,
for input 2 = 2 × 2 = 4  ,
for input 3 = 3 × 3 = 9  ,
for input 5 = 5 × 5 = 25 ,
for input 8 = 8 × 8 = 64 ,
for input 9 = 9 × 9 = 81 ,

So, for every value of Input = 1 , 2 , 3 , 5 , 8 , 9 , we have Output = 1 , 4 , 9 , 25 , 64 , 81 , respectively .
That is ( 1 , 1 ) , ( 2 , 4 ) , ( 3 , 9 ) , ( 5 , 25 ) , ( 8 , 64 ) , ( 9 , 81 ) .

b. The relation between the inputs and outputs is outputs are the result of colors of inputs .
for input Blueberry = color is blue as output
for  input lemon = color is yellow as output
for input Apple = color is yellow , red and green as output
for input Grape = color is green as output.

In this case we can notice that , for any one input we can have more than one output .

EXPLORATION 2

Describing Relationships Between Quantities
Work with a partner. The diagrams show the numbers of tickets bought by customers for two different plays and the total costs (in dollars).

a. For each diagram, how many outputs does each input have?
b. Describe the prices of tickets for each play.
c. A person buys 4 tickets for each play. Can you determine the total cost of all 8 tickets? Explain.
Answer:
a. For Play A ,The number of inputs are equal to number of outputs ,
For Play B , The number of inputs are not equal to number of outputs .
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A , The price of the each ticket is $8 . For Play B , The price of each ticket is$4 or $8 . c. For Play A , each ticket is$8 , Then for 4 tickets = 4 × $8 =$32 .
For Play B , each ticket is $4 or$8 , Then for 4 tickets = 4 × $8 =$32 . or 4 × $4 =$16 .

Explanation:
a. For Play A ,
The number of inputs are equal to number of outputs , 4 inputs = 4 outputs
That is ( 1 , 8 ) , ( 2 , 16 ) , ( 3 , 24 ) , ( 4 , 32 ) .
For Play B ,
The number of inputs are not equal to number of outputs , 3 inputs are not equal to 7 outputs
That is , for input 1 = 4 , 8 as outputs ,
for input 2 = 8 , 12 , 16 as outputs ,
for input 3 = 12 , 16 , 20 , 24 as outputs ,
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A ,
The price of the each ticket is $8 . For Play B , The price of each ticket is$4 or $8 . c. Given , A person buys 4 tickets for each play. For Play A , each ticket is$8 , Then for 4 tickets = 4 × $8 =$32 .
And for 8 tickets = 8 × $8 =$64 .
For Play B , each ticket is $4 or$8 , Then for 4 tickets = 4 × $8 =$32 . or 4 × $4 =$16 .
And for 8 tickets =8 × $4 =$32  or 8 × $8 =$64 .

Try It

List the ordered pairs shown in the mapping diagram.
Question 1.

Answer: Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Question 2.

Answer: Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Determine whether the relation is a function.
Question 3.

Answer: The relation is not a function

Explanation:
The each input has  more than two outputs , Even one of those inputs are unclear of outputs
So , The relation is not a function .

Question 4.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
PRECISION
Describe how relations and functions are different.
Answer: Relations are nothing but the ordered pairs with Inputs and Outputs . On the other hand , Functions are The relation that pairs with one input with exactly one output  are called Functions.

IDENTIFYING FUNCTIONS List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 6.

Answer: The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) .
Each input has exactly one output ,
So, The relation is a function .

Question 7.

Answer: The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) and relation is not a function .

Explanation:
As shown , The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) .
The input 0 has more than one output ,
So, The relation is not a function .

Question 8.
OPEN-ENDED
Copy and complete the mapping diagram at the left to represent a relation that is a function. Then describe how you can not modify the mapping diagram so that the relation is a function.

Answer: ordered pairs are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) . To have the relation as a function we must have only one output for one input.

Explanation:
The ordered pairs of the diagram are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) .
Each Input must have only one output in order to be the relation is a function ,
If ,The mapping diagram has the right to left representation or each input has more than one output , then the relation is not a function .
So , To have the relation as a function we must have only one output for one input.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
The mapping diagram represents the costs of reserving a hotel room for different numbers of nights.

a. Is the cost a function of the number of nights reserved?
b. Describe the relationship between the cost and the number of nights reserved.
Answer: a. Yes , The cost is a function of the number of nights reserved .
b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by $85 with increase in the next reservation , Explanation: a. From the diagram we have , Ordered pairs are ( 1 , -$85 ) , ( 2 , $170 ) , ( 3 ,$255 ) , ( 4 , $340 ) . each input has exactly one output , So , the relation is a function and , Yes , The cost is a function of the number of nights reserved . b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by$85 with increase in the next reservation,
that is , input 1 = $85 as output Input 2 =$85 + $85 =$170 as output
Input 3 = $170 +$85 = $255 as output Input 2 =$255 + $85 =$340 as output

