2.3 multiplying and dividing integers

46 Chapter 2 Operations with Integers

Multiplying and Dividing Integers2.3

How can you multiply and divide integers?

Work with a partner. Use repeated addition to fi nd 3 ⋅ (−2).

3 ⋅ (−2) means to add 3 groups of −2.

0−1−2−3−4−5−6−7 1

−2 −2 −2 Now you can write

3 ⋅ (−2) = + +

= .

So, 3 ⋅ (−2) = .

ACTIVITY: Multiplying Integers with Different Signs11

Work with a partner. Use a table to fi nd −3 ⋅ (−2).

Describe the pattern of the products in the table. Then complete the table.

So, −3 ⋅ (−2) = .

−3 ⋅ 3 = −9

−3 ⋅ 2 = −6

−3 ⋅ 1 = −3

−3 ⋅ 0 =

−3 ⋅ −1 =

−3 ⋅ −2 =

ACTIVITY: Multiplying Integers with the Same Sign22

Integers In this lesson, you will● multiply integers.● divide integers.● solve real-life problems.

Look for PatternsHow can you use the pattern to complete the table?

Math Process

Work with a partner. Use integer counters to fi nd −15 ÷ 3.

Because there are negative counters in each group, −15 ÷ 3 = .

ACTIVITY: Dividing Integers Using Manipulatives33

Show how you can separate thecounters into 3 equal groups.

Begin with 15negative counters.

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15. IN YOUR OWN WORDS Is the product (or quotient) of two integers positive, negative, or zero? How can you tell?

16. STRUCTURE Write general rules for multiplying and dividing (a) two integers with the same sign and (b) two integers with different signs.

Section 2.3 Multiplying and Dividing Integers 47

Inductive ReasoningWork with a partner. Complete the table.

Use what you learned about multiplying and dividing integers to complete Exercises 8 –23 on page 50.

Work with a partner. Rewrite the product 3 ⋅ (−4) = −12 as a quotient in two different ways. What can you conclude?

First Way Second Way

−12 ÷ ( ) = −12 ÷ ( ) =

When you divide a integer by a integer, you get a

integer. When you divide a integer by a

integer, you get a integer.

ACTIVITY: Rewriting a Product as a Quotient44

Maintain OversightHow do you know what the sign will be when you divide two integers?

Math Process

ExerciseType of Product

or QuotientProduct or Quotient

Product or Quotient: Positive, Negative, or Zero

5. 6 ⋅ 3 Integers with the same sign

6. 2 ⋅ (−5)

7. −6 ⋅ 5

8. −5 ⋅ (−3)

9. 0 ⋅ (−5)

10. −6 ÷ 2

11. −21 ÷ (−7)

12. 10 ÷ (−2)

13. 12 ÷ (−6)

14. 0 ÷ (−15)


Lesson2.3


Multiply or divide.

a. −5 ⋅ (−6) b. −18 ÷ (−6)

−5 ⋅ (−6) = 30 −18 ÷ (−6) = 3

The product is 30. The quotient is 3.

EXAMPLE Multiplying and Dividing Integers with the Same Sign11

The product and quotient are positive.

Multiplying and Dividing Integers with the Same Sign

Words The product (or quotient) of two integers with the same sign is positive.

Numbers 8 ⋅ 2 = 16 −8 ⋅ (−2) = 16 8 ÷ 2 = 4 −8 ÷ (−2) = 4

Multiplying and Dividing Integers with Different Signs

Words The product (or quotient) of two integers with different signs is negative.

Numbers 8 ⋅ (−2) = −16 −8 ⋅ 2 = −16 8 ÷ (−2) = −4 −8 ÷ 2 = −4

Lesson Tutorials

The integers have the same sign.

Exercises 8 –23

Multiply or divide.

a. 3(−4) b. 75

— −25

3(−4) = −12 75

— −25

= −3

The product is −12. The quotient is −3.

Multiply or divide.

1. 5 ⋅ 5 2. −7(−8) 3. 12 ⋅ (−2)

4. 14 ÷ 2 5. −32

— −4

6. −40 ÷ (8)

EXAMPLE Multiplying and Dividing Integers with Different Signs22

The product and quotient are negative.

The integers have different signs.

RememberDivision by 0 is undefi ned.



Multiply.

a. (−3) ⋅ (2) ⋅ (−7)

(−3) ⋅ (2) ⋅ (−7) = −6 ⋅ (−7)

= 42

b. (−4) ⋅ (−5) ⋅ (−9)

(−4) ⋅ (−5) ⋅ (−9) = 20 ⋅ (−9)

= −180

Evaluate the expression.

