Curriculum  /  Math  /  5th Grade  /  Unit 6: Multiplication and Division of Decimals  /  Lesson 3

Multiplication and Division of Decimals

Lesson 3

Math

Unit 6

5th Grade

Lesson 3 of 23

Objective

Construct viable arguments and critique the reasoning of others regarding the placement of the decimal point in computations that involve multiplying a whole number by a decimal.

Common Core Standards

Core Standards

  • 5.NBT.B.7 — Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Foundational Standards

  • 5.NBT.A.1
  • 5.NBT.A.2
  • 5.NBT.B.5
  • 5.NF.B.4

Criteria for Success

  1. Reason about the placement of the decimal point in cases involving multiplication of a decimal by a single-digit whole number, including any of the following lines of reasoning:
    1. Thinking about the product of the smallest base-ten units of each factor (e.g., one times a hundredth is a hundredth, so $$4 \times 2.31$$ will have an entry in the hundredths place),
    2. Thinking of decimals as fractions or as a whole number divided by 10 or 100 (e.g., to compute $$4 \times 2.31$$, students can use fractions: $$4\times\frac{231}{100}=\ \frac{924}{10}=9.24$$),
    3. Reasoning that when one carries out the multiplication without the decimal point, one has multiplied each decimal factor by 10 or 100, so they will need to divide by those numbers in the end to get the correct answers (e.g., $$4\times2.31{\rightarrow}4\times\left(2.31\times100\right)=4\times231=924{\rightarrow}924\div100=9.24$$), and
    4. Using estimation to reason about the size of numbers (e.g., $$4\times2.31\approx4\times2=8$$, so 9.24 is a more reasonable product than 924, 92.4, or 0.924) (MP.3).
  2. Understand some of the limitations and/or common errors regarding the placement of the decimal point, such as:
    1. Placing a decimal point incorrectly when a product ends in a 0, and
    2. Using estimation when the product is difficult to estimate (though this may be easier to see with more complex computations later in the unit) (MP.6).
  3. Given a whole number multiplication equation, determine the decimal product of a related decimal multiplication computation.

Tips for Teachers

Students may already know the general pattern that the number of decimal places in the product is equal to the sum of decimal places in each factor. However, to ensure they can explain why this pattern exists, push these students to reason about the placement of the decimal point beyond just the mention of a procedural rule. 

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Anchor Tasks

25-30 minutes

Problem 1

Elena, Noah, and Larae used different methods to compute $$9 \times 1.48$$. All three computations were correct.

Elena's method Noah's method Larae's method

$$1.48 \times 100 = 148$$

$$9 \times 148 = 1,332$$

$$1,332\div 100 = 13.32    $$

 $$1.48 = \frac{148}{100}$$

$$9 \times \frac{148}{100} = \frac{1,332}{100}$$

$$\frac{1,332}{100} = 13.32 $$

$$\begin{align} 9\times1.48&=9 \ \text{ones} \times 148 \ \text{hundredths} \\ \\ &=1,332 \ \text{hundredths} \\ \\ &=13.32\end{align} $$

a.   Which method makes the most sense to you? Why?

b.   What might Elena do to compute $$0.24 \times 5$$? What might Noah do? What might Larae do? Will the three methods result in the same product?

Guiding Questions

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Student Response

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References

Open Up Resources Grade 6 Unit 5 Lesson 6 — Activity 6.2 Using Properties of Numbers to Reason about Multiplication

Grade 6 Unit 5 Lesson 6 is made available by Open Up Resources under the CC BY 4.0 license. Copyright © 2017 Open Up Resources. Download for free at openupresources.org. Accessed Sept. 16, 2019, 11:59 a.m..

Modified by Fishtank Learning, Inc.

Problem 2

Xavier thinks he can use estimation to compute 9 × 1.48. He says:

“1.48 is close to 1, so 9 × 1.48 is close to 9 × 1, which is 9. So, since 9 × 148 = 1332, 13.32 makes more sense than 1.332, 133.2, or 1332.”

a.   Do you agree or disagree with Xavier’s reasoning? Explain.

b.   How would Xavier compute 0.24 × 5? 

Guiding Questions

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Student Response

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Problem 3

Use the whole number facts and one of the methods from Anchor Tasks #1 or #2 to find the related decimal product. Show or explain your reasoning for the placement of the decimal point.

a.   If 144 × 6 = 864, what is 14.4 × 6?

b.   If 7 × 398 = 2786, what is 7 × 3.98?

c.   If 15 × 8 = 120, what is 0.15 × 8?

Guiding Questions

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Student Response

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Problem Set

15-20 minutes

Discussion of Problem Set

  • How can whole-number multiplication help you with decimal multiplication?
  • Where did you place the decimal point in #1(a)? What about #1(b)?
  • How did you determine which expressions were equivalent in #2? Which would be most helpful to you to find the product? 
  • Look at #3. How did you determine where to place the decimal point in each part? (It may be worthwhile to discuss part (c) in more depth since estimation is not as useful a strategy here.)

Target Task

5-10 minutes

Problem 1

Which of the following is equivalent to 9 × 0.26?

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Student Response

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Problem 2

Find the product of 5.84 × 7. Then explain your reasoning for the placement of the decimal point.

Student Response

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Additional Practice

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Word Problems and Fluency Activities

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Next

Multiply a whole number by a decimal, reasoning about the placement of the decimal point.

Lesson 4
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Multiplying Decimals

Topic B: Dividing Decimals

Topic C: Decimal Real-World Problems

Topic D: Measurement Conversion

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