Lesson 18

Use Whole Number Facts

Warm-up: True or False: Group Dynamics (10 minutes)

Narrative

The purpose of this True or False is for students to demonstrate strategies and understandings they have for using the associative property of multiplication. The numbers in this warm-up are whole numbers. In this lesson, students will use whole number products to find the value of the product of a whole number and a decimal and this requires using the associative property of multiplication.

Launch

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

Activity

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

Student Facing

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

• $$30 \times 2 \times 10 = 6 \times 10$$
• $$30 \times 2 \times 10 = 20 \times 3 \times 10$$
• $$60 \times 10 = 30 \times 20$$

Student Response

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Activity Synthesis

• Display first equation.
• “How can you show this is false without finding the value of both sides?” (I know $$30 \times 2$$ is not 6 and then multiplying both sides by 10 will not make them equal.)
• Display second equation.
• “How can you show this is true without finding the value of both sides?” (I know $$30 \times 2$$ and $$20 \times 3$$ are equal and then they are both multiplied by 10.)

Activity 1: Agree or Disagree (15 minutes)

Narrative

The purpose of this activity is for students to use place value reasoning and properties of operations to relate products of a whole number and a decimal to products of a whole number and either 0.1 or 0.01 (MP7). Students may decide that an equation is false without finding the value of both sides. For example, in the first problem, they may determine that 28 is unreasonable because 0.7 is less than one whole so the answer will be less than 4. When students find the value that makes equations true, they think about place value and the associative property of multiplication.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory, Organization

• Groups of 2

Activity

• 4 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who:
• compare the size of the product to the size of the factors to determine reasonableness. For example, explain that $$4 \times 0.7$$ is not equal to 28 because the product should be less than 4.
• use the associative property to represent the product of a whole number and a decimal as a product of two whole numbers and a decimal such as $$3 \times 0.7 = (3 \times 7) \times 0.1$$.

Student Facing

1. Decide whether each equation is true or false and explain or show why.

1. $$4 \times 0.7 = 28$$
2. $$5 \times 0.8 = 0.40$$
3. $$6 \times 0.03 = (6 \times 3) \times 0.01$$
4. $$8 \times 0.07 = (8 \times 7) \times 0.1$$
2. Fill in the blank to make each equation true.

1. $$3 \times 0.7 = 3 \times 7 \times \underline{\hspace{0.9cm}}$$
2. $$3 \times 0.07 = 3 \times 7 \times \underline{\hspace{0.9cm}}$$
3. $$5 \times \underline{\hspace{0.9cm}} = (5 \times 4) \times 0.1$$

Student Response

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Activity Synthesis

• Display the equation $$8 \times 0.07 = (8 \times 7) \times 0.1$$ from the first problem.
• “Is the equation true or false?” (false)
• Display expression: $$(8 \times 7) \times0.1$$
• “How could we revise this expression to make the equation true?” (Change it to read $$56\times0.01$$, $$8\times7\times0.01$$, or $$0.56$$.)
• Display equation: $$8 \times \underline{\hspace{0.9cm}} = (8 \times 7) \times 0.1$$
• “What can I write in the blank to make the equation true?” (0.7 since that’s 8 groups of 7 tenths or $$8 \times 7 \times 0.1$$.)

Activity 2: Interpret Diagrams and Expressions (20 minutes)

Narrative

The purpose of this activity is for students to use a diagram to support understanding two different ways to calculate the product of a whole number and a decimal number. The first strategy is one that students saw in the previous lesson, namely using whole number arithmetic to find the number of hundredths in the product and then multiplying that number by 0.01. The second strategy is useful specifically when the decimal has both tenths and hundredths. Using the distributive property, students can find the product of the whole number and the tenths and then the product of the whole number and the hundredths and combine these (MP7). In the next several activities, students will use both of these strategies as they build their understanding of how to find the product of a whole number and a decimal.

• Groups of 2

Activity

• 2 minutes: quiet think time
• 10 minutes: partner work time

Student Facing

1. Explain or show how the diagram represents each expression.
1. $$3 \times 0.12$$
2. $$(3 \times 12) \times 0.01$$
3. $$(3 \times 0.1) + (3 \times 0.02)$$
2. Find the value of $$(3 \times 12) \times 0.01$$. Explain or show your reasoning.
3. Find the value of $$(3 \times 0.1) + (3 \times 0.02)$$. Explain or show your reasoning.

Student Response

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If students don’t explain how the diagram represents an expression, ask, “How does the diagram show multiplication of decimals?” Then, connect the student’s response to the expressions.

Activity Synthesis

• Invite students to share their calculations of the value of $$3 \times 0.12$$ using the different expressions.
• “How did the expression $$(3 \times 12) \times 0.01$$ help to find the value of $$3 \times 0.12$$?” (I was able to just multiply whole numbers and then notice that the product is that many hundredths.)
• “How did the expression $$(3 \times 0.1) + (3 \times 0.02)$$ help to find the value of $$3 \times 0.12$$?” (I multiplied the tenths and then the hundredths and added them together.)
• “Which strategy do you prefer?” (I like the first strategy because I can just use what I know about whole number products and it will always work.)

Lesson Synthesis

Lesson Synthesis

“Today we used our understanding of place value to multiply decimals.”

Display:

$$25 \times 0.3$$

$$25 \times 0.03$$

“Describe the process you would use to find the value of these expressions.” (Find $$25 \times 3 = 75$$ and then multiply that by 0.1 or 0.01. The first one is 75 tenths or 7.5 and the second one is 75 hundredths or 0.75.)

“How can we multiply any whole number by an amount of tenths or hundredths?” (Find the whole number multiplied by the number of tenths or hundredths and multiply that result by 0.1 or 0.01.)

Cool-down: Fill in the Blank (5 minutes)

Cool-Down

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