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# Multiplication & Division To Solve Problems

## Overview

Students solve division problems by changing them into multiplication problems. They then use the relationship between multiplication and division to determine the sign when dividing positive and negative numbers in general.

# Key Concepts

The rules for determining the sign of a quotient are the same as those for a product: If the two numbers have the same sign, the quotient is positive; if they have different signs, the quotient is negative. This can be seen by rewriting a division problem as a multiplication of the inverse.

For example, consider the division problem −27 ÷ 9. Here are two ways to use multiplication to determine the sign of the quotient:

- The quotient is the value of
*x*in the multiplication problem 9 ⋅*x*= −27. Because 9 is positive, the value of*x*must be negative in order to get the negative product. - The division −27 ÷ 9 is equivalent to the multiplication −27 ⋅ $\frac{1}{9}$. Because this is the product of a negative number and a positive number, the result must be negative.

# Goals and Learning Objectives

- Use the relationship between multiplication and division to solve division problems involving positive and negative numbers.
- Understand how to determine whether a quotient will be positive or negative.

# Division Is Multiplying by the Inverse

# Lesson Guide

Have students work in pairs on the Opening problem. Make sure that they understand how to rewrite a division problem as a multiplication problem.

Remind students that since any integer *x* can be written as $\frac{x}{1}$, its multiplicative inverse is $\frac{1}{x}$.

# Answers

- $12\xf7\left(-3\right)=12\cdot \left(\frac{1}{-3}\right)$
- $12\cdot \left(\frac{1}{-3}\right)=\frac{12}{-3}=-4$
- $12\xf7\left(-3\right)=-4$

## Opening

# Division Is Multiplying by the Inverse

You can use what you know about multiplication to find the answer to a division problem.

Consider this division problem: 12 ÷ (−3).

You know that dividing is the same as multiplying by the inverse (4 ÷ 2 is the same as 4⋅$\frac{1}{2}$).

- How could you rewrite 12 ÷ (−3) as a multiplication problem using the inverse of −3?
- What is the answer?
- What is the answer to the original division problem?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will create rules for determining the sign of the answer to a division problem.

## Opening

Create rules for determining the sign of the answer to a division problem.

# Divide Positive and Negative Numbers

# Lesson Guide

Students should work in pairs.

SWD: If students seem unsure of the task, model the steps for solving the problem with them before asking them to do the problems themselves.

# Interventions

**Student makes computation errors.**

- Look at the numbers in the problem. Does your answer seem reasonable?
- What method can you use to divide fractions?

# Answers

- −10 ÷ 2 = −10 ⋅ $\frac{1}{2}$ = −5
- −8 ÷ (−$\frac{1}{4}$) = −8 ⋅ (−4) = 32

## Work Time

# Divide Positive and Negative Numbers

Use the fact that dividing is the same as multiplying by the inverse to rewrite these problems as multiplication problems.

Then solve the multiplication problems in order to find the answer to the original division problems.

- −10 ÷ 2
- −8 ÷ −$\frac{1}{4}$

## Hint:

You need to find the inverse of 2 to rewrite the first division problem as a multiplication problem. What is the inverse of 2?

# Create Problems

# Lesson Guide

Students should continue to work in pairs.

# Mathematics

When writing their own division problems, students should use integers, decimals, and fractions and relate the division to multiplication.

There are a few ways to relate division problems to multiplication:

- Rewrite the division problem as a multiplication problem with an unknown factor. For example, the quotient of –3.6 ÷ (–0.12) is the missing factor in the multiplication equation –0.12 •
*x*= –3.6. - Rewrite the division problem as multiplication by the reciprocal of the divisor. For example, the quotient of –3.6 ÷ (–0.12) is the same as the product $-3.6\cdot \left(-\frac{1}{0.12}\right)$.

# Interventions

**Student does not understand how to relate a division problem to a multiplication problem.**

- Let
*x*represent the quotient. How can you write a multiplication problem that includes*x*? - Remember that multiplication and division are inverse operations. What does this mean? How can this help you write the division problem as a multiplication problem?

# Possible Answers

- Answers will vary. Possible division problems: $-3.6\xf7\left(-0.12\right)$

$5\xf7\frac{2}{5}$

$-8\xf72$ - Answers will vary. Possible multiplication problems and solutions: $-3.6\cdot \frac{1}{-0.12}=30$

$5\cdot \frac{5}{2}=\frac{25}{2}$

$-8\cdot \frac{1}{2}=-4$

## Work Time

# Create Problems

- Use a variety of numbers, including integers, fractions, and decimals, to write three division problems. Use both positive and negative numbers.
- Use the fact that dividing is the same as multiplying by the inverse to rewrite the problems as multiplication problems. Then solve the multiplication problems in order to find the answers to your original division problem.

## Hint:

Make sure to:

- Put positive and negative numbers in different positions in the problems.

# Complete the Table

# Lesson Guide

Give students time to complete the table. Students will use their completed tables to prepare a presentation that will be given during Ways of Thinking.

Look for students with different explanations for completing the table.

