1 Lesson 1.2.1 Multiplying with Integers. 2 Lesson 1.2.1 Multiplying with Integers California Standard: Number Sense 2.3 Solve addition, subtraction,

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Lesson 1.2.1Lesson 1.2.1

Multiplying with IntegersMultiplying with Integers

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Lesson

1.2.1Multiplying with IntegersMultiplying with Integers

California Standard:Number Sense 2.3Solve 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

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You’ve seen how to use a number line to show what happens when you add or subtract positive and negative integers.

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1.2.1

In this Lesson you’ll see how it can be useful for doing multiplication problems too.

–4… …plus 9… …equals 5.

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Multiplication Is All About Grouping Things

Multiplication is really a way of adding together groups of objects.

Lesson

1.2.1

You can do the same kind of grouping and counting on the number line.

For instance, 2 × 3 just means “2 groups of 3.”

There are 6 blocks in total, so 2 × 3 = 6.

=+

Doing “3 groups of 2” gives the same result.

There are still 6 blocks,

so 3 × 2 = 6.

+ =+

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Example 1

Solution follows…

Lesson

1.2.1

Show the answer to 2 × 3 using a number line.

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Example 1

Lesson

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You can show the answer with 2 arrows, each of length 3:

Solution

You could show the same answer with 3 arrows of length 2:

3 × 2 is three times as far from 0 as 2 is.

2 × 3 is twice as far from 0 as 3 is.

Show the answer to 2 × 3 using a number line.

3 3

6

6

2 22

7

1.

2.

3.


Guided Practice

Solution follows…

Lesson

1.2.1

What multiplication is shown on each number line?

3 × 7

10 × 2

4 × 4

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Multiplying by a Negative Changes the Direction

Even if you’re multiplying by a negative, you’re still dealing with groups.

Lesson

1.2.1

So 3 × (–2) still means “3 groups of –2.”

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.

You can see from this number line that 3 × (–2) = –6.

2 22

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Example 2

Calculate 4 × (–1).

Solution follows…

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Example 2

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You can see from this number line that 4 × (–1) = –4.

Solution–1–1 –1 –1

Calculate 4 × (–1).

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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 follows…

Lesson

1.2.1

The change in temperature is –3 °F each hour for five hours. So you need to solve 5 × (–3).

Solution

This shows that 5 × (–3) = –15, so at 5 a.m. the temperature was –15 °F.

–3–3 –3 –3–3

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Guided Practice

Solution follows…

Lesson

1.2.1

4.

What multiplication is shown on each number line?

4 × (–10)

What conclusion can you make from Exercises 4 and 5?

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.

10 × (–4)

6.

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Guided Practice

Solution follows…

Lesson

1.2.1

7 × (–4)

–7 × 3

30 × (–2)

–2 × 30

Calculate the following multiplications:

–28

–21

–60

–6010.

9.

8.

7.

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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?


Guided Practice

Solution follows…

Lesson

1.2.1

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

–16 × 5 = –80 feet

–6 × 7 = –42 gallons

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?

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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.

But if you multiply one negative number by another, their “–” signs cancel each other out.

Lesson

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–4 × 2 = –8

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Example 4

Calculate –3 × (–2).

Solution follows…

Lesson

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You know that 3 × (–2) means “3 groups of –2”, and 3 × (–2) = –6.

Solution

The extra negative sign in –3 × (–2) changes the sign again.

The answer must be positive: –3 × –2 = 6

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If you’ve got several negative integers to multiply, you can do it bit by bit.

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Example 5

Calculate –3 × (–2) × (–5).

Solution follows…

Lesson

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[–3 × (–2)] × –5

Solution

= 6 × (–5)

= –30

Work it out in smaller parts

Now, positive × negative = negative

First multiply two of the numbers: –3 × (–2) = 6

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You can multiply any two integers using the following rules:

Lesson

1.2.1

Rules for multiplying integers

positive × positive = positive

positive × negative = negative

negative × positive = negative

negative × negative = positive

2 × 3 = 6

(–2) × 3 = –6

(–2) × (–3) = 6

2 × (–3) = –6

For example:

For example:

For example:

For example:

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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.

Multiplying with IntegersMultiplying with IntegersLesson

1.2.1

You can use these rules even if you’re multiplying more than two numbers together.

Just count the number of “–” signs in the question.

If there’s an even number of negative factors, they’ll cancel out in pairs, and the answer will be positive.

Rules for multiplying integers

positive × positive = positive

positive × negative = negative

negative × positive = negative

negative × negative = positive

21Solution continues…


Example 6

Solve –2 × 5 × (–4) × (–10).

Solution follows…

Lesson

1.2.1

Work out the “size” of the number by finding:

2 × 5 × 4 × 10 = 400

This is an odd number, so the answer will be negative.

–2 × 5 × (–4) × (–10) has three minus signs.

Solution

So the answer must be –400.

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Example 6

Lesson

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Solution (continued)

= –400 Now, positive × negative = negative

= 40 × (–10) Negative × negative = positive

–2 × 5 × (–4) × (–10)

= –10 × (–4) × –10 Negative × positive = negative

To prove that the answer is –400, you can break the question down into smaller parts:

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Guided Practice

Solution follows…

Lesson

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–8 × (–3)

–2 × 9

2 × (–3) × (–5)

–27 × (–13) × (–7) × (–17)

Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.)

Negative (odd number of negative factors)

Positive (even number of negative factors)


Positive (even number of negative factors)14.

15.

16.

17.

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Guided Practice

Solution follows…

Lesson

1.2.1

–6 × 11 × (–19) × (–83)

–1 × 2 × (–3) × 4 × (–5)

225 × (–311) × (–277) × (–1008) × 47 × (–119)

Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.)

Negative (odd number of negative factors)


Negative (odd number of negative factors)18.

19.

20.

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1. 5 × 7

2. –3 × 12

3. 11 × 4


Independent Practice

Solution follows…

Lesson

1.2.1

In Exercises 1–3, use a number line to solve the multiplication.

35

–36

44

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4. 6 × (–6)

5. 21 × (–2)

6. –8 × (–3)



Solution follows…

Lesson

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In Exercises 4–6, use a number line to solve the multiplication.

–36

21 × (–2) = –21 × 2 = –42

–8 × (–3) = 8 × 3 = 24

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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?



Solution follows…

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–$150

negative

possible answers: 1 × (–28); –1 × 28; 2 × (–14); –2 × 14; 4 × (–7); –4 × 7

7.

8.

9.

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Round UpRound 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.

Lesson

1.2.1Multiplying with IntegersMultiplying with Integers

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1 Lesson 1.2.1 Multiplying with Integers. 2 Lesson 1.2.1 Multiplying with Integers California Standard: Number Sense 2.3 Solve addition, subtraction,