Lesson 15 Introduction Understand Multiplication as Scaling...138 Curriculum Associates, LLC Cop ying is not permitted. Comparing Factors and Products Lesson 15 Understand Multiplication

Introduction

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Think It Through

Lesson 15 Understand Multiplication as Scaling

Lesson 15Understand Multiplication as Scaling

What does scaling mean?

Think of how you use words and phrases such as “double,” “triple,” “half of,” or “take one tenth.” These words and phrases describe scaling, or changing the size of a quantity. Stretching and shrinking are two different ways to scale a quantity.

The table below shows some ways that a quantity of 6 can be scaled.

Words Symbols

6 doubled is 12. 2 3 6 5 12

6 tripled is 18. 3 3 6 5 18

Half of 6 is 3. 1 ·· 2 3 6 5 3

A tenth of 6 is 6 ·· 10 . 1 ·· 10 3 6 5 6 ·· 10

stretching

shrinking

Below is a rectangle with an area of 6 square units.

The model for 2 3 6 has an area that is double the size of the original rectangle.

The model for 1 ·· 2 3 6 has an area that is half the size

of the original rectangle.

Think How can you use models to show what scaling means?

Circle the factor that describes how the rectangle is being stretched or shrunk.

5.NF.B.5a

5.NF.B.5b

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Products aren’t always greater than their factors. The table below shows products of different factors multiplied by 6.

Factor 1 ·· 10 1 ·· 3 1 ·· 2 5 ·· 6 1 4 ·· 3 2 2 1 ·· 2 3

Product of factor and 6

6 ··· 10 2 3 5 6 8 12 15 18

Notice that the products are sometimes less than 6, sometimes greater than 6, and sometimes equal to 6.

What do the products less than 6 have in common? The other factor is less than 1.

What do the products greater than 6 have in common? The other factor is greater than 1.

• If you multiply 6 by a factor less than 1, the product will be less than 6.

• If you multiply 6 by a factor greater than 1, the product will be greater than 6.

• If you multiply 6 by 1, or a factor equivalent to 1, the product will be 6.

Reflect1 Describe the products you get if you multiply 8 by factors less than 1. Describe the

products you get if you multiply 8 by factors greater than 1. Give some examples that justify your answers.

Think How does the size of the factors affect the product?

Look at the products that are less than 6, then look at those that are greater than 6. What do you notice about the factors?

Guided Instruction

Think About


Comparing Factors and Products


Lesson 15

2 You can show 1 ·· 3 3 3 ·· 4 on a number line. If you break up 3 ·· 4 into 3 equal parts,

each part is 1 ·· 4 .

0 1 21

434

1 ·· 3 3 3 ·· 4 5

Is the product less than, greater than, or equal to 3 ·· 4 ?

3 Show 2 ·· 3 3 3 ·· 4 on a number line. If you break up 3 ·· 4 into 3 equal parts, each part is 1 ·· 4 .

Since you multiply by 2 ·· 3 , you need 2 of those parts. Shade and label 2 ·· 3 of 3 ·· 4 .

0 1 23

4

2 ·· 3 3 3 ·· 4 5


4 You multiplied 3 ·· 4 by two different factors. How do both factors relate to the number 1?

What happens when you multiply a given fraction by a factor less than 1?

Let’s Explore the Idea A number line can help you see what happens when a fraction is multiplied by a factor less than 1.

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Let’s Talk About It A number line can also help you see what happens when a fraction is multiplied by a factor greater than 1.

5 Shade the number line to show 4 ·· 3 3 3 ·· 4 .

0 1 23

4

4 ·· 3 3 3 ·· 4 5


6 Shade and label the number line to show 7 ·· 3 3 3 ·· 4 .

0 1 23

4

7 ·· 3 3 3 ·· 4 5


7 Think about how each of your answers compared to 3 ·· 4 . What can you say about the

product of a given fraction and a factor greater than 1?

Try It Another Way Explore multiplying 3 ·· 4 by a fraction using an area model.

The model to the right represents 3 ·· 4 .

34

8 Show 1 ·· 2 3 3 ·· 4 using the area model.

9 1 ·· 2 3 3 ·· 4 5

10 Is the product less than, greater than, or equal to 3 ·· 4 ?

11 Could you have answered problem 10 without drawing a model? Explain.

Guided Practice

Connect


Lesson 15


Ideas About Factors and Products

Talk through these problems as a class, then write your answers below.

12 Analyze Use reasoning to order the following expressions from least to greatest. Don’t calculate any of the products. Explain your reasoning.

7 ·· 9 3 348,980 12 ··· 11 3 348,980 50 ·· 50 3 348,980

13 Explain Gillian said that the product of a given number and a fraction is always less than the given number. Explain what is wrong with Gillian’s statement and give an example that does not follow her rule.

14 Compare Represent the expression 4 ·· 4 3 8 ·· 5 with a model. Write a sentence

comparing the product with 8 ·· 5 . Explain your reasoning.

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Independent Practice

Apply

Lesson 15


Ideas About Factors and Products

15 Put It Together You can compare the size of a product to the size of the factors in a multiplication equation if you know whether the factors are greater than, less than, or equal to 1.

Part A Write a multiplication equation (different from any in this lesson) in which the product is greater than both of the factors. Draw a model to support your answer.

Part B Write a multiplication equation (different from any in this lesson) in which both the factors are fractions and the product is less than both of the factors. Draw a model to support your answer.

Part C Write a multiplication equation (different from any in this lesson) in which the product is equal to one of the factors.

Lesson 15 Introduction Understand Multiplication as Scaling...138 Curriculum Associates, LLC Cop ying is not permitted. Comparing Factors and Products Lesson 15 Understand Multiplication

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