# Lesson 17

Multiply Decimals and Whole Numbers

## Warm-up: True or False: Place Value Products (10 minutes)

### Narrative

The purpose of this True or False is to consider relationships between place values. This recalls work that students did earlier in the unit and seeing these relationships expressed using multiplication naturally prepares students for the work of the next several lessons where they will learn to find products of decimals. Students will revisit multiplicative relationships of place values in a later unit.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$100 \times 0.01 = 1$$
• $$10 \times 0.1 = 0.01$$
• $$10 \times 0.01 = 0.1$$

### Activity Synthesis

• Display: $$100 \times 0.1$$, $$10 \times 0.1$$, $$10 \times 0.01$$
• “What is the same and different about these expressions?” (They all have multiples of 10 in them. They all have decimals. The values are all different.)

## Activity 1: Multiply Decimals By Whole Numbers (15 minutes)

### Narrative

The purpose of this activity is for students to find decimal products in a way that makes sense to them. Many approaches are possible including:

• thinking about the meaning of place value and multiplying by place value
• using the hundredths grids
• using fractions or a number line

For the last problem, students may use their understanding of arithmetic, the distributive property, and their work on the first two problems (MP7) or they may make a new calculation. The goal of the synthesis is to share and connect different strategies for finding the values of the products.

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding the value of each expression. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Organization, Attention

### Required Materials

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• Make copies of hundredths grid available to students.
• Display the image from student workbook.
• “How many do you see?” (2 big squares, 60 small shaded squares, 6 shaded rows)
• 1 minute: partner discussion
• Display expression: $$2 \times 0.3$$
• “How does the diagram represent the expression?” (There are 2 groups of 0.3 shaded.)

### Activity

• 5–7 minutes: partner work time
• Monitor for students who:
• use diagrams to find the products
• use multiplication of whole numbers and place value understanding to find the products

### Student Facing

Find the value of each expression in a way that makes sense to you. Explain or show your reasoning. Use the grids, if needed.

1. $$2 \times 0.7$$

2. $$2 \times 0.08$$

3. $$2 \times 0.78$$

### Activity Synthesis

• Invite students to share how they found the value of $$2 \times 0.7$$.
• Display student work or image from solution.
• “How does the diagram show $$2 \times 0.7$$?” (There are two groups of 0.7 shaded.)
• “How can you use the diagram to find the value of $$2 \times 0.7$$?” (I can see that I have $$2 \times 7$$ or 14 tenths and that’s 1 whole and 4 more tenths.)
• “How is finding the value of $$2 \times 0.08$$ the same as finding the value of $$2 \times 0.7$$? How is it different?” (I could shade part of each of the large squares and then see the total. This time I shaded individual squares rather than rows of squares. In both cases, I can use multiplication to find the product.)

## Activity 2: Using Whole Number Products (20 minutes)

### Narrative

In the previous activity students found products of a whole number and some tenths or hundredths using hundredths grids or a strategy that made sense to them. The goal of this activity is to find these products with a greater focus on place value and the associative property of multiplication (MP7). For example, $$5 \times 0.07$$ means 5 groups of 7 hundredths. That means that its value is 35 hundredths or 0.35. This way of thinking about products allows students to use what they know about finding whole number products in order to find products of a whole number and a decimal number (MP8).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: Reading, Writing, Representing.

### Required Materials

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• Make hundredths grids available for students.

### Activity

• “Take a few minutes to find the value of the expressions in the first problem.”
• 1–2 minutes: quiet think time
• 5 minutes: partner work time
MLR3 Clarify, Critique, Correct
• “What do you think Kiran means? What is unclear?”
• 1 minute: quiet think time
• 2 minutes: partner discussion
• “With your partner, work together to write a revised explanation.”
• Display and review the following criteria:
• Write an explanation for each step.
• Use specific words and phrases such as equal or groups of.
• Use complete sentences.
• Write expressions or equations as examples.
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
• “What is the same and different about the explanations?” (Kiran did not use any numbers or equations. He did not explain why his strategy works.)

### Student Facing

1. Find the value of each expression. Explain or show your reasoning.

1. $$3 \times 0.5$$
2. $$5 \times 0.3$$
3. $$7 \times 0.02$$
2. Kiran wrote this explanation to describe the strategy he used to multiply a whole number by some tenths:

“I just turn the numbers into whole numbers, multiply them, and call them tenths.” (Pause for teacher directions.)

3. Can you use Kiran’s reasoning to find the value of $$6 \times 0.07?$$ Explain your reasoning.

### Student Response

If a student does not find the correct value of the expressions, show them $$3 \times 5$$, $$5 \times3$$, $$7 \times2$$ and ask, “How are these expressions the same as and different from the expressions in the problem?”

### Activity Synthesis

• “Work with your partner to adapt Kiran’s reasoning to find the value of $$6 \times 0.07.$$
• 2-3 partner work time
• Invite students to share their responses and reasoning for the value of $$6 \times 0.07$$.
• “How is this the same as finding the value of $$6 \times 7$$? How is it different?” (I found $$6 \times 7$$ but needed to remember that it’s hundredths so the product is 42 hundredths.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found products of a whole number and some tenths and a whole number and some hundredths.”

“What questions do you have about multiplying whole numbers and decimals?” (Can I always use whole number multiplication to find these products? What do I do if the numbers are larger or more complicated? Is there an algorithm like we used for multiplying whole numbers?)

Give students time to record their answers in a math journal before they share their thinking.

Record responses for all to see. Keep display visible throughout the section and refer back to it in future lessons to see if any questions have been answered. Add to and adapt the display, as necessary.