## Engage NY Eureka Math 6th Grade Module 3 Lesson 19 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 19 Exercise Answer Key

Exploratory Challenge

The Length of a Line Segment is the Distance Between its End Points

Exercise 1.
Locate and label (4, 5) and (4, – 3). Draw the line segment between the end points given on the coordinate plane. How long is the line segment that you drew? Explain.

The length of the line segment is also 8 units. I found that the distance between (4, – 3) and (4, 5) is 8 units. Because the end points are on opposite sides of zero, I added the absolute values of the second coordinates together, so the distance from end to end is 8 units.

Exercise 2.
Draw a horizontal line segment starting at (4, – 3) that has a length of 9 units. What are the possible coordinates of the other end point of the line segment? (There is more than one answer.)
(- 5, – 3) or (13, – 3)

Which point did you choose to be the other end point of the horizontal line segment? Explain how and why you chose that point. Locate and label the point on the coordinate grid.
The other end point of the horizontal line segment is (- 5, – 3). I chose this point because the other option, (13, – 3), is located off of the given coordinate grid.
Note: Students may choose the end point (13, – 3), but they must change the number scale of the x-axis to do so.

Exercise 3.
Extending Lengths of Line Segments to Sides of Geometric Figures
The two line segments that you have just drawn could be seen as two sides of a rectangle. Given this, the end points of the two line segments would be three of the vertices of this rectangle.
a. Find the coordinates of the fourth vertex of the rectangle. Explain how you find the coordinates of the fourth vertex using absolute value.
The fourth vertex is (- 5, 5). The opposite sides of a rectangle are the same length, so the length of the vertical side starting at (- 5, – 3) has to be 8 units long. Also, the side from (- 5, – 3) to the remaining vertex is a vertical line, so the end points must have the same first coordinate. |- 3| = 3, and 8 – 3 = 5, so the remaining vertex must be five units above the x-axis.
Note: Students can use a similar argument using the length of the horizontal side starting at (4, 5), knowing it has to be 9 units long.

b. How does the fourth vertex that you found relate to each of the consecutive vertices in either direction?
Explain.
The fourth vertex has the same first coordinate as (- 5, – 3) because they are the end points of a vertical line segment. The fourth vertex has the same second coordinate as (4, 5) since they are the end points of a horizontal line segment.

c. Draw the remaining sides of the rectangle.

Using Lengths of Sides of Geometric Figures to Solve Problems

Exercise 4.
Using the vertices that you have found and the lengths of the line segments between them, find the perimeter of the rectangle.
8 + 9 + 8 + 9 = 34; the perimeter of the rectangle is 34 units.

Exercise 5.
Find the area of the rectangle.
9 × 8 = 72; the area of the rectangle is 72 units2.

Exercise 6.
Draw a diagonal line segment through the rectangle with opposite vertices for end points. What geometric figures are formed by this line segment? What are the areas of each of these figures? Explain.
The diagonal line segment cuts the rectangle into two right triangles. The areas of the triangles are 36 units2 each because the triangles each make up half of the rectangle, and half of 72 is 36.

Extension (If time allows): Line the edge of a piece of paper up to the diagonal in the rectangle. Mark the length of the diagonal on the edge of the paper. Align your marks horizontally or vertically on the grid, and estimate the length of the diagonal to the nearest integer. Use that estimation to now estimate the perimeter of the triangles.
The length of the diagonal is approximately 12 units, and the perimeter of each triangle is approximately 29 units.

Exercise 7
Construct a rectangle on the coordinate plane that satisfies each of the criteria listed below. Identify the coordinate of each of its vertices.
→ Each of the vertices lies in a different quadrant.
→ Its sides are either vertical or horizontal.
→ The perimeter of the rectangle is 28 units.
Answers will vary. The example to the right shows a rectangle with side lengths 10 and 4 units. The coordinates of the rectangle’s vertices are (- 6,3), (4, 3), (4, – 1), and (- 6, – 1).

Using absolute value, show how the lengths of the sides of your rectangle provide a perimeter of 28 units.
|- 6| = 6, |4| = 4, and 6 + 4 = 10, so the width of my rectangle is 10 units.
|3| = 3, |-1| = 1, and 3 + 1 = 4, so the height of my rectangle is 4 units.
10 + 4 + 10 + 4 = 28, so the perimeter of my rectangle is 28 units.

### Eureka Math Grade 6 Module 3 Lesson 19 Problem Set Answer Key

Question 1.
One end point of a line segment is (-3, -6). The length of the line segment is 7 units. Find four points that could serve as the other end point of the given line segment.
(- 10, – 6); (4, – 6); (- 3, 1); (- 3, – 13)

Question 2.
Two of the vertices of a rectangle are (1, – 6) and (- 8, – 6). If the rectangle has a perimeter of 26 units, what are the coordinates of its other two vertices?
(1, – 2) and (- 8, – 2), or (1, – 10) and (- 8, – 10)

Question 3.
A rectangle has a perimeter of 28 units, an area of 48 square units, and sides that are either horizontal or vertical. If one vertex is the point (- 5, – 7) and the origin is In the interior of the rectangle, find the vertex of the rectangle that is opposite (- 5, – 7).
(1, 1)

### Eureka Math Grade 6 Module 3 Lesson 19 Exit Ticket Answer Key

Question 1.
The coordinates of one end point of a line segment are (- 2, – 7). The line segment is 12 units long. Give three
possible coordinates of the line segment’s other end point.
(10, – 7); (- 14, – 7); (- 2, 5); (- 2, – 19)

Question 2.
Graph a rectangle with an area of 12 units2 such that its vertices lie in at least two of the four quadrants in the coordinate plane. State the lengths of each of the sides, and use absolute value to show how you determined the lengths of the sides.
Answers will vary. The rectangle can have side lengths of 6 and 2 or 3 and 4. A sample is provided on the grid on the right. 6 × 2 = 12

### Eureka Math Grade 6 Module 3 Lesson 19 Opening Exercise Answer Key

Question 1.
In the coordinate plane, find the distance between the points using absolute value.

The distance between the points is 8 units. The points have the same first coordinates and, therefore, lie on the same vertical line. |- 3| = 3, and |5| = 5, and the numbers lie on opposite sides of 0, so their absolute values are added together; 3 + 5 = 8. We can check our answer by just counting the number of units between the two points.

## Engage NY Eureka Math 6th Grade Module 3 End of Module Assessment Answer Key

### Eureka Math Grade 6 Module 3 End of Module Assessment Answer Key

Question 1.
Mr. Kindle invested some money in the stock market. He tracks his gains and losses using a computer program. Mr. Kindle receives a daily email that updates him on all his transactions from the previous day. This morning, his email read as follows:
Good morning, Mr. Kindle,
Yesterday’s investment activity included a loss of $800, a gain of$960, and another gain of $230. Log in now to see your current balance. a. Write an integer to represent each gain and loss.  Description Integer Representation Loss of$800 Gain of $960 Gain of$230

 Description Integer Representation Loss of $800 – 800 Gain of$960 960 Gain of $230 230 b. Mr. Kindle noticed that an error had been made on his account. The “loss of$800” should have been a “gain of $800.” Locate and label both points that represent “a loss of$800” and “a gain of $800” on the number line below. Describe the relationship of these two numbers when zero represents no change (gain or loss). Answer: – 800 and 800 are opposites. c. Mr. Kindle wanted to correct the error, so he entered – (-$800) into the program. He made a note that read, “The opposite of the opposite of $800 is$800.” Is his reasoning correct? Explain.
Yes, he is correct. The opposite of 800 is – 800, and the opposite of that is 800.

