Multiplication and Division of
Positive and Negative Numbers
Objective To develop and apply rules for multiplying and
dividing positive and negative numbers.
d
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Teaching the Lesson
Key Concepts and Skills
• Apply previous knowledge of fact families. [Operations and Computation Goal 2]
• Use the order of operations to
solve problems. [Patterns, Functions, and Algebra Goal 3]
• Apply properties of multiplication
(Commutative Property and
Multiplication Property of –1). [Patterns, Functions, and Algebra Goal 4]
Key Activities
Students study patterns to derive rules
for multiplying and dividing positive and
negative numbers.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Top-It with Positive
and Negative Numbers
Student Reference Book, pp. 337
and 338
Math Masters, p. 478
per partnership: complete deck of
number cards (the Everything Math
Deck, if available), calculator (optional)
Students practice adding
and subtracting positive and
negative numbers.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Masters, page 478. [Operations and Computation Goal 1]
Key Vocabulary
Math Boxes 6 4
Multiplication Property of –1
Math Journal 2, p. 217
Students practice and maintain skills
through Math Box problems.
Materials
Math Journal 2, pp. 214–216
Study Link 6 3
calculator (optional) slate (optional)
Study Link 6 4
Math Masters, p. 190
Students practice and maintain skills
through Study Link activities.
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 71–74, 100–102
548
Unit 6
Number Systems and Algebra Concepts
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Using a Multiplication Story
Math Masters, p. 191
Students use situations from a story to
remember a multiplication property for
positive and negative numbers.
ENRICHMENT
Exploring Patterns in Products
and Quotients
Math Masters, p. 192
Students explore patterns in products and
quotients of positive and negative numbers.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, p. 234
Students practice finding products and
quotients of positive and negative numbers.
Mathematical Practices
SMP4, SMP6, SMP7, SMP8
Content Standards
Getting Started
6.NS.5
Mental Math and Reflexes
Math Message
Students solve subtraction problems with
signed numbers.
Complete the problems on journal page 214.
Suggestions:
Study Link 6 3 Follow-Up
-5 - (-5) 0
9 - (-1) 10
-12 - 5 -17
25 - 12 13
5 - (-23) 28
-15 - (-20) 5
Go over the answers with the class.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
ELL
(Math Journal 2, p. 214)
Have students work with a partner to look for patterns in each set
of number sentences. Ask volunteers to share their findings.
Sample answers: One factor remains the same while the other
factor decreases by 1. The products in each set also decrease in a
pattern. Write key ideas and examples on the board to support
English language learners.
Be sure that students have circled the correct words in the
multiplication rules in Problems 1–4. Discuss the Multiplication
Property of -1 in Problem 5. (The product of a number and -1 is
the opposite of the number.) Write additional examples on the
board. Point out that the Commutative Property (turn-around rule)
is also true for multiplication with negative numbers. For example,
5 ∗ (-2) = -2 ∗ 5 = -10. Ask students to write a few examples of
the turn-around rule for multiplication with negative numbers.
Spend a few minutes practicing multiplication. Pose problems that
students can solve mentally. They should write the products on
their slates or on scratch paper and then check their answers with
a partner.
Student Page
Date
6 4
䉬
2
Multiplication Patterns
In each of Problems 1–4, complete the patterns in Part a. Check your answers
with a calculator. Then circle the word in parentheses that correctly completes
the statement in Part b.
1. a.
6 ⴱ 4 24
6 ⴱ 3 18
6ⴱ2
6ⴱ1
Suggestions:
8 ∗ (-6) -48
Time
LESSON
6ⴱ0
6 -_
2 ∗_
4
-_
5
3
5
(-8) 64
1.2 ∗ (-5) -6
-3 ∗ 5 -15
0.5 ∗ (-2) -1
2. a.
b.
2ⴱ36
1ⴱ33
0ⴱ30
1 ⴱ 3 2 ⴱ 3 4. a.
b.
(positive or negative) number.
b.
(positive or negative) number.
4 ⴱ 1 4
4 ⴱ 0 0
4 ⴱ (1) 4
Solve.
1 ⴱ 6 1 ⴱ (7.7) 1
1 ⴱ (2) 1 ⴱ m Negative º Positive Rule:
When a negative number
is multiplied by a positive
number, the product is a
3
6
4 ⴱ (3) Positive º Negative Rule:
When a positive number
is multiplied by a negative
number, the product is a
5
10
4 ⴱ (2) 5. a.
(positive or negative) number.