So, The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation. Question 10. DIG DEEPER! The graph represents the number of contestants in each round of a talent competition. a. Is the number of contestants a function of the round number? b. Predict the number of contestants in the talent competition during Round 7. Explain your reasoning. Answer: a. The number of contestants is a function of the round number. b. The number of contestants in the talent competition during Round 7 are 2. Explanation: a. From the given graph , The ordered pairs are ( 1 , 128 ) , ( 2 , 64 ) , ( 3 , 32 ) , ( 4 , 16 ) . Each input has only one output , The relation is a function . So , the number of contestants is a function of the round number. b. Firstly , The relation between the input and output is, With every increase in the round number the number of contestants are decreasing by half the number of the previous round , That is, for input 1 = 128 as output For input 2 = 128 – 64 = 64 as output For input 3 = 64 – 32 = 32 as output For input 4 = 32 – 16 = 16 as output For input 5 = 16 – 8 = 8 as output For input 6 = 8 – 4 = 4 as output For input 7 = 4 – 2 = 2 as output, So, The number of contestants in the talent competition during Round 7 are 2 . ### Relations and Functions Homework & Practice 7.1 Review & Refresh Choose an appropriate data display for the situation. Explain your reasoning. Question 1. the number of runners in each age group at a marathon Answer: In a marathon ,the people of all age group are participating for a promotion on healthy lifestyle, The number of runners in each group has kids, adults and old people to spread the awareness of leading a healthy life by running daily in the morning . Running or jogging in the morning can help us to maintain our body mass index at an optimal level which is good for heart. The Marathon is conducted by the government of health ministry to be example for the future generations. Question 2. the high temperature and the attendance at a water park each day Answer: Generally, The water park is normally crowded depending on the season and the temperature, In summer the attendance in the waterpark is at the utmost point because of the high temperature and the seasonal vacation. Going to the water park in summer is super fun due to the number of water slides , water rides will be a nice place to the whole family trip and as well as friends . In order to be there at a less crowded time spring is also a nice time to visit the water park . Graph the linear equation. Question 3. y = 2x – 3 Answer: Explanation: Given , y = 2x – 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 1 , then y = 2(1) – 3 = 2 – 3 = -1 . co-ordinates are (1 , -1) if x = 2 , then y = 2(2) – 3 = 4 – 3 = 1 , co-ordinates are (2 , 1) The co-ordinates (1 , -1) , (2 , 1) form a straight line . So, y = 2x – 3 is a linear equation. Question 4. y = – 0.5x Answer: Explanation: Given , y = -0.5x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = -0.5(0) = 0 . co-ordinates are (0 , 0) if x = 2 , then y = -0.5(2) = -1 , co-ordinates are (2 , -1) The co-ordinates (0 , 0) , (2 , -1) form a straight line . So, y = -0.5x is a linear equation. Question 5. y = – 3x + 4 Answer: Explanation: Given , y = – 3x + 4 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 3(0) + 4 = 4 . co-ordinates are (0 , 4) if x = 1 , then y = – 3(1) + 4 = -3 + 4 = 1 , co-ordinates are (1 , 1) if x = 2 , then y = – 3(2) + 4 = -6 + 4 = -2 , co-ordinates are (2 , -2) The co-ordinates (0 , 4) , (1 , 1) , (2 , -2) form a straight line . So, y = – 3x + 4 is a linear equation. Question 6. Which word best describes two figures that have the same size and the same shape? A. congruent B. adjacent C. parallel D. similar Answer: A. congruent Explanation: Two figures which have the same size and shape are congruent. Concepts, Skills, &Problem Solving INTERPRETING DIAGRAMS Describe the relationship between the inputs and outputs in the diagram. Then complete the diagram. Is there more than one possible answer? Explain your reasoning. (See Exploration 1, p. 275.) Question 7. Answer: The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output. Explanation: The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output , for input 1 = -1 as output for input 2 = -1 + (-4) = -5 as output for input 3 = -5 + (-4) = -9 as output for input 4 = -9 + (-4) = -13 as output for input 5 = -13 + (-4) = -17 as output for input 6 = -17 + (-4) = -21 as output. So, The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output . In this case , we are witnessing only one output for one input. Question 8. Answer: The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. Explanation: The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. For input basketball = b as output For input baseball = b as output For input football = f as output For input soccer = s as output For input swimming = s as output, So, The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. In this case we have more than one output for input. LISTING ORDERED PAIRS List the ordered pairs shown in the mapping diagram. Question 9. Answer: Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) . Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) . Question 10. Answer: Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ). Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ). Question 11. Answer: Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ). Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ). IDENTIFYING FUNCTIONS Determine whether the relation is a function. Question 12. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 0 has two outputs which are 10 and 20 . So , The relation is not a function . Question 13. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 14. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 15. YOU BE THE TEACHER Your friend determines whether the relation shown in the mapping diagram is a function. Is your friend correct? Explain your reasoning. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 4 has four outputs which are 5, 6 , 7 and 8. So , The relation is not a function . REASONING Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain. Question 16. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( 1 , 1 ), ( 3 , 3 ), ( -1 , -1 ), ( -3 , -3 ). Each input has exactly one output , So , The relation is a function . Question 17. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( 0 , 8 ),( 2 , 8 ),( 4 , 8 ),( 6 , 8 ),( 8 , 8 ),( -2 , 8 ),( -4 , 8 ). Each input has exactly one output , So , The relation is a function. Question 18. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( -2 , 1 ),( -2 , 2 ),( -2 , 3 ),( -2 , 4 ),( -2 , 5 ),( -2 , 6 ). Each input has more than one output , So , The relation is not a function. Question 19. MODELING REAL LIFE The normal pressure at sea level is 1 atmosphere of pressure(1 ATM). As you dive below sea level, the pressure changes. The mapping diagram represents the pressures at different depths. a. Complete the mapping diagram. b. Is pressure a function of depth? c. Describe the relationship between pressure and depth. d. List the ordered pairs. Then plot the ordered pairs in a coordinate plane. What do you notice about the points? e. RESEARCH What are common depths for beginner scuba divers? What are common depths for experienced scuba divers? Answer: The detailed explanation of all the answers are given below . Explanation: a. The mapping diagram is b. Yes , the pressure is a function of depth, Because depth is related to pressure in the given mapping diagram. c. The relationship between pressure and depth is, for every 10m increase in Depth of input there is an increase in 1 ATM pressure . d. The ordered pairs are ( 0 , 1 ) , ( 10 , 2 ) , (20 , 3 ) , ( 30 , 4 ) , ( 40 , 5 ), ( 50 , 6 ). The plot of the ordered pairs in a coordinate plane is From the graph, we have seen that, if the depth of the diving of scuba drivers increases then the water pressure increases with increase in depth. So, the graph have straight line . e. The common depths for beginner scuba divers is 30 feet to 60 feet or 9 to 18 meters , The common depths for experienced scuba divers is more than 60 feet or more than 18 meters . Question 20. DIG DEEPER! The table shows the cost of purchasing 1, 2, 3, or 4 T-shirts from a souvenir shop. a. Is the cost a function of the number of T-shirts purchased? b. Describe the relationship between the cost and the number cost per T-shirt of T-shirts purchased. How does the change as you purchase more T-shirts? Answer: The detailed explanation of all the answers are given below . Explanation: a. Yes , The cost is a function of the number of T-shirts purchased, Because the cost of the purchased T-shirts is varying with the number of T-shirts purchased. b. The relationship between the cost and the number cost per T-shirt of T-shirts purchased is, Input is the cost of 1 T-shirt is$10 as output , Then for 2 T-shirts cost will be $20 If 2 T-shirts will be purchased at same time, cost will be decreased by$2 so it will be $10 + 8 =$18 for 2 T-shirts.
As per the single T-shirt cost , For 3 T-shirts will be $30, So in the table given that 3 T-shirts will cost$24 , because it cost $18 + 6 =$24 for 3 T-shirts.
It goes same for 4 T-shirts , For 4 T-shirts will be $40, because it cost$24 + 4 = $28 for 4 T-shirts. The change as you purchase more T-shirts is For every increase in purchase of the number of T-shirts is decrease in the cost of total T-shirts purchased. Question 21. REPEATED REASONING The table shows the outputs for several inputs. Use two methods to predict the output for an input of 200. Answer: The output for an input of 200 is 1025. Explanation: Method 1. The relation between inputs and outputs is as follows, y = 25 + 5x As input increases by 1 , output increases by 5 units, To find output of 200 as input , put x = 200 in the equation, y = 25 + 5(200) = 25 + 1000 = 1025. So , y = 1025. Method 2. As the table shown, for every increase in input there is an increase in 5 numbers in output, So , For 1 input = 25 + 5 = 30 as output For 2 input = 30 + 5 = 35 as out put For 3 input = 35 + 5 = 40 as out put For 4 input = 40 + 5 = 45 as out put By doing this for number 200 as input we have , 1025 as output. ### Lesson 7.2 Representations of Functions EXPLORATION 1 Using a Table to Describe Relationships Work with a partner. Make a table that shows the relationship between the figure number x and the area A of each figure. Then use an equation to find which figure has an area of 81 square units when the pattern continues. Answer: a. The equation is y = 2x – 1, For figure has an area of 81 square units is 41. b. The equation is y = x², For figure has an area of 81 square units is 9. Explanation: a. figure shows the 1 square unit of each box for and it has a pattern of 2x – 1 figure 1 = 1 square unit figure 2 =3 square units figure 3 = 5 square unit and so on So, the equation is y = 2x – 1 , it is in the form of y = mx + c, Given to which figure has an area of 81 square units substitute y = 81, we have y = 2x – 1 81 = 2x – 1 2x = 82 x = 41 So, For figure has an area of 81 square units is 41. b. As shown above , we know that , figure 1 = 1 square unit figure 2 =4 square units figure 3 = 9 square unit and so on Here we have a pattern of power of its own number, So, the Equation will be y = x² Given to which figure has an area of 81 square units substitute y = 81, we have x = 9 So, For figure has an area of 81 square units is 9. EXPLORATION 2 Using a Graph Work with a partner. Use a graph to test the truth of each statement. If the statement is true, write an equation that shows how to obtain one measurement from the other. a. “You can find the horsepower of a race-car car engine if you know its volume in cubic inches” b. “You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches.” Answer: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600) We can not find the horsepower of a race-car car engine if you know its volume in cubic inches b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) , Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches Explanation: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600) We can not find the horsepower of a race-car car engine if you know its volume in cubic inches b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) , Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches Try It Question 1. Write a function rule for “The output is one-fourth of the input.” Answer: y = $$\frac{x}{4}$$ Explanation: Let us say x is input and y is output , then The output is one-fourth of the input, will be , y = $$\frac{x}{4}$$. Find the value of y when x = 5. Question 2. y = 4x – 1 Answer: y = 19. Explanation: Given, y = 4x – 1 substitute x = 5 , we get y = 4(5) – 1 y = 20 – 1 = 19 So, y = 19. Question 3. y = 10x Answer: y = 50 Explanation: Given, y =10x substitute x = 5 , we get y = 10(5) y = 50 So, y = 50. Question 4. y = 7 – 3x Answer: y = -8. Explanation: Given, y = 7 – 3x substitute x = 5 , we get y = 7 – 3(5) y = 7 – 15 = -8 So, y = -8. Graph the function. Question 5. y = x + 1 Answer: Explanation: Given , y = x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3) The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line . Question 6. y = – 3x Answer: Explanation: Given , y = – 3x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = -3(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = -3(1) = -3 . co-ordinates are (1 , -3) if x = 2 , then y = -3(2) = -6 , co-ordinates are (2 , -6) if x = 3 , then y = -3(3) = -9 , co-ordinates are (3 , -9) The co-ordinates (0 , 0) , (1 , -3) , (2 , -6) ,(3 , -9) form a straight line . Question 7. y = 3x + 2 Answer: Explanation: Given , y = 3x + 2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =3(0) + 2 = 2 . co-ordinates are (0 , 2) if x = 1 , then y = 3(1) + 2= 5 . co-ordinates are (1 , 5) if x = 2 , then y =3(2) + 2 = 7 , co-ordinates are (2 , 7) The co-ordinates (0 , 2) , (1 , 5) , (2 , 7) form a straight line . Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. WRITING FUNCTION RULES Write a function rule for the statement. Question 8. The output is three times the input. Answer: y = 3x Explanation: Let us say x is input and y is output , then The output is three times the input. will be , So , y = 3x . Question 9. The output is eight more than one-seventh of the input. Answer: y = 8 + $$\frac{x}{7}$$ . Explanation: Let us say x is input and y is output , then The output is eight more than one-seventh of the input., will be , So, y = 8 + $$\frac{x}{7}$$ . EVALUATING A FUNCTION Find the value of y when x = 5. Question 10. y = 6x Answer: y = 30 Explanation: Given, y = 6x substitute x = 5 , we get y = 6(5) =30 So, y = 30 Question 11. y = 11 – x Answer: y = 6 Explanation: Given, y = 11 – x substitute x = 5 , we get y = 11 – 5 = 6 So, y = 6. Question 12. y = $$\frac{1}{5}$$x + 1 Answer: y = 2. Explanation: Given, y = $$\frac{1}{5}$$x + 1 substitute x = 5 , we get y = $$\frac{x}{5}$$ + 1 y= $$\frac{5}{5}$$ + 1 y = 1 + 1 = 2 So, y = 2 . GRAPHING A FUNCTION Graph the function. Question 13. y = – 2x Answer: Explanation: Given , y = – 2x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 2(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = – 2(1)= -2 . co-ordinates are (1 , -2) if x = 2 , then y =- 2(2) = -4 , co-ordinates are (2 , -4) if x = 3 , then y =- 2(3) = -6 , co-ordinates are (3 , -6) The co-ordinates (0 , 0) , (1 , -2) , (2 , -4) , (3 , -6) form a straight line . Question 14. y = x – 3 Answer: Explanation: Given , y = x – 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 – 3 = -3 . co-ordinates are (0 , -3) if x = 1 , then y = 1 – 3= -2 . co-ordinates are (1 , -2) if x = 2 , then y = 2 – 3 = -1 , co-ordinates are (2 , -1) if x = 3 , then y = 3 – 3 = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , -3) , (1 , -2) , (2 , -1) , (3 , 0) form a straight line . Question 15. y = 9 – 3x Answer: Explanation: Given , y = 9 – 3x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 9 – 3(0) = 9 . co-ordinates are (0 , 9) if x = 1 , then y = 9 – 3(1) = 6 . co-ordinates are (1 , 6) if x = 2 , then y = 9 – 3(2) = 3 , co-ordinates are (2 , 3) if x = 3 , then y = 9 – 3(3) = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , 9) , (1 , 6) , (2 , 3) , (3 , 0) form a straight line . Question 16. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Answer: As mentioned in the explanation below a & d , b & c are different . Explanation: Given , a. what output is 4 more than twice the input 3? Let us say that , y is output and x is input and given as 3 , then, we have y = 4 + 2(3) = 10. b. What output is twice the sum of the input 3 and 4? Let us say that , y is output and x is input and given as 3 , then, we have y = 2( 3 + 4 ) = 14. c. what output is the sum of 2 times the input 3 and 4? Let us say that , y is output and x is input and given as 3 , then, we have y = 2( 3 + 4 ) = 14. d. what output is 4 increased by twice the input 3? Let us say that , y is output and x is input and given as 3 , then, we have y = 4 + 2(3) = 10. So, a & d , b & c are different . Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 17. The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. How many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people? Justify your answer using a graph. Answer: So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people Explanation: Given, The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. we need to find how many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people, For every 10,000 people we have 23 care takers Then for 250,000 people we have $$\frac{23 × 250,000}{10,000}$$ = 23 × 25 = 575 So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people Question 18. DIG DEEPER! A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. Write and graph a function that describes the relationship between carbon dioxide produced and distance traveled. Answer: y = 22x + 18 is the linear equation Explanation: Given, A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. So, we have y = 22x + 18 is in the form of y = mx +c To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 22(0) + 18 = 18 . co-ordinates are (0 , 18) if x = 1 , then y =22(1) + 18= 40 . co-ordinates are (1 , 40) if x = 2 , then y =22(2) + 18 = 62 , co-ordinates are (2 , 62) if x = 3 , then y =22(3) + 18 = 84 , co-ordinates are (3 , 84) The co-ordinates (0 , 18) , (1 , 40) , (2 , 62) , (3 , 84) form a straight line . The graph is ### Representations of Functions Homework & Practice 7.2 Review & Refresh Determine whether the relation is a function. Question 1. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 2. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 3. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 2 has two outputs which are 0 and -4 . So , The relation is not a function . Find the slope of the line. Question 4. Answer: slope = 1. Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -2 to -4 =2 change in x or ▲x is change from 1 to 3 = 2 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{2}{2}$$ slope = 1. Question 5. Answer: slope = $$\frac{5}{2}$$ . Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -4 to 1 = 5 change in x or ▲x is change from -1 to -3 = 2 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{5}{2}$$ . Question 6. Answer: slope = $$\frac{1}{3}$$ . Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -4 to -3 = 1 change in x or ▲x is change from 1 to 4 = 3 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{1}{3}$$ . Concepts, Skills, & Problem Solving USING A GRAPH Use a graph to test the truth of the statement. If the statement is true, write an equation that shows how to obtain one measurement from the other measurement. (See Exploration 2, p. 281.) Question 7. “You can find the weight of a cell phone in ounces if you know its screen size in inches.” Answer: we can does not find the weight of a cell phone in ounces if you know its screen size in inches. From the given table , Ordered pairs are (4 , 4) , (4.7 , 4.8) , (5 , 4.8) , (5.5 , 6.4) First find the slope m of the line containing the two given points (4, 4) and (4.7, 4.8) m = (y2-y1) / (x2-x1) m= (4.8 – 4) / (4.7 – 4) m = 0.8/0.7 . So, we can does not find the weight of a cell phone in ounces if you know its screen size in inches. Question 8. “You can find the age of a child in years if you know the age of the child in months.” Answer: YES, y = 0.08x + 0.04 is a linear equations Explanation: From the given table , Ordered pairs are (9 , 0.75) , (12 , 1) , (15 , 1.25) , (24 , 2) First find the slope m of the line containing the two given points (12 ,1) and (24, 2) m = (y2-y1) / (x2-x1) m= (2 – 1) / (24 – 12) m = 1/12 m = 0.08. substitute the slope in the (12 ,1) to get point slope to form a line. y-y1 = m (x-x1) y – 1 = 0.08(x – 12) y –1 = 0.08x – 0.96 y = 0.08x –0.96 + 1 y =0.08 x + 0.04 So, y = 0.08x + 0.04 is a linear equation WRITING FUNCTION RULES Write a function rule for the statement. Question 9. The output is half of the input. Answer: y = $$\frac{x}{2}$$. Explanation: Let us say x is input and y is output , then The output is half of the input, will be , y = $$\frac{x}{2}$$. Question 10. The output is eleven more than the input. Answer: y = x + 11 Explanation: Let us say x is input and y is output , then The output is eleven more than the input, will be , y = x + 11 Question 11. The output is three less than the input. Answer: y = x – 3 Explanation: Let us say x is input and y is output , then The output is three less than the input, will be , y = x – 3 Question 12. The output is the cube of the input. Answer: y = x³ Explanation: Let us say x is input and y is output , then The output is the cube of the input, will be , y = x³ Question 13. The output is six times the input. Answer: y = 6x Explanation: Let us say x is input and y is output , then The output is six times the input, will be , y = 6x Question 14. The output is one more than twice the input. Answer: y = 2x + 1 Explanation: Let us say x is input and y is output , then The output is one more than twice the input, will be , y = 2x + 1 EVALUATING A FUNCTION Find the value of y for the given value of x. Question 15. y = x + 5; x = 3 Answer: y = 8 Explanation: Given, y = x + 5 substitute x = 3 , we get y = 3 + 5 So, y = 8. Question 16. y = 7x; x = – 5 Answer: y = -35. Explanation: Given, y = 7x substitute x = -5 , we get y = 7(-5) So, y = -35. Question 17. y = 1 – 2x; x = 9 Answer: y = -17 Explanation: Given, y = 1 – 2x substitute x = 9 , we get y = 1 – 2(9) y = 1 – 18 So, y = -17. Question 18. y = 3x + 2; x = 0.5 Answer: y = 5.5 Explanation: Given, y = 3x + 2 substitute x = 0.5 , we get y = 3(0.5) + 2 y = 3.5 + 2 So, y = 5.5 . Question 19. y = 2x3; x = 3 Answer: y = 54 Explanation: Given, y = 2x3 substitute x = 3 , we get y = 2(3)³ y = 2 × 27 = 54 So, y = 54. Question 20. y = $$\frac{x}{2}$$ + 9; x = – 12 Answer: y = 3 Explanation: Given, y = $$\frac{x}{2}$$ + 9 substitute x = -12 , we get y = $$\frac{-12}{2}$$ + 9 y = -6 + 9 So, y = 3 . GRAPHING A FUNCTION Graph the function. Question 21. y = x + 4 Answer: Explanation: Given , y = x + 4 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 4 = 4 . co-ordinates are (0 , 4) if x = 1 , then y = 1 + 4 = 5 . co-ordinates are (1 , 5) if x = 2 , then y = 2 + 4 = 6 , co-ordinates are (2 , 6) if x = 3 , then y = 3 + 4 = 7 , co-ordinates are (3 , 7) The co-ordinates (0 , 4) , (1 , 5) , (2 , 6) , (3 , 7) form a straight line . Question 22. y = 2x Answer: Explanation: Given , y = 2x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 2(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = 2(1) = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2(2) = 4 , co-ordinates are (2 , 4) if x = 3 , then y = 2(3) = 6 , co-ordinates are (3 , 6) The co-ordinates (0 , 0) , (1 , 2) , (2 , 4) , (3 , 6) form a straight line . Question 23. y = – 5x + 3 Answer: Explanation: Given , y = – 5x + 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- 5(0) + 3 = 3 . co-ordinates are (0 , 3) if x = 1 , then y = – 5(1) + 3 = -2 . co-ordinates are (1 , -2) if x = 2 , then y = – 5(2) + 3 = -7 , co-ordinates are (2 , -7) if x = 3 , then y = – 5(3) + 3 = -12 , co-ordinates are (3 , -12) The co-ordinates (0 , 3) , (1 , -2) , (2 , -7) , (3 , -12) form a straight line . Question 24. y = $$\frac{x}{4}$$ Answer: Explanation: Given , y = $$\frac{x}{4}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = $$\frac{0}{4}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{1}{4}$$ = 0.25 . co-ordinates are (1 , 0.25) if x = 2 , then y = $$\frac{2}{4}$$ = 0.5 , co-ordinates are (2 , 0.5) if x = 3 , then y = $$\frac{3}{4}$$ = 0.75 , co-ordinates are (3 , 0.75) The co-ordinates (0 , 0) , (1 , 0.25) , (2 , 0.5) , (3 , 0.75) form a straight line . Question 25. y = $$\frac{3}{2}$$x + 1 Answer: Explanation: Given , y = $$\frac{3}{2}$$x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =$$\frac{3}{2}$$(0) + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = $$\frac{3}{2}$$(1) + 1= 2.5 . co-ordinates are (1 , 2.5) if x = 2 , then y = $$\frac{3}{2}$$(2) + 1 = 4 , co-ordinates are (2 , 4) if x = 3 , then y = $$\frac{3}{2}$$(3) + 1 = 5.5 , co-ordinates are (3 , 5.5) The co-ordinates (0 , 1) , (1 , 2.5) , (2 , 4) , (3 , 5.5) form a straight line . Question 26. y = 1 + 0.5x Answer: Explanation: Given , y = 1 + 0.5x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =1 + 0.5(0) = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 0.5(1) = 1.5 . co-ordinates are (1 , 1.5) if x = 2 , then y = 1 + 0.5(2) = 2 , co-ordinates are (2 , 2) if x = 3 , then y = 1 + 0.5(3) = 2.5 , co-ordinates are (3 , 2.5) The co-ordinates (0 , 1) , (1 , 1.5) , (2 , 2) , (3 , 2.5) form a straight line . MATCHING Match the graph with the function it represents. A. y = $$\frac{x}{3}$$ B. y = x + 1 C. y = – 2x + 6 Question 27. Answer: B. y = x + 1. Explanation: Given , y = x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3) The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line . Question 28. Answer: c. y = – 2x + 6 Explanation: Given , y = – 2x + 6 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 2(0) + 6 = 6 . co-ordinates are (0 , 6) if x = 1 , then y = – 2(1) + 6 = 4 . co-ordinates are (1 , 4) if x = 2 , then y = – 2(2) + 6 = 2 , co-ordinates are (2 , 2) if x = 3 , then y = – 2(3) + 6 = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , 6) , (1 , 4) , (2 , 2) , (3 , 0) form a straight line . Question 29. Answer: A. y = $$\frac{x}{3}$$ Explanation: Given , y = $$\frac{x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =$$\frac{0}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{1}{3}$$= 0.3 . co-ordinates are (1 , 0.3) if x = 2 , then y = $$\frac{2}{3}$$= 0.6 , co-ordinates are (2 , 0.6) The co-ordinates (0 , 0) , (1 , 0.3) , (2 , 0.6) form a straight line . Question 30. YOU BE THE TEACHER Your friend graphs the function represented by the input-output table. Is your friend correct? Explain your reasoning. Answer: Yes , He is correct Explanation: Ordered pairs are (-1 , -4) , (1 , -2) , (3 ,0) , (5 , 2) these points form a straight line when graphed. Yes , He is correct Question 31. MODELING REAL LIFE A dolphin eats 30 pounds of fish per day. a. Write and graph a function that relates the number p of pounds of fish that a dolphin eats in d days. b. How many total pounds of fish does a dolphin eat in 30 days? Answer: Explanation: a. Given , A dolphin eats 30 pounds of fish per day. by each passing day eating fish is increased by the day passes . So, y = 30x is the function, The graph represents the function as b. Given , A dolphin eats 30 pounds of fish per day. then for 30 days , 30 × 30 = 900 pounds So, A dolphin eats 900 pounds of fish in 30 days Question 32. MODELING REAL LIFE You fill a fish tank with 55 gallons of water on Saturday. The water evaporates at a rate of 1.5 gallons per day. You plan to add water when the tank reaches 49 gallons. When will you add water? Justify your answer. Answer: As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday. Explanation: Given data ,, implies that slope of the function m = -1.5 The y intercept b= 55, Then the equation will be y = 55 – 1.5x Given , You plan to add water when the tank reaches 49 gallons. determine x for y = 49 , So, 49 = 55 – 1.5x , 1.5x = 55 – 49 1.5x = 6 x = $$\frac{6}{1.5}$$ x = 4. As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday. USING AN EQUATION Find the value of x for the given value of y. Question 33. y = 5x – 7; y = – 22 Answer: x = -3 Explanation: Given, y = 5x – 7 x = $$\frac{y + 7}{5}$$ substitute y = -22 , we get x = $$\frac{-22 + 7}{5}$$ x = $$\frac{- 15}{5}$$ x = -3 So, x = -3 . Question 34. y = 9 – 7x; y = 37 Answer: x = -4 Explanation: Given, y = 9 – 7x x = $$\frac{9 – y}{7}$$ substitute y = 37 , we get x = $$\frac{9 – 37}{7}$$ x = $$\frac{- 28}{7}$$ x = -4 So, x = -4 . Question 35. y = $$\frac{x}{4}$$ – 7; y = 2 Answer: x = 36 Explanation: Given, y = $$\frac{x}{4}$$ – 7 x = 4( y + 7) substitute y = 2 , we get x = 4( 2 + 7) x = 4(9) x = 36 So, x = 36 . Question 36. PROBLEM SOLVING You decide to make and sell bracelets. The cost of your materials is$84.00. You charge $3.50 for each bracelet. a. P Write a function that represents the profit for selling b bracelets. b. Which variable is independent? dependent? Explain. c. You will break even when the cost of your materials equals your income. How many bracelets must you sell to break even? Answer: a. A function that represents the profit for selling b bracelets is p = 3.5b – 84. b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable. c. To break even you must sell 24 bracelets. Explanation: a. Given , The cost of your materials is$84.00. You charge $3.50 for each bracelet, Let p be the profit , b be the number of bracelets sold, So, profit = income – cost . p = 3.5b – 84. Thus , A function that represents the profit for selling b bracelets is p = 3.5b – 84. b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable. c. set the income expression from part a equal to the cost of 84 and solve for b , So, income = cost . 3.5b = 84 , b = $$\frac{84}{3.5}$$ b = 24. To break even you must sell 24 bracelets. Question 37. MODELING REAL LIFE A furniture store is having a sale where everything is 40% off. a. Write and graph a function that represents the amount of discount on an item at regular price. b. You buy a bookshelf that has a regular price of$85. What is the sale price of the bookshelf?
Answer: a. The function is y = 0.4x and the graph is given below.
b. The sale price of the bookshelf s $51. Explanation: a. A function that represents the amount of discount on an item at regular price is , Given , 40% = 0.4 , To find the percent of the number , we should multiply the number by the percent in the decimal form , so, the equation is d = 0.4p , let us convert it in to a function form , y = 0.4x we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0.4(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = 0.4(1)= 0.4 . co-ordinates are (1 , 0.4) if x = 2 , then y =0.4(2) = 0.8 , co-ordinates are (2 , 0.8) if x = 3 , then y = 0.4(3) = 1.2 , co-ordinates are (3 , 1.2) The co-ordinates (0 , 0) , (1 , 0.4) , (2 , 0.8) , (3 , 1.2) form a straight line . The graph is b. Given , You buy a bookshelf that has a regular price of$85.
The sale price of the bookshelf is ,
substituting the given price in p = 85 ,
it will be the discount d = 0.4 (85) = 34
Then the sale price is $85 –$34 = $51. So, The sale price of the bookshelf s$51.

Question 38.
REASONING
You want to take a two-hour air boat tour. Which is a better deal, Snake Tours or Gator Tours? Use functions to justify your answer.

Answer: By using functions , $50 >$40 , So, Gator tours are cheaper than the snake tours .

Explanation:
Given , You want to take a two-hour air boat tour.
Let x be the hours of  air boat tour and y be the cost of air boat tour ,
Snake tours , y = 25x
putt x = 2 ,
So , y = 25 (2) = 50 .
y = 50.

