RATIONAL NUMBERS - RUSD Math - Homerusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u1_-_answer_key… · Grade 6 Unit 1: Rational Numbers LESSON 5: ... Copyright © 2015 Pearson Education,

ANSWERS FOR EXERCISES

MATH GRADE 6 UNIT 1

RATIONAL NUMBERS

Copyright © 2015 Pearson Education, Inc. 42

Grade 6 Unit 1: Rational Numbers

ANSWERSLESSON 2: THE OPPOSITE OF A NUMBER

ANSWERS

6.NS.6.a 1. a. 21

b. –7.45

6.NS.6.a 2. 0

6.NS.6.a 3. B

6.NS.6.a 4. C

6.NS.6.a 5. A 33

6.NS.6.a 6.NS.6.c

6.

6.NS.6.a 6.NS.6.c

7.

6.NS.6.a 6.NS.6.c

8.

6.NS.6.a 6.NS.6.c

9.

0

–10

–20

–30

10

20

16

–16

30

58

–58

7–7 0

–4 4

7–7 0

–5 5

7–7 0

–1 1



ANSWERSLESSON 2: THE OPPOSITE OF A NUMBER

6.NS.6 10.

6.NS.6 11. The number 0 is rational because , where a = 0 and b ≠ 0.

6.NS.6.a 6.NS.6.c

12.

6.NS.6.a 6.NS.6.c

13.

6.NS.6.a 6.NS.6.c

14.

Challenge Problem

6.NS.6.a 6.NS.7.c

15. The distance between any number x and its opposite is 2|x|.

Negative Number

Integer Rational Number

None of These

a. 2 13

b.60

c. 8.2

d. –9.5

e. –9

ab= 0

0–1–2–3 1 2 3

2.5–2.5

0 0.02 0.04 0.06–0.02–0.04–0.06

–0.01 0.01

0–1–2–3 1 2 3

3 17–3 1

7



ANSWERSLESSON 3: ABSOLUTE VALUE

ANSWERS

6.NS.7.c 1. B 3

6.NS.7.c 2. 0

6.NS.7.c 3. or

6.NS.7.c 4. 4.3

6.NS.5 5. C January

6.NS.7.c 6. C 0.09

6.NS.7.c 7. 7.5

6.NS.7.c 8. or

6.NS.5 6.NS.7.b 6.NS.7.d

9. a. The highest boiling point was 111°C.

b. The lowest boiling point was 105°C.

c. The difference between the highest and lowest boiling points is 6°.

6.NS.5 6.NS.7.b 6.NS.7.d

10. The distance between the highest and lowest freezing points is 8°.

First, I figured out the highest and lowest freezing points recorded by the class. The highest freezing point is 4° higher than 0°C, or 4°C. The lowest freezing point is 4° lower then 0°C, or –4°C.

Next, I created a number line to show where the two numbers are located. Then I counted the number of jumps from 4°C to –4°C.

(continues)

5 13

163

3 12

72



ANSWERSLESSON 3: ABSOLUTE VALUE

6.NS.5 6.NS.7.b 6.NS.7.d

10. (continued)

0

–1

–2

–3

1

2

3

44

–4 –4

Challenge Problem

6.NS.7.c 11. In general, there are two values of a that have the same absolute value: a and its opposite. The only exception is a = 0, since 0 is its own opposite.



ANSWERSLESSON 4: OPPOSITE AND ABSOLUTE VALUE

ANSWERS

6.NS.7.c 1. B –8

6.NS.6.a 6.NS.7.c

2. Number Opposite Absolute Value

3

4

0

–11

–8

–3 3

–4 4

0 0

11 11

8 8

6.NS.7.d 3. B 4

6.NS.7.d 4. D 0

6.NS.6.a 5. A 55

6.NS.7.c 6. 12 and –12

6.NS.7.c 7. –15

6.NS.7.c 8. The answer is –22

6.NS.6.a 6.NS.7.c

9.A number and its opposite

are both positive. Never true

The absolute value of a number is never negative.

Always true

The absolute value of a number is 0.

Sometimes true

The opposite of a number’s absolute value is greater

than 0. Never true



ANSWERSLESSON 4: OPPOSITE AND ABSOLUTE VALUE

Challenge Problem

6.NS.6.a 6.NS.7.c

10. Emma is incorrect, because the opposite of any negative number is equal to its absolute value. Zero is, therefore, the greatest number with an opposite that is equal to its absolute value, since zero is its own opposite.



ANSWERSLESSON 5: ORDERING AND COMPARING

ANSWERS

6.NS.7.a 1. a. 2 > –3

b. 2 < 5

c. –4 > –6

d. –7 < 1

e. –2< 0

6.NS.7.a 2. C –12, –1.20, 1.02, 12, 120

6.NS.7.a 3. The order of the dives from deepest to shallowest is third dive, second dive, fourth dive, and first dive.

