Lesson 3 | Multiplication and Division of Decimals | 5th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Number and Operations in Base Ten

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

The foundational standards covered in this lesson

5.NBT.A.1

Number and Operations in Base Ten

5.NBT.A.1 — Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.A.2

Number and Operations in Base Ten

5.NBT.A.2 — Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.B.5

Number and Operations in Base Ten

5.NBT.B.5 — Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NF.B.4

Number and Operations—Fractions

5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Compute products of single-digit whole numbers and decimals to hundredths using general methods, including using an area model and/or algorithm (like the partial products algorithm or the standard algorithm).
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).
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, such as computing $$8 \times 0.09$$ (MP.6).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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 understand 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

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

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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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

Estimate each of the following products. Then find the actual product, using your estimate to determine the location of the decimal point if possible.

a. 14.4 × 6

b. 7 × 3.98

c. 0.15 × 8

Guiding Questions

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Problem Set

Problem Set

Answer Keys

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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)?
Look at #2(f). What made this problem more difficult than the previous ones? What about #2(h)?
How did you determine where to place the decimal point in the product in #4? Did anyone estimate? If so, was your estimate too large or too small? How do you know? Did anyone get a closer estimate?
Look at #5. What was Leslie’s mistake? How can you tell just by looking at the value in her claim that it was unreasonable?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Which of the following is equivalent to 9 × 0.26?

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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

An example response to the Target Task at the level of detail expected of the students.

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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.

Extra Practice Problems

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

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Lesson 2

Lesson 4

Lesson Map

Topic A: Multiplying Decimals

Multiply a single-digit whole number by a decimal in cases that involve basic facts. Estimate the product of a single-digit whole number with a decimal by rounding numbers to their largest place.

5.NBT.B.7

Multiply a single-digit whole number by a decimal.

5.NBT.B.7

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

5.NBT.B.7

Multiply a multi-digit whole number by a decimal.

5.NBT.B.7

Multiply a decimal by a decimal in cases that involve basic facts. Estimate the product of two decimals by rounding numbers to their largest place.

5.NBT.B.7

Multiply a decimal by a decimal.

5.NBT.B.7

Multiply with decimals in all cases, reasoning about the placement of the decimal point.

5.NBT.B.7

Topic B: Dividing Decimals

Divide a decimal by a single-digit whole number in cases that involve basic facts. Estimate quotients with single-digit divisors by rounding numbers to compatible numbers.

5.NBT.B.7

Divide a decimal by a single-digit whole number.

5.NBT.B.7

Divide a decimal by a single-digit whole number that requires decomposition in the smallest place.

5.NBT.B.7

Divide a decimal by a two-digit whole number.

5.NBT.B.7

Divide a decimal by a two-digit whole number that requires decomposition in the smallest place.

5.NBT.B.7

Divide a whole number or a decimal by 1 tenth or 1 hundredth.

5.NBT.B.7

Divide a whole number or a decimal by a decimal in cases that involve basic facts. Estimate quotients with decimal divisors by rounding numbers to compatible numbers.

5.NBT.B.7

Divide a whole number or a decimal by a decimal.

5.NBT.B.7

Divide with decimals in all cases, reasoning about the placement of the decimal point.

5.NBT.B.7

Topic C: Decimal Expressions and Real-World Problems

Solve real-world problems that require interpretation of the remainder involving all possible cases, including further decomposition into a decimal.

5.NBT.B.7

Solve real-world problems involving multiplication and division with decimals.

5.NBT.B.7 5.OA.A.2

Write and evaluate numerical expressions involving multiplication and division with decimals.

5.NBT.B.7 5.OA.A.1 5.OA.A.2

Topic D: Measurement Conversion and Real-World Problems

Express measurements in a whole number of larger units or mixed units in terms of a smaller unit.

5.MD.A.1

Express measurements in a fractional or decimal number of larger units in terms of a smaller unit.

5.MD.A.1

Express measurements in a whole number of smaller units in terms of a larger unit.

5.MD.A.1

Express measurements in a fractional or decimal number of smaller units in terms of a larger unit.

5.MD.A.1

Solve real-world problems involving measurement conversions.

5.MD.A.1

Multiplication and Division of Decimals

Objective

Common Core Standards

Core Standards

Number and Operations in Base Ten

Foundational Standards

Number and Operations in Base Ten

Number and Operations in Base Ten

Number and Operations in Base Ten

Number and Operations—Fractions

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem Set

Discussion of Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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