lesson 1: above and below sea level...

Copyright © 2015 Pearson Education, Inc. 5

Grade 6 Unit 1: Rational Numbers

EXERCISESLESSON 1: ABOVE AND BELOW SEA LEVEL

EXERCISES

1. Write your wonderings about positive and negative numbers. Share your wonderings with a classmate.

2. Write a goal stating what you plan to accomplish in this unit.

3. Based on your previous work in math, write three things that you will do during this unit to increase your success.

For example, consider ways you will participate in classroom discussions, your study habits, how you will organize your time, what you will do when you have a question, and so on.



EXERCISESLESSON 2: THE OPPOSITE OF A NUMBER

EXERCISES

1. Find the opposite of each number.

a. –21b. 7.45

2. Find the opposite of the number.

0

3. What is the opposite of –58

?

A 85

B 58

C –58

D –85

4. What is the opposite of the opposite of –58

?

A 85

B 58

C –58

D –85

5. What is the opposite of the opposite of 33?

A 33

B 133

C 33%

D –33




6. Plot the number 4 and its opposite on the number line.

7–7 0

7. Plot the number –5 and its opposite on the number line.

7–7 0


7–7 0


0

–10

–20

–30

10

20

30




10. Indicate whether each number is a negative number, an integer, a rational number, or none of these. There may be more than one correct category for each number.

Negative Number

Integer Rational Number

None of These

a. 2 13

b.60

c. 8.2

d. –9.5

e. –9

11. Is 0 a rational number? Explain how you know.

12. Plot the number –2.5 and its opposite on the number line.

0–1–2–3 1 2 3

13. Plot the number 0.01 and its opposite on the number line.

0 0.02 0.04 0.06–0.02–0.04–0.06

14. Plot the number −3 17 and its opposite on the number line.

0–1–2–3 1 2 3

Challenge Problem

15. Find an expression that represents the distance between any number x and its opposite.



EXERCISESLESSON 3: ABSOLUTE VALUE

EXERCISES

1. What is the absolute value of –3?

A 0

B 3

C –0.3

D –3

2. What is the absolute value of 0?

3. What is the absolute value of 5 13 ?

4. What is the absolute value of –4.3?

5. Which month has the lowest average daily minimum temperature?Average Daily Minimum Temperature in Omsk, Russia

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Temperature (°C) –21.8 –21.3 –13.0 –1.3 5.4 11.5 13.9 10.9 5.6 –2.0 –11.2 –18.6

A November

B December

C January

D February

6. Which number is the correct evaluation of the expression?

|–0.09|

A –0.9

B 0.9

C 0.09

D –0.09



EXERCISES

7. Evaluate each expression.

| –7.5 |

8. Evaluate each expression.

3 1

2

9. The boiling points of a liquid in a science class experiment varied by an absolute value of 3° from the actual boiling point of 108°C.

a. What was the highest boiling point of the liquid?b. What was the lowest boiling point of the liquid?c. What is the difference between the highest and lowest boiling points of

the liquid?

10. The freezing point of a liquid in a science class experiment varied by an absolute value of 4° from the actual freezing point of 0°C.

What is the distance between the highest and lowest freezing points of the liquid? Explain how you know.

Challenge Problem

11. In general, how many values of a have the same absolute value |a|? Can you think of any exceptions to this rule? Explain.

LESSON 3: ABSOLUTE VALUE



EXERCISESLESSON 4: OPPOSITE AND ABSOLUTE VALUE

EXERCISES

1. Evaluate this expression.

–|–8|

A –9

B –8

C 0.2

D 8

2. Fill in the blanks in the table.

Number Opposite Absolute Value

3

4

0

–11

–8

3. Which number has the least absolute value? –4.12, 4, –15.3, –8.5

A –4.12

B 4

C –15.3

D –8.5

4. Which number has the least absolute value? –203, 0, 55, –1

A 55

B –203

C –1

D 0



EXERCISESLESSON 4: OPPOSITE AND ABSOLUTE VALUE

5. Which number has the least opposite of the number? –203, 0, 55, –1

A 55

B –203

C –1

D 0

6. Jan’s parents have a broken calculator that always reports the absolute value of a number.

The calculator’s display shows the answer is 12. What are the possible answers?

