- Author:
- April Hamma
- Subject:
- Mathematics
- Level:
- Middle School
- Grade:
- 6
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Text/HTML

# Decimal Multiplication and Division Remix

## Overview

Students solve decimal multiplication and division problems related to the basic fact 3 × 7 = 21.

Students match cards that represent word problems, visual models, and numerical solutions to problems that include the numbers 0.8 and 0.2 for all four operations.

# Key Concepts

No new mathematics is introduced in this lesson. Students apply their knowledge about decimal operations.

# Goals and Learning Objectives

- Use reasoning and mental math to solve problems.
- Solve word problems involving simple addition, subtraction, multiplication, and division with decimals.

# Solve Decimal Problems Using Mental Math

# Lesson Guide

Students should solve the problems in the Opening themselves and then compare and discuss the answers with their partner. As a class, discuss strategies for solving the problems.

SWD: These problems provide good practice for students and good opportunities for you to assess students' ability to multiply and divide decimals. If there are students who are struggling, provide them with direct instruction for the first problem.

# Mathematics

Some strategies to discuss for the multiplication problems, using 0.3 x 0.7 as an example, include the following:

- Think about or write the problem as a whole number times tenths or hundreds. For example, 0.3 x 0.7 is $\frac{1}{10}$ x 3 x $\frac{1}{10}$ x 7 = $\frac{1}{100}$ x 21 = 0.21.
- Multiply 3 x 7 to get 21 and use estimation to place the decimal point. For .3 x 0.7, rounding the factors gives 0 x 1, or 0, so the answer is 0.21.
- Multiply 3 x 7 to get 21 and count decimal places to place the decimal point. For .3 x 0.7, the factors each have one decimal place for a total of two decimal places. The answer must have two decimal places, so it is 0.21.

A key part of using the “counting decimal places” method is to first ignore the decimal point and multiply the factors. For example, 0.5 x 0.4 = 0.2 appears to violate the rule because the factors have a total of two decimal places, while the product has only one. However, if we multiply 5 x 4, we get 20, so applying the rule gives 0.20, which is the same as 0.2. Emphasize that students should always use estimation to check that their answer is reasonable.

Some strategies for the division problems include the following:

- Multiply both numbers by a power of 10 so they are whole numbers. (This is the same as moving the decimal point the same number of places in both numbers.) Then divide mentally. For example, 21 ÷ 0.7 = 210 ÷ 7 = 30.
- Ignore the decimal point and divide. Then use estimation to place the decimal point. For example, to find 2.1 ÷ 3 , divide 21 by 3 to get 7. The answer should be close to 2 ÷ 3, or $\frac{2}{3}$, so it is 0.7.
- Use reasoning. For example, for 210 ÷ 0.7, use the fact that 210 ÷ 7 = 30. Dividing by a number that is equal to a tenth of 7 will result in a product 10 times as big, so 210 ÷ 0.7 = 300.

# Answers

- Solutions:

2.1

2.1

0.21

0.021

30

0.7

7

300

- Answers will vary.
- Answers will vary.

## Opening

# Solve Decimal Problems Using Mental Math

- Solve these problems using mental math.

0.3 × 7

3 × 0.7

0.3 × 0.7

0.03 × 0.7

21 ÷ 0.7

2.1 ÷ 3

2.1 ÷ 0.3

210 ÷ 0.7

- Compare your solutions with those of your classmates.
- Discuss the strategies you used to find the solutions.

# Match Solutions and Diagrams

# Lesson Guide

Students should work with a partner on the activity. Make sure that students understand the card sort activity, emphasizing that both partners should be able to explain the solution in their own words.

ELL: Word problems tend to be difficult for ELLs because they are highly language dependent. In addition, many word problems require formal operations or the ability to think abstractly and to manipulate concepts through language. Have students work with a partner so they can share and exchange ideas as they are working through word problems.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

Students must be able to relate a real-world situation, a visual model, and a numerical solution to each other in order to successfully match the cards.

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Students must be able to explain to their partner their reasoning for matching cards. In turn, partners must be able to critique and challenge, if appropriate.

# Answers

## Work Time

# Match Solutions and Diagrams

Work with a partner.

- Take turns matching a problem card, diagram card, and solution card.
- Explain to your partner how you know that the cards match.
- Your partner should either agree with your explanation or challenge it if your explanation is not correct, clear, and complete.
- For each matched set, explain what the solution card represents.

Ask yourself:

- Think about the operations you would use to solve the problem, and then find a diagram card that shows that operation.
- Think about what the problem asks for as you decide what the solution card represents.

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for the following students to present during Ways of Thinking:

- Students who have interesting explanations for the solution card
- Students who include a mathematical expression for their explanation to the solution card
- Student pairs who begin with incorrect card sets, but reason and critique each other to correct their sets

Choose some students to present the problems they created for the Challenge Problems as well.

# Challenge Problem

## Answers

- Problems will vary.

## Work Time

# Prepare a Presentation

Choose one of your matches. Explain why you think the cards match.

# Challenge Problem

- Write a word problem with decimals that requires using two different operations to solve.

# Make Connections

# Lesson Guide

Include presentations that address each problem. Student pairs should summarize their Work Time discussions.

Have students share the problems they wrote for the Challenge Problem.

ELL: When calling on students, be sure to call on ELLs and to encourage them to participate actively, even if their pace might be slower or they might be shyer due to their weaker command of the language.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

Students must be able to provide an explanation that supports the model, numerical solution, and real-world situation.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' approaches to matching the cards.

As your classmates present, ask questions such as:

- Can you explain the strategies you used to match the cards?
- How does the diagram on the diagram card reflect the problem?
- How does the diagram show the parts of the problem and the solution?

# Operations With Decimals

# A Possible Summary

When multiplying and dividing decimals, you can use similar methods for multiplying and dividing whole numbers. For multiplication, you can rewrite the problem as if multiplying two whole numbers, then either estimate or count decimal places to complete the calculation. For division, you can divide as if they are whole numbers, then use estimation to place the decimal point correctly.

Real-world situations can be solved by modeling them with mathematics. They can be modeled, or written as an equation, in order to understand how to solve them.

SWD: Refer to the Hint questions as a checklist of the information students need to include in their summaries. Create a digital or paper resource with the questions for students to keep in front of them as they write their summaries. For some students, navigating back and forth between the questions, Hints, and their written work will be confusing and distracting.

## Formative Assessment

# Summary of the Math: Operations With Decimals

Write a summary about solving problems with decimals.

Check your summary.

- Do you compare how you perform operations with decimals to how you perform operations with whole numbers?
- Do you discuss how estimation helps you operate with decimals?

# Reflect On Your Work

# Lesson Guide

Have students write a brief reflection before the end of class. Review the reflections to find out their understanding of how performing operations with decimals is similar to and different from performing operations with whole numbers.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starters below if you find them to be helpful.

**Ways that performing operations with decimals is like performing operations with whole numbers are …**

**Ways that performing operations with decimals is different from performing operations with whole numbers are …**