7. (6)(2)(−1) 8. (−7)(3)(−7) 9. (8)(−3)(−2) 10. (6)(2)(−9)

EXAMPLE Multiplying Integers33

Exercises 24–29

The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for 4 years.

a. Find the total change in the number of taxis.

b. What is the mean monthly change in the number of taxis?

a. The bar graph shows that the number of taxis in service decreases by 60 each year. Use a model to solve the problem.

total change = change per year ⋅ number of years

= −60 ⋅ 4

= −240

The total change in the number of taxis is −240.

b. Use a model to solve the problem.

mean change per month = total change ÷ number of months

= −240 ÷ 48

= −5

The mean change in the number of taxis is −5 taxis per month.

11. A cardinal population decreases by 24 cardinals each year for 3 years. Find the total change in the cardinal population. Then fi nd the mean monthly change.

EXAMPLE Real-Life Application44

Use −60 for the change per year because the number decreases each year.

0 1 2 3 4

Year

60 fewer taxis

Taxis in Service

Nu

mb

er o

f ta

xis


9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=


Exercises2.3

1. WRITING What can you conclude about the signs of two integers whose product (or quotient) is (a) positive and (b) negative?

2. OPEN-ENDED Write two integers whose product and quotient are negative.

Tell whether the product or quotient is positive or negative without multiplying or dividing. Explain your reasoning.

3. 4(−8) 4. −5(−7) 5. −42 ÷ (14)

Tell whether the statement is true or false. Explain your reasoning.

6. The product of three positive integers is positive.

7. The product of three negative integers is positive.

Multiply or divide.

8. 6(4) 9. −6 ⋅ 9 10. 3 ⋅ 9 11. 8(−11)

12. −1(−13) 13. −9(9) 14. −10 ⋅ 11 15. 0 ⋅ (−4)

16. 4 ÷ (−2) 17. 21

— −7

18. −20 ÷ 4 19. −18 ÷ (−3)

20. −33 ÷ 11 21. −49

— −7

22. 0 —

−2 23.

60 —

−6

Multiply.

24. 3 ⋅ (−8) ⋅ (−2) 25. 6(−9)(−1)

26. −3(−5)(−4) 27. (−5)(−7)(−20)

28. −6 ⋅ 3 ⋅ (−2) 29. 3 ⋅ (−12) ⋅ 0

30. JOGGING You burn 10 calories each minute you jog. What integer represents the change in your calories after you jog for 20 minutes?

31. WETLANDS About 60,000 acres of wetlands are lost each year in the United States. What integer represents the change in wetlands after 4 years?

32. SNOWBOARDING A snowboarder descends a 1200-foot hill in 3 minutes. What is the mean change in elevation per minute?

33. ALLIGATORS An alligator population in a nature preserve in the Everglades decreases by 60 alligators over 5 years. What is the mean yearly change in the alligator population?

11 22

33

Help with Homework



ERROR ANALYSIS Describe and correct the error in evaluating the expression.

34. −2(−7) = −14✗

35. 0 ÷ (−5) = −5✗

NUMBER SENSE Find the next two numbers in the pattern.

36. −12, 60, −300, 1500, . . . 37. −128, 64, −32, 16, . . .

38. GOLF The table shows a golfer’s score for each round of a tournament.

a. What was the golfer’s total score?

b. What was the golfer’s mean score per round?

39. INLINE SKATES In June, the price of a pair of inline skates is $165. The price changes each of the next 3 months.

a. Copy and complete the table.

b. Describe the change in the price of the inline skates for each month.

c. The table at the right shows the amount of money you save each month to buy the inline skates. Do you have enough money saved to buy the inline skates in August? September? Explain your reasoning.

40. Write fi ve different integers that have a mean of −10. Explain how you found your answer.

Amount Saved

June $35

July $55

August $45

September $18

Graph the values on a number line. Then order the values from least to greatest.

41. −6, 4, ∣ 2 ∣ , −1, ∣ −10 ∣ 42. 3, ∣ 0 ∣ , ∣ −4 ∣ , −3, −8 43. ∣ 5 ∣ , −2, −5, ∣ −2 ∣ , −7

44. MULTIPLE CHOICE What is the prime factorization of 84?

○A 22 × 32 ○B 23 × 7 ○C 33 × 7 ○D 22 × 3 × 7

Month Price of Skates

June

July

August

September

165 = $165

165 + (−12) = $____

165 + 2(−12) = $____

165 + 3(−12) = $____

k

Scorecard

Round 1 −2

Round 2 −6

Round 3 −7

Round 4 −3


2.3 multiplying and dividing integers

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