Question 2.
At 6:00 a.m., Buffalo, NY, had a temperature of 10°F. At noon, the temperature was – 10°F, and at midnight, it was – 20°F.
a. Write a statement comparing – 10°F and – 20°F.

– 10°F is warmer than – 20°F.

b. Write an inequality statement that shows the relationship between the three recorded temperatures. Which temperature is the warmest?
– 20 < – 10 < 10
10°F is the warmest temperature.

c. Explain how to use absolute value to find the number of degrees below zero the temperature was at noon.
|- 10| = 10
The temperature at noon was 10° below zero.

d. In Peekskill, NY, the temperature at 6:00 a.m. was – 12°F. At noon, the temperature was the exact opposite of Buffalo’s temperature at 6:00 a.m. At midnight, a meteorologist recorded the temperature as – 6°F in Peekskill. He concluded that “For temperatures below zero, as the temperature increases, the absolute value of the temperature decreases.” Is his conclusion valid? Explain and use a vertical number line to support your answer.

|- 12| = 12
|- 10| = 10
|- 6| = 6
The absolute values are decreasing.

Yes, his conclusion is valid. Absolute value is a number’s distance from zero. As the temperature increases from – 12 to – 10 to – 6 they get closer to zero, So their distance from zero is decreasing.

Question 3.
Choose an integer between 0 and – 5 on a number line, and label the point P. Locate and label each of the following points and their values on the number line.

a. Label point A: the opposite of point P.
3

b. Label point B: a number less than point P.
– 5

c. Label point C: a number greater than point P.
0

d. Label point D: a number halfway between point P and the integer to the right of point P.
– 2.5

Question 4.
Julia is learning about elevation in math class. She decided to research some facts about New York State to better understand the concept. Here are some facts that she found.

• Mount Marcy is the highest point in New York State. It is 5,343 feet above sea level.
• Lake Erie is 210 feet below sea level.
• The elevation of Niagara Falls, NY, is 614 feet above sea level.
• The lobby of the Empire State Building is 50 feet above sea level.
• New York State borders the Atlantic Coast, which is at sea level.
• The lowest point of Cayuga Lake is 435 feet below sea level.

a. Write an integer that represents each location in relationship to sea level.

 Mount Marcy Lake Erie Niagara Falls, NY Empire State Building Atlantic Coast Cayuga Lake

 Mount Marcy 5,343 Lake Erie – 210 Niagara Falls, NY 614 Empire State Building 50 Atlantic Coast 0 Cayuga Lake – 435

b. Explain what negative and positive numbers tell Julia about elevation.
A  negative number means the elevation is below sea level. A positive number means the elevation is above sea level.

c. Order the elevations from least to greatest, and then state their absolute values. Use the chart below to record your work.

d. Circle the row in the table that represents sea level. Describe how the order of the elevations below sea level compares to the order of their absolute values. Describe how the order of the elevations above sea level compares to the order of their absolute values.
The elevations below sea level have absolute values that are their opposites, so the order is opposite. – 435 < – 210  but 435 > 210. The elevations above sea level are the same as their absolute values, so the order is the same.
50 < 614 < 5,343

Question 5.
For centuries, a mysterious sea serpent has been rumored to live at the bottom of Mysterious Lake. A team of historians used a computer program to plot the last five positions of the sightings.

a. Locate and label the locations of the last four sightings: A (- 9$$\frac{1}{2}$$, 0), B(- 3, – 4.75), C(9, 2), and D(8, – 2.5).

b. Over time, most of the sightings occurred in Quadrant Ill. Write the coordinates of a point that lies in Quadrant III.
(- 6, – 3)

c. What is the distance between point A and the point (9$$\frac{1}{2}$$, 0)? Show your work to support your answer.

d. What are the coordinates of point E on the coordinate plane?
(5, 2)

e. Point F is related to point E. Its x-coordinate is the same as point E’s, but its y-coordinate is the opposite of point E’s. Locate and label point F. What are the coordinates? How far apart are points E and F? Explain how you arrived at your answer.
The coordinates of F are (5, -2). Points E and F are 4 units apart. Since their x-coordinates are the same, I just counted the number of units from 2 to – 2 (between their y- Coordinates), and that is 4.

## Engage NY Eureka Math 6th Grade Module 3 Mid Module Assessment Answer Key

### Eureka Math Grade 6 Module 3 Mid Module Assessment Answer Key

Question 1.
The picture below is a flood gauge that is used to measure how far (in feet) a river’s water level is above or below its normal level.
a. Explain what the number 0 on the gauge represents, and explain what the numbers above and below 0 represent.
The number 0 represents the normal average water level in the river. The numbers below 0 indicate low water and the numbers above 0 indicate high water level.

b. Describe what the picture indicates about the river’s current water level.

The river’s water level is about 2 feet below normal.

c. What number represents the opposite of the water level shown in the picture, and where is it located on the gauge? What would it mean if the river water was at that level?
The water level is currently at approximately – 2.0 feet. The opposite of – 2 is 2. 2 is on the opposite side of 0, or above zero. If the river was at 2, the water level would be higher than normal.

d. If heavy rain is in the forecast for the area for the next 24 hours, what reading might you expect to see on this gauge tomorrow? Explain your reasoning.
I would expect to see the water level closer to 0 or even higher. Heavy rain should cause the amount of water in the river to increase, So its level would move up the number line.

Question 2.
Isaac made a mistake in his checkbook. He wrote a check for $8.98 to rent a video game but mistakenly recorded it in his checkbook as an$8.98 deposit.
a. Represent each transaction with a rational number, and explain the difference between the transactions.
A check will decrease his account balance
So it can be represented by – 8.98
A deposit will increase his account balance
So it can be represented by 8.98.

b. On the number line below, locate and label the points that represent the rational numbers listed in part (a). Describe the relationship between these two numbers. Zero on the number line represents Isaac’s balance before the mistake was made.

The numbers that represent the two transactions are opposites.

c. Use absolute value to explain how a debit of $8.98 and a credit of$8.98 are similar.
The check and deposit have the same absolute value (8.98) So they will change his account balance by the same amount of money, but they change the balance in opposite directions.

Question 3.
A local park’s programs committee is raising money by holding mountain bike races on a course through the park. During each race, a computer tracks the competitors’ locations on the course using GPS tracking. The table shows how far each competitor is from a checkpoint.

a. The checkpoint is represented by 0 on the number line. Locate and label points on the number line
for the positions of each listed participant. Label the points using rational numbers.

b. Which of the competitors is closest to the checkpoint? Explain.
Florence is closest to the checkpoint because her distance to the checkpoint is 0.1 miles which is less than any of the other girls distances.

c. Two competitors are the same distance from the checkpoint. Are they in the same location? Explain.
Rebecca and Lita are both of 0.5 miles from the checkpoint, they are just on opposite sides of the check point.

d. Who is closer to finishing the race, Nancy or Florence? Support your answer.
Florence is closer to finishing the race because the number representing her position (- 0.1) is to the right of (-$$\frac{2}{10}$$) on the number line which is Nancy’s position.

