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Common Core State Standards

822 Unit 10 Reflections and Symmetry

Advance PreparationFor Part 1, make and cut apart copies of Math Masters, page 320. Place them near the Math Message.

For the second optional Readiness activity in Part 3, use masking tape to create a life-size number line

(–10 to 10) on the floor.

Teacher’s Reference Manual, Grades 4–6 pp. 71–74, 100 –102

Key Concepts and Skills• Compare and order integers.  

[Number and Numeration Goal 6]

• Add signed numbers.  

[Operations and Computation Goal 2]

• Identify a line of reflection.  

[Geometry Goal 3]

Key ActivitiesStudents review positive and negative

numbers on the number line, thinking of

them as reflected across the zero point.

They discuss and practice addition of

positive and negative numbers as accounting

problems, keeping track of “credits” and

“debits.” They play the Credits/Debits Game.

Ongoing Assessment: Informing Instruction See page 825.

Key Vocabularyopposite (of a number) � credit � debit

MaterialsStudent Reference Book, pp. 60 and 238

Study Link 10�5

Math Masters, pp. 320 and 468

transparencies of Math Masters, pp. 318

and 321 (optional) � per partnership:

1 transparent mirror, deck of number cards

(the Everything Math Deck, if available)

Solving Fraction, Decimal, and Percent ProblemsMath Journal 2, p. 283

Students solve problems involving

fractions, decimals, and percents.

Math Boxes 10�6Math Journal 2, p. 284

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Data and Chance Goal 4]

Study Link 10�6Math Masters, p. 322

Students practice and maintain skills

through Study Link activities.

READINESS

Exploring Skip Counts on a Calculatorcalculator

Students skip count on a calculator to

explore patterns in negative numbers.

READINESS

Using a Number Line to Add Positive and Negative Numbersmasking tape

Students use a number line to add

positive and negative integers.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Positive andNegative Numbers

Objective To introduce addition involving negative integers.

�

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Lesson 10�6 823

Getting Started

LESSON

10�6

Name Date Time

Positive and Negative Numbers

Place your transparent mirror on the dashed line that passes through 0

on the number line above. Look through the mirror. What do you see?

What negative number image do you see . . .

above 1? above 2? above 8? �8�2�1

–10

10987654321

–1–2–3–4–5–6–7–8–90

60

Math Masters, page 320

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Student Reference Book, p. 60; Math Masters, p. 320)

One way to think about a number line is to imagine the whole numbers reflected across the zero point. Each of these positive numbers picks up a negative sign as it crosses to the other side of zero. The opposite of a positive number is a negative number.

Conversely, imagine the negative numbers reflected across the zero point. The sign of each number changes from negative to positive as it crosses to the other side of zero. The opposite of a negative number is a positive number.

NOTE In this “flipping” of the number line, the zero point stays motionless,

like the fulcrum of a lever. Zero is the only number that equals its opposite.

When students place the transparent mirror on the line passing through the zero point on Math Masters, page 320, the negative numbers appear (reversed) across from the corresponding positive numbers.

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�8

�9

�7�

6�5�

4�3�

2�1

108

97

65

43

21

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00

Math MessageTake a copy of Math Masters, page 320. Follow the directions and answer the questions. Share 1 transparent mirror with a partner.

Study Link 10�5 Follow-UpReview answers. Have students share some of the patterns they created on their own. An overhead transparency of Study Link 10�5 (Math Masters, page 318) may be helpful.

Mental Math and ReflexesPose problems involving comparisons of integers. Suggestions:

Are you better off if you have $3 or owe $10? Have $3

Owe $4 or owe $9? Owe $4

Owe $20 or owe $100? Owe $20

Which is colder?

–3°C or 10°C? –3°C–9°C or 19°C? –9°C–7°C or –11°C? –11°C

Which is greater?

–10 or 8? 8

5 or –1? 5

10 or –10? 10

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824 Unit 10 Reflections and Symmetry

Note

Note

Fractions

Rename as fractions: 0, 12, 15.3, 3.75, and 25%.

0 = 0 _ 1 12 = 12 _ 1 15.3 = 153 _ 10 3.75 = 375 _ 100 25% = 25 _ 100

Negative Numbers and Rational Numbers

People have used counting numbers (1, 2, 3, and so on) for thousands of years. Long ago people found that the counting numbers did not meet all of their needs. They needed numbers for in-between measures such as 2 1 _ 2 inches and 6 5 _ 6 hours.

Fractions were invented to meet these needs. Fractions can also be renamed as decimals and percents. Most of the numbers you have seen are fractions or can be renamed as fractions.

Every whole number (0, 1, 2, and so on) can be renamed as a fraction. For example, 0 can bewritten as 0 _ 1 . And 8 can be written as 8 _ 1 .