–90, –60.5, –45, –30

6.NS.7.a 4. –50.5 > –50.7

6.NS.7.a 6.NS.7.c

5. B Mia

6.NS.7.a 6. B

6.NS.7.a A.REI.10

7. D –0.6 > –6

6.NS.7.b 8. Carlos won the tournament: Carlos (–5), Jan (–4), Emma (–3), Jason (–1), Denzel (1), Mia (6).

6.NS.7.b 9. The score with the greatest opposite wins. The absolute value of Mia’s score is greater than the absolute value of Carlos’s score.

Explanations will vary. Here is one possible answer.

Golfer Score Opposite Absolute Value

Jason –1 1 1Emma –3 3 3Denzel 1 –1 1Carlos –5 5 5

Mia 6 –6 6Jan –4 4 4

− < −414



ANSWERSLESSON 5: ORDERING AND COMPARING

6.NS.7.b 10. D 2007: –75° south latitude

Challenge Problem

6.NS.7.a 6.EE.2

11. a. Inequalities will vary. However, the resulting inequality will be true, whether a is negative or positive.

b. The resulting inequality is true if a is positive, but false if a is negative.



ANSWERSLESSON 6: PUTTING IT TOGETHER

ANSWERS

6.NS.5 2. Negative Numbers in Everyday Life

Elevations of locations below sea level

Temperatures below zero degrees

Diving under water

Depths of roots below the soil

Golf scores

Latitudes south of the equator, longitudes west of the prime meridian

Money that you owe (debt)

An overdrawn bank account balance

Discounts on items for sale

In video games: loss of life, damage, penalty, or using up a resource

Negative statistics in sports (e.g., errors in baseball, technical fouls in basketball, face-offs lost in hockey)

Race times in sports competitions (e.g., downhill skiing, swimming, and running) showing the current racer compared to the leader (negative if the current racer completes the race faster than the leader)

Lap times in Formula 1 racing—the difference between the previous lap and the lap just completed (negative if the lap just completed was faster than the previous lap)

Penalty minutes in hockey

Differences in time of day between time zones




6.NS.5 6.NS.7

3. Definitions and examples will vary. Here are some examples.

Word or Phrase Definition Examples

integer A whole number Can be positive, negative, or zero

1 –5,411 0 256 –8

positive number

A number that is greater than zero

10 6.5 112

Numbers to the right of zero on the number line

–10 0 10 20 30

negative number

A number that is less than zero

Written with a minus sign in front of the number

–23 –0.24 −14

Numbers to the left of zero on the number line

–30 –20 10 0 10

opposite of a number

The opposite of a number is the same distance away from zero but on the opposite side of the number line.

The opposite of the opposite of a number is the number itself.

Number: Opposite: 3 –3 0 0 –6 6 –2.6 2.6

12

−12

–20

opposite –15 15 number

–10 0 10 20

distance to zero = 15

distance to zero = 15

absolute value

The distance a number is from zero on a number line

Absolute value is always either 0 or positive.

Number: Absolute Value: 4 4 –4 4 0 0 –5.2 5.2

−

12

12

Written as a number inside two lines: |–3|



ANSWERSLESSON 9: THE COORDINATE PLANE

ANSWERS

6.NS.6.b 6.NS.6.c

1.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.6.b 6.NS.6.c

2.

2

2

6

8

A

D

C

EB

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.6.c 3. D (–3, 2)

6.NS.6.c 4. C (–2, 2)

6.NS.6.c 5. B (2, –2)

6.NS.6.c 6.

2

2

4

6

8

10

NORTH

SOUTH

WES

T

EAST

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10



ANSWERSLESSON 9: THE COORDINATE PLANE

6.NS.6.c 7. C Quadrant III

6.NS.6.c 8. D

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y

6.NS.6.c 9. C

2

4

–6

–4

–2642–2–6 –4 x

y

Challenge Problem

6.NS.6.b 10.Sign (x-coordinate) Sign (y-coordinate) Quadrant

++ I

– IV

–+ II

– III



ANSWERSLESSON 10: DRAWING FIGURES

ANSWERS

6.NS.8 1. B Point B

6.NS.6.b 2. A Point A and C

6.NS.6.b 3. A Point A

6.NS.6.c 6.NS.8

4. The distance between point A and point D is 7 units.

Strategies will vary. Here are two possible answers.

Example 1:

2

6

4

–6

–4

–2642–2–6 –4 x

y

Points A and D share the same y-value and are on opposite sides of the y-axis. It’s as if they are on a horizontal number line on opposite sides of zero. The distance between the two points is the distance between their x-coordinates, 7 units.

Example 2:

2

6

4

–6

–4

–2642–2–6

A

absolutevalue = 3

absolutevalue = 4

D

–4 x

y

I plotted points A and D in the coordinate plane and then found the distance to zero, which is the absolute value. For point A, the distance is 3 units. For point D, the distance is 4 units.

The x-coordinate of point A is –3. | –3 | = 3 units The x-coordinate of point B is 4. | 4 | = 4 units

Then, I added the two absolute values together to find the total distance of 7 units.




