Lesson 4 | Multiplication and Division of Decimals | 5th Grade Mathematics

Objective

Multiply a multi-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 multi-digit whole numbers and decimals using general methods.
Reason about the placement of the decimal point in cases involving multiplication of a decimal by a multi-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 43 × 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 43 × 2.31, students can use fractions: $$43\times\frac{231}{100}=\ \frac{9,933}{100}=99.33$$),
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., $$43\times2.31\ {\rightarrow}43\times\left(2.31\times100\right)=43\times231=9,933{\rightarrow9,933\div100=99.33}$$), and
4. Using estimation to reason about the size of numbers (e.g., $$43\times2.31\approx40\times2=80$$, so 99.33 is a more reasonable product than 9933, 993.3, 9.933, or 0.9933) (MP.3).

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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

a. Find each product. Be prepared to explain your reasoning.

60 × 3 = _________
60 × 0.3 = _________
60 × 0.03 = _________

b. What do you notice about Part (a)? What do you wonder?

c. Use what you noticed in Part (b) to solve 600 × 0.3 and 600 × 0.03.

Guiding Questions

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Problem 2

Solve. Show or explain your reasoning for the placement of the decimal point.

a. 0.05 × 68

b. 36 × 6.5

c. 602 × 5.99

Guiding Questions

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Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

Look at #1. 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.)
Look at #2. What do you notice about the factors in each problem and the size of the product? How is that related to what we saw with analogous fraction computations?
Look at #4(d). Where did you place the decimal point? Why?
Did Greg have enough money to pay for his gas in #5? If so, how much change would he get? If not, how much more would he need? Why might estimation give Greg the false impression that he has enough money?
Look at #6(a). What is similar about this problem and #6(c)? What is different?
Look at #6(d). Did anyone solve using the standard algorithm? Did you write 20.81 as the “top” number in the algorithm? Why?
What error did Michelle make in #7?
How does being fluent in whole-number multi-digit multiplication help you multiply decimals? After finding the analogous whole number product, how can you then determine where to place the decimal point?

Target Task

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

Problem 1

Solve. Show or explain your work.

$$1.07 \times 54$$

Problem 2

Arthur computes 485 × 3.4 and gets a product of 164.9.

What was Arthur’s mistake?
Find the correct product.
Show your work or explain your answer.

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.

Extra Practice Problems

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

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Lesson 3

Lesson 5

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

Anchor Tasks

Problem 1

Guiding Questions

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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