Gator tour , y = 35 + $$\frac{5}{2}$$x
Put x = 2 ,
So, y = 35 + $$\frac{5}{2}$$ x
y = 35 + 2.5x
y = 35 + 2.5 (2)
y = 35 + 5
y = 40 .

Finally $50 >$40 , So, Gator tours are cheaper than the snake tours

Question 39.
REASONING
The graph of a function is a line that passes through the points (3, 2), (5, 8), and (8, y). What is the value of y?
Answer: The value of y is 17 , so, The third given point is (8, 17)

Explanation:
First find the slope m of the line containing the two given points (3,2) and (5,8)
m = (y2-y1) / (x2-x1)
m= (8 – 2) / (5 – 3)
m = 6 / 2
m = 3
Then use the slope and one of the given points (3,2) to find the y-intercept
y = mx +
2 = 3(3) + b
2 = 9 + b
-7 = b
The equation is   y = 3x -7
Then find the third point (8, y) by replacing x by 8
y = 3x -7
y = 3(8) -7
y = 24 -7
y = 17

so the third given point is (8, 17)

Question 40.
CRITICAL THINKING
Make a table where the independent variable is the side length of a square and the dependent variable is the perimeter. Make a second table where the independent variable is the side length of a square and the dependent variable is the area. Graph both functions in the same coordinate plane. Compare the functions.
Answer: The graph for the perimeter is linear , The graph for the Area is Quadratic .

Explanation:
Let us say , s be the side length of the square ,
Then the perimeter is P = 4s ,
The function will be y= 4x,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 4(1) = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = 4(2) =8 , co-ordinates are (2 , 8)
if x = 3 , then y = 4(3) = 0 , co-ordinates are (3 , 12)
The co-ordinates (0 , 0) , (1 , 4) , (2 ,8) , (3 , 12) form a straight line .

Table will be ,

Let us say , s be the side length of the square ,
Then the Area is A = s² ,
The function will be y=x²,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0² = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 1² = 1 . co-ordinates are (1 , 1)
if x = 2 , then y = 2² =4 , co-ordinates are (2 , 4)
if x = 3 , then y = 3² = 9 , co-ordinates are (3 , 9)
The co-ordinates (0 , 0) , (1 , 1) , (2 ,4) , (3 , 9) form a straight line .

Second table is
Then the graph is
The graph for the perimeter is linear , The graph for the Area is Quadratic .

Question 41.
PUZZLE
The blocks that form the diagonals of each square are shaded. Each block has an area of one square unit. Find the “green area” of Square 20. Find the “green area” of Square 21. Explain your reasoning.

Answer:  The green area of the Square 20 is 46 square units and The green area of the Square 21 is 48 square units.

Explanation:
Given , Each block has an area of one square unit,
Square 1 has  the diagonals of each square are shaded. the “green area” is 3 + 3 = 6 square units ,
Square 2 has  the diagonals of each square are shaded. the “green area” is 4 + 4 = 8 square units ,
Square 3 has  the diagonals of each square are shaded. the “green area” is 5 + 5 = 10 square units ,
Square 4 has  the diagonals of each square are shaded. the “green area” is 6 + 6 = 12 square units,
Square 5 has  the diagonals of each square are shaded. the “green area” is 7 + 7 = 14 square units ,
Here , The number of squares are increasing by one block with the square numbers.
So for the , Square 20 has  the diagonals of each square are shaded. the “green area” is 23 + 23 = 46 square units,
And Square 21 has  the diagonals of each square are shaded. the “green area” is 24 + 24 = 48 square units.

### Lesson 7.3 Linear Functions

EXPLORATION 1

Writing and Graphing Functions
Work with a partner. Each table shows a familiar pattern from geometry.

• Determine what the variables x and y represent. Then write a function rule that relates y to x.
• Is the function a linear function? Explain your reasoning.

Answer: All of them are explained below

Explanation:
The variables x and y represents a rectangle
a. From the given table , Ordered pairs are (1 , 10) , (2 , 12) , (3 , 14) , (4 , 16)
First find the slope m of the line containing the two given points (1 ,-1) and (2, -2)
m = (y2-y1) / (x2-x1)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
substitute the slope in the (1 , 10) to get point slope to form a line.
y-y1 = m (x-x1)
y – 10 = -1/2 ( x –1)
2(y – 10) = -x  + 1
2y – 20 = -x+ 1
2y = -x  + 21
y = $$\frac{-1}{2}$$ (x – 21)
So ,  y = $$\frac{-1}{2}$$ (x – 21) is linear function.

b. The variables x and y represent a circle
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y2-y1) / (x2-x1)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

c. The variables x and y represents a trapezoid
a. From the given table , Ordered pairs are (1 , 5) , (2 , 6) , (3 , 7) , (4 , 8)
First find the slope m of the line containing the two given points (1 ,5) and (2, 6)
m = (y2-y1) / (x2-x1)
m= (6 – 5) / (2 – 1)
m = 1 .
substitute the slope in the (1 ,5) to get point slope to form a line.
y-y1 = m (x-x1)
y – 5 = 1(x – 1)
y – 5 = x – 1
y = x – 1 + 5
y = x + 4
So, y = x + 4 is a linear equation

d. The variables x and y represents a cube
a. From the given table , Ordered pairs are (1 , 28) , (2 , 40) , (3 , 52) , (4 , 64)
First find the slope m of the line containing the two given points (1 ,28) and (2, 40)
m = (y2-y1) / (x2-x1)
m= (40 – 28) / (2 – 1)
m = 12 .
substitute the slope in the (1 ,28) to get point slope to form a line.
y-y1 = m (x-x1)
y – 28 = 12(x – 1)
y – 28 = 12x – 12
y = 12x – 12 + 28
y = 12x + 16
So, y = 12x + 16 is a linear equation.

Try It

Question 1.
Use the graph to write a linear function that relates y to x.

Answer: the linear function is y = $$\frac{-1}{2}$$x -1.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , 1) , (-2 , 0 ) , (0 , -1) , ( 2, -2 )
First find the slope m of the line containing the two given points (0 ,-1) and (2, -2)
m = (y2-y1) / (x2-x1)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = $$\frac{-1}{2}$$x -1.

Question 2.
Use the table to write a linear function that relates y to x.

Answer: the linear function is y = (0)x + 2.

Explanation:
Ordered pairs are (-2 , 2) , (-1 , 2) , (0 , 2) , (1 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,2) and (1, 2)
m = (y2-y1) / (x2-x1)
m= (2 – 2) / (1 – 0)
m = 0
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = (0)x + 2.

Question 3.
WHAT IF?
The rate of descent doubles. Repeat parts (a) and (b).
Answer: a. the linear function is y = -1x + 65.
b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Explanation:
a. From the Given table , The rate of descents is 5
If it doubles , then The rate of descents is 10.
The the ordered pairs will be (0 , 65) , (10 ,55) , (20 , 45) .
First find the slope m of the line containing the two given points (0 ,65) and (10, 55)
m = (y2-y1) / (x2-x1)
m= (55 – 65) / (10 – 0)
m = -10 / 10
m = -1
Because the line crosses the y axis at ( 0, 65 ) , The y intercept is 65.
So , the linear function is y = -1x + 65.

b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
WRITING A LINEAR FUNCTION
Use the graph to write a linear function that relates y to x.

Answer:  The linear function is y = -4x -2 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 6) , (-1 , 2 ) , (0 , -2) , ( 1, -6 )
First find the slope m of the line containing the two given points (0 ,-2) and (1, -6)
m = (y2-y1) / (x2-x1)
m= (-6 – (-2)) / (1 – 0)
m = -4 .
Because the line crosses the y axis at ( 0, -2) , The y intercept is -2.
So , the linear function is y = -4x -2 .

Question 5.
INTERPRETING A LINEAR FUNCTION
The table shows the revenue R (in millions of dollars) of a company when it spends A (in millions of dollars) on advertising.

a. Write and graph a linear function that relates R to A.
b. Interpret the slope and the y-intercept.
Answer:  a. The linear function is y = 2x + 2. and the graph is shown below
b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Explanation:
a. From the given table ,
The the ordered pairs will be (0 , 2) , (2 ,6) , (4 , 10) , (6 , 14) , (8 ,18) .
The graph is
First find the slope m of the line containing the two given points (0 ,2) and (2, 6)
m = (y2-y1) / (x2-x1)
m= (6 – 2) / (2 – 0)
m = 4 / 2
m = 2
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = 2x + 2.

b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
Manager A earns $15 per hour and receives a$50 bonus. The graph shows the earnings of Manager B. (a) Which manager has a greater hourly wage? (b) After how many hours does Manager B earn more money than Manager A?

Answer: a. Manager B has the greater hourly wage than Manager A .
b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A . Explanation: a. Manager A earns$15 per hour and receives a $50 bonus. The ordered pairs will be (0 , 0) , (1 , 15) , (2 , 30) , (3 , 45) The graph shows the earnings of Manager B. Ordered pairs from the graph are (0 , 0) , (1 , 25) , (2 , 50) , (3 , 75) So, Manager B has the greater hourly wage than Manager A . b. As manager A receives a$50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Question 7.
Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day. The table shows the amount (in gigabytes) of data that your friend has left days after the start of each month. Who runs out of data first? Justify your answer.

Answer:  you will be run out of data first

Explanation:
a. Given , Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day.
Let x be the number of days and y be the total data in gigabytes.
So, y = -0.08x + 2 ,
You will be out of data if , -0.08x + 2 = 0 ,
-0.08x + 2 = 0
2 = 0.08x
x = $$\frac{2}{0.08}$$
x = 25.
Hence ,you will be run out of data in 25 days.
b. Daily data usage for the friend will be given by the slope of the graph.
The the ordered pairs will be (0 , 3) , (7 ,2.3) , (14 , 1.6) .
First find the slope m of the line containing the two given points (7 ,2.3) and (14, 1.6)
m = (y2-y1) / (x2-x1)
m= (1.6 – 2.3) / (14 – 7)
m = -0.7 / 7
m = -0.1
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = -0.1x + 3.
Your friend will be out of data if ,
-0.1x + 3 = 0
3 = 0.1x
x = $$\frac{3}{0.1}$$
x = 30 .
Hence ,Friend will be run out of data in 30 days

So , you will be run out of data first

### Linear Functions Homework & Practice 7.3

Review & Refresh

Write a function rule for the statement. Then graph the function.
Question 1.
The output is ten less than the input.
Answer: y = x – 10.

Explanation:
Let us say x is input and y is output , then
The output is ten less than the input, will be ,
y = x – 10.

Question 2.
The output is one-third of the input.
Answer: y = $$\frac{x}{3}$$

Explanation:
Let us say x is input and y is output , then
The output is one-third of the input, will be ,
y = $$\frac{x}{3}$$ .

Solve the system.
Question 3.
y = x + 5
y = – 3x + 1
Answer: X = 0 , Y = 5

Explanation:
Y=3X+5 ——————-(1)
Y=X+5 ——————(2)
Substitute Y=X+5 in equation (1)
X+5=3X+5
Solve it for X
X+3X=55
4X=0
X=0/4=0
X = 0
Substitute X=0 in equation (1)
Y=0+5
Y=5

Question 4.
x + y = – 4
6x + 2y = 4
Answer:  X = 3 , Y= -7 .

Explanation:
2Y=−6X+4 ——————-(1)
Y= –X-4 ——————(2)
Substitute Y= –X-4 in equation (1)
2Y = −6X+4
2 ( X – 4 ) = −6X + 4
-2X – 8 = -6X + 4
6X -2X = 8 + 4
4X = 12
X = 3
Substitute X=3 in equation (2)
Y=– 3 – 4
Y= -7 .

Question 5.
– 4x + 3y = 14
y = 2x + 8
Answer:  X = -5 , Y = -2 .

Explanation:
3Y = 4X+14 ——————-(1)
Y = 2X + 8 ——————(2)
Substitute Y= 2X + 8 in equation (1)
3Y = 4X+14
3(2X + 8) = 4X+14
6X + 24 = 4X + 14
6X – 4X = 14 – 24
2X = -10
X = -5
Substitute X= -5 in equation (2)
Y= 2(-5) + 8
Y= -10 + 8
Y = -2.

Concepts, Skills, &Problem Solving

WRITING AND GRAPHING FUNCTIONS The table shows a familiar pattern from geometry. (a) Determine what the variables x and y represent. Then write a function rule that relates y to x. (b) Is the function a linear function? Explain your reasoning. (See Exploration 1, p. 289.)
Question 6.

Answer: a. The variables x and y represent a right angle triangle
b. y = 2x  is linear function.

Explanation:
In order to write the function we have to write the ordered pairs
Ordered pairs are  (1 , 2) , (2 , 4) ,  (3 , 6 ) , (4 , 8), (5 , 10 ) .
a. the variables x and y represent a right angle triangle
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 2) , (2 , 4)
m = (y2-y1) / (x2-x1)
m= (4 – 2) / (2– 1)
m = 2/1
m = 2
b. substitute the slope in the (2 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 4 = 2 ( x – 2)
y – 4 = 2x – 4
y = 2x – 4 + 4
y = 2x
So ,  y = 2x  is linear function.

Given side of triangle is 4 then x= 4/2 = 2
x = 2 and y = 4.

Question 7.

Answer: y = 3.14x  is linear function. and The variables x and y represent a circle

Explanation:
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y2-y1) / (x2-x1)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

WRITING LINEAR FUNCTIONS Use the graph or table to write a linear function that relates y to x.
Question 8.

Answer: The linear function is y = $$\frac{4}{3}$$x +2

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-3 , -2) , (0 , 2 ) , (3 , 6) , ( 6, 10 )
First find the slope m of the line containing the two given points (3 ,6) and (6, 10)
m = (y2-y1) / (x2-x1)
m= (10 – 6) / (6 – 3)
m = 4/3 .
Because the line crosses the y axis at ( 0, 2) , The y intercept is 2.
So , the linear function is y = $$\frac{4}{3}$$x +2 .

Question 9.

Answer: The linear function is y = (0)x +3 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 3) , (-1 , 3 ) , (0 , 3) , ( 1, 3 ) , (2 , 3)
First find the slope m of the line containing the two given points (1 ,3) and (2, 3)
m = (y2-y1) / (x2-x1)
m= (3 – 3) / (2 – 1)
m = 0 .
Because the line crosses the y axis at ( 0, 3) , The y intercept is 3.
So , the linear function is y = (0)x +3 .

Question 10.

Answer: The linear function is y = $$\frac{-1}{4}$$x + 0.

Explanation:
Ordered pairs are (-8 , 2) , (-4 , 1) , (0 , 0) , (4 , -1)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (-8 ,2) and (-4, 1)
m = (y2-y1) / (x2-x1)
m= (1 – 2) / (-4 – (-8))
m = -1/4
Because the line crosses the y axis at ( 0, 0 ) , The y intercept is 0.
So , the linear function is y = $$\frac{-1}{4}$$x + 0.

Question 11.

Answer: The linear function is y = $$\frac{2}{3}$$x + 5.

Explanation:
Ordered pairs are (-3 , 3) , (0 , 5) , (3 , 7) , (6 , 9)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 ,7) and (6, 9)
m = (y2-y1) / (x2-x1)
m= (9 – 7) / (6 – 3)
m = 2/3
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y = $$\frac{2}{3}$$x + 5.

Question 12.
INTERPRETING A LINEAR FUNCTION
The table shows the length y (in inches) of a person’s hair after x months.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
Answer: a. The linear function is y = 0.5x + 11.
b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Explanation:
a. Given ,
The ordered pairs will be (0 , 11) , (3 ,12.5) , (6 , 14) .
The graph is
First find the slope m of the line containing the two given points (3 ,12.5) and (6 , 14)
m = (y2-y1) / (x2-x1)
m= (14 – 12.5) / (6 – 3)
m = 1.5 / 3
m = 0.5
Because the line crosses the y axis at ( 0, 11 ) , The y intercept is 11.
So , the linear function is y = 0.5x + 11.

b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Question 13.
INTERPRETING A LINEAR FUNCTION
The table shows the percent (in decimal form) of battery power remaining x hours after you turn on a laptop computer.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope, the x-intercept, and the y-intercept.
c. After how many hours is the battery power at75%?
Answer: a. The linear function is y = -0.2x + 1.
b. given below the explanation.
c. Battery will be 75% after 1.25 hours.

Explanation:
a. Given ,
The ordered pairs will be (0 , 1) , (2 ,0.6) , (4 , 0.2) .
The graph is
First find the slope m of the line containing the two given points (2 ,0.6) and (4 , 0.2)
m = (y2-y1) / (x2-x1)
m= (0.2 – 0.6) / (4 – 2)
m = -0.4 / 2
m = -0.2
Because the line crosses the y axis at ( 0, 1 ) , The y intercept is 1.
So , the linear function is y = -0.2x + 1.

b. Slope is -0.2 which means that as time increases by 1 hour, Battery power remaining decreases by 20% .
y intercept is 1, which means initially the battery power remaining before usage was 100%.
x intercept is 5 which means the battery remaining will be 0 after 5 hours.

c. battery percent will be 75% of 0.75 if ,
-0.2x + 1 = 0.75
0.2x = 1 – 0.75
x = 0.25/0.2
x = 1.25
Battery will be 75% after 1.25 hours.

Question 14.
MODELING REAL LIFE
The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x. The graph shows the number of calories burned by hiking.

a. Which activity burns more calories per minute?
b. You perform each activity for 45 minutes. How many total calories do you burn? Justify your answer.
Answer: a. hiking burns more calories than kayaking .
b. In kayaking, 202.5 calories are burnt per minute. and In hiking , 225 calories are burnt per minute.