6.NS.6.c 6.NS.8

5.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.6.c 6.NS.8 6.G.3

6.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.8 7. A 10 units

6.NS.6.c 6.NS.8

8.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10




6.NS.6.c 6.NS.8 6.G.3

9. Strategies will vary. Here is one possible answer.

The two given vertices form the end points of the diagonal of the square: (3, 1) and (3, –4). Because they have the same first coordinate, I can picture them as being on a vertical number line with one point at 1 and the other point at –4. The distance between them is the length of the diagonal.

The distance from 1 to 0 is its absolute value: 1.

The distance from –4 to 0 is its absolute value: 4.

Adding the two absolute values together, I find the length of the diagonal.

1 + 4 = 5

The diagonal is 5 units long.

1

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

2 3 4 5 6 x

y

–1–2–3–4–5–6

0

–1

–2

–3

–4

1

2

1

1

4

3

2

1

–4

3




6.NS.6.c 6.NS.8

10. The hospital is 9 blocks east of police station 1 and 9 blocks west of police station 2.

Strategies will vary. Here is one possible answer.

The two police stations share the same y-value as the hospital. The distance between them is the distance along the x-axis.

From police station 1 (–10, –6) to police station 2 (8, –6), I counted 18 units. Half of 18 is 9. So, the hospital is 9 units from each of the police stations.

From police station 2 (8, –6) to the hospital, I counted 9 units to the left (west) on the x-axis and arrived at (–1, –6).

From police station 1 (–10, –6) to the hospital, I counted 9 units to the right (east) on the x-axis and arrived at (–1, –6).

The hospital is at (–1, –6).

Challenge Problem

6.NS.6.c 6.NS.8 6.G.3

11. a. The x-coordinate must be 0, and the y-coordinate can be any number except 0.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

(continues)

2

2

4

6

8

10

NORTH

SOUTH

WES

T

EAST

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

PoliceStation 1

9 9

Hospital

PoliceStation 2




6.NS.6.c 6.NS.8 6.G.3

11. (continued)

b. The x-coordinate must be either 4 or –4, and the y-coordinate can be any number except 0.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

c. The x-coordinate is either less than –4 or greater than 4, and the y-coordinate can be any number except 0.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10



ANSWERSLESSON 11: CREATING MIRROR IMAGES

ANSWERS

6.NS.6.b 1. A

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10

x

y

–2–4–6–8–10

6.NS.6.b 2. B

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.6.b 6.NS.6.c 6.NS.8

3. a. (–5, 6), (–9, 4), (–7, 11)

b. Reflecting a point across the y-axis results in the opposite of the x-coordinate. The y-coordinate remains the same.

6.NS.6.b 6.NS.6.c 6.NS.8

4.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10




6.NS.6.b 6.NS.6.c 6.NS.8

5.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

6.NS.6.b 6.NS.6.c 6.NS.8 6.G.3

6. Ask a classmate to check your work.

Here is one possible solution.

a.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–4–6

6 units

6 units

6 units

–8–10

I made a quick sketch. I counted and found that the given points, (1, 5) and (1, –1), are 6 units apart.

The other sides of the square need to be the same length. From (1, 5), I counted 6 units to the left in the coordinate plane to find point (–5, 5). That means the other point is at (–5, –1).

Points: (–5, 5) and (–5, –1)

b. The line of reflection is x = –2. All four points are the same distance (3 units) from the line of reflection.




6.NS.6.c 6.NS.8 6.G.3

7.

2

4

–6

–4

–242–2–6 –4 x

y

6

6

66 44–

Challenge Problem

6.NS.6.b 6.NS.6.c 6.NS.8 6.G.3

8. Ask a classmate to check your work.




ANSWERS

6.NS.6.b 6.NS.6.c 6.NS.8

3. Definitions and examples will vary. Here are some examples.

Word or Phrase Definition Examples

non-integer A number that is not a whole number

Can be positive or negative

1.5 256.78 18

–54.1 –8 13

coordinate plane

A plane made of a horizontal number line (x-axis) and a vertical number line (y-axis) that cross at the coordinates (0, 0)

also called the Cartesian plane

coordinates A set of two numbers that show where a point is located in the coordinate plane: (x, y)

To plot coordinates in the coordinate plane:

1. Use the first number (x-coordinate) to move right or left along the horizontal number line (x-axis). 2. Use the second number (y-coordinate) to move up or down along the vertical number line.Examples: (3, 5) is located at x = 3 and y = 5. (–4, –2) is located at x = –4 and y = –2.

1 2 3 4–5 –4 –3 –2 –1–1

1

2

3

4

5

–2

–3

–4

–5

5 x

y

origin(0, 0)

Iquadrant

horizontal axis

vert

ical

axi

s

IIquadrant

IVquadrant

IIIquadrant

1 2 3 4–5 –4 –3 –2 –1–1

1

2

3

4

5

–2

–3

–4

–5

5 x

y

(–4, –2)

(3, 5)

Documents

RATIONAL NUMBERS - RUSD Math - Homerusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u1_-_answer_key… · Grade 6 Unit 1: Rational Numbers LESSON 5: ... Copyright © 2015 Pearson Education,