1. Additional Example 1A: Multiplying Integers Using Repeated Addition
Use a number line to find each product.
2 · (–6)
Use the Commutative Property.
+ (–6) (–6)
Think: Add -6 two times.
2 · (–6) = –12

2. Additional Example 1B: Multiplying Integers Using Repeated Addition
Use a number line to find each product.
–8 · 3
–8 · 3 = 3 · (–8)
Use the Commutative Property.
+ (–8) (–8) (–8)
–24–23–22–21–20–19–18–
Think: Add –8 three times.
–8 · 3 = –24

3. Use a number line to find each product.
Check It Out: Example 1A
Use a number line to find each product.
3 · –4
+(–4) +(–4)
–12
+(–4)

4. Use a number line to find each product.
Check It Out: Example 1B
Use a number line to find each product.
–5 · 3
–15
–30 –25 –20 –15 –10 –
–5 · 3

5. Multiplication and division are inverse operations
Multiplication and division are inverse operations. They “undo” each other. Notice how these operations undo each other in the patterns shown.
Remember!

6. The patterns below suggest that when the signs of integers are different, their product or quotient is negative. The patterns also suggest that the product or quotient of two negative integers is positive.

7. If the signs are: Your answer will be: the same positive different
MULTIPLYING AND DIVIDING TWO INTEGERS
If the signs are:
Your answer will be:
the same
positive
different
negative

8. Additional Example 2: Multiplying Integers
Find each product.
A. –6 · (–5)
–6 · (–5)
Both signs are negative, so
the product is positive. 30
B. –4 · 7
–4 · 7
The signs are different, so
the product is negative.
-28

9. Check It Out: Example 2
Find each product.
A. –2 · (–8) 16
B. 5 · (–5)
–25

10. Check It Out: Example 2
Find each product.
C. –11 · 11
–121
D. –100 · (–28)
2,800

11. Additional Example 3: Dividing Integers
Find each quotient.
A. 35 ÷ (–5)
35 ÷ (–5)
Think: 35 ÷ 5 = 7. –7
The signs are different, so the
quotient is negative.
B. –32 ÷ (–8)
–32 ÷ (–8)
Think: 32 ÷ 8 = 4.
The signs are the same, so the
quotient is positive. 4

12. Additional Example 3: Dividing Integers
Find the quotient.
C. –48 ÷ 6
–48 ÷ 6
Think: 48 ÷ 6 = 8. –8
The signs are different, so the
quotient is negative.

13. Check It Out: Example 3
Find the quotient.
A. –12 ÷ 3 –4
B. –24 ÷ (–6) 4

14. Check It Out: Example 3
Find the quotient.
C. 18 ÷ (–9) –2

15. Zero divided by any number is zero, but you cannot find an answer for division by zero. For example –6 ÷ 0 ≠ 0, because 0 · 0 ≠ –6. We say that division by zero is undefined.

16. Additional Example 4: Diving Application
Roxy went scuba diving. She descended 14 feet from her original position. She then rose, going up in 2 increments of 4 feet each. Her final position was 16 feet below sea level. Find Roxy's original position.
Work backward to find Roxy's original position. Use
negative integers to represent positions below sea level.
Start with Roxy's final position. Because Roxy went up
in 2 increments of 4 feet each, subtract to find her
position before rising. 16

17. Additional Example 4 Continued
Roxy went scuba diving. She descended 14 feet from her original position. She then rose, going up in 2 increments of 4 feet each. Her final position was 16 feet below sea level. Find Roxy's original position.
Use the order of
operations.
–16 – (2 · 4) = –16 – 8
Add the opposite
of 8.
= –16 + (–8) = –24 ft
Then add the distance she descended.
– = –10 ft
Roxy's original position was 10 feet below sea level. 17

18. Find position before descending: –20 + (3 5) = –20 + 15 = –5 ft
Check It Out: Example 4
Sid went scuba diving. He rose 12 feet from his original position. He then descended, going down in 3 increments of 5 feet each. His final position was 20 feet below sea level. Find Sid’s original position.
Find position before descending:
–20 + (3 5) = – = –5 ft
Find position before rising:
–5 – 12 = –17 ft
Sid’s original position was 17 ft below sea level. 18

19. Lesson Quiz: Part I
Use a number line to find the product.
1. –8 · 2
Find each product or quotient.
2. –3 · 5 · (–2)
3. –75 ÷ 5
4. –110 ÷ (–2)
–16 30
–15 55

20. Lesson Quiz: Part II
5. The temperature in Bar Harbor, Maine, was –3 °F. It then dropped during the night to be four times as cold. What was the temperature then?
–12°F