NAME _____________________________________ DATE ____________________________
Ch 3 Lesson 2 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.
Example 1
Example 2
Find 4 + (–6).
Find –2 + (–3).
Use a number line.
• Start at 0.
• Move 4 units right.
• Then move 6 units left.
Use a number line.
• Start at 0.
• Move 2 units left.
• Move another 3 units left.
Exercises Add.
1. –5 + (–2) = -7
2. 8 + 1
3. –7 + 10
4. 16 + (–11)
5. –22 + (–7)
6. –50 + 50
7. –10 + (–10)
8. 100 + (–25)
9. –35 + (–20)
Ch 3 Lesson 3 Reteach
Subtract Integers
To subtract an integer, add its opposite.
Example 1
Find 6 – 9.
6 – 9 = 6 + (–9)
= –3
To subtract 9, add –9.
Simplify.
Example 2
Find –10 – (–12).
–10 – (–12) = –10 + 12
=2
To subtract –12, add 12.
Simplify.
Example 3
Evaluate a – b if a = –3 and b = 7.
a – b = –3 – 7
Course 2 • Chapter
Replace a with –3 and b with 7.
PERIOD ____________
NAME _____________________________________ DATE ____________________________
= –3 + (–7)
= –10
PERIOD ____________
To subtract 7, add –7.
Simplify.
Exercises
Subtract.
1. 7 – 9 =
-2
3. –10 – 4
= -14
2. 20 – (–6) two negatives make + so 20 + 6 = 26
4. 0 – 12
5. –7 – 8
6. 13 – 18
7. –20 – (–5)
8. –8 – (–6)
9. 25 – (–14)
10. –75 – 50
11. 15 – 65
12. 19 – (–10)
Evaluate each expression if m = –2, n = 10, and p = 5.
13. m – 6 =
-2 -6=
-8
14. 9 – n
15. p – (–8) __________________________
16. p – m
17. m – n
18. –25 – p
Ch 3 Lesson 4 Reteach
__________________ 9-10= -1
Multiply Integers
The product of two integers with different signs is negative.
The product of two integers with the same sign is positive.
Example 1
Find 5(–2).
5(–2) = –10
The integers have different signs. The product is negative.
Example 2
Find –3(7).
–3(7) = –21
The integers have different signs. The product is negative.
Example 3
Find –6(–9).
–6(–9) = 54
The integers have the same sign. The product is positive.
Example 4
Find (–7)2.
(–7)2 = (–7)(–7)
= 49
There are 2 factors of –7.
The product is positive.
Example 5
Course 2 • Chapter
NAME _____________________________________ DATE ____________________________
Find –2(–3)(4).
–2(–3)(4)
= 6(4)
= 24
PERIOD ____________
Multiply –2 and –3.
Multiply 6 and 4.
Exercises
Multiply.
1. –5(8)
2. –3(–7)
3. 10(–8)
4. –8(3)
5. –12(–12)
6. (–8)2
7. –5(7)
8. 3(–2)
9. –6(–3)
11. –4(–4)
12. 2(–3)(5)
-5 x 8= -40
10. 5(–4)(5)
-8 x -8= 64
Ch 3 Lesson 5 Reteach
Divide Integers
The quotient of two integers with different signs is negative.
The quotient of two integers with the same sign is positive.
Example 1
Find 30 ÷ (–5).
30 ÷ (–5)
The integers have different signs.
30 ÷ (–5) = –6
The quotient is negative.
Example 2
Find –100 ÷ (–5).
–100 + (–5)
The integers have the same sign.
–100 + (–5) = 20
The quotient is positive.
Exercises
Rules x ÷ : +number x/÷ + number = +answer
+ number x/÷ - number = - answer
- number x/÷ - number = + answer
- number x/÷ + number = - answer
Divide.
1. –12 ÷ 4
3.
= -3
2. –14 ÷ (–7) = 2
18
−2
line in a fraction means to divide top # by the bottom #
18 ÷ -2 = -9
−80
−20
5. –10 ÷ 10
6.
7. 350 ÷ (–25)
8. –420 ÷ (–3)
Course 2 • Chapter
4. –6 ÷ (–3)
NAME _____________________________________ DATE ____________________________
9.
540
45
10.
PERIOD ____________
−256
16
ALGEBRA Evaluate each expression if d = –24, e = –4, and f = 8.
11. 12 ÷ e 12 ÷ -4= -3
12. 40 ÷ f
13. d ÷ 6
14. d ÷ e -24 ÷ -4= 6
15. f ÷ e
16. e2 ÷ f -4^2
÷ 8=
-4 x -4 ÷ 8=
16
÷ 8= 2
Ch 4 Lesson 3 Reteach Add and Subtract Like Fractions
Like fractions are fractions that have the same denominator. To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Simplify if necessary.
Example 1
𝟑
𝟏
Find 𝟒 + 𝟒. Write in simplest form.