Question 4.
Andrea and Marta are testing three different coolers to see which keeps the coldest temperature. They placed a bag of ice in each cooler, closed the coolers, and then measured the air temperature inside each after 90 minutes. The temperatures are recorded in the table below:

Marta wrote the following inequality statement about the temperatures:
– 4.3 < – 2.91 < 5.7.
Andrea claims that Marta made a mistake in her statement and that the inequality statement should be written as
– 2.91 < – 4.3 < 5.7.
a. Is either student correct? Explain.
Marta is correct because the order of the numbers in her inequality is the same as the order of the numbers on the number line moving from left to right (or from down to up).

b. The students want to find a cooler that keeps the temperature inside the cooler more than 3 degrees below the freezing point of water (0°C) after 90 minutes. Indicate which of the tested coolers meets this goal, and explain why.
More than 3 degrees below 0°C means less than – 3°C. The only cooler to keep the temperature less than – 3°C is Cooler c. Cooler c held a temperature of – 4.3°C which is to the left of – 3°C on the number line.

Question 5.
Mary manages a company that has been hired to flatten a plot of land. She took several elevation samples from the land and recorded those elevations below:

a. The landowner wants the land flat and at the same level as the road that passes in front of it. The road’s elevation is 830 feet above sea level. Describe in words how elevation samples B, C, and E compare to the elevation of the road.
Samples B and C are higher than 830 feet and so higher than the road. sample E is lower than 830 feet and so lower than the road.

b. The table below shows how some other elevation samples compare to the level of the road:

Write the values in the table in order from least to greatest.
_________< _________< _________< _________< _________< _________
– 4.5    <   – 0.9    <   – 0.5    <   1.3    <   2.2    <   3.1

c. Indicate which of the values from the table in part (b) is farthest from the elevation of the road. Use absolute value to explain your answer.
– 4.5 (sample k) is furthest from the elevation of the road because its absolute value (4.5) is greater than the absolute values of the other sample in the table.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 17 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 17 Example Answer Key

Example 1.
Locate and label the points {(3, 2), (8, 4), (- 3, 8), (- 2, – 9), (0, 6), (- 1, – 2), (10, – 2)) on the grid above.

Example 2.
Drawing the Coordinate Plane Using an Increased Number Scale for One Axis
Draw a coordinate plane on the grid below, and then locate and label the following points:
{(- 4, 20), (- 3, 35), (1, – 35), (6, 10), (9, – 40)}.

Example 3.
Drawing the Coordinate Plane Using a Decreased Number Scale for One Axis
Draw a coordinate plane on the grid below, and then locate and label the following points:
{(0. 1, 4), (0. 5, 7), (- 0.7, – 5), (- 0.4, 3), (0.8, 1))}.

Example 4.
Drawing the Coordinate Plane Using a Different Number Scale for Both Axes
Determine a scale for the x-axis that will allow all x-coordinates to be shown on your grid.
The grid is 16 units wide, and the x-coordinates range from – 14 to 14. If I let each grid line represent 2 units, then the x-axis will range from – 16 to 16.

Determine a scale for the y-axis that will allow all y-coordinates to be shown on your grid.
The grid is 16 units high, and the y-coordinates range from – 4 to 3. 5. I could let each grid line represent one unit, but if I let each grid line represent $$\frac{1}{2}$$ of a unit, the points will be easier to graph.

Draw and label the coordinate plane, and then locate and label the set of points.
{(- 14, 2), (- 4, – 0. 5), (6,- 3. 5), (14, 2. 5), (0, 3.5), (- 8, – 4)}

### Eureka Math Grade 6 Module 3 Lesson 17 Problem Set Answer Key

Question 1.
Label the coordinate plane, and then locate and label the set of points below.
{(0.3, 0.9), (- 0.1, 0.7), (- 0.5, – 0.1), (- 0.9, 0.3), (0, – 0.4)}

Question 2.
Label the coordinate plane, and then locate and label the set of points below.
{(90, 9), (- 110, – 11), (40, 4), (- 60, – 6), (- 80, – 8)}

Extension:

Question 3.
Describe the pattern you see in the coordinates In Problem 2 and the pattern you see in the points. Are these patterns consistent for other points too?
The x-coordinate for each of the given points is 10 times its y-coordinate. When I graphed the points, they appear to make a straight line. I checked other ordered pairs with the same pattern, such as (- 100, – 10), (20, 2), and even (0, 0), and it appears that these points are also on that line.

### Eureka Math Grade 6 Module 3 Lesson 17 Exit Ticket Answer Key

Question 1.
Determine an appropriate scale for the set of points given below. Draw and label the coordinate plane, and then locate and label the set of points.
{(10, 0. 2), (- 25, 0.8), (0, – 0.4), (20, 1), (- 5, – 0. 8)}
The x-coordinates range from – 25 to 20. The grid is 10 units wide. If I let each grid line represent 5 units, then the x-axis will range from – 25 to 25.

The y-coordinates range from – 0.8 to 1. The grid is 10 units high. If I let each grid line represent two-tenths of a unit, then the y-axis will range from – 1 to 1.

### Eureka Math Grade 6 Module 3 Lesson 17 Opening Exercise Answer Key

Question 1.
Draw all necessary components of the coordinate plane on the blank 20 × 20 grid provided below, placing the origin at the center of the grid and letting each grid line represent 1 unit.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 18 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 18 Example Answer Key

Example 1. The Distance Between Points on an Axis
Consider the points (- 4, 0) and (5, 0).
What do the ordered pairs have in common, and what does that mean about their location in the coordinate plane?
Both of their y-coordinates are zero, so each point lies on the x-axis, the horizontal number line.

How did we find the distance between two numbers on the number line?
We calculated the absolute values of the numbers, which told us how far the numbers were from zero. If the numbers were located on opposite sides of zero, then we added their absolute values together. If the numbers were located on the same side of zero, then we subtracted their absolute values.

Use the same method to find the distance between (- 4, 0) and (5, 0).
|- 4| = 4 and |5| = 5. The numbers are on opposite sides of zero, so the absolute values get combined: 4 + 5 = 9. The distance between (- 4, 0) and (5, 0) is 9 units.

Example 2.
The Length of a Line Segment on an Axis
Consider the line segment with end points (0,- 6) and (0, – 11).
What do the ordered pairs of the end points have in common, and what does that mean about the line segment’s location in the coordinate plane?
The x-coordinates of both end points are zero, so the points lie on the y-axis, the vertical number line. If its end points lie on a vertical number line, then the line segment itself must also lie on the vertical line.

Find the length of the line segment described by finding the distance between its end points (0,- 6) and (0, – 11).
|- 6| = 6 and |- 11| = 11. The numbers are on the same side of zero, which means the longer distance contains the shorter distance, so the absolute values need to be subtracted: 11 – 6 = 5. The distance between (0, – 6) and (0, – 11) is 5 units, so the length of the line segment with end points (0, – 6) and (0, – 11) is 5 units.