7. Jan’s parents have a broken calculator that always reports the absolute value of a number.

The calculator’s display shows the answer is 15. What is the answer if you know the number is negative?

8. Martin’s parents have a broken calculator that always reports the opposite of a number.

The calculator’s display shows the answer is 22. What are the possible answers?

9. Classify each statement as always true, sometimes true, or never true.

A number and its opposite are both positive.

The absolute value of a number is never negative.

The absolute value of a number is 0.

The opposite of a number’s absolute value is greater

than 0.

Challenge Problem

10. Emma is learning about absolute value in her math class. She reasons, “The only number with an opposite that is equal to its absolute value is 0.”

Do you agree with Emma? For which numbers, if any, is the absolute value equal to the opposite? Explain.



EXERCISES

1. Place the appropriate symbol (<, =, or >) between each pair of numbers.

a. 2 –3b. 2 5c. –4 –6d. –7 1e. –2 0

2. Consider these numbers.

12, –1.20, –12, 120, 1.02

Which of the following shows these numbers arranged from least to greatest?

A 120, –12, 12, –1.20, 1.02

B 1.02, –1.20, –12, 12, 120

C –12, –1.20, 1.02, 12, 120

D –1.20, 1.02, –12, 12, 120

3. Mia’s mother is a scuba diver. She wants to explore a shipwreck at a depth of 118 ft below sea level. To prepare, she made several practice dives. On the first dive, she went –30 ft; on the second dive, she went –60.5 ft; on the third dive, she went –90 ft; and on the fourth dive, she went –45 ft.

Arrange the dives in order from the deepest to the shallowest.

4. Place the appropriate symbol (<, =, or >) between the pair of numbers.

–50.5 –50.7

5. This table shows the amount of money each student has. A negative amount means the student owes money.

Who has the least amount of money?

A Martin

B Mia

C Denzel

D Emma

StudentMartin –$2.00

Mia –$2.50Denzel $2

$.00

Emma 0.00

Amount of Money

EXERCISESLESSON 5: ORDERING AND COMPARING




6. Which inequality is correct?

A 014

< −

B − < −414

C − < −14

4

D 0 4< −

7. Which inequality is correct?

A –0.6 > 0

B –0.6 < –6

C –0.6 > 0.6

D –0.6 > –6

8. In a miniature golf tournament, the player with the lowest score wins. Look at this table of scores.

Golfer ScoreJason –1Emma –3Denzel 1Carlos –5

Mia 6Jan –4

Arrange the golfers’ names and scores in order from first to sixth place. Who won the miniature golf tournament?

9. This is a table of scores from a miniature golf tournament. The player with the lowest score wins.

Golfer ScoreJason –1Emma –3Denzel 1Carlos –5

Mia 6Jan –4

Which score wins the tournament, the score with the greatest absolute value or with the greatest opposite? Explain.




10. Every winter, Mr. and Mrs. Valdez travel south for the holidays. In which year did they travel the farthest south?

A 2004: –30° south latitude

B 2005: –15° south latitude

C 2006: –22° south latitude

D 2007: –75° south latitude

Challenge Problem

11. Write any true inequality.

a. Add any number a to both sides of the inequality. What do you notice? Make sure you experiment with both positive and negative values of a.

b. Multiply both sides of your inequality by any number a. What do you notice? Make sure you experiment with both positive and negative values of a.



EXERCISESLESSON 6: PUTTING IT TOGETHER

EXERCISES

1. Read your Self Check and think about your work so far in this unit.

Write down three things you have learned during the unit.

Share your work with a classmate. Does your classmate understand what you wrote?

2. During the lessons about positive and negative numbers, you discovered some situations in which negative numbers are used in the real world. For example, a Math News video showed elevations of points around the world that are below sea level.

Add more examples to this table. Include examples that you recall from the lessons and exercises. Add other examples from your own experiences, stories, the news, and so on.

Negative Numbers in Everyday Life

Elevations of locations below sea level

3. Use your notes from class and your thoughts about the unit to start a math vocabulary list in your notebook.

Use a chart similar to the one shown to organize your list. Include the vocabulary word or phrase, a definition in your own words, and one or more examples. When appropriate, your example should include a diagram, a picture, or a step-by-step problem-solving approach.

