Reteach

Add Integers

To add integers with the same sign, add their absolute values. The sum is:

• positive if both integers are positive.

• negative if both integers are negative.

To add integers with different signs, subtract their absolute values. The sum is:

• positive if the positive integer's absolute value is greater.

• negative if the negative integer's absolute value is greater.

To add integers, it is helpful to use a number line.

CD't1D Find 4 + (-6). ~ Find -2 + (-3).

Use a number line. Use a number line.

• Start at O. • Start at O.

• Move 4 units right. • Move 2 units left.

• Then move 6 units left. • Move another 3 units left. I -6 I i -3 i I I

I " I

: +4: :'I

1~2I

• I II I I I • .i 1 • '1' II ~ II·

-3 -2 -1 0 1 2 3 4 5 -6-5-4-3-2-1 0 1 2 3

4 + (-6) =-2 -2+ (-3)=-5

Add.

1. -5 + (-2)

Write an addition expression to describe each situation. Then find

each sum.

13. HAWKA hawk is in a tree 100 feet above the ground. It flies down to the

ground.

Reteach

Subtract Integers

6 - 9 = 6 + (-9) To subtract 9, add -9.

= -3 Simplify .

••• Find -10 - (-12).

-10 - (-12) = -10 + 12 To subtract -12, add 12.

= 2 Simplify.

~ Evaluate a - b i~a = -3 and b = 7.

a - b = -3 - 7 Replace a with -3 and b with 7.

= -3 + (-7) To subtract 7, add -7.

= -10 Simplify.

Subtract.

1. 7 - 9

Reteach

Multiply Integers

The product of two integers with different signs is negative.

The product of two integers with the same sign is positive.

~ Find 5(-2).

~ Find -3(7).

~ Find -6(-9).

~ Find (-7)2.

(-7)2 = (-7)(-7) There are 2 factors of -7.

= 49 The product is positive.

~. Evaluate abc if a = -2, b = -3, and c = 4.

abc = -2(-3)(4) Replace a with -2, b with -3, and c with 4.

= 6(4) MUltiply -2 and -3.

= 24 Multiply 6 and 4.

crmD.. ... -.........•...........

Reteach

Divide Integers

The quotient of two integers with different signs is negative.

The quotient of two integers with the same sign is positive.

~ Find30+(-5).

30 + (-5) The integers have different signs.

30 + (-5) = -6 The quotient is negative.

~ Find -100 + (-5).

-100 + (-5) The integers have the same si~n.

-100 + (-5) = 20 The quotient is positive.

3. 18 -2

6•

-80 -20

9• 540

• -256

45 10

16

11. 12 + e 12. 40 + f

13. d + 6 14. d + e

15. f + e 16. e2 + f

17. ~d 18. ef + 2

19. f ~48 20. d ~ e

Reteach

Add and Subtract Unlike Fractions

To add or subtract fractions with different denominators,

• Rename the fractions using the least common denominator (LCD).

• Add or subtract as with like fractions.

• If necessary, simplify the sum or difference.

~Find:+~.

Method 1 Use a Model.

2 3

I

+ 1 I

I

4 - - - - - - T - - - - - - - r - - -- - --.

I I I _______

IL

IL

II

11

12

Method 2 Use the LCD.

-+-=-.-+-.-2 124 1 3

3 4 3 4 4 3 8 3 11

= 1:2+ 1:2or 1:2

Add or subtract. Write in simplest form.

1. 2+41 3 2.

3- 1

8 2

3. 175+ (- ~) 4. "5-3

2 1

5. ~ + (- :2) 6·

11 3

1:2-4

7. ~ - (- ~) 8. 7- 1

9 2

9. 3

+1:27

10."5+3

32

10

Reteach

Multiply Fractions

To multiply fractions, multiply the numerators and multiply the denominators .

.§..x~=5x3=~=-.1

6 5 6x5 30 2

To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply the

fractions.

~ Find ~ X :. Write in simplest form.

~ x .!= 2 x 4 +- Multiply the numerators.

3 5 3 x 5 +- Multiply the denominators.

~ Find ~ X ~. Write in simplest form.

1.. x 21.. = 1.. x -2.. Rename 2- 1as an improper fraction, ~2'

3 2 3 2 2

1 x 5

3x2

5

6

Multiply. Write in simplest form.

1. g2 xg2

2.2"1

xs7

4."9

5 x 4 5. 1~ x (- ~)

11. -2

1 x"64

12. 1

x 2 3

3 8 4

Reteach

To divide by a fraction, multiply by its multiplicative inverse or reciprocal. To divide by a mixed number,

rename the mixed number as an improper fraction .

.~. Find ~ + :. Write in simplest form.

st + ~ = ~o+ ~ Rename 3t as an improper fraction,

= 310 . 2"9 MUltiply by the reciprocal of 9'

2 which is 2'

9

= 1-.% Divide out common factors. 5 }

1 1

= 15 MUltiply.

Divide. Write in simplest form.

12.1 22.5 _l ~l '374 '576

3• 2' 5

C"l

4. 5 + (- ~) 5. ~ + 10 6. 7~ + 2 0

'tl

'g:~.