Example 3.
Length of a Horizontal or Vertical Line Segment That Does Not Lie on an Axis
Consider the line segment with end points (- 3, 3) and (- 3, – 5).
What do the end points, which are represented by the ordered pairs, have in common? What does that tell us about the location of the line segment on the coordinate plane?
Both end points have x-coordinates of – 3, so the points lie on the vertical line that intersects the x-axis at – 3. This means that the end points of the line segment, and thus the line segment, lie on a vertical line.

Find the length of the line segment by finding the distance between its end points.
The end points are on the same vertical line, so we only need to find the distance between 3 and – 5 on the number line.
|3| = 3 and |- 5| = 5, and the numbers are on opposite sides of zero, so the values must be added: 3 + 5 = 8. So, the distance between (- 3, 3) and (- 3, – 5) is 8 units.

### Eureka Math Grade 6 Module 3 Lesson 18 Exercise Answer Key

Exercise

Find the lengths of the line segments whose end points are given below. Explain how you determined that the line segments are horizontal or vertical.
a. (- 3, 4) and (- 3, 9)
Both end points have x-coordinates of – 3, so the points lie on a vertical line that passes through – 3 on the x-axis. |4| = 4 and |9| = 9, and the numbers are on the same side of zero. By subtraction, 9 – 4 = 5, so the length of the line segment with end points (- 3, 4) and (- 3, 9) is 5 units.

b. (2, – 2) and (- 8, – 2)
Both end points have y-coordinates of – 2, so the points lie on a horizontal line that passes through – 2 on the y-axis. |2| = 2 and |- 8| = 8, and the numbers are on opposite sides of zero, so the absolute values must be added. By addition, 8 + 2 = 10, so the length of the line segment with end points (2, – 2) and (- 8, – 2)is 10 units.

c. (- 6, – 6) and (- 6, 1)
Both end points hove x-coordinates of – 6, so the points lie on a vertical line. |- 6| = 6 and |1| = 1, and the numbers are on opposite sides of zero, so the absolute values must be added. By addition, 6 + 1 = 7, so the length of the line segment with end points (- 6, – 6) and (- 6, 1) is 7 units.

d. (- 9, 4) and (- 4, 4)
Both end points have y-coordinates of 4, so the points lie on a horizontal line. |- 9| = 9 and |- 4| = 4, and the numbers are on the same side of zero. By subtraction, 9 – 4 = 5, so the length of the line segment with end points (- 9, 4) and (- 4, 4) is 5 units.

e. (0, – 11) and (0, 8)
Both end points hove x-coordinates of 0, so the points lie on the y-axis. |- 11| = 11 and |8| = 8, and the numbers are on opposite sides of zero, so their absolute values must be added. By addition, 11 + 8 = 19, so the length of the line segment with end points (0, – 11) and (0, 8) is 19 units.

### Eureka Math Grade 6 Module 3 Lesson 18 Problem Set Answer Key

Question 1.
Find the length of the line segment with end points (7, 2) and (- 4, 2), and explain how you arrived at your solution.
11 units. Both points have the same y-coordinate, so I knew they were on the same horizontal line. I found the
distance between the x-coordinates by counting the number of units on a horizontal number line from – 4 to zero and then from zero to 7, and 4 + 7 = 11.
or
I found the distance between the x-coordinates by finding the absolute value of each coordinate. |7| = 7 and |- 4| = 4. The coordinates lie on opposite sides of zero, so I found the length by adding the absolute values together. Therefore, the length of a line segment with end points (7, 2) and (- 4, 2) is 11 units.

Question 2.
Sarah and Jamal were learning partners in math class and were working independently. They each started at the point (- 2, 5) and moved 3 units vertically in the plane. Each student arrived at a different end point. How is this possible? Explain and list the two different end points.
It is possible because Sarah could have counted up and Jamal could have counted down or vice versa. Moving 3 units in either direction vertically would generate the following possible end points: (- 2, 8) or (- 2, 2).

Question 3.
The length of a line segment is 13 units. One end point of the line segment is (- 3,7). Find four points that could be the other end points of the line segment.
(- 3, 20), (- 3, – 6), (- 16, 7) or (10, 7)

### Eureka Math Grade 6 Module 3 Lesson 18 Exit Ticket Answer Key

Question 1.
Determine whether each given pair of end points lies on the same horizontal or vertical line. If so, find the length of the line segment that joins the pair of points. If not, explain how you know the points are not on the same horizontal or vertical line.

a. (0, – 2) and (0, 9)
The end points both have x-coordinates of 0, so they both lie on the y-axis, which is a vertical line. They lie on opposite sides of zero, so their absolute values have to be combined to get the total distance. |- 2| = 2 and |9| = 9, so by addition, 2 + 9 = 11. The length of the line segment with end points (0, – 2) and (0, 9) is 11 units.

b. (11, 4) and (2, 11)
The points do not lie on the same horizontal or vertical line because they do not share a common x- or y-coordinate.

c. (3, – 8) and (3, – 1)
The end points both have x-coordinates of 3, so the points lie on a vertical line that passes through 3 on the x-axis. The y-coordinates lie on the some side of zero. The distance between the points is determined by subtracting their absolute values, |- 8| = 8 and |- 1| = 1. So, by subtraction, 8 – 1 = 7. The length of the line segment with end points (3, – 8) and (3, – 1) is 7 units.

d. (- 4, – 4) and (5, – 4)
The end points have the same y-coordinate of – 4, so they lie on a horizontal line that passes through – 4 on the y-axis. The numbers lie on opposite sides of zero on the number line, so their absolute values must be added to obtain the total distance, |- 4| = 4 and |5| = 5. So, by addition, 4 + 5 = 9. The length of the line segment with endpoints (- 4, – 4) and (5, – 4) is 9 units.

### Eureka Math Grade 6 Module 3 Lesson 18 Opening Exercise Answer Key

Four friends are touring on motorcycles. They come to an intersection of two roads; the road they are on continues straight, and the other is perpendicular to it. The sign at the intersection shows the distances to several towns. Draw a map/diagram of the roads, and use it and the information on the sign to answer the following questions:

What is the distance between Albertsville and Dewey Falls?
Students draw and use their maps to answer. Albertsville is 8 miles to the left, and Dewey Falls is 6 miles to the right. Since the towns are in opposite directions from the intersection, their distances must be combined by addition, 8 + 6 = 14, so the distance between Albertsville and Dewey Falls is 14 miles.

What is the distance between Blossville and Cheyenne?
Blossville and Cheyenne are both straight ahead from the intersection in the direction that they are going. Since they are on the same side of the intersection, Blossville is on the way to Cheyenne, so the distance to Cheyenne includes the 3 miles to Blossville. To find the distance from Blossville to Cheyenne, I have to subtract; 12 – 3 = 9. So, the distance from Blossville to Cheyenne is 9 miles.

On the coordinate plane, what represents the intersection of the two roads?
The intersection is represented by the origin.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 16 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 16 Example Answer Key

Example 1.
Extending Opposite Numbers to the Coordinate Plane
Extending Opposite Numbers to the Coordinates of Points on the Coordinate Plane Locate and label your points on the coordinate plane to the right. For each given pair of points in the table below, record your observations and conjectures in the appropriate cell. Pay attention to the absolute values of the coordinates and where the points lie in reference to each axis.