Numbers like -2.75 and -100 may not look like negative fractions, but they can be renamed as negative fractions.

-2.75 = - 11 _ 4 , and

-100 = - 100 _ 1

However, even fractions did not meet every need. For example, problems such as 5 - 7 and 2 3 _ 4 - 5 1 _ 4 have answers that are less than 0 and cannot be named as fractions. (Fractions, by the way they are defined, can never be less than 0.) This led to the invention of negative numbers. Negative numbers are numbers that are less than 0. The numbers - 1 _ 2 , -2.75, and -100 are negative numbers. The number -2 is read “negative 2.”

Negative numbers serve several purposes:

♦ To express locations such as temperatures below zero on a thermometer and depths below sea level

♦ To show changes such as yards lost in a football game

♦ To extend the number line to the left of zero

♦ To calculate answers to many subtraction problems

The opposite of every positive number is a negative number, and the opposite of every negative number is a positive number. The number 0 is neither positive nor negative; 0 is also its own opposite.

The diagram at the right shows this relationship.

The rational numbers are all the numbers that can be written or renamed as fractions or as negative fractions.

Student Reference Book, p. 60

Student Page

End/Start ofTransaction Start Change Next Transaction

New business, $0 $0 $0 start at $0

Credit (payment) $0

add +$5

+$5

of $5 comes in

Credit of $3 +$5 add +$3 +$8

Debit of $6 +$8 add -$6 +$2

Debit of $8 (Be

+$2

add -$8

-$6 sure to share

strategies.)

Debit of $3 -$6 add -$3 -$9

Credit of $5 -$9

add +$5

-$4

(At last!)

Credit of $6 -$4 add +$6 +$2

Read and discuss page 60 of the Student Reference Book with the class. The diagram on the page is another way of showing that the opposite of every positive number is a negative number, and the opposite of every negative number is a positive number.

� Using Credits and Debits WHOLE-CLASS ACTIVITY

to Practice Addition of Positive and Negative Numbers(Math Masters, p. 321)

Display a transparency of Math Masters, page 321. Tell students that in this lesson they pretend that they are accountants for a new business. They figure out the “bottom line” as you post transactions.

Discuss credits (money received for sales, interest earned, and other income) as positive additions to the bottom line, and debits (cost of making goods, salaries, and other expenses) as negative additions to the bottom line. Explain that you will label credits with a “+” and debits with a “–” to keep track of them as positive and negative numbers.

To support English language learners, clarify any misconceptions about the use of the words credits, debits, and bottom line in this lesson as compared with students’ observations of the use of credit and debit cards at stores.

Be consistent throughout this lesson in “adding” credits and debits as positive and negative numbers, because Lesson 11-6 uses the same format to show “subtraction” of positive and negative numbers—the effect on the bottom line of “taking away” what were thought to be credits or debits.

Following is a suggested series of transactions. Entries in black are reported to the class; entries in color are appropriate student responses. To support English language learners, discuss the meaning of the words transaction and change.

ELL

Adjusting the Activity Have students experiment with their

calculators to find out how to enter negative

numbers and expressions with negative

numbers. On the TI-15 students use the (–)

key, and on the Casio fx-55, students

use the key.

AUDITORY � KINESTHETIC � TACTILE � VISUAL

Links to the FutureStudents explore subtraction of positive and

negative integers in Lesson 11-6. Addition

and subtraction of signed numbers is a

Grade 5 Goal.

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Name Date Time

Credits/Debits Record Sheets 132

4

22

21

20

19

18

17

16

15

14

13

12

11

10

98

76

54

32

10

�1

�2

�3

�4

�5

�6

�7

�8

�9

�10

�11

�12

�13

�14

�15

�16

�17

�18

�19

�20

�21

�22

238

Reco

rd Sh

eet

StartCh

ange

End

, and

n

ext start

1+

$10

23456789

10

Gam

e 1

Reco

rd Sh

eet

StartCh

ange

End

, and

n

ext start

1+

$10

23456789

10

Gam

e 2

Math Masters, p. 468

Game Master

Lesson 10�6 825

Note

Games

Beth has a “Start” balance of +$20. She draws a black 4. This is a credit of $4, so she records +$4 in the “Change” column.She adds $4 to the bottom line: $20 + $4 = $24. She records +$24 in the “End” column, and +$24 in the “Start” column on the next line.

Alex has a “Start” balance of +$10. He draws a red 12. This is a debit of $12, so he records -$12 in the “Change” column. He adds -$12 to the bottom line: $10+(-$12) = -$2. Alex records -$2 in the “End” column. He also records -$2 in the “Start” column on the next line.

Credits/Debits Game

Materials □ 1 complete deck of number cards

□ 1 Credits/Debits Game Record Sheet for each player (Math Masters, p. 468)

Players 2

Skill Addition of positive and negative numbers

Object of the game To have more money afteradjusting for credits and debits.