Explanation:
a. The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x.
So, The ordered pairs of the graph are (0 , 0) , (1 , 4.5) , (2 , 9) , (3, 13.5)
Here , In kayaking burns 4.5 calories per minute .
For hiking ,
The ordered pairs of the graph are (0 , 0) , (1 , 5) , (2 , 10) , (3, 15)
Here , In hiking burns 5 calories per minute.
Thus , hiking burns more calories than kayaking .

b. Given , perform each activity for 45 minutes.
Liner function of the kayaking is y = 4.5x
substitute x = 45 in equation
y = 4.5 (45)
y = 202.5
In kayaking, 202.5 calories are burnt per minute.
Linear function of the hiking is y = 5x
substitute x = 45 in equation
y = 5 (45)
y = 225
In hiking , 225 calories are burnt per minute.

Question 15.
DIG DEEPER!
You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50. The table shows your distance at various times throughout the race. For what distances will you win the race? Explain.

Answer: you will win the race for distances greater than 190 feet

Explanation:
The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50.
The slope of the line is 14 so , your friend runs at the rate of 14 ft per second
To find your rate , the ordered pairs are (2 , 38) , (4 , 76) , (6 , 114) , (8 , 152)
First find the slope m of the line containing the two given points (2 ,38) and (4 , 76)
m = (y2-y1) / (x2-x1)
m= (76 – 38) / (4 – 2)
m = 38 / 2
m = 19
You are running at the rate of 19 ft per second.
To get the linear equation , substitute the slope in the (2 , 38) to get point slope to form a line.
Then we have , y = 19x
Now if x = 10 , to run faster then ,
y = 19(10)
y = 190 .
Your friend linear equation is y = 14x + 50 .
if x = 10 ,then
y = 14(10) + 50
y = 140 + 50
y = 190.
So , for x > 10 , means you will run farther than your friend which means you would win the race .
Therefore, you will win the race for distances greater than 190 feet.

Question 16.
REASONING
You and your friend are saving money to buy bicycles that cost $175 each. You have$45 to start and save an additional $5 each week. The graph shows the amount y(in dollars) that your friend has after x weeks. Who can buy a bicycle first? Justify your answer. Answer: your friend will but the bicycle first. Explanation: Given , your friend savings are the ordered pairs are (0,15) and (3,39) First find the slope m of the line containing the two given points (0,15) and (3,39) m = (y2-y1) / (x2-x1) m= (39 – 15) / (3 – 0) m = 24 / 3 m = 8 Because the line crosses the y axis at ( 0, 15 ) , The y intercept is 15. So , the linear function is y = 8x + 15. to buy bicycles that cost$175 each
if y = 175 , then
175 = 8x + 15
8x = 175 – 15
x = 160/8
x = 20
So, your friend need 20 weeks to buy the bicycle
Given, You have $45 to start and save an additional$5 each week
So , the linear function will be y = 5x + 45.
to buy bicycles that cost $175 each if y = 175 , then 175 = 5x + 45 5x = 175 – 45 x = 130/5 x = 26 So, you need 26 weeks to buy the bicycle. Hence, your friend will but the bicycle first. Question 17. CRITICAL THINKING Is every linear equation a linear function? Explain your reasoning. Answer: All linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value). Question 18. PROBLEM SOLVING The heat index is calculated using the relative humidity and the temperature. For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F. On a summer day, the relative humidity is 75%, the temperature is 94°F, and the heat index is 124°F. Estimate the heat index when the relative humidity is 75% and the temperature is 100°F. Use a function to justify your answer. Answer: Heat index is 148°F Explanation: The form of linear equation is y = mx + c and the slope of the function is given by m = (y2-y1) / (x2-x1) Let y be the heat index and x be the temperature Given , (94, 124) For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F that is m = 4 Since the line passes through (94, 124) means 124 = 4x + c 124 = 4(94) + c 124 = 376 + c c = 124 – 376 c = -252 Linear function for the heat index is y = 4x – 252 put x = 100 So, y = 4(100) – 252 y = 400 – 252 y = 148. Finally, Heat index is 148°F. ### Lesson 7.4 Comparing Linear and Non Linear Functions EXPLORATION 1 Comparing Functions Work with a partner. Each equation represents the height h (in feet) of a falling object after t seconds. • Graph each equation. Explain your method. • Decide whether each graph represents a or function. • Compare the falling objects. Answer: Explained below Explanation: a. Given, h = 300 – 15t can be written as y = 300 – 15x h = 300 – 15t , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =300 – 15(0) = 300 . co-ordinates are (0 , 300) if x = 1 , then y = 300 – 15(1) = 285 . co-ordinates are (1 , 285) if x = 2 , then y = 300 – 15(2) = 270 , co-ordinates are (2 , 270) if x = 3 , then y = 300 – 15(3) = 255 , co-ordinates are (3 , 255) The co-ordinates (0 , 300) , (1 , 285) , (2 , 270) , (3 , 255) form a straight line . The graph is Given , h = 300- 16t2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =300- 16(0)2 = 300 . co-ordinates are (0 , 300) if x = 1 , then y =300- 16(1)2 = 284 . co-ordinates are (1 , 284) if x = 2 , then y = 300- 16(2)2 = 236 , co-ordinates are (2 , 236) if x = 3 , then y = 300- 16(3)2 = 252 , co-ordinates are (3 , 252) The co-ordinates (0 , 300) , (1 , 284) , (2 , 236) , (3 , 252) does not form a straight line . The graph is b. For, h = 300 – 15t , The graph is linear so the so it is a function, For h = 300- 16t2 , The graph is linear so the so it is a function. c. Sky diver has the slow fall while compared to the bowling ball , because parachute can be controlled with the wind and can be divert the destination point, and bowling ball cannot be controlled while falling. Try It Does the table represent a linear or nonlinear function? Explain. Question 1. Answer: y = 2x – 12 is linear function. Explanation: Ordered pairs are (2 , -8) , (4 , -4) , (6 , 0) , (8 , 4) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (6 ,0) and (8, 4) m = (y2-y1) / (x2-x1) m= (4 – 0) / (8– 6) m = 4/2 m = 2 substitute the slope in the (8 , 4) to get point slope to form a line. y-y1 = m (x-x1) y – 4 = 2 ( x – 8) y – 4 = 2x – 16 y = 2x – 16 + 4 y = 2x – 12 So , y = 2x – 12 is linear function. Question 2. Answer: y = –$$\frac{5}{3}$$x + 25 is linear function. Explanation: Ordered pairs are (0 , 25) , (3 , 20) , (7 , 15) , (12 , 10) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (0 ,25) and (3, 20) m = (y2-y1) / (x2-x1) m= (20 – 25) / (3– 0) m = -5/3 Because the line crosses the y axis at ( 0, 25 ) , The y intercept is 25. So , the linear function is y = –$$\frac{5}{3}$$x + 25. So , y = –$$\frac{5}{3}$$x + 25 is linear function. Does the equation represent a linear or nonlinear function? Explain. Question 3. y = x + 5 Answer: y = x + 5 is a linear function Explanation: Given , y = x + 5 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 5 = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 1 + 5 = 6 . co-ordinates are (1 , 6) if x = 2 , then y = 2 + 5 = 7 , co-ordinates are (2 , 7) The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line . Each x input has only one y output so it is a function . And it forms a straight line when graphed . So, y = x + 5 is a linear function. Question 4. y = $$\frac{4x}{3}$$ Answer: y = $$\frac{4x}{3}$$ is a linear function. Explanation: Given , y = $$\frac{4x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = $$\frac{4(0)}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{4(1)}{3}$$ = $$\frac{4}{3}$$ = 1.3. co-ordinates are (1 , 1.3) if x = 2 , then y = $$\frac{4(2)}{3}$$ = $$\frac{8}{3}$$ = 2.6 , co-ordinates are (2 , 2.6) The co-ordinates (0 , 0) , (1 ,1.3 ) , (2 , 2.6) form a straight line . Each x input has only one y output so it is a function . And it forms a straight line when graphed . So, y = $$\frac{4x}{3}$$ is a linear function. Question 5. y = 1 – x2 Answer: y = 1 – x2 is not a linear function. Explanation: Given , y = 1 – x2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 1 – 02 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 – 12 = 0 . co-ordinates are (1 , 0) if x = 2 , then y = 1 – 22 = -3 , co-ordinates are (2 , -3) The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line . Each x input has only one y output so it is a function . And it does not forms a straight line when graphed . So, y = 1 – x2 is not a linear function. Does the graph represent a linear or nonlinear function? Explain. Question 6. Answer: The graph represents a nonlinear function. Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 2) , (-1 , 0) , (-2 , -2 ) , (-3 , -4), (0 , 1 ) , (2 , -2) , ( 3, -4 ) The inputs have more than one output , And points form a straight line So , the graph is non linear function Question 7. Answer: The graph is a linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 0) , (-1 , -1) , (-2 , -2 ) , (-3 , -3), (1 , 1 ) , (2 , 2) , ( 3, 3 ) The inputs have exactly one output , And points form a straight line So , the graph is a linear function. Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. IDENTIFYING FUNCTIONS Does the table or graph represent a linear or nonlinear function? Explain. Question 8. Answer: It is not a linear function Explanation: Ordered pairs are (3 , 0) , (-1 , 2) , (-5 , 4) , (-9 , 6) Each input has exactly one output and it does not form a straight line when graphed So, it is not a linear function . Question 9. Answer: The graph is non linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , -1) , (-1 , 0) , (-2 , 3 ) , (1 , 0 ) , (2 , 3) . The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 10. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain your reasoning. Answer: 5xy = -2 does not belong with the other three. Explanation: 15y = 6x , y = $$\frac{2}{5}$$x , 10y = 4x . These are evaluated as 5y = 2x 5xy = -2 , is different from 5y = 2x. Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 11. The loudness of sound is measured in (dB). The graph shows the loudness y of a sound (in decibels) x meters from the source of the sound. Is the relationship between loudness and distance linear or nonlinear? Approximate the loudness of the sound 12 meters from the source. Answer: The relationship between loudness and distance is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph. Explanation: As shown in the graph , the plot of the points does not form a straight line , Its a parabolic decay , The amount of loudness decreases with the increase in distance, So, The relationship between loudness and distance is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph. Question 12. A video blogger is someone who records a video diary. A new website currently hosts 90 video bloggers and projects a gain of 10 video bloggers per month. The table below shows the actual numbers of video bloggers. How does the projection differ from the actual change? Answer: Projections are more than the actual values Explanation: So, Projections are more than the actual values ### Comparing Linear and Non Linear Functions Homework & Practice 7.4 Review & Refresh Write a linear function that relates y to x. Question 1. Answer: The linear function is y = x – 2 In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , -2) , (1 , -1 ) , (-1 , -3) , ( 2, 0), (3 , 1) , (4 , 2) , ( 5, 3) First find the slope m of the line containing the two given points (2 ,0) and (3, 1) m = (y2-y1) / (x2-x1) m= (1 – 0) / (3 – 2) m = 1 . Because the line crosses the y axis at ( 0, -2 ) , The y intercept is -2. So , the linear function is y = x – 2 . Question 2. Answer: The linear function is y =$$\frac{-1}{1.5}$$x + 5. Explanation: Ordered pairs are (0 , 5) , (1.5 , 4) , (3 , 3) , (4.5 , 2) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (1.5 ,4) and (3, 3) m = (y2-y1) / (x2-x1) m= (3 – 4) / (3 – 1.5) m = -1 /1.5 Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5. So , the linear function is y =$$\frac{-1}{1.5}$$x + 5. The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation. Question 3. A (- 3, 1), B (- 1, 3), C (- 1, 1); k = 3 Answer: The New right angle triangle is larger than the original one So , its a increase . Explanation: Given , (- 3, 1), (- 1, 3), (- 1, 1) these pairs form a right angle triangle K = 3 , For the dilation figure multiply the 3 with the given ordered pairs , then (- 3, 1) × 3 = ( -9 , 3) (- 1, 3) × 3 = ( -3 , 9) (- 1, 1) × 3 = (-3 , 3) From these new ordered pairs we form a new right angle triangle The figure is The New right angle triangle is larger than the original one So , its a increase . Question 4. J (2, 4), K (6, 10), L (8, 10), M (8, 4); k = $$\frac{1}{4}$$ Answer: It is a reduction Explanation: Given , (2, 4), (6, 10), (8, 10) ,(8,4) these pairs forms a figure K = 0.25 , For the dilation figure multiply the 3 with the given ordered pairs , then (2, 4) × 0.25 = (0.5, 1) (6, 10) × 0.25 = (1.5, 2.5) (8, 10) × 0.25 = (2, 2.5) (8, 4) × 0.25 = (2 , 1) From these new ordered pairs we form a new figure The figure is The New figure is smaller than the original , So, It is a reduction . Concepts, Skills, & Problem Solving COMPARING FUNCTIONS Graph each equation. Decide whether each graph represents a linear or nonlinear function. (See Exploration 1, p. 295.) Question 5. h = 5 + 6t Equation 1 h = 5 + 6t2 Equation 2 Answer: h = 5 + 6t Equation 1 is a linear function h = 5 + 6t2 Equation 2 is a non linear function . Explanation: Given , h = 5 + 6t , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 5 + 6(0) = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 5 + 6(1) = 11 . co-ordinates are (1 , 11) if x = 2 , then y = 5 + 6(2) = 17 , co-ordinates are (2 , 17) if x = 3 , then y = 5 + 6(3) = 23 , co-ordinates are (3 , 23) The co-ordinates (0 , 5) , (1 , 11) , (2 , 17) , (3 , 23) form a straight line . Given , h = 5 + 6t2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 5 + 6(0)² = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 5 + 6(1)² = 11 . co-ordinates are (1 , 11) if x = 2 , then y = 5 + 6(2)² = 26 , co-ordinates are (2 , 26) if x = 3 , then y = 5 + 6(3)² = 59 , co-ordinates are (3 , 59) The co-ordinates (0 , 5) , (1 , 11) , (2 , 26) , (3 , 59) does not form a straight line . The graph of both equations is So, h = 5 + 6t Equation 1 is a linear function h = 5 + 6t2 Equation 2 is a non linear function . Question 6. y = – $$\frac{x}{3}$$ Equation 1 y = – $$\frac{3}{x}$$ Equation 2 Answer: y = – $$\frac{x}{3}$$ Equation 1 is a linear function y = – $$\frac{3}{x}$$ Equation 2 is a non linear function. Given , y =- $$\frac{x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- $$\frac{0}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = – $$\frac{1}{3}$$ = – 0.3 . co-ordinates are (1 , – 0.3 ) if x = 2 , then y = – $$\frac{2}{3}$$ = – 0.6 , co-ordinates are (2 ,-0.6) if x = 3 , then y = – $$\frac{3}{3}$$ = -1 , co-ordinates are (3 , -1) The co-ordinates (0 , 0) , (1 , -0.3) , (2 , -0.6) , (3 , -1) form a straight line . Given , y =- $$\frac{3}{x}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- $$\frac{3}{0}$$ = no number if x = 1 , then y = – $$\frac{3}{1}$$ = – 3 . co-ordinates are (1 , – 1 ) if x = 2 , then y = – $$\frac{3}{2}$$ = – 1.5 , co-ordinates are (2 ,-1.5) if x = 3 , then y = – $$\frac{3}{3}$$ = -1 , co-ordinates are (3 , -1) The co-ordinates (1 , -1) , (2 , -1.5) , (3 , -1) form a straight line . The graph of both the equations is So, y = – $$\frac{x}{3}$$ Equation 1 is a linear function y = – $$\frac{3}{x}$$ Equation 2 is a non linear function. IDENTIFYING FUNCTIONS FROM TABLES Does the table represent a linear or nonlinear function? Explain. Question 7. Answer: linear function is y = 4x + 4. Explanation: Ordered pairs are (0 , 4) , (1 , 8) , (2 , 12) , (3 , 16) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (2 , 12) and (3 , 16) m = (y2-y1) / (x2-x1) m= (16 – 12) / (3– 2) m = 4/1 m = 4 Because the line crosses the y axis at ( 0, 4 ) , The y intercept is 4. So , the linear equation is y = 4x + 4. And it is a linear function. The graph is Question 8. Answer: y = 4x – 6 is linear function. Explanation: Ordered pairs are (6 , 21) , (5 , 15) , (4 , 10) , (3 , 6) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (4 , 10) and (3 , 6) m = (y2-y1) / (x2-x1) m= (6 – 10) / (3– 4) m = -4/-1 m = 4 substitute the slope in the(4 , 10) to get point slope to form a line. y-y1 = m (x-x1) y – 10 = 4 ( x – 4) y – 10 = 4x – 16 y = 4x – 16 + 10 y = 4x – 6 So , y = 4x – 6 is linear function. The graph is IDENTIFYING FUNCTIONS FROM EQUATIONS Does the equation represent a linear or nonlinear function? Explain. Question 9. 2x + 3y = 7 Answer: The function is linear when m = $$\frac{-2}{3}$$ and c = $$\frac{7}{3}$$ Explanation: Given ,2x + 3y = 7 3y = 7 – 2x y = $$\frac{-2}{3}$$x+ $$\frac{7}{3}$$ So, The function is linear when m = $$\frac{-2}{3}$$ and c = $$\frac{7}{3}$$ Question 10. y + x = 4x + 5 Answer: The function is linear when m = 3 and c = 5 . Explanation: Given , y + x = 4x + 5 y = 4x – x + 5 y = 3x + 5 So, The function is linear when m = 3 and c = 5 . Question 11. y = $$\frac{8}{x^{2}}$$ Answer: The function is linear when m = 8 and c = 0 . Explanation: Given , y = $$\frac{8}{x^{2}}$$ slope m = 8 c = 0 So, The function is linear when m = 8 and c = 0 . IDENTIFYING FUNCTIONS FROM GRAPHS Does the graph represent a linear or nonlinear function? Explain. Question 12. Answer: The graph is linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 1) , (2 , 0) , (4 , -1 ) , (-2 , 2), ( -4, 3 ) The inputs have exactly one output , And points form a straight line So , the graph is linear function Question 13. Answer: The graph is non linear function. Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 0) , (-1 , -1) , (-4 , -2 ) , (1 , 1), ( 4, 2 ) The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 14. IDENTIFYING A FUNCTION The graph shows the volume V (in cubic feet) of a cube with an edge length of x feet. Does linear nonlinear the graph represent a linear or nonlinear function? Explain. Answer: The graph is non linear function n order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (1 , 1) , (2 , 8) , (3 , 27 ) , (4 , 64) The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 15. MODELING REAL LIFE The frequency y (in terahertz) of a light wave is a function of its wavelength x (in nanometers). Is the function relating the wavelength of light to its frequency linear or nonlinear? Answer: The function is a non linear function Explanation: table is as follows change in x is constant but change in y is not constant , it is increasing So, the function is a non linear function . Question 16. DIG DEEPER! The table shows the cost (in dollars) of pounds of sun flower seeds. a. What is the missing -value that makes the table represent a linear function? b. Write a linear function that represents the cost of x pounds of seeds. Interpret the slope. c. Does the function have a maximum value? Explain your reasoning. Answer: a. 3 pounds =$4.2
b. y = 1.4x  is linear function.
c.  If y has maximum value then the x also has maximum value.