Examples 2 – 3: Navigating the Coordinate Plane

### Eureka Math Grade 6 Module 3 Lesson 16 Exercise Answer Key

Exercises

In each column, write the coordinates of the points that are related to the given point by the criteria listed in the first column of the table. Point S(5, 3) has been reflected over the x- and y-axes for you as a guide, and its images are shown on the coordinate plane. Use the coordinate grid to help you locate each point and its corresponding coordinates.

Exercise 1.
When the coordinates of two points are (x, y) and (- x, y), what line of symmetry do the points share? Explain.
They share the y-axis because the y-coordinates are the same and the x-coordinates are opposites, which means the points will be the same distance from the y-axis but on opposite sides.

Exercise 2.
When the coordinates of two points are (x, y) and (x, – y). what line of symmetry do the points share? Explain.
They share the x-axis because the x-coordinates are the same and the y-coordinates are opposites, which means the points will be the same distance from the x-axis but on opposite sides.

### Eureka Math Grade 6 Module 3 Lesson 16 Problem Set Answer Key

Question 1.
Locate a point In Quadrant IV of the coordinate plane. Label the point A, and write its ordered pair next to it.

a. Reflect point A over an axis so that its image is in Quadrant III. Label the image B, and write its ordered pair next to it. Which axis did you reflect over? What is the only difference in the ordered pairs of points A and B?
B(- 5, – 3); reflected over the y-axis
The ordered pairs differ only by the sign of their x-coordinates: A(5, – 3) and B(- 5, – 3).

b. Reflect point B over an axis so that its image is in Quadrant II. Label the image C, and write its ordered pair next to it. Which axis did you reflect over? What is the only difference in the ordered pairs of points B and C? How does the ordered pair of point C relate to the ordered pair of point A?
C(- 5, 3); reflected over the x-axis
The ordered pairs differ only by the signs of their y-coordinates: B(- 5, – 3) and C(- 5, 3).
The ordered pair for point C differs from the ordered pair for point A by the signs of both coordinates:
A(5, – 3) and C(- 5, 3).

c. Reflect point C over an axis so that its image is in Quadrant I. Label the image D, and write its ordered pair next to it. Which axis did you reflect over? How does the ordered pair for point D compare to the ordered pair for point C? How does the ordered pair for point D compare to points A and B?
D(5, 3); reflected over the y-axis again
Point D differs from point C by only the sign of its x-coordinate: D(5, 3) and C(- 5, 3).
Point D differs from point B by the signs of both coordinates: D(5, 3) and B(- 5, – 3).
Point D differs from point A by only the sign of the y-coordinate: D(5, 3) and A(5, – 3).

Question 2.
Bobbie listened to her teacher’s directions and navigated from the point (- 1,0) to (5, – 3). She knows that she has the correct answer, but she forgot part of the teacher’s directions. Her teacher’s directions included the following:
“Move 7 units down, reflect about the   ?   -axis, move up 4 units, and then move right 4 units.”
Help Bobbie determine the missing axis in the directions, and explain your answer.
The missing line is a reflection over the y-axis. The first line would move the location to (- 1, – 7). A reflection over the y-axis would move the location to (1, – 7) in Quadrant IV, which is 4 units left and 4 units down from the end point (5, – 3).

### Eureka Math Grade 6 Module 3 Lesson 16 Exit Ticket Answer Key

Question 1.
How are the ordered pairs (4, 9) and (4, – 9) similar, and how are they different? Are the two points related by a reflection over an axis in the coordinate plane? If so, indicate which axis is the line of symmetry between the points. If they are not related by a reflection over an axis in the coordinate plane, explain how you know.
The x-coordinates are the same, but the y-coordinates are opposites, meaning they are the same distance from zero on the x-axis and the same distance but on opposite sides of zero on the y-axis. Reflecting about the x-axis interchanges these two points.

Question 2.
Given the point (- 5, 2), write the coordinates of a point that is related by a reflection over the x- or y-axis. Specify which axis is the line of symmetry.
Using the x-axis as a line of symmetry, (-3, -2); using the y-axis as a line of symmetry, (5,2)

### Eureka Math Grade 6 Module 3 Lesson 16 Opening Exercise Answer Key

Question 1.
Give an example of two opposite numbers, and describe where the numbers lie on the number line. How are opposite numbers similar, and how are they different?
Answers may vary. 2 and – 2 are opposites because they are both 2 units from zero on a number line but in opposite directions. Opposites are similar because they have the same absolute value, but they are different because opposites are on opposite sides of zero.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 15 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 15 Example Answer Key

Example 1. Extending the Axes Beyond Zero
The point below represents zero on the number line. Draw a number line to the right starting at zero. Then, follow directions as provided by the teacher.

→ Use a straightedge to extend the x-axis to the left of zero to represent the real number line horizontally, and complete the number line using the same scale as on the right side of zero.
→ Describe the y-axis. What types of numbers should it include?
The y-axis is a vertical number line that includes numbers on both sides of zero (above and below), and so it includes both positive and negative numbers.
→ Use a straightedge to draw a vertical number line above zero.

Provide students with time to draw.
→ Extend the y-axis below zero to represent the real number line vertically, and complete the number line using
the same scale as above zero.

Example 2: Components of the Coordinate Plane
All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?
The origin is the point where the x- and y-axes intersect. The coordinates of the origin are (0, 0).

To describe locations of points in the coordinate plane, we use _________________________ of numbers. Order is important, so on the coordinate plane, we use the form ( ). The first coordinate represents the point’s location from zero on the ________-axis, and the second coordinate represents the point’s location from zero on the ________-axis.
To describe locations of points in the coordinate plane, we use   ordered   pairs of numbers. Order is important, so on the coordinate plane, we use the form (   x, y   ). The first coordinate represents the point’s location from zero on the    x   -axis, and the second coordinate represents the point’s location from zero on the   y   -axis.

### Eureka Math Grade 6 Module 3 Lesson 15 Exercise Answer Key

Exercise 1.
Use the coordinate plane below to answer parts (a) – (c).
a. Graph at least five points on the x-axis, and label their coordinates.
Points will vary.

b. What do the coordinates of your points have in common?
Each point has a y-coordinate of 0.

c. What must be true about any point that lies on the x-axis? Explain.
If a point lies on the x-axis, its y-coordinate must be 0 because the point is located 0 units above or below the x-axis. The x-axis intersects the y-axis at 0.

Exercise 2.
Use the coordinate plane to answer parts (a) – (c).
a. Graph at least five points on the y-axis, and label their coordinates.
Points will wary.

b. What do the coordinates of your points have in common?
Each point has an x-coordinate of 0.

c. What must be true about any point that lies on the y-axis? Explain.
If a point lies on the y-axis, its x-coordinate must be 0 because the point is located 0 units left or right of the y-axis. The y-axis intersects 0 on the x-axis.

Exercise 3.
If the origin is the only point with 0 for both coordinates, what must be true about the origin?
The origin is the only point that is on both the x-axis and the y-axis.

Exercise 4.
Locate and label each point described by the ordered pairs below. Indicate which of the quadrants the points lie in.
a. (7, 2)

b. (3, – 4)

c. (1, – 5)

d. (- 3, 8)

e. (- 2, – 1)

Exercise 5.
Write the coordinates of at least one other point in each of the four quadrants.
Answers will vary, but both numbers must be positive.