5 ⴱ 2 10
5ⴱ15
5ⴱ00
5 ⴱ (2) 3. a.
Positive º Positive Rule:
When a positive number
is multiplied by a positive
number, the product is a
12
6
0
5 ⴱ (1) 97
b.
When a negative number
is multiplied by a negative
number, the product is a
8
12
6
7.7
12
m
Negative º Negative Rule:
(positive or negative) number.
b.
Multiplication Property of ⴚ1:
For any number a,
1 ⴱ a a ⴱ 1 OPP(a), or a. The
number a can be a negative number, so
OPP(a) or a can be a positive number. For
example, if a 5, then a OPP(5) 5.
Math Journal 2, p. 214
Lesson 6 4
549
Student Page
Date
Time
LESSON
6 4
䉬
Fact Families for Multiplication and Division
▶ Deriving Rules for Division
of Positive and Negative Numbers
A fact family is a group of four basic, related multiplication and division facts.
Example: The multiplication and division fact family for
6, 3, and 18 is made up of the following facts:
6 ⴱ 3 18
3 ⴱ 6 18
18 / 6 3
18 / 3 6
a.
5 ⴱ (3) 3 ⴱ 5 15 / (3) 15 / 5 d.
15
15
5
3
b.
8 ⴱ 6 48 / (8) 48 / 6 Negative / Negative Rule:
When a negative number
is divided by a negative
number, the quotient is a
a.
c.
3.
c.
Students use fact families to derive rules for dividing positive and
negative numbers.
5 ⴱ (5) 25
5 ⴱ 5 25
25 / 5 5
25 / 5 5
e.
Negative / Positive Rule:
When a negative number
is divided by a positive
number, the quotient is a
b.
7 ⴱ (5) d.
Positive / Negative Rule:
When a positive number
is divided by a negative
number, the quotient is a
(positive or negative) number.
2.
48
48
6
8
6 ⴱ (8) Review the example at the top of journal page 215. Then do
Problem 1a as a class. Students can apply multiplication rules to
solve the first two problems: 5 ∗ (-3) = -15 and -3 ∗ 5 = -15.
The three numbers used to build this fact family are 5, -3, and
-15. Students can use this information to complete the division
facts in the family.
(positive or negative) number.
12
12
3 ⴱ (4) 4
12 / (3) 3
12 / (4) 20
2 ⴱ (10) 10 ⴱ (2) 20
20 / (10) 2
20 /(2) 10
4 ⴱ (3) 35
5 ⴱ (7) 35
35 / (5) 7
35 / (7) 5
(positive or negative) number.
Adjusting the Activity
Solve. Check your answers with a calculator.
b.
6
9
e.
81 / 9 a.
ⴱ (4) 24 (Think: What number multiplied by 4 is equal to 24?)
ⴱ 9 81
9
8
c.
6 ⴱ
f.
48 / (6) 48
8
9
d.
g.
9
Math Journal 2, p. 215
Multiplication fact family for 7, -8, and -56:
A U D I T O R Y
-
7
K I N E S T H E T I C
T A C T I L E
-56
54 / (-9) -6
-49 / 7 -7
∗
-60 / (-12) 5
(-8)2 / (-8) -8
-8
V I S U A L
Review the rules for division. Discuss the relationship between
multiplication and division as shown in Problem 3; that is, any
division problem can be solved in terms of multiplication. For
example, to solve 35 / (-5), think: What number, multiplied by -5,
is equal to 35? -7
-
7 ∗ (-8) = -56 -56 / 7 = -8
-8 ∗ 7 = -56
Spend a few minutes practicing division.
Suggestions:
,/
+
ELL
Consider using fact triangles with both positive and negative numbers to
show patterns. For example, draw a fact triangle for 7, -8, and -56 on the
board. Draw large + (positive) and - (negative) signs at the corners for
emphasis. Discuss the pattern of signs for products and quotients. Repeat for
other examples, such as -6, 8, and -48 and -3, -4, and 12. (See margin.)
ⴱ (3) 27
27 / (3) PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
(Math Journal 2, p. 215)
As you already know, when a positive number is divided by a positive number, the
quotient is a positive number. Problems 1 and 2 will help you discover the rules for
division with negative numbers. Complete the fact families. Check your answers
with a calculator. Then complete each rule.
1.
INDEPENDENT
ACTIVITY
-56 / -8 = 7
Students complete journal page 215 independently. Partners
then compare and check answers.