3
4
1
4
3+1
4
4
=4
+ =
Add the numerators.
Write the sum over
the denominator.
Simplify.
=1
Example 2
𝟐
𝟑
𝟏
𝟑
1
2−1
=
3
3
1
=
3
Find − . Write in simplest form.
2
3
−
Subtract the numerators.
Write the difference over
the denominator.
Exercises
Add or subtract. Write in simplest form.
Rule: add the top #; the bottom number remains the same as long as both fractions have the same bottom #.
5
1
5+1 6 6÷2
3
7
2
1. 8 + 8 = 8 = 8 = 8÷2 = 4
2. 9 − 9 7 b/c 9 – 2 b/c 9 = 5/9
Calculator directions:
5 b/c 8 + 1 b/c 8 = enter enter
1
3
3. − 4 + 4
5
5
5. 9 + 9
7
5
4. 8 − 8
3
1
6. − 8 − 8
Ch 4 Lesson 4 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.
Course 2 • Chapter
NAME _____________________________________ DATE ____________________________
PERIOD ____________
• If necessary, simplify the sum or difference.
Example
𝟐
𝟏
Find 𝟑 + 𝟒.
Method 1 Use a model.
2
3
1
+4
11
12
Method 2 Use the LCD.
2
1
2 4
1 3
+4 = 3•4+4•3
3
=
8
3
+ 12
12
or
Rename using the LCD, 12.
11
12
Add the fractions.
Exercises
Add or subtract. Write in simplest form.
Rule: Multiply or divide fractions to get an equivalent fraction. They must have same bottom number to add or subtract.
1
2
1. +
3.
3
4
7
+
15
1 b/c 2 + 3 b/c 4 enter enter
5
6
(− )
5
2. −
3
8
1
2
2
5
1
3
4. −
5
11
5. 9 + (− 12)
3
6. 12 − 4
Ch 4 Lesson 6 Reteach
Multiply Fractions
To multiply fractions, multiply the numerators and multiply the denominators.
5 3 5 × 3 15 1
× =
=
=
6 5 6 × 5 30 2
To multiply mixed numbers, rename each mixed number as an improper fraction. Then multiply the fractions.
2
1 8 5 40
1
2 ×1 = × =
=3
3
4 3 4 12
3
Example 1
𝟐
𝟒
Find 𝟑 × 𝟓. Write in simplest form.
2
3
4
2×4
× 5 = 3×5
=
8
15
Example 2
𝟏
← Multiply the
numerators.
Simplify.
← Multiply the
denominators.
𝟏
Find 𝟑 × 𝟐 𝟐. Write in simplest form.
1
3
1
2
1
5
3
2
1×5
= 3×2
5
=6
×2 = ×
1
5
2
2
Rename 2 as an improper fraction, .
Multiply.
Simplify.
Exercises
Multiply. Write in simplest form.
Course 2 • Chapter
NAME _____________________________________ DATE ____________________________
2
2
2𝑥2
4
1
1. 3 × 3 = 3𝑥3 = 9
7
1
2. 2 × 8
5
9
3
3. − 3 × 5
2
4. × 4
3
5 𝑥 −3
15
3
4
5. 1 × (− )
= (− ) = −1
3
5
3𝑥5
15
C: 1unit 2 b/c 3 x -3 b/c 5= 1
3
2
1
7. 4 × 1 3
PERIOD ____________
1
1
8. −3 3 × (−2 2)
Ch 4 Lesson 8 Reteach
6. 3 × 1
1
6
1
9. 4 5 × 7
Divide Fractions
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.
Example
𝟏
𝟐
Find 𝟑 𝟑 ÷ 𝟗. Write in simplest form.
1
2
10
2
÷9
3
10 9
= 3 •2
5
3
10 9
= •
3 2
1
1
1
33 ÷ 9 =
Rename 3 as an improper fraction.
3
2
9
9
2
Multiply by the reciprocal of , which is .
Divide out common factors.
= 15
Multiply.
Exercises Divide. Write in simplest form.
2
3
1
4
2
3
1. ÷ = . 𝑥
4
1
=.
2𝑥1
3𝑥4
=
2
12
=
1
6
2
5
2. ÷
5
6
1
2
3. − ÷
1
5
2 b/c 3 ÷ 1 b/c 4 = enter enter
1
4. 5 ÷ (− 2)
5
1
7. 6 ÷ 3 2
5
1
5. 8 ÷ 10
6. 7 3 ÷ 2
1
8. 36 ÷ 1 2
1
9. −2 2 ÷ (−10)
Ch 5 Lesson 1 Reteach Algebraic Expressions
To evaluate an algebraic expression you replace each variable with its numerical value, then use the order of
operations to simplify.
Example 1
Evaluate 6x – 7 if x = 8.