Directions

You are an accountant for a business. Your job is tokeep track of the company’s current balance. The current balance is also called the “bottom line.” As credits and debits are reported, you will record them and then adjust the bottom line.

1. Shuffle the deck and lay it number-side down between the players.

2. The black-numbered cards are the “credits,” and the blue- or red-numbered cards are the “debits.”

3. Each player begins with a bottom line of +$10.

4. Players take turns. On your turn, do the following:

♦ Draw a card. The card tells you the dollar amount and whether it is a credit or debit to the bottom line. Record the credit or debit in your “Change” column.

♦ Add the credit or debit to adjust your bottom line.♦ Record the result in your table.

5. At the end of 10 draws each, the player with more money is the winner of the round.

Each player uses oneRecord Sheet.

If both players have negative dollar amounts at the end of the round, the player whose amount is closer to 0 wins.

Student Reference Book, p. 238

Student Page

� Playing the Credits/Debits Game PARTNER ACTIVITY

(Student Reference Book, p. 238; Math Masters, p. 468)

Students play the Credits/Debits Game to practice adding positive and negative numbers. They record their work on Math Masters, page 468.

Ongoing Assessment: Informing Instruction

As students play, watch for those who are beginning to devise shortcuts for

finding answers. For example, most students will probably count up and back on

a number line. Some students may notice that when two positive numbers are

added, the result is “more positive”; when two negative numbers are added, the

result is “more negative”; and when a positive and a negative number are added,

the result is the difference of the two (ignoring the signs) and has the sign of the

number that is “bigger” in the sense of being farther from 0.

Do not try too hard to get explanations; these will evolve over time as students

have more experience with positive and negative numbers.

2 Ongoing Learning & Practice

� Solving Fraction, Decimal, INDEPENDENTACTIVITY

and Percent Problems(Math Journal 2, p. 283)

Students solve problems involving equivalent fractions, decimals, percents, and discounts.

� Math Boxes 10�6 INDEPENDENTACTIVITY

(Math Journal 2, p. 284)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-3. The skill in Problem 5 previews Unit 11 content.

PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMMMLEBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBBBLBLBBLBLBLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROOROROROOOPPPPPPP MMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELEEELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBLELEELEMMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBBBLLOOOOROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLLVINVINVINVINVINNNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLLOOOO VVVLLLLLLLLLLVVVVVVVVVOOSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

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826 Unit 10 Reflections and Symmetry

�

Math Boxes LESSON

10�6

Date Time

3. Solve each open sentence.

a. 67.3 + p = 75.22 p =

b. 6.86 - a = 2.94 a =

c. x + 5.69 = 7.91 x =

d. 4.6 - n = 0.32 n = 4.28

2.22

3.92

7.92

b. Explain how you designed your spinner.

Sample answer: If it lands on

blue 27 out of the 36 spins,

then 27

__ 36 , or 3 _ 4 , of the board

should be blue. Likewise,

9 __ 36 , or 1 _ 4 , of the board should

be red.82–86

2. Complete.

Rule: -

1

_ 4 in out

8

_

16

4

_

16

8

_

8

3 _

4 2

_

4

7 _ 12

15

_

20

6 _ 8

10

_ 12

55 57

93 142 143 140

34–37

4. Angle RUG is an acute

(acute or obtuse) angle.

Measure of ∠RUG = 20 °

R

GU

5. Sebastian and Joshua estimated the

weight of their mother. What is the most

reasonable estimate? Fill in the circle

next to the best answer.

A. 50 pounds

B. 150 pounds

C. 500 pounds

D. 1,000 pounds

1. a. Make a spinner.

Color it so that

if you spin it

36 times, you

would expect it

to land on blue

27 times and

red 9 times.

red

blue

Sample answer:

10

_ 20

274-285_EMCS_S_MJ2_G4_U10_576426.indd 284 2/18/11 9:16 AM

Math Journal 2, p. 284

Student Page

Review: Fractions, Decimals, and PercentsLESSON

10� 6

Date Time

1. Fill in the missing numbers in

the table of equivalent fractions,

decimals, and percents.

2. Kendra set a goal of saving $50 in 8 weeks. During the first

2 weeks, she was able to save $10.

a. What fraction of the $50 did she save in the first 2 weeks?

b. What percent of the $50 did she save?

c. At this rate, how long will it take her to reach her goal? weeks

3. Shade 80% of the square.

a. What fraction of the square did you shade?

b. Write this fraction as a decimal.

c. What percent of the square is not shaded?t

4. Tanara’s new skirt was on sale at 15% off the original price.

The original price of the skirt was $60.

a. How much money did Tanara save with the discount?

b. How much did she pay for the skirt?