▶ Multiplying and Dividing Positive
INDEPENDENT
ACTIVITY
and Negative Numbers
(Math Journal 2, p. 216)
Review the rules for multiplication and division at the top of
journal page 216. Have students complete the journal page
independently. Partners then compare and check answers.
Have students generate real-life situations that could be modeled
by some of the problems on journal page 216. For example,
Problem 1 might model this situation: Your parents deduct $7.00
from your monthly allowance for 8 months to help you save for a
new bicycle. -$7 ∗ 8 = -$56
550
Unit 6
Number Systems and Algebra Concepts
Student Page
Date
2 Ongoing Learning & Practice
▶ Playing Top-It with Positive
º, / of Positive and Negative Numbers
6 4
䉬
A Division Property
A Multiplication Property
97
For all numbers a and b, if the values
of a and b are both positive or both
negative, then the quotient a / b is a
positive number. If one of the values is
positive and the other is negative, then
the quotient a / b is a negative number.
For all numbers a and b, if the values
of a and b are both positive or both
negative, then the product a ⴱ b is a
positive number. If one of the values is
positive and the other is negative, then
the product a ⴱ b is a negative number.
PARTNER
ACTIVITY
and Negative Numbers
Time
LESSON
Solve. Use a calculator to check your answers.
(Student Reference Book, pp. 337 and 338;
Math Masters, p. 478)
Distribute a complete deck of number cards (the Everything Math
Deck, if available) to each partnership.
7 ⴱ 8 56
700 (10) 70
5. 0.5 ⴱ (15) 7.5
84
7. 3 ⴱ 4 ⴱ (7) 6
9. 54 / 9 2.
3.
1
4. 2
3
11. 5
13.
Students use cards to practice comparing, adding, and subtracting
positive and negative numbers.
15.
17.
19.
21.
Ongoing Assessment:
Recognizing Student Achievement
Math Masters
Page 478
4
(5)
6.
8.
3
4
90 15 6
9 11
99 /
3 ⴱ (4 6) 6
(9 ⴱ 4) 6 30
(6 ⴱ 3) (6 ⴱ 5) 48
27.
1
2
ⴱ 3.3 3.3
ⴱ (8) ⴱ (3) 48
3
12.
45 / (5) / (3) 72 / (8) 1
16. 2
1
0
y (7 3) (8 ⴱ y ) 6 26
3,285
38
4
36 / (12) 14.
3 (y) ⴱ (3) 10.
(3) 4
3
9
23
22.
4
35
(4 ⴱ (7)) (4 ⴱ (3)) 16
24.
y / (6) 18.
32 (5 3) 20.
(75 / 5) (20) Evaluate each expression for y 4.
23.
25.
Use the Top-It record sheet to assess students’ abilities to add and subtract
positive and negative numbers. Students are making adequate progress if
they can calculate and compare sums and differences of signed numbers.
73 ⴱ (45) 1.
26.
23
y (y 2) 2
28.
(8 ⴱ 6) (8 ⴱ y) 80
Math Journal 2, p. 216
[Operations and Computation Goal 1]
▶ Math Boxes 6 4
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 217)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-2. The skills in Problems 5 and 6
preview Unit 7 content.
▶ Study Link 6 4
INDEPENDENT
ACTIVITY
(Math Masters, p. 190)
Student Page
Date
Time
LESSON
Home Connection Students practice multiplication and
division of positive and negative numbers.
Math Boxes
6 4
䉬
1.
Write the reciprocal.
a.
3
2
2
b. 3
3 Differentiation Options
3.
2.
1
5
5
c.
2 4
d.
9.64
7
7
18
25
241
Divide. Simplify if possible.
3
a. 5
1 5
b. 6
2 c.
d.
93
Lines a and b are parallel. Without using
a protractor, find the degree measure of
each numbered angle. Write each
measure on the drawing.
4.
12
25
5
14
1
4
1
3
18
225
2
4
5
1
2
6 2
2
5
1
5
20 10 93
1
If 1 kilogram (kg) is about 2 pounds (lb),
5
then 5 kg is about
lb.
Circle the best answer.
11
A. 25
READINESS
▶ Using a Multiplication Story
SMALL-GROUP
ACTIVITY
58ⴗ 1
122ⴗ
5–15 Min
122ⴗ
4
58ⴗ
a
58ⴗ 5 6 122ⴗ
122ⴗ 7 58ⴗ
b
2
3
(Math Masters, p. 191)
7
C.
10
D.
11
371
163
5.
To provide experience with rules for multiplying positive and
negative numbers, have students read Math Masters, page 191.