Word or Phrase Definition Examplesinteger A whole number Can be positive, negative, or zero

1 –5,411 0 256 –8

Add these words to your vocabulary list.

integer

positive number

negative number

opposite of a number

absolute value

4. Complete any exercises that you have not finished from earlier lessons in this unit.



EXERCISESLESSON 9: THE COORDINATE PLANE

EXERCISES

1. Plot this point in the coordinate plane.

(–7, 0)

2. Plot these points on the coordinate plane.

a. Point A (0, 3)b. Point B (–1, 4)c. Point C (–9, –5)d. Point D (8, –2)

e. Point E (4, 5)

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10




3. What are the coordinates of point M?

A (2, 3)

B (–3, –2)

C (2, –3)

D (–3, 2)

4. What are the coordinates of the heart?

A (2, 2)

B (2, –2)

C (–2, 2)

D (–2, –2)

5. What are the coordinates of the square?

A (2, 2)

B (2, –2)

C (–2, 2)

D (–2, –2)

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y

M

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y




6. The townspeople of Cahuana have decided to build a hospital midway between the two police stations. On the town grid, the town center is located at (0, 0). One police station is located 10 blocks west and 6 blocks south of the town center. The other police station is located 8 blocks east and 6 blocks south of the town center.

Plot the locations of the two police stations in the coordinate plane.

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

7. The town center of Cahuana is located at (0, 0) on the town grid. One police station is located 10 blocks west and 6 blocks south of the town center. The other police station is located 8 blocks east and 6 blocks south of the town center.

The townspeople of Cahuana plan to build their new hospital halfway between the two police stations.

In which quadrant of the coordinate plane will the new hospital be located?

A Quadrant I

B Quadrant II

C Quadrant III

D Quadrant IV2

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




8. The table shows the x- and y-coordinates of three points.

x –2 1 3y 2 –2 2

Which coordinate plane shows these three points

A

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y B

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y

C

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y D

1

2

3

4

–4

–3

–2

–1 4321–2–3–4 x

y




9. The table shows the x- and y-coordinates of three points.

x –4 3 –1y –5 6 6

Which coordinate plane shows these three points?

A

2

6

4

–6

–4

–2642–2–6 –4 x

y B

2

6

4

–6

–4

–2642–2–6 –4 x

y

C

2

4

–6

–4

–2642–2–6 –4 x

y D

2

6

4

–6

–4

–2642–2–6 –4 x

y

Challenge Problem

10. The signs of each point in the coordinate system can be determined by its quadrant. Complete the table by filling in the missing y-coordinates, and assigning a quadrant to each set of coordinate signs.

Sign (x-coordinate) Sign (y-coordinate) Quadrant

++

–III



EXERCISESLESSON 10: DRAWING FIGURES

EXERCISES

1. Which point has the y-coordinate with the greatest absolute value?

Point A (–3, 1) Point B (–2, –5) Point C (3, –1) Point D (4, 1)

A Point A

B Point B

C Point C

D Point D

2. Which two points have opposite coordinates?


A Point A and C

B Point B and D

C Point C and D

D Point D and B

3. Which point has the x-coordinate with the greatest opposite value?


A Point A

B Point B

C Point C

D Point D



EXERCISES

4. Refer to the coordinate plane to find the distance between points A and D.

Point A (–3, 1) Point D (4, 1)

2

6

4

–6

–4

–2642–2–6 –4

y

5. Draw a line segment that meets the following conditions.

One endpoint of the line segment has the coordinates (–5, –5).

The other endpoint of the line segment is located in Quadrant II in the coordinate plane.

The line segment is 9 units long.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

LESSON 10: DRAWING FIGURES




6. Four vertices of a rectangle are located at (–5, 3), (7, 3), (7, –3), and (–5, –3).

Plot these points in the coordinate plane. Draw the rectangle.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

7. Look at this figure.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

What is the diameter of the circle?