5. Star Video and Vic’s Video Mart sell videos at about the same regular prices. Both

stores are having sales. Star Video is selling its videos at 1_3

off the regular price._

Vic’s Video Mart is selling its videos at 25% off the regular price. Which store has

the better sale? Explain your answer.

Star Video has the better sale since1_3

= 331_3%, which

is more than 25%. So they’re taking more off their

regular prices.

$9$51

80_100

0.820%

10_50

20%10

Fraction Decimal Percent

0.4 40%0.6 60%

0.75 75%75100

610

61 62

4

_ 10

274-285_EMCS_S_MJ2_G4_U10_576426.indd 283 2/15/11 6:15 PM

Math Journal 2, p. 283

Student Page

Math Boxes

Problem 1 �

Writing/Reasoning Have students write a response to the following: The weights in Problem 5 are expressed in pounds. Make a table to show equivalent weights in ounces for 50; 150; 500; and 1,000 pounds. Then explain how you converted the weights.

Pounds Ounces50 800150 2,400500 8,000

1,000 16,000

Sample answer: I know that there are 16 ounces in a pound, so I multiplied each weight by 16 to get the number of ounces.

Ongoing Assessment: Recognizing Student Achievement

Use Math Boxes, Problem 1 to assess students’ ability to express the

probability of an event as a fraction. Students are making adequate progress

if they design a spinner that is 1

_ 4 red and

3

_ 4 blue. Many students will design

a spinner that has 3 consecutive parts red and 9 consecutive parts blue.

Some students will explore other possibilities—for example, 2 consecutive

red parts, followed by 4 blue parts, 1 red part, and 5 blue parts.

[Data and Chance Goal 4]

� Study Link 10�6 INDEPENDENTACTIVITY

(Math Masters, p. 322)

Home Connection Students compare and order positive and negative numbers and add positive and negative integers.

3 Differentiation Options

READINESS SMALL-GROUP ACTIVITY

� Exploring Skip Counts 5–15 Min

on a CalculatorTo explore patterns in negative numbers, have students skip count on the calculator. Ask students to start with 30 and count back by 10s on their calculator as they say the numbers aloud. Stop at –10 and ask the following questions:

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STUDY LINK

10�6 Positive and Negative Numbers

60

Name Date Time

Write � or � to make a true number sentence.

1. 3 14 2. �7 7 3. 19 20 4. �8 �10

List the numbers in order from least to greatest.

5. 5, �8, �12�, ��

14�, 1.7, �3.4

least greatest

6. �43, 22, �174�, 5, �3, 0

least greatest

7. Name four positive numbersless than 2.

8. Name four negative numbersgreater than �3.

Use the number line to help you solve Problems 9–11.

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12��1�2

1�34��

12��

14�

225�174�0�3�43

51.7�12���

14��3.4�8

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�15 �14 �13 �12 �11 �10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

9. a. 4 � 9 � b. 4 � (�9) � c. (�4) � (�9) �

10. a. 5 � 3 � b. (�5) � 3 � c. (�5) � (�3) �

11. a. � 2 � 13 b. � (�2) � 13 c. � (�2) � (�13)�151115�8�28

�13�513

Practice

12. 1.02 � 12.88 � 13. 7.26 � 1.94 �

14. � 5.84 � 8.75 15. 3.38 � � 2.620.762.915.3213.90

Sample answers:

Math Masters, p. 322

Study Link Master

Lesson 10�6 827

–5 –4 –3 –2 –1 0 1 2

-4 + 3 = -1

● What does the calculator display show after zero? –10

● How do you read this number? negative ten

● Can you predict what number will come next? –20

Have students continue counting back, stopping at –50.

Repeat the routine counting back with other numbers such as 2, 5, 25, and 100. Remind students to clear their calculators after each count.

READINESS SMALL-GROUP ACTIVITY

� Using a Number Line to 5–15 Min

Add Positive and Negative NumbersTo explore addition of positive and negative integers using a number line model, have students act out addition problems by walking on a life-size number line from –10 to 10.

� The first number tells students where to start.

� The operation sign (+ or –) tells which way to face:

+ means face the positive end of the number line.

– means face the negative end of the number line.

� If the second number is negative, then walk backward. Otherwise, walk forward.

� The second number (ignoring its sign) tells how many steps to walk.

� The number where the student stops is the answer.

Example: –4 + 3

� Start at –4.

� Face the positive end of the number line.

� Walk forward 3 steps.

� You are now at –1. So –4 + 3 = –1.

Suggestions:

● –6 + –3 = ? (Start at –6. Face in the positive direction. Walk backward 3 steps. End up at –9.)

–10 –9 –8 –7 –6 –5 –4 –3

● 4 + –6 = ? (Start at 4. Face in the positive direction. Walk backward 6 steps. End up at –2.)

–2 –1 0 1 2 3 4 5

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