Answers will vary, but the x-coordinate must be negative, and the y-coordinate must be positive.

Answers will vary, but both numbers must be negative.

Answers will vary, but the x-coordinate must be positive, and the y-coordinate must be negative.

Exercise 6.
Do you see any similarities in the points within each quadrant? Explain your reasoning.
The ordered pairs describing the points in Quadrant I contain both positive values. The ordered pairs describing the points in Quadrant III contain both negative values. The first coordinates of the ordered pairs describing the points in Quadrant II are negative values, but their second coordinates are positive values. The first coordinates of the ordered pairs describing the points in Quadrant IV are positive values, but their second coordinates are negative values.

### Eureka Math Grade 6 Module 3 Lesson 15 Problem Set Answer Key

Question 1.
Name the quadrant in which each of the points lies. If the point does not lie in a quadrant, specify which axis the point lies on.
a. (- 2, 5)

b. (8, – 4)

c. (- 1, – 8)

d. (9. 2, 7)

e. (0, – 4)
None; the point is not in a quadrant because it lies on the y-axis.

Question 2.
Jackie claims that points with the same x- and y-coordinates must lie in Quadrant I or Quadrant Ill. Do you agree or disagree? Explain your answer.
Disagree; most points with the same x- and y-coordinates lie in Quadrant I or Quadrant III, but the origin (o, 0) is on the x- and y-axes, not in any quadrant.

Question 3.
Locate and label each set of points on the coordinate plane. Describe similarities of the ordered pairs in each set, and describe the points on the plane.
a. {(- 2, 5), (- 2, 2), (- 2, 7), (- 2, – 3), (- 2, – 0. 8))
The ordered pairs all have x-coordinates of – 2, and the points lie along a vertical line above and below – 2 on the x-axis.

b. {(- 9, 9), (- 4, 4), (- 2, 2), (1, – 1), (3, – 3), (0, 0)}
The ordered pairs each have opposite values for their x- and y-coordinates. The points in the plane line up diagonally through Quadrant II, the origin, and Quadrant IV.

c. {(- 7, – 8), (5, – 8), (0, – 8), (10, – 8), (- 3, – 8)}
The ordered pairs all have y-coordinates of – 8, and the points lie along a horizontal line to the left and right of – 8 on the y-axis.

Question 4.
Locate and label at least five points on the coordinate plane that have an x-coordinate of 6.
a. What is true of the y-coordinates below the x-axis?
The y-coordinates are all negative values.

b. What is true of the y-coordinates above the x-axis?
The y-coordinates are all positive values.

c. What must be true of the y-coordinates on the x-axis?
The y-coordinates on the x-axis must be 0.

### Eureka Math Grade 6 Module 3 Lesson 15 Exit Ticket Answer Key

Question 1.
Label the second quadrant on the coordinate plane, and then answer the following questions:
a. Write the coordinates of one point that lies in the second quadrant of the coordinate plane.

b. What must be true about the coordinates of any point that lies in the second quadrant?
The x-coordinate must be a negative value, and the y-coordinate must be a positive value.

Question 2.
Label the third quadrant on the coordinate plane, and then answer the following questions:
a. Write the coordinates of one point that lies in the third quadrant of the coordinate plane.

b. What must be true about the coordinates of any point that lies in the third quadrant?
The x- and y-coordinates of any point in the third quadrant must both be negative values.

Question 3.
An ordered pair has coordinates that have the same sign. In which quadrant(s) could the point lie? Explain.
The point would have to be located either in Quadrant I where both coordinates are positive values or in Quadrant III where both coordinates are negative values.

Question 4.
Another ordered pair has coordinates that are opposites. In which quadrant(s) could the point lie? Explain.
The point would have to be located in either Quadrant II or Quadrant IV because those are the two quadrants where the coordinates have opposite signs. The point could also be located at the origin (0, 0) since zero is its own opposite.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 14 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 14 Example Answer Key

Example 1: The Order in Ordered Pairs
The first number of an ordered pair is called the ___________ .
first coordinate.

The second number of an ordered pair is called the ___________ .
second coordinate.

Example 2.
Using Ordered Pairs to Name Locations
Describe how the ordered pair is being used in your scenario. Indicate what defines the first coordinate and what defines the second coordinate in your scenario.
Ordered pairs are like a set of directions; they indicate where to go in one direction and then indicate where to go in the second direction.
→ Scenario 1: The seats in a college football stadium are arranged into 210 sections, with 144 seats in each
section. Your ticket to the game indicates the location of your seat using the ordered pair of numbers
(123,37). Describe the meaning of each number in the ordered pair and how you would use them to find your seat.

→ Scenario 2: Airline pilots use measurements of longitude and latitude to determine their location and to find airports around the world. Longitude is measured as 0 – 180° east or 0 – 180° west of a line stretching from the North Pole to the South Pole through Greenwich, England, called the prime meridian. Latitude is measured as 0 – 90° north or 0 – 90° south of the earth’s equator. A pilot has the ordered pair (90° west, 30° north). What does each number in the ordered pair describe? How would the pilot locate the airport on a map? Would there be any confusion if a pilot were given the ordered pair (90°, 30°)? Explain.

→ Scenario 3: Each room in a school building is named by an ordered pair of numbers that indicates the number of the floor on which the room lies, followed by the sequential number of the room on the floor from the main staircase. A new student at the school is trying to get to science class in room 4 – 13. Describe to the student what each number means and how she should use the number to find her classroom. Suppose there are classrooms below the main floor. How might these rooms be described?

### Eureka Math Grade 6 Module 3 Lesson 14 Exercise Answer Key

The first coordinates of the ordered pairs represent the numbers on the line labeled X, and the second coordinates represent the numbers on the line labeled y.

Exercise 1.
Name the letter from the grid below that corresponds with each ordered pair of numbers below.

a. (1, 4)
Point F

b. (0, 5)
Point A

c. (4, 1)
Point B

d. (8.5, 8)
Point L

e. (5, – 2)
Point G

f. (5, 4.2)
Point H

g. (2,- 1)
Point C

h. (0, 9)
Point E

Exercise 2.
List the ordered pair of numbers that corresponds with each letter from the grid below.

a. Point M
(5, 7)

b. Point S
(- 2, 3)

c. Point N
(6, 0)

d. Point T
(- 3, 2)

e. Point P
(0, 6)

f. Point U
(7, 5)

g. Point Q
(2, 3)

h. Point V
(- 1, 6)

I. Point R
(0, 3)

### Eureka Math Grade 6 Module 3 Lesson 14 Problem Set Answer Key

Question 1.
Use the set of ordered pairs below to answer each question.
{(4, 20), (8, 4), (2, 3), (15, 3), (6, 15), (6, 30), (1, 5), (6, 18), (0, 3)}

a. Write the ordered pair(s) whose first and second coordinate have a greatest common factor of 3.
(15, 3) and (6, 15)

b. Write the ordered pair(s) whose first coordinate is a factor of its second coordinate.
(4, 20), (6, 30), (1, 5), and (6, 18)

c. Write the ordered pair(s) whose second coordinate is a prime number.
(2, 3), (15, 3), (1, 5), and (0, 3)

Question 2.
Write ordered pairs that represent the location of points A, B, C, and D, where the first coordinate represents the horizontal direction, and the second coordinate represents the vertical direction.

