1. Multiplying with Integers
Lesson 1.2.1

2. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
California Standard:
Number Sense 2.3
Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations.
What it means for you:
You’ll see what happens when you multiply positive and negative whole numbers.
Key words:
integer
product
factor

3. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
You’ve seen how to use a number line to show what happens when you add or subtract positive and negative integers.
–4…
…plus 9…
…equals 5.
In this Lesson you’ll see how it can be useful for doing multiplication problems too.

4. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Multiplication Is All About Grouping Things
Multiplication is really a way of adding together groups of objects. = +
There are 6 blocks in total, so 2 × 3 = 6.
For instance, 2 × 3 just means “2 groups of 3.”
There are still 6 blocks, so 3 × 2 = 6.
Doing “3 groups of 2” gives the same result. + =
You can do the same kind of grouping and counting on the number line.

5. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 1
Show the answer to 2 × 3 using a number line.
Solution follows…

6. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 1
Show the answer to 2 × 3 using a number line.
Solution
You can show the answer with 2 arrows, each of length 3: 3
2 × 3 is twice as far from 0 as 3 is. 6
You could show the same answer with 3 arrows of length 2: 2
3 × 2 is three times as far from 0 as 2 is. 6

7. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
What multiplication is shown on each number line?
3 × 7
10 × 2
4 × 4
Solution follows…

8. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Multiplying by a Negative Changes the Direction
Even if you’re multiplying by a negative, you’re still dealing with groups.
So 3 × (–2) still means “3 groups of –2.” 2
You can see from this number line that 3 × (–2) = –6.
Just like in Example 1, there are 3 arrows of length 2 on the number line, but this time the negative sign means they’re pointing left.

9. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 2
Calculate 4 × (–1).
Solution follows…

10. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 2
Calculate 4 × (–1).
Solution –1 –1 –1 –1
You can see from this number line that 4 × (–1) = –4.

11. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 3
The outside temperature at midnight was 0 °F. Every hour after that, the temperature dropped by 3 °F. What was the temperature at 5 a.m.?
Solution
The change in temperature is –3 °F each hour for five hours. So you need to solve 5 × (–3). –3
This shows that 5 × (–3) = –15, so at 5 a.m. the temperature was –15 °F.
Solution follows…

12. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
What multiplication is shown on each number line?
4 × (–10)
10 × (–4) 6.
What conclusion can you make from Exercises 4 and 5?
4 × (–10) is the same as 10 × (–4). It doesn’t matter which number the negative sign belongs to — the answer is the same.
Solution follows…

13. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
Calculate the following multiplications: 7.
7 × (–4)
–7 × 3
30 × (–2)
–2 × 30
–28 8.
–21 9.
–60
10.
–60
Solution follows…

14. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
11. A submarine changes its depth in the water by –25 feet per minute. What is its total change in depth in four minutes?
–25 × 4 = –100 feet
12. A bird is flying toward the ground. Its height changes by –16 feet per second. What is the bird’s total change in height in 5 seconds?
–16 × 5 = –80 feet
13. The amount of fuel in a racing car changes by –6 gallons per lap. What is the change in its fuel load over 7 laps?
–6 × 7 = –42 gallons
Solution follows…

15. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
A Negative Times a Negative Equals a Positive
You’ve already seen that multiplying a positive integer by a negative integer results in a negative solution.
–4 × 2 = –8
But if you multiply one negative number by another, their “–” signs cancel each other out.

16. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 4
Calculate –3 × (–2).
Solution
You know that 3 × (–2) means “3 groups of –2”, and 3 × (–2) = –6.
The extra negative sign in –3 × (–2) changes the sign again.
The answer must be positive: –3 × –2 = 6
Solution follows…

17. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
If you’ve got several negative integers to multiply, you can do it bit by bit.

18. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 5
Calculate –3 × (–2) × (–5).
Solution
[–3 × (–2)] × –5
Work it out in smaller parts
= 6 × (–5)
First multiply two of the numbers: –3 × (–2) = 6
= –30
Now, positive × negative = negative
Solution follows…

19. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
You can multiply any two integers using the following rules:
Rules for multiplying integers
For example:
positive × positive = positive
2 × 3 = 6
For example:
positive × negative = negative
2 × (–3) = –6
For example:
negative × positive = negative
(–2) × 3 = –6
For example:
negative × negative = positive
(–2) × (–3) = 6

20. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
You can use these rules even if you’re multiplying more than two numbers together.
Just count the number of “–” signs in the question.
Rules for multiplying integers
positive × positive = positive
positive × negative = negative
negative × positive = negative
negative × negative = positive
If there’s an even number of negative factors, they’ll cancel out in pairs, and the answer will be positive.
If there’s an odd number of negative factors, you’ll end up with one that doesn’t cancel out, so the final answer will be negative.

21. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 6
Solve –2 × 5 × (–4) × (–10).
Solution
–2 × 5 × (–4) × (–10) has three minus signs.
This is an odd number, so the answer will be negative.
Work out the “size” of the number by finding:
2 × 5 × 4 × 10 = 400
So the answer must be –400.
Solution continues…
Solution follows…

22. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Example 6
Solution (continued)
To prove that the answer is –400, you can break the question down into smaller parts:
–2 × 5 × (–4) × (–10)
= –10 × (–4) × –10
Negative × positive = negative
= 40 × (–10)
Negative × negative = positive
= –400
Now, positive × negative = negative

23. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.)
14.
–8 × (–3)
–2 × 9
2 × (–3) × (–5)
–27 × (–13) × (–7) × (–17)
Positive (even number of negative factors)
15.
Negative (odd number of negative factors)
16.
Positive (even number of negative factors)
17.
Positive (even number of negative factors)
Solution follows…

24. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Guided Practice
Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.)
18.
–6 × 11 × (–19) × (–83)
–1 × 2 × (–3) × 4 × (–5)
225 × (–311) × (–277) × (–1008) × 47 × (–119)
Negative (odd number of negative factors)
19.
Negative (odd number of negative factors)
20.
Positive (even number of negative factors)
Solution follows…

25. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Independent Practice
In Exercises 1–3, use a number line to solve the multiplication.
1. 5 × 7
2. –3 × 12
× 4 35
–36 44
Solution follows…

26. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Independent Practice
In Exercises 4–6, use a number line to solve the multiplication.
4. 6 × (–6)
× (–2)
6. –8 × (–3)
–36
21 × (–2) = –21 × 2 = –42
–8 × (–3) = 8 × 3 = 24
Solution follows…

27. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Independent Practice 7.
Ms. Ross is overdrawn on her bank account. Her balance is –$30. Mr. Banks is overdrawn on his bank account by 5 times the amount Ms. Ross is overdrawn. What is Mr. Banks’s account balance?
Sara multiplied two negative integers together. She then multiplied her answer by another negative number. Is her final result positive or negative?
Pablo multiplied two integers together. The answer that he got was –28. What integers might he have multiplied together?
–$150 8.
negative 9.
possible answers: 1 × (–28); –1 × 28; 2 × (–14); –2 × 14; 4 × (–7); –4 × 7
Solution follows…

28. Multiplying with Integers
Lesson
1.2.1
Multiplying with Integers
Round Up
It’s important to know what happens when you multiply by negative integers, because they appear in lots of math topics.
You’ll need the rules for multiplying again when you learn about dividing with negative integers in the next Lesson.