Explanation:
a. As per the table 1 pound = $1.4 2 pounds =$2.8
3pounds = $4.2 4 pounds =$5.6
So, the price is increasing with weight of the seeds.

b. Ordered pairs are (2 , 2.8) , (3 , 4.2) , (4 , 5.6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 4.2) and (4 , 5.6)
m = (y2-y1) / (x2-x1)
m= (5.6 – 4.2) / (4 – 3)
m = 1.4
substitute the slope in the (3 , 4.2) to get point slope to form a line.
y-y1 = m (x-x1)
y – 4.2 = 1.4 ( x – 3)
y – 4.2 = 1.4x – 4.2
y = 1.4x – 4.2 + 4.2
y = 1.4x
So ,  y = 1.4x  is linear function.

c. As shown in the table , and the function if y increases then x also increases with respect to the y
So, if y has maximum value then the x also has maximum value.

Question 17.
MODELING REAL LIFE
A birch tree is 9 feet tall and grows at a rate of 2 feet per year. The table shows the height h (in feet) of a willow tree after x years.

a. Does the table represent a linear or nonlinear function? Explain.
b. Which tree is taller after 10 years? Explain.
Answer: There is no linear relationship between x and y .

Explanation:
Table is as follows
Change in y is constant but change in x is increasing , not a constant
Hence, there is no linear relationship between x and y .

Question 18.
CRITICAL THINKING
In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
a. Determine whether the function relating the year to the number of viewers is linear or nonlinear for each show.
b. Which show has more viewers in its sixth year?
Answer: a. The function relating the year to the number of viewers is linear
b. Both shows  has same number of viewers in the sixth year .

Explanation:
a. Given, In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
For show A
So , In first year = 7
2 year = 90% of 7 = 6.3
3 year = 90% of 6.3 = 5.6
4 year = 90% of 5.6 = 5.04
5 year = 90% of 5 = 4.5
6 year = 90% of 4.5 = 4.05
So the ordered pairs are (1 , 7) , (2 , 6.3) , (3 , 5.6) , (4 , 5), (5 , 4.5) , (6 , 4)

For show B
In first year = 5 , As the viewers reduces by 2,00,000 in 5M
2 year = 5 – 0.2 = 4.8
3 year = 4.8 – 0.2 = 4.6
4 year = 4.6 – 0.2 = 4.4
5 year = 4.4 – 0.2 = 4.2
6 year = 54.2 – 0.2 = 4
So the ordered pairs are (1 , 5) , (2 , 4.8) , (3 , 4.6) , (4 , 4.4), (5 , 4.2) , (6 , 4)
As the year increases the viewers are also decreasing constantly as per the individual shows
So, The function relating the year to the number of viewers is linear .

b. As shown in part a , the ordered pairs having (6,4) represents the number of viewers to the year
So, Both shows  has same number of viewers in the sixth year .

Question 19.
NUMBER SENSE
The ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. Graph the ordered pairs and describe the pattern. Is the function linear or nonlinear?
b. Write an equation that represents the function.
Answer: a. The graph is shown below and function is linear
b. The linear equation is y = 7x – 1.

Explanation:
Given, ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. the graph is
Each input has exactly one output and it forms a straight line So, the graph is linear
b. First find the slope m of the line containing the two given points (3 ,8) and (4, 15)
m = (y2-y1) / (x2-x1)
m= (15 – 8) / (4– 3)
m = 7/1
m = 7
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear equation is y = 7x – 1.

### Lesson 7.5 Analyzing and Sketching Graphs

EXPLORATION 1

Matching Situations to Graphs
Work with a partner. Each graph shows your speed during a bike ride. Match each situation with its graph. Explain your reasoning.

a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. Analyze Relationships
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed.
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Answer: a – C ,
b – A ,
c – D ,
d – B ,

Explanation:
a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. The graph C has the perfect graph representing the situation of given question.
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
Because The graph A has the bike speed representing the situation for the time .
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed. Thus, The graph D is the final answer for the question
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Because of the speed with respect to time the graph B is the correct answer for the question.

EXPLORATION 2

Interpreting a Graph
Work with a partner. Write a short paragraph that describe show the height changes over time in the graph shown. What situation can this graph represent?

Answer: The Graph can be representing a situation for low and high tides of the Ocean

Explanation:
As shown in the figure, The graph is plotted between the height and time,
We can take an example of an Ocean for its waves , As the time passes at the morning of a normal day, The waves of the ocean start rising higher at a period of time, and for the time being maintaining a peak height then drops to a lower height at a particular intervals of time , this process takes place for a while and vise versa.
Thus, the Graph can be representing a situation for low and high tides of the Ocean

Try It

Question 1.
The graph shows the location of a pelican relative to your location.

a. Describe the path of the pelican.
b. Write an explanation for the decrease in the vertical distance of the pelican.
Answer: Both of them are explained below.

Explanation:
a. The path of the pelican is flying in the air , As they always fly in line and the amazing thing is the deeper the prey the higher they dive.
The graph shows the relationship between the horizontal distance that is the height from the land, vertical distance is the point from where its destination point is located, so at the starting point of the flight it has more distance from the ground means flying at a higher level , as the time passes it reaches to the closer point of its destination point so the altitude of the flight decreases with the decrease in the vertical distance and at a particular distance reaches its point of destination.

b. The decrease in the vertical distance of the pelican. is due to its flight to the destination point as it requires to stop flying to reach it, so in order to have a smooth landing on the ground , the bird gradually decreases its speed by decreasing its altitude.

Question 2.
A fully-charged battery loses its charge at a constant rate until it has no charge left. You plug it in, and it fully recharges at a constant rate. Then it loses its charge at a constant rate until it has no charge left. Sketch a graph that represents this situation.
Answer:  The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the battery charge ,
A fully-charged battery loses its charge at a constant rate until it has no charge left. So, line segment starts from 100 and decreases until it touches the x-axis.
You plug it in, and it fully recharges at a constant rate. Thus, line segment increases at a constant rate until it reaches 100
Then it loses its charge at a constant rate until it has no charge left. line segment decreases again at a constant rate until it again touches the x-axis .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
ANALYZING GRAPHS
The graph shows the growth rate of a plant over time.

a. Describe the change in growth rate.
b. Write an explanation for the decrease in growth rate and the increase in growth rate.
Answer: the answers are given below

Explanation:
a. the change in growth rate of a plant over the time is given by its size and height , So as the time passes the growth rate is constant from the the start and from a particular time the growth rate has been dropping slightly due to external or internal reasons of a plant and again at some time the growth rate is increasing at a constant rate until it reaches to its perfect growth of a plant.

b. The decrease in growth rate of the plant is due to some external causes like weather, rain, sunlight , watering, and the soil may effect its growth rate and the increase in growth rate is probably due to its soil fertility and sufficient sunlight providing sufficient chlorophyll.

Question 4.
SKETCHING GRAPHS
As you snowboard down a hill, you gain speed at a constant rate. You come to a steep section of the hill and gain speed at a greater constant rate. You then slow down at a constant rate until you come to a stop. Sketch a graph that represents this situation.
Answer: The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the speed ,
As you snowboard down a hill, you gain speed at a constant rate, line segment decreases at a constant rate
You come to a steep section of the hill and gain speed at a greater constant rate, line segment becomes steeper i.e., the line segment decreases at a high constant rate.
You then slow down at a constant rate until you come to a stop, line segment becomes flatter i.e., the constant rate of decrease becomes less until it touches its x-axis

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
Two rowing teams are in a race. The graph shows their distances from the finish line over time. Describe the speed of each team throughout the race. Then determine which team finishes first.

Answer: Team B will finishes race first.

Explanation:
Team A , The relationship between the time and distance from the finish line is given in the graph,
At starting point Team A has maintained a fair speed at the Beginning of the race and has been a little slow while reaching out to the destination point, and for a while they have been balancing the speed with the distance representing a curving point in the graph and directly dropping to the finish line drastically creating a slope, until it reaches in the x-axis line.
Team B , The relationship between the time and distance from the finish line is given in the graph,
As same as the Team A , Team B has a perfect start but it has been a way different them Team A because Team B has a game plan to win the race, as shown in the graph they have maintained a constant speed while reaching out to the destination and also having a smooth drift at a level of decreasing their distance from the finish line.

Team B will  finishes the race first because they are having a constant and smooth decreasing speed which comes to an end gradually at the finishing line.

Question 6.
DIG DEEPER!
The graphs show the movements of two airplanes over time. Describe the movement of each airplane.

Answer: Detailed explanation is given below.

Explanation:
As shown in the graph , x-axis is time and y-axis be the height above ground
Airplane A, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.

Airplane B, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.
It is as same as the airplane A.

### Analyzing and Sketching Graphs Homework & Practice 7.5

Review & Refresh

Does the table or equation represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = -0.5x + 11.5 is a linear function.

Explanation:
Ordered pairs are (-5 , 14) , (-1 , 12) , (3 , 10) , (7 , 8)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 10) and (7 , 8)
m = (y2-y1) / (x2-x1)
m= (8 – 10) / (7 – 3)
m = -2/4
m = -0.5
substitute the slope in the(3 , 10) to get point slope to form a line.
y-y1 = m (x-x1)
y – 10 = -0.5 ( x – 3)
y – 10 = -0.5x + 1.5
y = -0.5x + 1.5 + 10
y = -0.5x + 11.5
So , the linear equation is y = -0.5x + 11.5
And it is a linear function.

The graph is

Question 2.
y = x2 + 8
Answer: The function is linear  when m= 1 and c = 8.

Explanation:
Given , y = x2 + 8 ,
slope m = 1
c = 8
So, the function is linear when m = 1 and c= 8.

Graph the linear equation.
Question 3.
– 4x + y = – 1
Answer: The graph is

Explanation:
we can write – 4x + y = – 1 as y = 4x – 1
Given , y = 4x – 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) – 1 = -1 . co-ordinates are (0 , -1)
if x = 1 , then y = 4(1) – 1 = 3 . co-ordinates are (1 , 3)
if x = 2 , then y = 4(2) – 1 = 7 , co-ordinates are (2 , 7)
if x = 3 , then y = 4(3) – 1= 11 , co-ordinates are (3 , 11)
The co-ordinates (0 , -1) , (1 , 3) , (2 , 7) , (3 , 11) form a straight line .

Question 4.
2x – 3y = 12
Answer: The graph is

Explanation:
we can write  2x – 3y = 12 as y = $$\frac{2x-12}{3}$$ or y = $$\frac{2}{3}$$x – 4
Given , y =$$\frac{2}{3}$$x – 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = $$\frac{2}{3}$$0 – 4= – 4 . co-ordinates are (0 , -4)
if x = 1 , then y = $$\frac{2}{3}$$1 – 4 = 0.66 – 4 = -3.3 . co-ordinates are (1 , -3.3)
if x = 2 , then y = $$\frac{2}{3}$$2 – 4 =0.66(2) – 4 =1.3 – 4 = -2.6, co-ordinates are (2 , -2.6)
if x = 3 , then y = $$\frac{2}{3}$$3 – 4 = 0.66(3) – 4 = 1.98 – 4 = -2.0  , co-ordinates are (3 , -2.0)
The co-ordinates (0 , -4) , (1 , -3.3) , (2 , -2.6) , (3 , -2) form a straight line .

Question 5.
5x + 10y = 30
Answer: The graph is

Explanation:
5x + 10y = 30 can be written as y = -0.5x + 3
take 5 common on both sides we get
x + 2y = 6
y = $$\frac{-x + 6}{2}$$
y = $$\frac{-x}{2}$$ + 6
y = -0.5x + 3
Given , y =-0.5x + 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = -0.5(1) + 3= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = -0.5(2) + 3 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = -0.5(3) + 3 = 4.5  , co-ordinates are (3 , 4.5)
The co-ordinates (0 , 3) , (1 , 2.5) , (2 , 4) , (3 , 4.5)does not form a straight line .

Concepts, Skills, &Problem Solving

MATCHING DESCRIPTIONS WITH GRAPHS The graph shows your speed during a run. Match the verbal description with the part of the graph it describes. (See Exploration 1, p. 301.)

Question 6.
You run at a constant speed.
Answer: C

Explanation:
Because the line segment of the graph at point C show that the running speed is constant for a particular time ,
Thus forming a straight horizontal line.

Question 7.
You slow down at a constant rate.
Answer: D

Explanation:
Because the line segment of the graph at point D show that the running speed is decreasing at a constant rate for a particular time ,
Thus forming a straight steep line down the time axis.

Question 8.
You increase your speed at a constant rate.
Answer: A

Explanation:
Because the line segment of the graph at point A show that the running speed is increasing at a constant rate at a starting point of the race on time ,
Thus forming a slope in the graph.

Question 9.
You increase your speed at a faster and faster rate.
Answer: B

Explanation:
Because the line segment of the graph at point B show that the running speed is increasing at a faster rate after starting the race and maintaining a gradual growth of the speed and after reaching the next point speed is doubled from before ,
Thus forming a slope with a curve in the graph.

ANALYZING GRAPHS Describe the relationship between the two quantities.
Question 10.

Answer: As the Time passes there will be increase in the volume.

Explanation:
The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

So, As the Time passes there will be increase in the volume.

Question 11.

Answer: As the times passes Dollars are maintaining imbalance.

Explanation:
The relationship between the time and dollars is given in the graph, As we all know money is never ever constant with time , As if it only increases or decreases or having both simultaneously , in this graph the line segment is having a steep and at some point of time it is maintaining a slight growth constantly with the time.

So, As the times passes Dollars are maintaining imbalance.

Question 12.

Answer: An engine power is directly proportional to the engine speed and its horse power

Explanation:
The relationship between the engine speed and horse power is given in the graph, Generally every automobile is is defined as the best for its horse power which is the heart of the engine and it highlights the speed of the vehicle, Here engine power is defined by the horse power and the engine speed the line segment is having a curve increment in the horse power due to the increase in engine speed.

So, An engine power is directly proportional to the engine speed and its horse power

Question 13.

Answer: As time increases the process of grams decaying will be faster.

Explanation:
The relationship between grams and time is given in the graph, its obvious that every product has its own expiry date, and if it crosses that its starts to decay, the graph implies that with the increase time the quality of the gram decreases or grams start to decay . The line segment in the graph shows that the gradually decrease indicating the spoiling rate of the grams  with rate of change of time.

So, As time increases the process of grams decaying will be faster.

Question 14.

Answer: At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Explanation:
The graph shows the relationship between the length of the hair and time taken to the growth of the hair, of course hair growth is not constant every time, here we have the graph with the line segment  not constant and having breaks at the times of interval.

So, At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Question 15.

Answer: In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Explanation:
The relationship between the balance of the loan with the time period of the loan to be cleared, The loan should be cleared in the time limit and should maintain a neat balance, every increase in time period the balance is debited from the loan , there will be decrease in the balance and gaps are occurred in the graph.

so, In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Question 16.
ANALYZING GRAPHS
Write an explanation for the relationship shown in the graph in Exercise 10.
Answer: The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

Question 17.
MODELING REAL LIFE
The graph shows the natural gas usage for a house.

a. Describe the change in usage from January to March.
b. Describe the change in usage from March to May.
Answer: a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.
b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

Explanation:
a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.

b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

SKETCHING GRAPHS Sketch a graph that represents the situation.
Question 18.
The value of a television decreases at a constant rate, and then remains constant.
Answer: The graph is

Explanation:
Draw the axis and label the x- axis as time and y- axis as value, then sketch the graph,
The value of the television decreases at a constant rate: line segment starts to decrease at a constant rate,
And then remains constant, after reaching a certain value : line segment becomes parallel to horizontal axis.