A (4, 1); B (1, – 3); C (6, 0); D (1, 4)

Extension:

Question 3.
Write ordered pairs of integers that satisfy the criteria in each part below. Remember that the origin is the point whose coordinates are (0, 0). When possible, give ordered pairs such that (i) both coordinates are positive, (ii) both coordinates are negative, and (iii) the coordinates have opposite signs in either order.

a. These points’ vertical distance from the origin Is twice their horizontal distance.
Answers will vary; examples are (5, 10), (- 2, 4), (- 5, – 10), (2, – 4).

b. These points’ horizontal distance from the origin Is two units more than the vertical distance.
Answers will vary; examples are (3, 1), (- 3, 1), (- 3,- 1), (3, – 1).

c. These points’ horizontal and vertical distances from the origin are equal, but only one coordinate is positive.
Answers will vary; examples are (3, – 3), (- 8,8).

### Eureka Math Grade 6 Module 3 Lesson 14 Exit Ticket Answer Key

Question 1.
On the map below, the fire department and the hospital have one matching coordinate. Determine the proper order of the ordered pairs In the map, and write the correct ordered pairs for the locations of the fire department and hospital. Indicate which of their coordinates are the same.
The order of the numbers is (x, y); fire department: (6, 7) and hospital: (10, 7); they have the same second coordinate.

Question 2.
On the map above, locate and label the location of each description below:
a. The local bank has the same first coordinate as the fire department, but its second coordinate is half of the fire department’s second coordinate. What ordered pair describes the location of the bank? Locate and label the bank on the map using point B.
(6, 3. 5); see the map image for the correct location of point B.

b. The Village Police Department has the same second coordinate as the bank, but its first coordinate is – 2. What ordered pair describes the location of the Village Police Department? Locate and label the Village Police Department on the map using point P.
(- 2, 3.5); see the map image for the correct location of point P.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 13 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 13 Example Answer Key

Example 1. Ordering Numbers In the Real World
A $25 credit and a$25 charge appear similar, yet they are very different.
Describe what is similar about the two transactions.
The transactions look similar because they are described using the same number. Both transactions have the same magnitude (or absolute value) and, therefore, result in a change of $25 to an account balance. How do the two transactions differ? Answer: The credit would cause an increase to an account balance and, therefore, should be represented by 25, while the charge would instead decrease an account balance and should be represented by – 25. The two transactions represent changes that are opposites. Example 2: Using Absolute Value to Solve Real-World Problems The captain of a fishing vessel is standing on the deck at 23 feet above sea level. He holds a rope tied to his fishing net that is below him underwater at a depth of 38 feet. Draw a diagram using a number line, and then use absolute value to compare the lengths of rope in and out of the water. Answer: The captain is above the water, and the fishing net is below the water’s surface. Using the water level as reference point zero, I can draw the diagram using a vertical number line. The captain is located at 23, and the fishing net is located at – 38. |23| = 23 and |- 38| = 38,so there is more rope underwater than above. 38 – 23 = 15 The length of rope below the water’s surface is 15 feet longer than the rope above water. Example 3: Making Sense of Absolute Value and Statements of Inequality A recent television commercial asked viewers, “Do you have over$10,000 in credit card debt?”

What types of numbers are associated with the word debt, and why? Write a number that represents the value from the television commercial.
Negative numbers; debt describes money that is owed; – 10,000

Give one example of “over $10,000 in credit card debt.” Then, write a rational number that represents your example. Answer: Answers will vary, but the number should have a value of less than – 10,000. Credit card debt of$11,000; – 11,000

How do the debts compare, and how do the rational numbers that describe them compare? Explain.
The example $11, 000 is greater than$10, 000 from the commercial; however, the rational numbers that represent these debt values have the opposite order because they are negative numbers. – 11,000 < – 10, 000. The absolute values of negative numbers have the opposite order of the negative values themselves.

### Eureka Math Grade 6 Module 3 Lesson 13 Exercise Answer Key

Exercise 1.
Scientists are studying temperatures and weather patterns in the Northern Hemisphere. They recorded
temperatures (in degrees Celsius) in the table below as reported in emails from various participants. Represent each reported temperature using a rational number. Order the rational numbers from least to greatest. Explain why the rational numbers that you chose appropriately represent the given temperatures.

– 13 < – 8 < – 6 < – 5 < – 4 < 0 < 2 < 12
The words “below zero” refer to negative numbers because they are located below zero on a vertical number line.

Exercise 2.
Jami’s bank account statement shows the transactions below. Represent each transaction as a rational number describing how it changes Jami’s account balance. Then, order the rational numbers from greatest to least. Explain why the rational numbers that you chose appropriately reflect the given transactions.

5.5 > 4.08 > – 1.5 > – 3 > – 3.95 > – 12.2 > – 20
The words “debit,” “charge,” and “withdrawal” all describe transactions in which money is taken out of Jami’s account, decreasing its balance. These transactions are represented by negative numbers. The words “credit” and “deposit” describe transactions that will put money into Jami’s account, increasing its balance. These transactions are represented by positive numbers.

Exercise 3.
During the summer, Madison monitors the water level in her parents’ swimming pool to make sure it is not too far above or below normal. The table below shows the numbers she recorded In July and August to represent how the water levels compare to normal. Order the rational numbers from least to greatest. Explain why the rational numbers that you chose appropriately reflect the given water levels.

$$-1 \frac{1}{4}<-\frac{3}{4}<-\frac{1}{2}<-\frac{3}{8}<\frac{1}{8}<\frac{1}{4}<\frac{1}{2}$$
The measurements are taken in reference to normal level, which is considered to be 0. The words “above normal” refer to the positive numbers located above zero on a vertical number line, and the words “below normal” refer to the negative numbers located below zero on a vertical number line.

Exercise 4.
Changes in the weather can be predicted by changes in the barometric pressure. Over several weeks, Stephanie recorded changes in barometric pressure seen on her barometer to compare to local weather forecasts. Her observations are recorded In the table below. Use rational numbers to record the indicated changes In the pressure in the second row of the table. Order the rational numbers from least to greatest. Explain why the rational numbers that you chose appropriately represent the given pressure changes.

### Eureka Math Grade 6 Module 3 Lesson 13 Problem Set Answer Key

Question 1.
Negative air pressure created by an air pump makes a vacuum cleaner able to collect air and dirt into a bag or other container. Below are several readings from a pressure gauge. Write rational numbers to represent each of the readings, and then order the rational numbers from least to greatest.

– 13 < – 7.8 < – 6.3 < – 1.9 < 2 < 7.8 < 25

Question 2.
The fuel gauge in Nic’s car says that he has 26 miles to go until his tank is empty. He passed a fuel station 19 miles ago, and a sign says there is a town only 8 miles ahead. If he takes a chance and drives ahead to the town and there isn’t a fuel station there, does he have enough fuel to go back to the last station? Include a diagram along a number line, and use absolute value to find your answer.
No, he does not have enough fuel to drive to the town and then back to the fuel station. He needs 8 miles’ worth of fuel to get to the town, which lowers his limit to 18 miles. The total distance between the fuel station and the town is 27 miles; |8| + |- 19| = 8 + 19 = 27. Nic would be9miles short on fuel. It would be safer to go back to the fuel station without going to the town first.