# Mathematics

Students can choose to complete the table by testing with various positive and negative numbers. Alternatively, they can use the commutative property and the multiplication rules that were proved in Lesson 12:

- A positive number times a negative number equals a negative number.
- A negative number times a negative number equals a positive number.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students work through quantitative problems, then must draw generalizations in order to develop rules for dividing positive and negative numbers.

**Mathematical Practice 7: Look for and make use of structure.**

Students should see patterns in order to complete the table.

# Answers

## Work Time

# Complete the Table

Complete this table to show what you think the rules are for dividing positive and negative numbers.

HANDOUT: Creating a Table for Rules of Operations with Integers

## Hint:

Look at the division problems you worked on today. What patterns do you see in your answers that can help you determine the rules?

# Prepare a Presentation

# Preparing for Ways of Thinking

Choose students who demonstrate the rules for dividing positive and negative numbers to present in Ways of Thinking. Presentations should make clear connections between the rules for dividing positive and negative numbers and their completed table.

Have students who solved the Challenge Problem present and explain their solutions during Ways of Thinking as well.

# Challenge Problem

## Answers

- $-15\xf73>-15\cdot 3$
- $-\frac{4}{5}\xf7-\frac{1}{2}>-\frac{4}{5}\cdot -\frac{1}{2}$
- $-1.8\cdot 0.3>-1.8\xf70.3$
- $-78\xf7-0.12>-78\cdot -0.12$

## Work Time

# Prepare a Presentation

Explain your rules for dividing positive and negative numbers. Support your explanation with your table and your work.

# Challenge Problem

Insert one $\cdot $ symbol and one $\xf7$ symbol in each to make the resulting inequality true. Try not to perform any calculations.

- $-15\text{[\hspace{1em}]}3>-15\text{[\hspace{1em}]}3$
- $-\frac{4}{5}\text{[\hspace{1em}]}-\frac{1}{2}>-\frac{4}{5}\text{[\hspace{1em}]}-\frac{1}{2}$
- $-1.8\text{[\hspace{1em}]}0.3>-1.8\text{[\hspace{1em}]}0.3$
- $-78\text{[\hspace{1em}]}-0.12>-78\text{[\hspace{1em}]}-0.12$

# Make Connections

# Mathematics

Have students present their rules for dividing positive and negative numbers. They should justify their answers with numerical examples. Students presenting the Challenge Problem should explain their reasoning and how they could determine the answer without calculating.

ELL: When calling on students, be sure to call on ELLs and to encourage them to participate as actively as all students—despite their possible weaker command of the language.

# Mathematical Practices

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

Students should listen closely, ask questions, and help to clarify any points that are confusing.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ explanations and rules for dividing positive and negative numbers.

## Hint:

As your classmates present, ask questions such as:

- How did you determine the sign of the answer for problems that involve dividing a positive number by a negative number?
- How did you determine the sign of the answer for problems that involve dividing a negative number by a positive number?
- How did you determine the sign of the answer for problems that involve dividing a negative number by a negative number?

# Positive and Negative Numbers

# Lesson Guide

Have students work individually on the problems. Students should apply the rules that they wrote for multiplying and dividing positive and negative numbers.

# Answers

- $4\cdot 8=32$
- $0.1\left(-0.1\right)=-0.01$
- $(-10)\cdot 0.1=-1$
- $\left(-4\right)\xf7\left(-8\right)=\frac{1}{2}$
- $(-0.1)\xf7\left(-0.1\right)=1$
- $40\xf7\left(-8\right)=-5$
- $\left(-\frac{3}{4}\right)\xf7\left(-\frac{3}{8}\right)=2$

## Work Time

# Positive and Negative Numbers

Calculate:

- $4\cdot 8$
- $0.1(-0.1)$
- $(-10)\cdot 0.1$
- $(-4)\xf7(-8)$
- $(-0.1)\xf7(-0.1)$
- $40\xf7(-8)$
- $(-\frac{3}{4})\xf7(-\frac{3}{8})$

# Multiplying and Dividing

# Lesson Guide

Have students discuss the rules with a partner before turning to a whole class discussion. Use this opportunity to correct or clarify misconceptions. Have students compare the rules presented here with the rules that they wrote.

ELL: When discussing the rules, make a point of writing questions on the board, along with students’ responses. This will assist ELLs by giving them written and oral access to the questions and the rules. Have students write all important information in their Notebook.

## Formative Assessment

# Summary of the Math: Multiplying and Dividing

**Read and Discuss**

- A positive number multiplied or divided by a positive number equals a positive number.
- A positive number multiplied or divided by a negative number equals a negative number.
- A negative number multiplied or divided by a positive number equals a negative number.
- A negative number multiplied or divided by a negative number equals a positive number.

## Hint:

Can you:

- Explain how to determine whether a quotient is positive or negative?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out what they learned about multiplying and dividing positive and negative numbers.

SWD: Students with disabilities often have difficulty with written expression; provide students with a variety of options to express their understanding of the key ideas learned in the lesson (audio recording a verbal summary, creating annotated graphics/drawings that explain the key ideas, etc.).

## Work Time

# Reflection

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

**Something that surprised me about multiplying and/or dividing with positive and negative numbers is …**