They use the characters and situations in a familiar story to help
them remember a multiplication property for multiplication of
positive and negative numbers.
1
5
B.
Express each decimal as a percent.
a.
b.
c.
d.
6.
104%
8.25% 0.0825
0.0035 0.35%
400% 4.0
1.04 Suppose you toss a penny and a nickel
together. How many equally likely
outcomes are there?
Complete the table.
Penny
60
4
Nickel
H
H
H
T
T
T
H
T
equally likely outcomes
150–153
Math Journal 2, p. 217
Lesson 6 4
551
Study Link Master
Name
Date
STUDY LINK
Time
䉬
A Multiplication Property
A Division Property
䉬 The product of two numbers with
the same sign is positive.
䉬 The quotient of two numbers with
the same sign is positive.
䉬 The product of two numbers with
different signs is negative.
䉬 The quotient of two numbers with
different signs is negative.
97
▶ Exploring Patterns in
60
2. 63 / 7 (Math Masters, p. 192)
9
3. 24 (4) 6
4. 9 º
5
10
6. 6 º 5 º 8 5. 50 /
6
7. 48 / (6 2) 2
13.
( )
54
º (7) º (4) 56
Try This
5
2
15. º 3
6
59
162
10. 6 º (12 15) 12. 56 / (7) / (4) 14.
360
18.
To further explore patterns in products and quotients,
students multiply and divide by signed numbers that have
repeating digits and examine the resulting patterns. They
use these patterns to predict products and quotients.
28
2
40 9
16. (8 º (3)) (8 º (9)) 2
17. 0.25 º (8) 54
240
8. (8 º 5) 12 1,150
9. 50 º (23) 11. (90 10) (45) 6
48
c. b / (14) 36
0.5
b. 11 º (b) d. b (b 16) 77
16
SMALL-GROUP
ACTIVITY
EXTRA PRACTICE
1
(34) (12) 12
19. Evaluate each expression for b 7.
a. (9 º b) 27 15–30 Min
Products and Quotients
Solve.
1. 12 º 5 INDEPENDENT
ACTIVITY
ENRICHMENT
º, / of Signed Numbers
64
▶ 5-Minute Math
15–30 Min
To offer more practice finding products and quotients of positive
and negative numbers, see 5-Minute Math, page 234.
Math Masters, p. 190
Teaching Master
Teaching Master
Name
LESSON
64
䉬
Date
Time
A Multiplication Story
In many fairy tales and children’s stories, there are good characters and bad characters.
For example, in the story “Little Red Riding Hood,” the grandmother is a good character;
the wolf is a bad character.
You can use these character situations to remember a multiplication property for positive
and negative numbers.
䉬 When something good () happens to a good () character, we think it is good ().
Name
LESSON
64
䉬
Date
Time
Patterns with Signed Numbers
1. Multiply each number in the far left column by each number in the top row. Look for patterns.
Use your calculator as few times as possible to complete the table.
ⴱ
11
1,111
111
11
121
1,221
12,221
111
1,221
12,321
123,321
1,111
12,221
123,321
1,234,321
11,111
122,221
1,233,321
12,344,321
2. Use the patterns from the table above to predict the products below. Then check each
䉬 When something bad () happens to a good () character, we think it is bad ().
prediction with your calculator.
ⴱ
䉬 When something good () happens to a bad () character, we think it is bad ().
11
Prediction
111,111
Answers vary.
Actual
111,111
1,222,221
111
12,333,321
3. Divide each number in the far left column by each number in the top row. Look for
patterns. Use your calculator as few times as possible to complete the table. Write your
own number pattern in last row.
Divisor
99
999
9,999
0.0
苶1
苶2
苶
0.0
苶0
苶1
苶2
苶
0.0
苶0
苶3
苶4
苶
12
0.1
苶2
苶
䉬 When something bad () happens to a bad () character, we think it is good ().
py g
g
p
兾
34
0.3
苶4
苶
0.0
苶3
苶4
苶
45
0.4
苶5
苶
0.0
苶4
苶5
苶
0.0
苶0
苶4
苶5
苶
67
0.6
苶7
苶
0.0
苶6
苶7
苶
0.0
苶0
苶6
苶7
苶
Answers vary.
Example: Solve 4 ⴱ 5 ? Think, “When something bad (4) happens to a
good (5) character, we think it is bad ().” So, 4 ⴱ 5 20.
Math Masters, p. 191
552
Unit 6
Number Systems and Algebra Concepts
Math Masters, p. 192