### Eureka Math Grade 6 Module 3 Lesson 13 Exit Ticket Answer Key

Question 1.
Loni and Daryl call each other from different sides of Watertown. Their locations are shown on the number line below using miles. Use absolute value to explain who is a farther distance (in miles) from Watertown. How much closer is one than the other?

Loni’s location is – 6, and |- 6| = 6 because – 6 is 6 units from 0 on the number line. Daryl’s location is 10, and |10| = 10 because 10 is 10 units from 0 on the number line. We know that 10 > 6, so Daryl is farther from Watertown than Loni.
10 – 6 = 4; Loni is 4 miles closer to Watertown than Daryl.

Question 2.
Claude recently read that no one has ever scuba dived more than 330 meters below sea level. Describe what this means in terms of elevation using sea level as a reference point.
330 meters below sea level is an elevation of – 330 feet. “More than 330 meters below sea level” means that no diver has ever had more than 330 meters between himself and sea level when he was below the water’s surface while scuba diving.

### Eureka Math Grade 6 Module 3 Lesson 13 Opening Exercise Answer Key

Question 1.
A radio disc jockey reports that the temperature outside his studio has changed 10 degrees since he came on the air this morning. Discuss with your group what listeners can conclude from this report.
The report is not specific enough to be conclusive because 10 degrees of change could mean an increase or a decrease in temperature. A listener might assume the report says an increase in temperature; however, the word “changed” is not specific enough to conclude a positive or negative change.

## Engage NY Eureka Math 6th Grade Module 3 Lesson 12 Answer Key

### Eureka Math Grade 6 Module 3 Lesson 12 Example Answer Key

Example 1: Comparing Order of Integers to the Order of Their Absolute Values
Write an Inequality statement relating the ordered integers from the Opening Exercise. Below each integer, write its absolute value.

Circle the absolute values that are in increasing numerical order and their corresponding integers. Describe the circled values.

The circled integers are all positive values except zero. The positive integers and their absolute values have the same order.

Rewrite the integers that are not circled in the space below. How do these integers differ from the ones you circled?
– 12, – 9, – 5, – 2, – 1
They are all negative integers.

Rewrite the negative integers in ascending order and their absolute values in ascending order below them.

Describe how the order of the absolute values compares to the order of the negative integers.
The orders of the negative integers and their corresponding absolute values are opposite.

Example 2: The Order of Negative Integers and Their Absolute Values
Draw arrows starting at the dashed line (zero) to represent each of the integers shown on the number line below. The arrows that correspond with 1 and 2 have been modeled for you.

As you approach zero from the left on the number line, the integers ___________, but the absolute values of those integers ___________. This means that the order of negative integers is ___________ the order of their absolute values.
As you approach zero from the left on the number line, the integers   increase   , but the absolute values of those integers    decrease  . This means that the order of negative integers is   opposite   the order of their absolute values.

### Eureka Math Grade 6 Module 3 Lesson 12 Exercise Answer Key

Complete the steps below to order these numbers:

{2.1, – 4$$\frac{1}{2}$$, – 6. 0.25, – 1.5, 0, 3.9, – 6.3, – 4, 2$$\frac{3}{4}$$, 3.99, – 9$$\frac{1}{4}$$}

a. Separate the set of numbers into positive rational numbers, negative rational numbers, and zero in the top cells below (order does not matter).

b. Write the absolute values of the rational numbers (order does not matter) in the bottom cells below.

c. Order each subset of absolute values from least to greatest.

d. Order each subset of rational numbers from least to greatest.

e. Order the whole given set of rational numbers from least to greatest.
– 9$$\frac{1}{4,}$$, – 6.3, – 6, – 4$$\frac{1}{2}$$, – 4, – 1.5, 0, 0.25, 2.1, 2$$\frac{3}{4}$$, 3.9, 3.99

Exercise 2.
a. Find a set of four integers such that their order and the order of their absolute values are the same.
Answers will vary. An example follows: 4, 6, 8, 10

b. Find a set of four integers such that their order and the order of their absolute values are opposite.
Answers will vary. An example follows: – 10, – 8, – 6, – 4

c. Find a set of four non-integer rational numbers such that their order and the order of their absolute values are the same.
Answers will vary. An example follows: 2$$\frac{1}{2}$$, 3$$\frac{1}{2}$$, 4$$\frac{1}{2}$$, 5$$\frac{1}{2}$$

d. Find a set of four non-integer rational numbers such that their order and the order of their absolute values are opposite.
Answers will vary. An example follows: – 5$$\frac{1}{2}$$, – 4$$\frac{1}{2}$$, – 3$$\frac{1}{2}$$, – 2$$\frac{1}{2}$$

e. Order all of your numbers from parts (a) – (d) in the space below. This means you should be ordering 16 numbers from least to greatest.
Answers will vary. An example follows:
– 10, – 8, – 6, – 5$$\frac{1}{2}$$, – 4$$\frac{1}{2}$$, – 4, – 3$$\frac{1}{2}$$, – 2$$\frac{1}{2}$$, 2$$\frac{1}{2}$$, 3$$\frac{1}{2}$$, 4, 4$$\frac{1}{2}$$, 5$$\frac{1}{2}$$, 6, 8, 10

### Eureka Math Grade 6 Module 3 Lesson 12 Problem Set Answer Key

Question 1.
Micah and Joel each have a set of five rational numbers. Although their sets are not the same, their sets of numbers have absolute values that are the same. Show an example of what Micah and Joel could have for numbers. Give the sets in order and the absolute values in order.
Examples may vary, If Micah had 1, 2, 3, 4, 5, then his order of absolute values would be the same: 1, 2, 3, 4, 5. If Joel had the numbers – 5, – 4, – 3, -2, -1, then his order of absolute values would also be 1, 2, 3, 4, 5.

Enrichment Extension: Show an example where Micah and Joel both have positive and negative numbers.
If Micah had the numbers: – 5, – 3, – 1, 2, 4, his order of absolute values would be 1, 2, 3, 4, 5. If Joel hod the numbers – 4, – 2, 1, 3, 5, then the order of his absolute values would also be 1, 2, 3, 4, 5.

Question 2.
For each pair of rational numbers below, place each number In the Venn diagram based on how it compares to the other.
a. – 4, – 8
b. 4,8
C. 7,- 3
d. – 9,2
e. 6,1
f. – 5,5
g. – 2,0

### Eureka Math Grade 6 Module 3 Lesson 12 Exit Ticket Answer Key

Question 1.
Bethany writes a set of rational numbers in increasing order. Her teacher asks her to write the absolute values of these numbers in Increasing order. When her teacher checks Bethany’s work, she Is pleased to see that Bethany has not changed the order of her numbers. Why is this?
All of Bethany’s rational numbers are positive or 0. The positive rational numbers have the same order as their absolute values. If any of Bethany’s rational numbers are negative, then the order would be different.

Question 2.
Mason was ordering the following rational numbers In math class: – 3. 3, – 15, – 8$$\frac{8}{9}$$.
a. Order the numbers from least to greatest.
– 15, – 8$$\frac{8}{9}$$, – 3.3

b. List the order of their absolute values from least to greatest.
3.3, 8$$\frac{8}{9}$$, 15