Course 2 • Chapter
NAME _____________________________________ DATE ____________________________
6x – 7 = 6(8) – 7
= 48 – 7
= 41
PERIOD ____________
Replace x with 8.
Use the order of operations.
Subtract 7 from 48.
Example 2
Evaluate 5m – 3n if m = 6 and n = 5.
5m – 3n = 5(6) – 3(5) Replace m with 6 and n with 5.
= 30 – 15
Use the order of operations.
= 15
Subtract 15 from 30.
Example 3
𝒂𝒃
𝟑
(7)(6)
Evaluate
𝑎𝑏
3
=
42
3
= 3
= 14
if a = 7 and b = 6.
Replace a with 7 and b with 6.
The fraction bar is like a grouping symbol.
Divide.
Example 4
Evaluate x3 + 4 if x = 3.
x3 + 4 = 3 3 + 4
Replace x with 3.
= 27 + 4
Use the order of operations.
= 31
Add 27 and 4.
Exercises
Evaluate each expression if a = 4, b = 2, and c = 7.
1. 3ac
2. 5b3
3. abc
𝑎𝑏
8
6. 2a – 3b
8. c – a
9. 20 – bc
10. 2bc
11. ac – 3b
12. 6a2
13. 7c
14. 6a – b
15. ab – c
4. 5 + 6c
7.
𝑏4
4
5.
Ch5 Lesson 2 Reteach Sequences
An arithmetic sequence is a list in which each term is found by adding the same number to the previous term.
1, 3, 5, 7, 9, …
Example 1
Course 2 • Chapter
NAME _____________________________________ DATE ____________________________
PERIOD ____________
Describe the relationship between terms in the arithmetic sequence 17, 23, 29, 35, … Then write the next three
terms in the sequence.
17, 23, 29, 35, ….
Each term is found by adding 6 to the previous term.
35 + 6 = 41
41 + 6 = 47
47 + 6 = 53
The next three terms are 41, 47, and 53.
Ch 5 L2Exercises
Describe the relationship between terms in the arithmetic sequences.
Write the next three terms in the sequence.
1. 2, 4, 6, 8, …
2. 4, 7, 10, 13, …
3. 0.3, 0.6, 0.9, 1.2, …
4. 200, 212, 224, 236, …
5. 1.5, 2.0, 2.5, 3.0, …
6. 12, 19, 26, 33, …
Lesson 4 Reteach
The Distributive Property
Distributive Property
Words
To multiply a sum or difference by number, multiply each term inside the
parentheses by the number outside the parentheses.
Symbols
Examples
a (b + c) = ab + ac
3( 2 + 5) = 3 · 2 + 3 · 5
a (b – c) = ab – ac
6(8 – 3) = 6 ∙ 8 - 6 ∙ 3
Examples
Use the Distributive Property to evaluate each expression.
1 5(x + 3)
5 (x + 3) = 5 ● x + 5  3
Expand using the Distributive Property
= 5x + 15
Simplify.
2 (4x – y)9
(4x – y) 9 = [4x + (–y)]9
= (4x)9 + (–y)9
= 36x + (–9y)
= 36x – 9y
Rewrite 4x – y as 4x + (–y).
Expand using the Distributive Property.
Simplify.
Definition of subtraction.
Example 3
MOVIES Alwyn is taking three of his friends to the movies. Tickets cost $8.90 per person. Find Alwyn’s total cost.
You can use the Distributive Property to find the total cost mentally
4 ($9 – $0.10) = 4 ($9) – 4 ($0.10)
= $36 – $0.40
= $35.60
Course 2 • Chapter
Distributive Property
Multiply.
Subtract.
NAME _____________________________________ DATE ____________________________
Ch 5 L 1 Algebraic Expressions Exercises
Evaluate each expression if a = 4, b = 2, and c = 7.
1. 3ac
2. 5b3
4. 5 + 6c
7.
𝑏4
4
3. abc
𝑎𝑏
8
6. 2a – 3b
8. c – a
9. 20 – bc
5.
Ch 5 L2 Sequences Exercises
Describe the relationship between terms in the arithmetic sequences.
Write the next three terms in the sequence.
1. 2, 4, 6, 8, …
4. 200, 212, 224, 236, …
2. 4, 7, 10, 13, …
3. 0.3, 0.6, 0.9, 1.2, …
5. 1.5, 2.0, 2.5, 3.0, …
6. 12, 19, 26, 33
Ch 5 L4 Distributive Property Exercises
Use the Distributive Property to evaluate or rewrite each expression.
1. 5(w + 4)
2. (x – 5)(–2)
3. 7 (6x – 2y)
4. –6(4 + 2m)
5. 8(2n + 7)
Simplify Fractions
9/12
6. (3v + 6w)2
15/3
14/5
Change Each fraction to a decimal
3/10
6/8
-1/8
7/8
-2/5
9/10
2 5/9
-8/9
Course 2 • Chapter
1/3
PERIOD ____________