@

9 ~ 0

7• 6

5 . 32"

1 8. 36 + 1~ 9. -2~ + (-10)

~

7 ~ ~

~ * ee"

$. l!J. 0:; g,

10. ~ + 1~ 11. 6~ + 12·4+ : ~ '"a:: "~ ~

~ C"l 0

13. 4~ + 2~ 14. 12 + (-2~) 15. 4+ ~ S

'tl

~. !"

5"'

"

Study Guide and Intervention

Fractions and Decimals

To write a decimal as a fraction, divide the numerator of the fraction by the denominator. Use a power of

ten to change a decimal to a fraction.

Write : as a decimal.

Method 1 Use pencil and paper.

0.555 ...

9)5.000

45

50 You can use bar notation 0.5 to

45 indicate that 5 repeats forever. 50 So, "9

5 =

-

0.5.

45

5

Write 0.32 as a fraction in simplest form.

0.32 = 32

100

_ 8

- 25

Write each fraction or mixed number as a decimal. Use bar notation

if the decimal is a repeating decimal.

6. 3

47

99

Practice: Skills

Write each fraction or mixed number as a decimal. Use bar notation

if the decimal is a repeating decimal.

19 5. 20

8. 1 5

9.2"519

8

10. 4 17

12. 17

37 24

13. 6 7

15. 1 17

32 48

Study Guide and Intervention

Fractions and Percents

A ratio is a comparison of two numbers by division. When a ratio compares a number to 100, it

can be written as a percent. To write a ratio or fraction as a percent, find an equivalent fraction with

a denominator of 100. You can also use the meaning of percent to change percents to fractions.

Write ~~ as a percent.

/X5",

19 95 _ 95nt Since 100 -7- 20 = 5, mUltiply the numerator and l'l7\ 'l7\7\ - 70 £.V .LVV denominator by 5.

'" X5/

92% =

92

100

_ 23 - 25

1. 14 27 100 2. 100

3 14

7. 100 8. 100

10. 1

11. 2513

12. 104

20

Study Guide and Intervention

Percents and Decimals

To write a percent as a decimal, divide the percent by 100 and remove the percent symbol. To write a

decimal as a percent, multiply the decimal by 100 and add the percent symbol.

42.5% = i~·g 42.5 x 10

100 x 10

425

= 1,000

= 0.425

0.625 = 0§65 Multiply by 100.

= 62.5% Add the % symbol.

Write each percent as a decimal.

1. 6%

Fractions, Decimals, and Percents

1. 12% 2. 125% 3. 96%

4. 1.7% 5. 5.75% 6. 45%

7. 0.36 8. 1.94 9. 0.0425

10. 0.5 11. 0.85 12. 5.2

64 5 18 13. 100 14. "8 15. 20

16. 41

17. '73 18.

7

50 10

Reteach

Solve Two-Step Equations

To solve a two-step equation, undo the addition or subtraction first. Then undo the multiplication or

division.

7v - 3 = 25 Write the equation.

+3 = +3 Undo the subtraction by adding 3 to each side.

7v = 28 Simplify.

7v =

28 7 7 v=4

7v - 3 = 25 Write the original equation.

7(4) - 3 ~ 25 Replace v with 4.

28 - 3 ~ 25 Multiply.

25 = 25 ..( The solution checks.

The solution is 4.

Solve -10 = 8 + 3x. Check your solution.

-10= 8 + 3x Write the equation.

-8= -8 Undo the addition by subtracting 8 from each side.

-18= 3x Simplify.

-18 3x Undo the multiplication by dividing each side by 3.

3 3

-6=x Simplify.

-10= 8 + 3x Write the original equation.

-10~8+3(-6) Replace x with -6.

-10~ 8 + (-18) Multiply.

-10 = -10 ..( The solution checks.

1. 4y + 1 = 13 2. 6x + 2 = 26 3. -3 = 5k + 7 4. 3"n2

+ 4 = -26

5.7= -3c - 2 6. -8p + 3 = -29 7. -5 = -5t - 5 8. -9r + 12= -24

9.11 + gn7 = 4 10. 35 = 7 + 4b 11. -15 + -p

4 = 9 12. 49 = 16+ 3y

5

13.2 = 4t - 14 14. -9x - 10= 62 15. 30= 12z - 18

Reteach

Solve Equations with Variables on Each Side

To solve an equation with variables on each side, use the Properties of Equality to write an equivalent

equation with the variables on one side.

Then solve the equation.

6x + 10 - 10 = x + 8 - 10

6x=x-2

4x = x + 27 Write the equation.

4x - x = x - x + 27 Subtraction Property of Equality

3x = 27 Simplify.

-3x 27

Division Property of Equality -

3 3

x=9 Simplify. Check your solution.

Solve 3x - 16 = 5x - 4.

3x -16 = 5x - 4

3x - 3x - 16 = 5x - 3x - 4 Subtraction Property of Equality

-16 = 2x - 4 Simplify.

-16 + 4 = 2x - 4 + 4 Addition Property of Equality

-12 = 2x Simplify.

--12

-2x

Division Property of Equality

2 2

-6 =x Simplify. Check your solution.

Express each equation as another equivalent equation. Justify your

answer.

Solve each equation. Check your solution.

3. 2x + 6 = x + 15

5. -12 - x = 8 - 3x

Chapter 3

Skills Practice

Solve Equations with Variables on Each Side

Express each equation as another equivalent equation. Justify your

answer.

6. ~ - 12 = -3 + x 2

11. ~- 3 = x + 8 4