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

# Multiplying a Negative by a Negative

# Reviewing The Properties of Multiplication

## Overview

Students use properties of multiplication to prove that the product of any two negative numbers is positive and the product of a positive number and a negative number is negative.

# Key Concepts

Multiplication properties can be used to develop the rules for multiplying positive and negative numbers.

Students are familiar with the properties from earlier grades:

- Associative property of multiplication: Changing the grouping of factors does not change the product. For any numbers
*a*,*b*, and*c*, (*a*⋅*b*) ⋅*c*=*a*⋅ (*b*⋅*c*). - Commutative property of multiplication: Changing the order of factors does not change the product. For any numbers
*a*and*b*,*a*⋅*b*=*b*⋅*a*. - Multiplicative identity property of 1: The product of 1 and any number is that number. For any number
*a*,*a*⋅ 1 = 1 ⋅*a*=*a*. - Property of multiplication by 0: The product of 0 and any number is 0. For any number
*a*,*a*⋅ 0 = 0 ⋅*a*= 0. - Property of multiplication by −1: The product of −1 and a number is the opposite of that number. For any number
*a*, (−1) ⋅*a*= −*a*. - Existence of multiplicative inverses: Dividing any number by the same number equals 1. Multiplying any number by its multiplicative inverse equals 1. For every number
*a*≠ 0,*a*÷*a*=*a*⋅ 1*a*= 1*a*⋅*a*= 1. - Distributive property: Multiplying a number by a sum is the same as multiplying the number by each term and then adding the products. For any numbers
*a*,*b*, and*c*,*a*⋅ (*b*+*c*) =*a*⋅*b*+*a*⋅*c*.

In this lesson, students will encounter a proof showing that the product of a positive number and a negative number is negative and two different proofs that the product of two negative numbers is positive. Two alternate proofs are as follows.

Proof that the product of two negative numbers is positive:

Represent the negative numbers as −*a* and −*b*, where *a* and *b* are positive.

(−a) ⋅ (−b) | Original expression |
---|---|

= ((−1) ⋅ a) ⋅ ((−1) ⋅ b) | Property of multiplication by −1 |

= (−1) ⋅ (a ⋅ (−1)) ⋅ b | Associative property of multiplication |

= (−1) ⋅ ((−1) ⋅ a) ⋅ b | Commutative property of multiplication |

= ((−1) ⋅ (−1)) ⋅ (a ⋅ b) | Associative property of multiplication |

= 1 ⋅ (a ⋅ b) | Property of multiplication by −1 |

= a ⋅ b | Multiplicative identity property of 1 |

Because *a* and *b* are positive, *a* ⋅ *b* is positive.

Proof that the product of a positive number and a negative number is negative:

Let *a* be the positive number. Let −*b* be the negative number, where *b* is positive.

a ⋅ (−b) | Original expression |
---|---|

= a ⋅ ((−1) ⋅ b) | Property of multiplication by −1 |

= (a ⋅ (−1)) ⋅ b | Associative property of multiplication |

= ((−1) ⋅ a) ⋅ b | Commutative property of multiplication |

= (−1) ⋅ (a ⋅ b) | Associative property of multiplication |

= −(a ⋅ b) | Property of multiplication by −1 |

Because *a* and *b* are positive, *a* ⋅ *b* is positive, so −(*a* ⋅ *b*) must be negative.

# Goals and Learning Objectives

- Review properties of multiplication.
- Explain why the product of two negative numbers is positive and the product of a negative number and a positive number is negative.

# Review Properties of Operations

# Lesson

Have students read the properties in the Opening. Remind them that they have seen and used all of these properties before. Ask volunteers to give a numerical example of each property.

Next, have students read the proof for multiplying negative numbers. They should notice that the first step is to state what you want to prove and then represent it by an equation. This step is common to many mathematical proofs, whether direct or indirect (by contradiction).

# Mathematics

The steps for proving that the assumption is false use many of the same properties as are presented in the table.

