Lesson 16Methods for Multiplying Decimals

Learning Goal

Let’s look at some ways we can represent multiplication of decimals.

Learning Targets

I can use area diagrams to represent and reason about multiplication of decimals.
I can use place value and fractions to reason about multiplication of decimals.

Warm Up: Multiplying by 10

Problem 1

In which equation is the value of $x$ the largest?

$x \cdot 10 = 810$

$x \cdot 10 = 81$

$x \cdot 10 = 8.1$

$x \cdot 10 = 0.81$

Problem 2

How many times the size of 0.81 is 810?

Activity 1: Fractionally Speaking: Powers of Ten

Problem 1

Work with a partner to answer the following questions. One person should answer the questions labeled “Partner A,” and the other should answer those labeled “Partner B.” Then compare the results.

Find each product or quotient. Be prepared to explain your reasoning.
Partner A
- $250 \cdot \frac{1}{10}$
- $250 \cdot \frac{1}{100}$
- $48 \div 10$
- $48 \div 100$
Partner B
- $250 \div 10$
- $250 \div 100$
- $48 \cdot \frac{1}{10}$
- $48 \cdot \frac{1}{100}$
Use your work in the previous problems to find $720 \cdot (0.1)$ and $720 \cdot (0.01)$ . Explain your reasoning.

Problem 2

Find each product. Show your reasoning.

$36 \cdot (0.1)$
$(24.5) \cdot (0.1)$
$(1.8) \cdot (0.1)$
$54 \cdot (0.01)$
$(9.2) \cdot (0.01)$

Problem 3

Jada says: “If you multiply a number by 0.001, the decimal point of the number moves three places to the left.” Do you agree with her? Explain your reasoning.

Activity 2: Using Properties of Numbers to Reason About Multiplication

Problem 1

Elena and Noah used different methods to compute $(0.23) \cdot (1.5)$ . Both computations were correct.

Elena and Noah's method of computing (0.23) times (1.5).

Analyze the two methods, then discuss these questions with your partner.

Which method makes more sense to you? Why?
What might Elena do to compute $(0.16) \cdot (0.03)$ ? What might Noah do to compute $(0.16) \cdot (0.03)$ ? Will the two methods result in the same value?

Problem 2

Compute each product using the equation $21 \cdot 47 = 987$ and what you know about fractions, decimals, and place value. Explain or show your reasoning.

$(2.1) \cdot (4.7)$
$21 \cdot (0.047)$
$(0.021) \cdot (4.7)$

Activity 3: Connecting Area Diagrams to Calculations with Decimals

Problem 1

You can use area diagrams to represent products of decimals. Here is an area diagram that represents $(2.4) \cdot (1.3)$ .

Find the region that represents $(0.4) \cdot (0.3)$ . Label it with its area of 0.12.
Label the other regions with their areas.
Find the value of $(2.4) \cdot (1.3)$ . Show your reasoning.

Problem 2

Here are two ways of calculating $2.4 \times 1.3$ . Analyze the calculations and discuss these questions with a partner:

Two vertical multiplication problems for 2.4 times 1.3. A shows the partial products method.

In Calculation A, where does the 0.12 and other partial products come from?
In Calculation B, where do the 0.72 and 2.4 come from?
In each calculation, why are the numbers below the horizontal line aligned vertically the way they are?

Problem 3

Find the product of $(3.1) \cdot (1.5)$ by drawing and labeling an area diagram. Show your reasoning.
Show how to calculate $(3.1) \cdot (1.5)$ using numbers without a diagram. Be prepared to explain your reasoning. If you are stuck, use the examples in a previous question to help you.
Use the applet to verify your answers and explore your own scenarios. To adjust the values, move the dots on the ends of the segments.
open applet in presentation mode

Print Version

Find the product of $(3.1) \cdot (1.5)$ by drawing and labeling an area diagram. Show your reasoning.
Show how to calculate $(3.1) \cdot (1.5)$ using numbers without a diagram. Be prepared to explain your reasoning. If you are stuck, use the examples in a previous question to help you.

Are you ready for more?

How many hectares is the property of your school? How many morgens is that?

Lesson Summary

Here are three other ways to calculate a product of two decimals such as $(0.04) \cdot (0.07)$ .

First, we can multiply each decimal by the same power of 10 to obtain whole-number factors.
Because we multiplied both 0.04 and 0.07 by 100 to get 4 and 7, the product 28 is $(100 \cdot 100)$ times the original product, so we need to divide 28 by 10,000.

$(0.04) \cdot 100 = (0.07)$ $(0.04) \cdot 100 = 7$ $4 \cdot 7 = 28$

Second, we can think of $(0.04) \cdot (0.07)$ as 4 hundredths times 7 hundredths and write:
We can rearrange whole numbers and fractions:
This tells us that $(0.04) \cdot (0.07) = 0.0028$ .

$(4 \cdot \frac{1}{100}) \cdot (7 \cdot \frac{1}{100})$ $(4 \cdot 7) \cdot (\frac{1}{100} \cdot \frac{1}{100})$ $28 \cdot \frac{1}{10, 000} = \frac{28}{10, 000}$

Third, we can use an area model. The product $(0.04) \cdot (0.07)$ can be thought of as the area of a rectangle with side lengths of 0.04 unit and 0.07 unit.

An area model with horizontal length of 7 units and a vertical length of 4 units. The horizontal length is labeled 0.07 and the vertical length is labeled 0.04.

In this diagram, each small square is 0.01 unit by 0.01 unit. The area of each square, in square units, is therefore $(\frac{1}{100} \cdot \frac{1}{100})$ , which is $\frac{1}{10, 000}$ .

Because the rectangle is composed of 28 small squares, the area of the rectangle, in square units, must be: $28 \cdot \frac{1}{10, 000} = \frac{28}{10, 000} = 0.0028$