Question 19.
The distance from the ground changes as your friend swings on a swing.
Answer: The graph is

Explanation:
Your friend starts close to the ground and then swings up. Then she falls back down close to the ground again and swings back . When she swings back, she gets higher than when she was swinging forward, she then starts to swing forward again getting close to the ground and then going up even higher than when she was swinging backward, she continues to getting higher and higher every time she swings forwards and backwards,

Question 20.
The value of a rare coin increases at a faster and faster rate.
Answer: The graph is

Explanation:
Draw the Axis and label them as x-axis as time and y – axis as distance,
The value of a rare coin increases at a faster and faster rate , so the curve moves upwards at an increasing rate.

Question 21.
You are typing at a constant rate. You pause to think about your next paragraph and then you resume typing at the same constant rate.
Answer: The graph is

Explanation:
A constant rate means that portion of the graph is linear , pausing means the number of words stays constant, typing again at the same constant rate means the last piece of the graph is linear again with the same slope as the first portion of the graph.

Question 22.
CRITICAL THINKING
The graph shows the speed of an object over time.

a. Sketch a graph that shows the distance traveled by the object over time.
b. Describe a possible situation represented by the graphs.
Answer: a. The distance and time are directly proportional to each other.
b. As time passes the speed and time are relatively balancing each other in the graph.

Explanation:
a. The graph is
In this graph the relationship between distance and time is shown, for example , let the object be a bike, the time taken to reach the destination for the bike is directly proportional to the distance travelled , So as time passes the distance is gradually increasing from the starting point.

So, the distance and time are directly proportional to each other.

b. Th graph shown , is the relationship between the speed and the time , let the object moving be Train,
it is running between the station so it has to be halted in the stations to be listed in the stoppings , So the line segment is started with a constant speed with the time and again at the time interval dropping the speed with respect to time it has maintaining the same speed .

So, As time passes the speed and time are relatively balancing each other in the graph.

Question 23.
MODELING REAL LIFE
The graph shows the average scores of two bowlers from the start of a season to the end of the season.

a. Describe each bowler’s performance.
b. Who had a greater average score most of the season? Who had a greater average score at the end of the season?
c. Write an explanation for the change in each bowler’s average score throughout the bowling season.
Answer: All the answers are explained below

Explanation:
a.  Bowler A : As the graph represent the relationship between the score and the week, bowler A has started with the good take off and having able to grasp the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B.

Bowler B : As the graph represent the relationship between the score and the week, bowler B has started with the good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week.

b.  Bowler A and Bowler B had a greater average score most of the season, but Bowler A had a greater average score at the end of the season

c. Bowler A has the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B. so it has a smaller change in average’s score in the bowling season .
While Bowler B has good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week. so he has a drastic change in average’s score in the bowling season .

Question 24.
DIG DEEPER!
You can use a supply and demand model to understand how the price of a product changes in a market. The supply curve of a particular product represents the quantity suppliers will produce at various prices. The demand curve for the product represents the quantity consumers are willing to buy at various prices.

a. Describe and interpret each curve.
b. Which part of the graph represents a surplus? Explain your reasoning.
c. The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Suppose that demand for a product suddenly increases, causing the entire demand curve to shift to the right. What happens to the equilibrium point?
Answer:  All of them are explained below .

Explanation:
a. The supply curve of a particular product represents the quantity suppliers will produce at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices increases gradually Quantity increases .
The demand curve for the product represents the quantity consumers are willing to buy at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices decreases with increase in Quantity .

b. The graph does not implies any surplus because each demand and supply is given by their respective curve over the prices and quantity

c. As shown in the graph, The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Given, that demand for a product suddenly increases, causing the entire demand curve to shift to the right. Then the equilibrium point will be pointed where the two curves meet after the change in the demand graph so change in the supply graph is also possible.

### Functions Connecting Concepts

Using the Problem-Solving Plan
Question 1.
The table shows the lengths x (in inches) and weights y(in pounds) of several infants born at a hospital. Determine whether weight is a function of length. Then estimate the weight of an infant that is 20 inches long.

Understand the problem.
You know the lengths and weights of several infants. You are asked to determine whether weight is a function of length and to estimate the weight of a 20-inch-long infant.

Make a plan.
Determine whether any of the lengths are paired with more than one weight. Then use a graphing calculator to find an equation that represents the data. Evaluate the equation when x = 20 to estimate the weight of a 20-inch-long infant.

Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer: Weight is the function of the length

Explanation:
From the table we have , Each length has only one weight , so weight is a function of length.
First find the slope m of the line containing the two given points (19.3 , 7.3) and (18.9 , 6.5)
m = (y2-y1) / (x2-x1)
m= (6.5 – 7.3) / (18.9 – 19.3)
m = 0.2
substitute the slope in the (19.3 , 7.3) to get point slope to form a line.
y-y1 = m (x-x1)
y – 7.3 = 0.2 ( x – 19.3)
y – 7.3 = 0.2x – 3.86
y = 0.2x – 3.86 + 7.3
y = 0.2x + 3.4
So ,  y = 0.2x + 3.4 is linear function.

For x = 20 ,
y = 0.2 (20) + 3.4
y = 4 + 3.4
y = 7.4

So, The weight of an infant that is 20 inches long. is 7.4.

Question 2.
Each mapping diagram represents a linear function. At what point do the graphs of the functions intersect? Justify your answer.

Answer:  The point of intersection is (-1, -4)

Explanation:
Function 1 – Ordered pairs are ( -8 , 24 ) , ( -3 , 4 ) , ( -1 , -4 ) , ( 1 , -12) .
Function 2 – Ordered pairs are ( 6 , 17 ) , ( 10 , 29 ) , ( 13 , 38 ) , ( 15 , 44 ) .
Graph the points we get, So, The point of intersection is (-1,-4).

Performance Task

Heat Index
At the beginning of this chapter, you watched a STEAM Video called “Apparent Temperature.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

Answer:

### Functions Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.

Input: Ordered pairs can be used to show inputs and outputs , inputs are represented by x

Output: Ordered pairs can be used to show inputs and outputs , Outputs are represented by y

Relation: A relation pairs inputs with outputs

Mapping diagram: A relation can be represented by ordered pairs or mapping diagrams.

Function: The relation that pairs each input with exactly one output is a function.

Function rule: it is an equation, that describes the relationship between inputs(independent variables) and outputs(dependent variables).

Linear function: A linear function is a function whose graph is a straight line i.e., non vertical line . A linear can be written in the form y = mx + c , where m is the slope and c is the y intercept

Non linear function: The graph of a linear function shows a constant rate of change, A non linear function does not have a constant rate of change, So its graph is a not a line.

Graphic Organizers
You can use an Example and Non-Example Chart to list examples and non-examples of a concept. Here is an Example and Non-Example Chart for functions.

Choose and complete a graphic organizer to help you study the concept.

1. linear functions
2. nonlinear functions
3. linear functions with positive slope
4. linear functions with negative slope

Answer: 1. linear functions

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

7.1 Relations and Functions (pp. 275–280)
Learning Target: Understand the concept of a function.

List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 1.

Answer: The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) and The relation is not a function .

Explanation:
As shown , The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) .
The input 7 has more than one output,
So, The relation is not a function .

Question 2.

Answer: ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ).
Each input has exactly one output ,
So, The relation is a function .

Question 3.

Answer: The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) and The relation is not a function

Explanation:
As shown , The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) .
The input -1 has more than one output ,
So, The relation is not a function .

Question 4.
For ordered pairs that represent relations, which coordinate represents the input? the output?

Answer: x coordinate is the input and y coordinate is the output

Explanation:
Ordered pairs from the given graph are ( 2 , 7 ) , ( 3 , 7 ) , ( 4 , 5 ) , ( 5 , 5 ) , ( 6 , 3 ) .
So , x coordinate is the input and y coordinate is the output

Question 5.
Draw a mapping diagram that represents the relation shown in the graph. Then determine whether the relation is a function. Explain.
Answer:

Explanation:
The mapping diagram is
each input has more than one output
So, relation is not a function.

Question 6.
The mapping diagram represents the lengths (in centimeters) of a rubber band when different amounts of force (in Newtons) are applied.

a. Is the length of a rubber band a function of the force applied to the rubber band?
b. Describe the relationship between the length of a rubber band and the force applied to the rubber band.
Answer:  a. Yes
b. For every increase in 0.7 in input there is an increment of 2 in output.

Explanation:
a. The ordered pairs are  ( 0 , 5 ) , ( 0.7 , 7 ) , ( 1.4 , 9 ) , ( 2.1 , 11 )
Each input has exactly one output
So, the length of a rubber band a function of the force applied to the rubber band.

b. For every increase in 0.7 in input there is an increment of 2 in output.

7.2 Representations of Functions (pp. 281–288)
Learning Target: Represent functions in a variety of ways.

Write a function rule for the statement.
Question 7.
The output is two less than the input.
Answer: y = x – 2

Explanation:
Let us say x is input and y is output , then
The output is two less than the input, will be
y = x – 2

Question 8.
The output is two more than one-fourth of the input.
Answer: y = $$\frac{x}{4}$$ + 2

Explanation:
Let us say x is input and y is output , then
The output is two more than one-fourth of the input, will be
y = $$\frac{x}{4}$$ + 2

Find the value of y for the given value of x.
Question 9.
y = 2x – 3; x = – 4
Answer: y = -8

Explanation:
Given, y = 2x
substitute x = -4 , we get
y = 2(-4)
y = -8.

Question 10.
y = 2 – 9x ; x = $$\frac{2}{3}$$
Answer: y = – 3.4

Explanation:
Given , y = 2 – 9x
substitute x = $$\frac{2}{3}$$ , we get
y = 2 – 9 (0.6)
y = 2 – 5.4
y = – 3.4

Question 11.
y = $$\frac{x}{3}$$ + 5; x = 6
Answer: y = 7.

Explanation:
Given, y = $$\frac{x}{3}$$ + 5
substitute x = 6 , we get
y = $$\frac{6}{3}$$ + 5
y = 2 + 5
y = 7.

Graph the function.
Question 12.
y = x + 3
Answer: The graph is

Explanation:
Given , y = x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 3 = 3 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 3  = 4 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 3 = 5 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 3 = 6 , co-ordinates are (3 , 7)
The co-ordinates (0 , 3) , (1 , 4) , (2 , 5) , (3 , 6) form a straight line .

Question 13.
y = – 5x
Answer: The graph is

Explanation:
Given , y = – 5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 5(1)  = – 5 . co-ordinates are (1 , – 5)
if x = 2 , then y = – 5(2) = -10 , co-ordinates are (2 , -10)
if x = 3 , then y =- 5(3) = -15 , co-ordinates are (3 , -15)
The co-ordinates (0 , 0) , (1 , -5) , (2 , -10) , (3 , -15) form a straight line .

Question 14.
y = 3 – 3x
Answer: The graph is

Explanation:
Given , y =3 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 3 – 3(0) = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = 3 – 3(1)  = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 3 – 3(2) = – 3 , co-ordinates are (2 , – 3)
if x = 3 , then y =3 – 3(3) = – 6 , co-ordinates are (3 , – 6)
The co-ordinates (0 , 3) , (1 , 0) , (2 , – 3) , (3 , – 6) form a straight line .

Question 15.
An online music store sells songs for $0.90 each. a. Write a function that you can use to find the cost of buying songs. b. What is the cost of buying 5 songs? Answer: a. C = 0.90s b.$4.5

Explanation:
a. The total cost is equal to the cost of each song times the number of songs, if each song is $0.90, Then the total cost C of s songs is C = 0.90s. b. Substituting s= 5 in C = 0.90s we get, C = 0.90(5) = 4.5. So, cost of 5 songs is$4.5.

7.3 Linear Functions (pp. 289–294)
Learning Target: Use functions to model linear relationships.

Use the graph or table to write a linear function that relates y to x.
Question 16.

Answer: The linear function is y = $$\frac{1}{3}$$x + 3.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (3 , 4) , (0 , 3 ) , (-3 , 2) , ( -6, 1 )
First find the slope m of the line containing the two given points (0 ,3) and (-3, 2)
m = (y2-y1) / (x2-x1)
m= (2 – 3) / (-3 – 0)
m = -1 / -3 .
m = 1/3 .
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = $$\frac{1}{3}$$x + 3.

Question 17.

Answer: The linear function is y = −(0)x -7.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , -7) , (0 , -7 ) , (2 , -7) , ( 4 , -7 )
First find the slope m of the line containing the two given points (0 ,-7) and (2, -7)
m = (y2-y1) / (x2-x1)
m= (-7 – (-7)) / (2 – 0)
m = 0 .
Because the line crosses the y axis at ( 0, -7 ) , The y intercept is -7.
So , the linear function is y = −(0)x -7.

Question 18.
The table shows the age x (in weeks) of a puppy and its weight y (in pounds).

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
c. After how many weeks will the puppy weigh 33 pounds?
Answer: a. y = $$\frac{3}{2}$$x + 3
b. 3 pounds
c. Age is 20 weeks

Explanation:
a. In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (6 , 12) , (8 , 15 ) , (10 , 18) , ( 12 , 21 )
First find the slope m of the line containing the two given points ((6 ,12) and (8 , 15)
m = (y2-y1) / (x2-x1)
m= (15 – 12) / (8 – 6)
m = 3/2 .
substitute the slope in the (6 ,12) to get point slope to form a line.
y-y1 = m (x-x1)
y – 12 = 3/2 ( x – 6)
2(y – 12) = 3(x – 6)
2y – 24 = 3x – 18
2y = 3x – 18 + 24
2y  = 3x + 6
So ,  2y  = 3x + 6 or y = $$\frac{3}{2}$$x + 3 is linear function.

b. The slope measures the rate of change of weight due to change in weeks, Here the slope of 3/2 means that as one week passes, weight of the puppy increases by 3/2 pounds.
y intercept measures the weight of the puppy, when it was born which is 3 pounds in this case measured by c.

c. put y = 33,
33 = $$\frac3}{2}$$x + 3
30 = $$\frac{3}{2}$$x
30 × 2 = 3x
x = 60/3
x = 20.
So, Age is 20 weeks.

7.4 Comparing Linear and Nonlinear Functions (pp. 295–300)
Learning Target: Understand differences between linear and nonlinear functions.

Does the table represent a linear or nonlinear function? Explain.
Question 19.

Answer: y = 3x – 8 is linear function.

Explanation:
Ordered pairs are (3 , 1 ) , (6 , 10) , (9 , 19) , (12 , 28)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 1 ) and (6 , 10)
m = (y2-y1) / (x2-x1)
m= (10 – 1) / (6– 3)
m = 9/3
m = 3
substitute the slope in the (3 , 1) to get point slope to form a line.
y-y1 = m (x-x1)
y – 1 = 3 ( x – 3)
y – 1 = 3x – 9
y = 3x – 9 + 1
y = 3x – 8
So ,  y = 3x – 8 is linear function.

Question 20.

Answer: y = -x + 4 is linear function.

Explanation:
Ordered pairs are (1 , 3 ) , (3 , 1) , (5 , 1) , (7 , 3)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3 ) and (3 , 1)
m = (y2-y1) / (x2-x1)
m= (1 – 3) / (3– 1)
m = -2/2
m = -1
substitute the slope in the (1 , 3) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3 = -1 ( x – 1)
y – 3 = -x + 1
y = -x + 1 + 3
y = -x + 4
So ,  y = -x + 4 is linear function.

Question 21.
Does the graph represent a linear or nonlinear function? Explain.

Answer: The graph represent a non linear function.

Explanation:
As shown in the graph linear function represents a  straight line to which not happened here,
So , the graph is non linear function

Question 22.
Does the equation y = 2.3x represent a linear or nonlinear function? Explain.
Answer: y = 2.3x is a linear function.

Explanation:
Given , y = 2.3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2.3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2.3(1) = 2.3 . co-ordinates are (1 , 2.3)
if x = 2 , then y = 2.3(2) = 4.6 , co-ordinates are (2 , 4.6)
The co-ordinates (0 , 0) , (1 , 2.3) , (2 , 4.6) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = 2.3x is a linear function.

7.5 Analyzing and Sketching Graphs (pp. 301–306)
Learning Target: Use graphs of functions to describe relationships between quantities.

Question 23.
Describe the relationship between the two quantities in the graph.

Answer: At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.

Explanation:
The relationship between the graph is population and time ,
At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.
So, the city population is not constant at all the time.

Sketch a graph that represents the situation.
Question 24.
You climb a climbing wall. You climb halfway up the wall at a constant rate, then stop and take a break. You then climb to the top of the wall at a greater constant rate.
Answer: The graph is

Explanation:
You start climbing a wall at a constant rate so the first portion of the graph needs to be linear with a positive slope, you then take a break which means your height is constant so the second part of the graph needs to be a horizontal line, you then start climbing again at a constant rate, so the last part of the graph needs to be linear with a positive slope.

Question 25.
The price of a stock increases at a constant rate for several months before the stock market crashes. The price then quickly decreases at a constant rate.
Answer: The graph is

Explanation:
The stock price is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, Then price begins to drop quickly so the second part of the graph needs to be linear with a steep negative slope.

Question 26.
The graph shows the sales of two companies during a particular year.

a. Describe the sales of each company.
b. Which company has greater total sales for the year?
c. Give a possible explanation for the change in each company’s sales throughout the year.
Answer: All The explanation is given below

a. Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, and decreasing with a slight negative steep and again increasing at a constant rate increasing the sales of the company

Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales and vise versa.

b. Company A has the greater total sales for the year compared to Company B, with maintaining the sales up to the mark without losses.

c. The change in each company’s sales throughout the year,  Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope,
Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales

### Functions Practice Test

Question 1.
List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.

Answer: The relation is a function

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 2 , 9 ) , ( 4 , 9 ) , ( 6 , 10 ) , ( 8 , 11 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 2.
Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.