A 10 units

B 5 units

C 2 units

D –7 units




8. Two vertices of a square are diagonal from each other and have the coordinates (3, 1) and (–2, –4).

Draw the square in the coordinate plane.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

9. Two vertices of a square are diagonal from each other and have the coordinates (3, 1) and (3, –4).

What is the length of the diagonal of square? Explain your strategy for finding the length.

10. In the town grid of Cahuana, the town center is located at (0, 0). Police station 1 is located 10 blocks west and 6 blocks south of the town center. Police station 2 is located 8 blocks east and 6 blocks south of the town center. The hospital is halfway between the two police stations.

How far is the hospital from each police station? Explain your strategy for finding the distances.

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




Challenge Problem

11. One side of a triangle is given by the line connecting the points (–4, 0) and (4, 0).

a. Describe the coordinates of the third vertex if the triangle is isosceles and the given side is the base. Give an example, and draw a figure to support your answer.

b. Describe the coordinates of the third vertex if the triangle is right and the given side is one of the legs. Give an example, and draw a figure to support your answer.

c. Write a statement to describe the coordinates of the third vertex if the triangle is obtuse and one of the given points is the vertex of the obtuse angle. Give an example, and draw a figure to support your answer.



EXERCISES

EXERCISES

1. Which graph shows the reflection of the original figure across the x-axis?

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

A

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10

x

y

–2–4–6–8–10

B

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10

x

y

–2–4–6–8–10

C

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

D

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

LESSON 11: CREATING MIRROR IMAGES



EXERCISESLESSON 11: CREATING MIRROR IMAGES

2. Which graph shows the reflection of the original figure across the y-axis?

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

A

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10

x

y

–2–4–6–8–10

B

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

C

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

D

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10




3. a. Write the coordinates of the vertices of the triangle shown after it is reflected across the y-axis.

2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

b. What happens to both the x- and y-coordinates of the triangle as a result of this reflection?


2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

In the coordinate plane, plot the figure that results from reflecting the trapezoid across the x-axis.





2

2

4

6

8

10

–2

–4

–6

–8

–10

4 6 8 10 x

y

–2–4–6–8–10

In the coordinate plane, plot the figure that results from reflecting the hexagon across the x-axis.

6. A figure can be reflected across any line in the coordinate plane, not just one of the axes. Suppose the points (1, 5) and (1, –1) give two vertices on one side of a square in the coordinate plane.

a. What are some possible coordinates for the other two vertices? b. The points you found for part a can be determined by reflecting (1, 5) and

(1, –1) across some line other than the x- or y-axis. Name this line, and explain how you know your reflection works.

7. Look at this triangle, which has vertices at (3, 1), (4, 3), and (6, 1) in the coordinate plane.

2

4

–6

–4

–242–2–6 –4 x

y

6

6

Plot the figure that results from reflecting twice: first reflect the triangle across the y-axis and then reflect it across the x-axis.




Challenge Problem

8. Lines of reflection need not be parallel to either the x- or y-axis. Draw a figure in the coordinate plane, and reflect the figure across the line y = x. What do you notice about the reflected figure? Pick some points in your original figure and compare the points to their reflections.

2

2

4

6

8

10

12

14

16

18

20

–2

–4

–6

–8

–10

–12

–14

–16

–18

–20

4 6 8 10 12 14 16 18 20 x

y

–2–4–6–8–10–12–14–16–18–20



EXERCISES

EXERCISES

1. Read your Self Check and think about your work in this unit.

Write down three things you have learned during the unit.


2. What steps do you follow when you plot the reflection of a figure in the coordinate plane?


3. Use your notes from class and your thoughts about the unit to add to your math vocabulary list in your notebook.

Include the vocabulary word or phrase, a definition, and one or more examples. When appropriate, your example should include a diagram, a picture, or a step-by-step problem-solving approach.

Word or Phrase Definition Examples

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

1 –5,411 0 256 –8

Add these words to your vocabulary list.

non-integer

coordinate plane

coordinates

4. Review the notes you took during the lessons about the coordinate plane. Add any additional ideas you have about the topic to your notes.

5. Complete any exercises from this unit you have not finished.

LESSON 12: PUTTING IT TOGETHER

lesson 1: above and below sea level...

Documents