Answer: The mapping diagram is

Explanation:
Ordered pairs from the given graph are ( -3 , 5 ) , ( -1 , 1 ) , ( -1 , 3 ) , ( 1 , 2 ) , ( 3 , 4 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 3.
Write a function rule for “The output is twice the input.”
Answer: y = 2x

Explanation:
Let us say x is input and y is output , then
The output is twice the input. will be
y = 2x

Question 4.
Graph the function y = 1 – 3x.
Answer: The graph is

Explanation:
Given , y = 1 – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 – 3(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 3(1)  = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 1 – 3(2) = -5 , co-ordinates are (2 , -5)
if x = 3 , then y =1 – 3(3) = -8 , co-ordinates are (3 , -8)
The co-ordinates (0 , 1) , (1 , -2) , (2 , -5) , (3 , -8) form a straight line .

Question 5.
Use the graph to write a linear function that relates y to x.

Answer: The linear function is y = 0.5x – 1

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , -3) , (-2 , -2 ) , (0 , -1) , ( 2 , 0 )
First find the slope m of the line containing the two given points (0 , -1) and ( 2 , 0 )
m = (y2-y1) / (x2-x1)
m= (0 – (-1)) / (2 – 0)
m = 1 / 2 .
m = 0.5 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = 0.5x – 1 .

Question 6.
Does the table represent a linear or nonlinear function? Explain.

Answer: The linear function is y = −4x + 8

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 8) , (2 , 0 ) , (4 , -8) , ( 6 , -16 )
First find the slope m of the line containing the two given points (0 , 8) and (2 , 0 )
m = (y2-y1) / (x2-x1)
m= (0 – 8) / (2 – 0)
m = -4
Because the line crosses the y axis at ( 0, 8 ) , The y intercept is 8.
So , the linear function is y = −4x + 8.

Question 7.
The table shows the number of y meters a water-skier travels in x minutes.

a. Write a function that relates y to x.
b. Graph the linear function.
c. At this rate, how many kilometers will the water-skier travel in 12 minutes?
d. Another water-skier travels at the same rate but starts a minute after the first water-skier. Will this water-skier catch up to the first water-skier? Explain.
Answer: All the answers are given below

Explanation:
Ordered pairs are  (1 , 600) , (2 , 1200 ) , (3 , 1800) , ( 4 , 2400 ) , (5 , 3000)
First find the slope m of the line containing the two given points(1 , 600) and (2 , 1200 )
m = (y2-y1) / (x2-x1)
m= (1200 – 600) / (2 – 1)
m = 600
So, the line is of the form y = 600x + c
put x= 3 and y = 1800 in the above equation we get,
1800 = 600(3) + c
c = 1800 – 1800
c = 0.
So, The line is y = 600x.

b. The graph is

c. put x = 12 in y = 600x
y = 600(12)
y = 7200
7200 meters, i.e., 7.2km

d. Another water skier travels at the same rate but starts a minute after the first water skier, Since both are travelling at the same rate , the water skier who was late will always be behind the first water skier.

Question 8.
The graph shows the prices of two stocks during one day.

a. Describe the changes in the price of each stock.
b. Which stock has a greater price at the end of the day?
c. Give a possible explanation for the change in the price of Stock B throughout the day.
Answer: Detailed Explanation is given below.

Explanation:
a. The changes in the price of each stock is Stock A has the constant increase in stock for a particular time and maintains a constant price forming a straight line in the graph, and again decreasing with a negative slope and vise versa, while Stock B is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again.

b. stock B has a greater price at the end of the day, having a positive increase in the slope

c. The change in the price of Stock B throughout the day, is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again, compared to stock A .

Question 9.
You are competing in a footrace. You begin the race by increasing your speed at a constant rate. You then run at a constant speed until you get a cramp and have to stop. You wait until your cramp goes away before you start increasing your speed again at a constant rate. Sketch a graph that represents the situation.
Answer: The graph is

Explanation:
You begin the race by increasing your speed at a constant rate so the first portion of the graph needs to be linear with a positive slope , you then run at a constant speed so the next portion of the graph needs to be horizontal line , you then stop and take a break , so your speed is zero, which means the next portion of the line needs to be
horizontal line on the x axis , you then increase your speed again at a constant rate sop that the last portion of the graph needs to be linear with a positive slope

### Functions Cumulative Practice

Question 1.
What is the slope of the line?

Answer: Not in the options but the answer is m = -4/3

Explanation:
Ordered pairs are  (-4 , 5) , (1 , -3 ),
First find the slope m of the line containing the two given points
m = (y2-y1) / (x2-x1)
m= (-3 – 5) / (2 – (-4))
m = -8/6
m = -4/3.

Question 2.
Which value of a makes the equation 24 = $$\frac{a}{3}$$ – 9 true?
F. 5
G. 11
H. 45
I. 99
Answer: I. 99

Explanation:
Substitute a = 99 , in the given equation we get,
24 = $$\frac{a}{3}$$ – 9
24 = $$\frac{99}{3}$$ – 9
24 = 33 – 9
24 = 24.
So, last option is the correct answer.

Question 3.
A mapping diagram is shown.

What number belongs in the box so that the equation describes the function represented by the mapping diagram?

Answer: m = 7 , y = 7x + 5

Explanation:
Ordered pairs are  (4 , 33) , (7 , 54 ), (10 , 75) , (13 , 96 ),
First find the slope m of the line containing the two given points (4 , 33) and (7 , 54 )
m = (y2-y1) / (x2-x1)
m= (54 – 33) / (7 – 4)
m = 21/3
m = 7.
So, y = 7x + 5

Question 4.
What is the solution of the system of linear equations?
3x + 2y = 5
x = y + 5
A. (3, – 2)
B. (- 2, 3)
C. (- 1, 4)
D. (1, – 4)
Answer: A. (3, – 2)

Explanation:
Given 3x + 2y = 5
Then substitute , x = y + 5 in the above equation
3( y + 5) + 2y = 5
3y + 15 + 2y = 5
5y + 15 = 5
5( y + 3) = 5
y + 3 = 1
y = 1 – 3
y = -2,
substitute y = -2 in x = y + 5 then
x = 3
So, (3 , -2)

Question 5.
The director of a research lab wants to present data to donors. The data show how the lab uses a large amount of donated money for research and only a small amount of money for other expenses. Which type of display best represents these data?
F. box-and-whisker plot
G. circle graph
H. line graph
I. scatter plot
Answer: I. scatter plot

Explanation:
Scatter plot is the best graph for this type of data where vertical axis will show the amount of money and Horizontal axis will show research and other expenses.

Question 6.
Which graph shows a nonlinear function?

Answer: option B

Explanation:
As all the other options are representing the linear function that is forming a straight line  expect for option B , it is representing a non linear equation.

Question 7.
Which equation of a line passes through the point (—2, 3) and has a slope of $$\frac{3}{4}$$?

Answer:  F. y – 3 = $$\frac{3}{4}$$(x + 2)

Explanation:
Given, y – 3 = $$\frac{3}{4}$$(x + 2)
it is in the form of y = mx + c
so, slope m = $$\frac{3}{4}$$
Substitute the given points in this equation that is x = -2 and y = 3
3 – 3 = $$\frac{3}{4}$$(-2 + 2)
0 = 0.
So, F is the correct option.

Question 8.
The tables show the sales (in millions of dollars) for two companies over a five-year period.

Part A Does the first table show a linear function? Explain your reasoning.
Part B Does the second table show a linear function? Explain your reasoning.
Answer: Part A, the first table shows a linear function,
Part B the second table shows a linear function.

Explanation:
Part A – ordered pairs are (1 , 2) , (2 , 4) , (3 , 6) , (4 , 8) , (5 , 10)
Each input has exactly one output and forms a straight line when graphed
So, it is a linear function.

Part B – ordered pairs are (1 , 1) , (2 , 1) , (3 , 2) , (4 , 3) , (5 , 5)
Each input has exactly one output and does not form a straight line when graphed
So, it is a linear function.

Question 9.
The equations y = – x + 4 and y = $$\frac{1}{2}$$x – 8 form a system of linear equations. The table shows the values of y for given values of x.

What can you conclude from the table?
A. The system has one solution, when x = 0.
B. The system has one solution, when x = 4.
C. The system has one solution, when x = 8.
D. The system has no solution.
Answer: C. The system has one solution, when x = 8.

Explanation:
Given , y = – x + 4 and y = $$\frac{1}{2}$$x – 8
for x = 8 we have
y = -8 + 4 = -4
y = 0.5(8) – 8 = 4 – 8 = -4
Both the equations have one solution for x = 8
So, The system has one solution, when x = 8.

Question 10.
The vertices of a triangle are A (- 1, 3), B (1, 2), and C (- 1, – 1). Dilate the triangle using a scale factor of 2. What is the y-coordinate of the image of B?

Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 1, 3),  ( 1, 2 ),  (- 1, -1) these pairs form a right angle triangle
K = 2 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 1, 3) × 2 = ( -2 , 6)
( 1, 2) × 2 = ( 2 , 4)
(- 1, -1) × 2 = (-2 , -2)
From these new ordered pairs we form a new  right angle triangle

The New right angle triangle is larger than the original one So , its a increase .

Conclusion:

Improve your performance in maths with the help of Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions. Test yourself by solving the questions given at the end of the chapter. It also helps the students to have the real life calculation go very smoothly and neat defining their quick responses to daily life tasks. Students who are facing issues with math can be at ease now.

## Big Ideas Math Answers Grade 7 Chapter 6 Percents

Big Ideas Math Book 7th Grade Answer Key Chapter 6 Percents is on the line to help students who are willing to be perfect in their Math skills and also to the parents guiding their children to have a best score in examinations. This chapter Percent have conceptualized lessons on Discounts , Simple interest , all along with the subject knowledge, which will also be useful to check whether their Practical skills are intact . Get started to stop those hurdling times to studying hard to grasping the solutions by learning these methods of solving modern math problem in an effective way from Big Ideas Math Answers Grade 7 Chapter 6 Percents.

## Big Ideas Math Book 7th Grade Answer Key Chapter 6 Percents

Students who are facing difficulties in solving Math problems can at ease now! Big ideas Math Book 7th Grade Answer key  Chapter 6 percents  gives the most accurate answers to all the questions you have related to this chapter . It contains different methods of solving each question in a easy way to perform in their exams .This chapter deals with percents , decimals and fractions , It also helps the students to have the real life calculation to go very smooth and neat defining their quick responses to daily life tasks .

STEAM Video/Performance Task

Getting Ready for Chapter 6

Lesson 1 : Fractions, Decimals, and Percents

Lesson 2 : The Percent Proportion

Lesson 3 : The Percent Equation

Lesson 4 : Percents of Increase and Decrease

Lesson 5 : Discounts and Markups

Lesson 6 : Simple Interest

Percents Connecting Concepts

### STEAM Video/Performance Task

STEAM Video

Tornado!
More tornadoes occur each year in the United States than in any other country. How can you use a percent to describe the portion of tornadoes in the United States that occur in your state?

Watch the STEAM Video “Tornado!” Then answer the following questions.
1. The map below shows the average annual number of tornadoes in each state. Which regions have the most tornadoes? the fewest tornadoes?

2. Robert says that only Alaska, Hawaii, and Rhode Island average less than 1 tornado per year. What percent of states average more than 1 tornado per year?

Performance Task

Tornado Alley
After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given information about the average annual numbers of tornadoes in several states over a 25-year period. For example:

You will be asked to solve various percent problems about tornadoes. Why is it helpful to know the percent of tornadoes that occur in each state?

### Getting Ready for Chapter 6

Chapter Exploration
Work with a partner. Write the percent of the model that is shaded. Then write the percent as a decimal.

Answer : The Percent and decimal for the given models are
1. percent = 30%, decimal = 0.3
2. percent = 100%, decimal = 1
3. percent = 33%, decimal = 0.33
4. percent = 50%, decimal = 0.5
5. percent = 40%, decimal = 0.4
6. percent = 64%, decimal = 0.64
7. percent = 60%, decimal = 0.6

Explanation:
All the models given here are 10 by 10 grid forming a Square of 100 equal sections.
This entire square represents a whole and the shaded part is fraction.
Each of these shaded squares represents 1/100. So by using this data we have ,
1.    =
From the shaded part and the whole, we have the fraction of $$\frac{30}{100}$$,
Then the percent will be $$\frac{30}{100}$$ = 30%,
By rewriting it in decimal form we have 0.3

2.  =
From the shaded part and the whole, we have the fraction of $$\frac{100}{100}$$,
Then the percent will be $$\frac{100}{100}$$ = 100%,
By rewriting it in decimal form we have 1

3.  =
From the shaded part and the whole, we have the fraction of $$\frac{33}{100}$$,
Then the percent will be $$\frac{33}{100}$$ = 33%,
By rewriting it in decimal form we have 0.33

4.  =
From the shaded part and the whole, we have the fraction of $$\frac{50}{100}$$,
Then the percent will be $$\frac{50}{100}$$ = 50%,
By rewriting it in decimal form we have 0.5

5.  =
From the shaded part and the whole, we have the fraction of $$\frac{40}{100}$$,
Then the percent will be $$\frac{40}{100}$$ = 40%,
By rewriting it in decimal form we have 0.4

6.  =
From the shaded part and the whole, we have the fraction of $$\frac{64}{100}$$,
Then the percent will be $$\frac{64}{100}$$ = 64%,
By rewriting it in decimal form we have 0.64

7.  =
From the shaded part and the whole, we have the fraction of $$\frac{60}{100}$$,
Then the percent will be $$\frac{60}{100}$$ = 60%,
By rewriting it in decimal form we have 0.6

8. WRITE A PROCEDURE Work with a partner. Write a procedure for rewriting a percent as a decimal. Use examples to justify your procedure.

Answer:
Let us say that the fraction be $$\frac{44}{100}$$,
Then its percentage will be 44%,
To rewrite it as decimal, we divide 44 by 100 we get 0.44 (a decimal number). So, to convert from percent to decimal divide by 100 and remove the “%” sign.
We get 0.44 as decimal of 44%.

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
percent of change
percent of decrease
discount
percent of increase
percent error
markup

Answer:
percent of change:
Percentage Change is all about comparing old values to new values.

percent of decrease:
percent of decrease is a measure of percent change, which is the extent to which something loses value. or
A negative percent of change indicates a decrease from the original value to the second value.

discount: A reduction of price  is known as discount .Sometimes discounts are in percent, such as a 10% discount, and then you need to do a calculation to find the price reduction.

percent of increase :
Percent increase is a measure of percent change, which is the extent to which something gains value. or
A positive percent of change indicates an increase from the original value to the second value.

percent error :
Percent error is the difference between estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

markup:
Markup is all about how much a retailer increases the price over what they paid for it to buy the product or item in order to which is how they make money to pay for all their costs and hopefully make a profit.

### Lesson 6.1 Fractions, Decimals, and Percents

EXPLORATION 1

Comparing Numbers in Different Forms
Work with a partner. Determine which number is greater. Explain your method.

Answer:
a. 7% sales tax is greater than 5% sales tax
b. 0.37 cup of flour is greater than 0.33 cup of flour
c. 0.625 inch wrench is greater than 0.375 inch wrench
d. 12.6 dollars are greater than 12.56 dollars
e. 5.83 fluid ounces is greater than 5.6 fluid ounces

Explanation:
a. 7% sales tax or $$\frac{1}{20}$$ sales tax
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{1}{20}$$ as  0.05 in decimal form,
To get get the percent , multiply 100 to 0.05, then we get 5%.
So, $$\frac{1}{20}$$ can be write as 5%,
Finally, by comparing two values 7% sales tax is greater than 5% sales tax

b. 0.37 cup of flour or $$\frac{1}{3}$$ cup for flour
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{3}$$ as  0.33 in decimal form
Finally, by comparing two values 0.37 cup of flour is greater than 0.33 cup of flour.

c. $$\frac{5}{8}$$ inch wrench or 0.375 inch wrench
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{5}{8}$$ by dividing 5 by 8, we have  0.625 in decimal form
Finally, by comparing two values 0.625 inch wrench is greater than 0.375 inch wrench.

d. $$12$$ $${\Large\frac{3}{5}}$$ dollars or 12.56 dollars
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction$$\frac{63}{5}$$ by dividing 63 by 5, we have 12.6 in decimal form
Finally, by comparing two values 12.6 dollars are greater than 12.56 dollars

e.  $$5$$ $${\Large\frac{5}{6}}$$ fluid ounces or 5.6 fluid ounces
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction $$\frac{35}{6}$$  by dividing 35 by 6, we have 5.83 in decimal form
Finally, by comparing two values 5.83 fluid ounces is greater than 5.6 fluid ounces

EXPLORATION 2

Work with a partner and follow the steps below.

• Write five different numbers on individual slips of paper. Include at least one decimal, one fraction, and one percent.
• On a separate sheet of paper, create an answer key that shows your numbers written from least to greatest.
• Exchange slips of paper with another group and race to order the numbers from least to greatest. Then exchange answer keys to check your orders.

Answer: 78%, 0.95, $$\frac{83}{45}$$, 6, 21

Explanation:
Let the numbers be 6,  21, 0.95, $$\frac{83}{45}$$, 78%
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{83}{45}$$ can be rewrite as 1.84 in decimal form
Then 78% can be rewrite as 0.78 in decimal form,
As we can see 0.78 is less than 0.95 , 0.95 is less than $$\frac{83}{45}$$,$$\frac{83}{45}$$ is less than 6, 6 is less than 21,
Finally, we have the ascending order as 78%, 0.95, $$\frac{83}{45}$$, 6, 21.

Try It

Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 1.
39%
Answer: 0.39

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 39% in decimal form is 0.39

Question 2.
12. 6 %
Answer: 0.126

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 12.6% in decimal form is 0.126

Question 3.
0.05
Answer: 5%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.05 can be rewrite as 5%

Question 4.
1.25
Answer:  125%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 1.25 can be rewrite as 125%

Write the fraction as a decimal and a percent.
Question 5.
$$\frac{5}{8}$$
Answer: 0.625 or 62.5%

Explanation:
To get the percent or decimal from  fraction $$\frac{5}{8}$$ we have to divide 5 by 8 ,
Then, we get 0.625,
To get the percent of 0.625 multiply by 100 , it will be 62.5%
So, $$\frac{5}{8}$$ can be written as  0.625 or 62.5%

Question 6.
$$\frac{1}{6}$$
Answer: 0.166 or 16.6%

Explanation:
To get the percent or decimal from  fraction $$\frac{1}{6}$$ we have to divide 1 by 6 ,
Then, we get 0.166,
To get the percent of 0.166 multiply by 100 , it will be 16.6%
So, $$\frac{1}{6}$$ can be written as  0.166 or 16.6%

Question 7.
$$\frac{11}{3}$$
Answer:  3.66 or 366%

Explanation:
To get the percent or decimal from  fraction $$\frac{11}{3}$$ we have to divide 11 by 3 ,
Then, we get 3.66,
To get the percent of 3.66 multiply by 100 , it will be 366%
So, $$\frac{11}{3}$$ can be written as  3.66 or 366%

Question 8.
$$\frac{3}{1000}$$
Answer: 0.003 or 0.3%

Explanation:
To get the percent or decimal from  fraction $$\frac{3}{1000}$$ we have to divide 3 by 1000 ,
Then, we get 0.003,
To get the percent of 0.003 multiply by 100 , it will be 0.3%
So, $$\frac{3}{1000}$$ can be written as  0.003 or 0.3%

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

CONVERTING BETWEEN PERCENTS AND DECIMALS Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 9.
46%
Answer: 0.46

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 46% in decimal form is 0.46

Question 10.
$$66 . \overline{6} \%$$
Answer: $$0 .66 \overline{6}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$66 . \overline{6} \%$$ in decimal form is $$0 .66 \overline{6}$$

Question 11.
0.18
Answer: 18%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.18 can be rewrite as 18%

Question 12.
$$2 . \overline{3}$$
Answer: $$233 . \overline{3} \%$$

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, $$2 . \overline{3}$$ can be rewrite as $$233 . \overline{3} \%$$

WRITING FRACTIONS AS DECIMALS AND PERCENTS Write the fraction as a decimal and a percent.
Question 13.
$$\frac{7}{10}$$
Answer: decimal = 0.7, percent = 70%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{10}$$ as  0.7 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.7 can be rewrite as 70%
So, $$\frac{7}{10}$$ in decimal = 0.7, percent = 70%

Question 14.
$$\frac{5}{9}$$
Answer: decimal = $$0 .\overline{5}$$, percent = $$55 . \overline{5} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{5}{9}$$ as  $$0 .\overline{5}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$0 .\overline{5}$$ can be rewrite as $$55 . \overline{5} \%$$
So, $$\frac{5}{9}$$ in decimal = $$0 .\overline{5}$$, percent = $$55 . \overline{5} \%$$

Question 15.
$$\frac{7}{2000}$$
Answer: decimal = 0.0035, percent = 0.35%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{2000}$$ as  0.0035 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.0035 can be rewrite as 0.35%
So, $$\frac{7}{2000}$$ in decimal = 0.0035, percent = 0.35%

Question 16.
$$\frac{17}{15}$$
Answer: decimal = $$1.1 \overline{3}$$ , percent = $$113 . \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{17}{15}$$ as $$1.1 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$1.1 \overline{3}$$ can be rewrite as $$113 . \overline{3} \%$$
So, $$\frac{17}{15}$$ in decimal = $$1.1 \overline{3}$$, percent = $$113 . \overline{3} \%$$

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 17.
An astronaut spends 53% of the day working, 0.1 of the day eating, $$\frac{3}{10}$$ of the day sleeping, and the rest of the day exercising. Order the events by duration from least to greatest. Justify your answer.

Answer:  An astronaut spends the day 7% by exercising, 10% by eating, 30% by sleeping, 53% by working.

Explanation:
An astronaut spends 53% of the day working,
0.1 of the day eating, in terms of percent we can write it as 10%,
$$\frac{3}{10}$$ of the day sleeping, in decimals we ca rewrite as 0.3 and in percent it will be 30%,
Let us say that the whole day be 100% , The sum of the works he is doing in percent we get,
53% + 10% + 30% = 93%, and
Given that the rest of the day exercising, so 100% – 93% = 7%, A whole day is completed with these works.
To put them in Order the events by duration from least to greatest, we have , 7%, 10%, 30%, 53%.
An astronaut spends the day 7% by exercising, 10% by eating, 30% by sleeping, 53% by working.

Question 18.
DIG DEEPER!
A band plays one concert in Arizona, one concert in California, and one concert in Georgia. In California, the band earned $$\frac{3}{2}$$ the profit that they earned in Arizona. Of the total profit earned by the band, 32% is earned in Arizona. How many times more money did the band earn at the most profitable concert than at the least profitable concert? Justify your answer.
Answer:

Explanation:

### Fractions, Decimals, and Percents Homework & Practice 6.1

Review & Refresh

Find the missing dimension. Use the scale 1 : 15.

Answer: The model height of the Figure skater is 4.5inches and The Actual length of pipe is 75 feet .

Explanation:
Given , to use the scale oof 1 : 15 , Let the model height  be x
The model height of Figure skater is $$\frac{1}{15}$$ = $$\frac{x}{67.5}$$
15x = 67.5
x = $$\frac{67.5}{15}$$
x = 4.5
So , The model height of the Figure skater is 4.5inches

Let the Actual length is y
The Actual length of pipe is $$\frac{1}{15}$$ = $$\frac{5}{y}$$
y = 15 × 5 = 75
So , The Actual length of pipe is 75 feet .

Simplify the expression.
Question 3.
2(3p – 6) + 4p
Answer: p = 1.2

Explanation:
Let us say that whole expression is equal to 0
2(3p – 6) + 4p = 0
[2(3p) – 2(6)] + 4p = 0
6p – 12 + 4p = 0
10p – 12 = 0
p = 12/10 = 1.2
So, p = 1.2

Question 4.
5n – 3(4n + 1)
Answer: n = -0.42

Explanation:
Let us say that whole expression is equal to 0
5n – 3(4n + 1) = 0
5n – [ 3(4n) + 3(1) ] = 0
5n – 12n – 3 = 0
– 3 – 7n = 0
7n = – 3
n = -3/7 = -0.42
So, n = -0.42

Question 5.
What is the solution of 2n – 4 > – 12?
A. n < – 10
B. n < – 4
C. n > – 2
D. n > – 4
Answer:  D . n > -4

Explanation:
Given, 2n – 4 > – 12
add 4 in both sides,
2n – 4 + 4 > – 12 + 4
2n > – 8
divide both sides by 2
2n/2 > -8/2
n > – 4 .

Concepts, Skills, & Problem Solving
COMPARING NUMBERS IN DIFFERENT FORMS Determine which number is greater. Explain your method. (See Exploration 1, p. 235.)
Question 6.
4$$\frac{2}{5}$$ tons or 4.3 tons
Answer: 4$$\frac{2}{5}$$ tons is greater than 4.3 tons

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction $$\frac{22}{5}$$  by dividing 22 by 5, we have 4.4 in decimal form
Finally, by comparing two values 4$$\frac{2}{5}$$ tons is greater than 4.3 tons

Question 7.
82% success rate or $$\frac{5}{6}$$ success rate
Answer: $$\frac{5}{6}$$ success rate is greater than 82% success rate

Explanation:
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{5}{6}$$  as  0.833 in decimal form,
To get get the percent , multiply 100 to 0.833, then we get 83.3%.
So, $$\frac{5}{6}$$  can be write as 83.3%,
Finally, by comparing two values $$\frac{5}{6}$$ success rate is greater than 82% success rate

CONVERTING BETWEEN PERCENTS AND DECIMALS Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 8.
26%
Answer: 0.26

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 26% in decimal form is 0.26

Question 9.
0.63
Answer: 63%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.63 can be rewrite as 63%

Question 10.
9%
Answer: 0.09

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 9% in decimal form is 0.09

Question 11.
0.6
Answer: 60%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.6 can be rewrite as 60%

Question 12.
44.7%
Answer: 0.447

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 44.7% in decimal form is 0.447

Question 13.
55%
Answer: 0.55

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 55% in decimal form is 0.55

Question 14.
$$39 . \overline{2} \%$$
Answer: $$0.39 \overline{2}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$39 . \overline{2} \%$$ in decimal form is $$0.39 \overline{2}$$

Question 15.
3.554
Answer: 355.4%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 3.554  can be rewrite as 355.4%

Question 16.
123%
Answer: 1.23

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 123% in decimal form is 1.23

Question 17.
0.041
Answer: 4.1%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.041 can be rewrite as 4.1%

Question 18.
0.122
Answer: 12.2%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.122 can be rewrite as 12.2%

Question 19.
$$49 . \overline{92} \%$$
Answer: $$0.49 \overline{92}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$49 . \overline{92} \%$$ in decimal form is $$0.49 \overline{92}$$

Question 20.
YOU BE THE TEACHER
Your friend writes $$49 . \overline{8} \%$$ as a decimal. Is your friend correct? Explain your reasoning.

Answer: He wrote the decimal for $$4. \overline{8} \%$$ instead of $$49 . \overline{8} \%$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$49 . \overline{8} \%$$  in decimal form is $$0.49 \overline{8}$$

WRITING FRACTIONS AS DECIMALS AND PERCENTS Write the fraction as a decimal and a percent.
Question 21.
$$\frac{29}{100}$$
Answer: decimal = 0.29, percent = 29%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{29}{100}$$ as  0.29 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.29 can be rewrite as 29%
So, $$\frac{29}{100}$$ in decimal = 0.29, percent = 29%

Question 22.
$$\frac{3}{4}$$
Answer: decimal = 0.75, percent = 75%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{3}{4}$$ as  0.75 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.75 can be rewrite as 75%
So, $$\frac{3}{4}$$ in decimal = 0.75, percent = 75%

Question 23.
$$\frac{7}{8}$$
Answer: decimal = 0.875, percent = 87.5%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{8}$$ as  0.875 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.875 can be rewrite as 87.5%
So, $$\frac{7}{8}$$ in decimal = 0.875, percent = 87.5%

Question 24.
$$\frac{2}{3}$$
Answer: decimal = $$0. \overline{6}$$, percent = $$66. \overline{6} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{2}{3}$$ as $$0. \overline{6}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then$$0. \overline{6}$$ can be rewrite as $$66. \overline{6} \%$$
So, $$\frac{2}{3}$$ in decimal =$$0. \overline{6}$$, percent = $$66. \overline{6} \%$$

Question 25.
$$\frac{7}{9}$$
Answer: decimal = 0.77, percent = 77.7%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{7}{9}$$ as  0.77 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.77 can be rewrite as 77.7%
So, $$\frac{7}{9}$$ in decimal = 0.77, percent = 77.7%

Question 26.
$$\frac{12}{5}$$
Answer: decimal = 2.4, percent = 240%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{12}{5}$$ as  2.4 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 2.4 can be rewrite as 240%
So, $$\frac{12}{5}$$ in decimal =2.4, percent = 240%

Question 27.
$$\frac{9}{2}$$
Answer: decimal = 4.5, percent = 450%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{9}{2}$$ as 4.5 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 4.5 can be rewrite as 450%
So, $$\frac{9}{2}$$ in decimal = 4.5, percent = 450%

Question 28.
$$\frac{1}{1000}$$
Answer: decimal = 0.0010, percent = 0.10%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{1000}$$ as  0.0010 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.0010 can be rewrite as 0.10%
So, $$\frac{1}{1000}$$ in decimal = 0.0010, percent = 0.10%

Question 29.
$$\frac{17}{6}$$
Answer: decimal = $$2.8 \overline{3}$$, percent = $$283 . \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{17}{6}$$ as $$2.8 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then$$2.8 \overline{3}$$ can be rewrite as $$283 . \overline{3} \%$$
So, $$\frac{17}{6}$$ in decimal = $$2.8 \overline{3}$$, percent = $$283 . \overline{3} \%$$

Question 30.
$$\frac{3}{11}$$
Answer: decimal = 0.27, percent = 27%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{3}{11}$$ as  0.27 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.27 can be rewrite as 27%
So, $$\frac{3}{11}$$ in decimal = 0.27, percent = 27%

Question 31.
$$\frac{1}{750}$$
Answer: decimal = $$0.001 \overline{3}$$, percent =$$0.1 \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{750}$$ as  $$0.001 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$0.001 \overline{3}$$ can be rewrite as $$0.1 \overline{3} \%$$
So, $$\frac{1}{750}$$ in decimal = $$0.001 \overline{3}$$, percent = $$0.1 \overline{3} \%$$

Question 32.
$$\frac{22}{9}$$
Answer: decimal = $$2. \overline{4}$$, percent = $$244 . \overline{4} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{22}{9}$$ as  $$2. \overline{4}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$2. \overline{4}$$ can be rewrite as $$244 . \overline{4} \%$$
So, $$\frac{22}{9}$$ in decimal = $$2. \overline{4}$$, percent = $$244 . \overline{4} \%$$

PRECISION Order the numbers from least to greatest.
Question 33.
66.1%, 0.66, $$\frac{2}{3}$$, 0.667
Answer: 0.66, 66.1%, $$\frac{2}{3}$$, 0.667

Explanation:
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{2}{3}$$ can be rewrite as $$0. \overline{6}$$ in decimal form
Then 66.1% can be rewrite as 0.661 in decimal form,
As we can see 0.66 is less than 66.1% , 66.1% is less than $$\frac{2}{3}$$, $$\frac{2}{3}$$  is less than 0.667,
Finally, we have the ascending order as 0.66, 66.1%, $$\frac{2}{3}$$, 0.667.

Question 34.
$$\frac{2}{9}$$, 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$
Answer: 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$, $$\frac{2}{9}$$

Explanation:
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{11}{50}$$ can be rewrite as 0.22 in decimal form,
$$\frac{2}{9}$$ can be rewrite as $$0.22 \overline{2}$$in decimal form,
Then 21% can be rewrite as 0.21 in decimal form,
As we can see 21%  is less than $$0.2 \overline{1}$$ ,$$0.2 \overline{1}$$ is less than $$\frac{11}{50}$$,  $$\frac{11}{50}$$  is less than $$\frac{2}{9}$$,
Finally, we have the ascending order as 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$, $$\frac{2}{9}$$

MATCHING Tell which letter shows the graph of the number.
Question 35.
$$\frac{7}{9}$$
Answer: decimal = 0.777 , it is in the graph at the point A

Explanation:
By using the method of converting fraction into decimal ,
We can rewrite $$\frac{7}{9}$$ as 0.777 in decimal form,
So, looking at the graph given, it is at point A.

Question 36.
0.812
Answer:  it is at the point C in the given graph.

Question 37.
$$\frac{5}{6}$$
Answer: decimal = 0.833 , it is in the graph at the point D

Explanation:
By using the method of converting fraction into decimal ,
We can rewrite $$\frac{5}{6}$$ as 0.833 in decimal form,
So, looking at the graph given, it is at point D.

Question 38.
79.5%
Answer: 0.795, it is in the graph at the point B

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 79.5% in decimal form is 0.795
0.795 is in the graph given , at the point B vb

Question 39.
PROBLEM SOLVING
The table shows the portion of students in each grade that participate in School Spirit Week. Order the grades by portion of participation from least to greatest.

Answer: The grades by portion of participation from least to greatest are 7 , 6 , 8.

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left. So, 65% in decimal form is 0.65.
By using the method of converting fraction into Percent  ,
We can rewrite $$\frac{5}{6}$$ as 0.6 in decimal form,
Then grade 6 = 0.64 , grade 7 = 0.6 , grade 8 = 0.65 ,
So, The grades by portion of participation from least to greatest are 7 , 6 , 8.

Question 40.
MODELING REAL LIFE
The table shows the portion of gold medals that were won by the United States in five summer Olympic games. In what year did the United States win the least portion of gold medals? the greatest portion? Justify your answers.

Answer: The United States win the least portion of gold medals in the year 2004 and the greatest portion of gold medals won in the year 2012

Explanation:
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{36}{301}$$  as  0.119 in decimal form,
$$\frac{23}{150}$$ as 0.153 in decimal form,
$$\frac{46}{307}$$ as 0.149 in decimal form,
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
$$12. \overline{3} \%$$ as $$0.12 \overline{3}$$ in decimal form.
So, according to their years and portions of gold medals we have ,
year 2000 – $$0.12 \overline{3}$$,
year 2004 – 0.119,
year 2008 – $$0. \overline{12}$$,
year 2012 – 0.153,
year 2016 – 0.149,
Finally, The United States win the least portion of gold medals in the year 2004 and the greatest portion of gold medals won in the year 2012.

Question 41.
PROBLEM SOLVING
You, your friend, and your cousin have a basketball competition where each person attempts the same number of shots. You make 70% of your shots, your friend makes of her shots, $$\frac{7}{9}$$ and your cousin makes $$0.7 \overline{2}$$ of his shots. How many times more shots are made by the first place finisher than the third place finisher?

Answer: The first place finisher made 0.077 more shorts than the third place finis