(−a) ⋅ (−b) | = | −c | Represent the assumption as an equation |
---|---|---|---|

(−1) ⋅ a ⋅ (−1) ⋅ b | = | (−1) ⋅ c | Property of multiplication by −1 |

(−1) ⋅ (−1) ⋅ a ⋅ b | = | (−1) ⋅ c | Commutative property of multiplication |

(−1)(−1) ⋅ (−1) ⋅ a ⋅ b | = | (−1)(−1) ⋅ c | Multiplication property of equality (divide by −1 or multiply by 1 − 1) |

1 ⋅ (−1) ⋅ a ⋅ b | = | 1 ⋅ c | Definition of multiplicative inverse |

(−1) ⋅ a ⋅ b | = | c | Multiplicative identity property of 1 |

(−1) ⋅ (ab) | = | c | Associative property of multiplication |

−(ab) | = | c | Property of multiplication by −1 |

SWD: Students have learned and retained information from prior learning experiences, but do not realize when to use that information. Teachers need to remind students of what they know, but also when to apply that knowledge. This strategy is sometimes referred to as “priming” background information. Priming background knowledge can be done in simple ways, such as merely stating “Remember when we learned...” or “See how this concept applies in this situation, too?” It may also involve a more intensive overview of a topic.

## Opening

# Review Properties of Operations

In the previous lesson, you built patterns to illustrate that the product of two negative numbers is a positive number. In today’s lesson, you will prove the product of two negative numbers is a positive number.

One way to prove the rule is through contradiction. Instead of trying to prove something is true, you prove that its opposite is false.

You want to prove that a negative times a negative equals a positive. The opposite is that a negative times a negative equals a negative.

So, if you can find something that contradicts the assumption that the product of two negative numbers is negative, then the product must instead be positive.

Discuss this proof with your class. Try to identify which property is used in each step:

- Assume that a negative times a negative equals a negative and find a contradiction.

(−a) ⋅ (−b) | = | −c | Represent the assumption as an equation |
---|---|---|---|

(−1) ⋅ a ⋅ (−1) ⋅ b | = | (−1) ⋅ c | _ |

(−1) ⋅ (−1) ⋅ a ⋅ b | = | (−1) ⋅ c | _ |

(−1)(−1) ⋅ (−1)⋅ a ⋅ b | = | (−1)(−1) ⋅ c | _ |

1 ⋅ (−1) ⋅ a ⋅ b | = | 1 ⋅ c | _ |

(−1) ⋅ a ⋅ b | = | c | _ |

(−1) ⋅ (ab) | = | c | _ |

−(ab) | = | c | _ |

- Since
*a*and*b*are positive, so is*ab*. However, since*c*is also positive, the last step says that the opposite of a positive number is also positive, which is a contradiction! - Since assuming that a negative times a negative leads to a contradiction, it must be false, and the opposite must be true. Therefore, a negative times a negative equals a positive.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will use the properties of operations to justify the rules for multiplying positive and negative integers.

## Opening

Use the properties of operations to justify the rules for multiplying positive and negative integers.

# Explain and Justify Proofs

# Lesson Guide

Students should work in small groups. Try to group students of varying ability levels so more capable students can help students who are struggling.

# Mathematics

Explain to students that in order to prove a rule in mathematics, the proof can only use properties or already proven rules.

If necessary, walk students through the proof from the Opening, highlighting the specific properties that were used.

SWD: Students with disabilities may have difficulty with cooperative learning tasks. Support students who have trouble resolving conflicts and/or disagreements and coach students to recognize the importance of considering alternate perspectives. Also, provide suggestions as to how students can navigate differences of opinion in learning situations.

ELL: When students work in groups, teachers are able to monitor individual student progress by listening and recording student conversations and peer problem solving. This type of collaborative work gives ELLs the opportunity to use mathematical language and to engage in conversation with their peers.

# Interventions

**Student has difficulty choosing a reason for a step.**

- Look at the previous step. What changed between that step and this step?
- Look at the properties in the Opening. Do any of the properties explain that change?

# Answers

−b + b | = | 0 | Additive identity property of 0 |
---|---|---|---|

a ⋅ (−b + b) | = | a ⋅ 0 | Multiplication property of equality |

a ⋅ (−b + b) | = | 0 | Property of multiplication by 0 |

a ⋅ (−b) + ab | = | 0 | Distributive property |

a ⋅ (−b) + ab + −(ab) | = | 0 + −(ab) | Addition property of equality |

a ⋅ (−b) | = | −(ab) | Property of additive inverses and additive identity property of 0 |

## Work Time

# Explain and Justify Proofs

The rule “a positive times a negative equals a negative” can be represented by the equation *a *⋅ (−*b*) = −(*ab*). One way to prove that the rule is true is to prove that the equation is true—which we do in the following proof.

- Work on the handout and justify each step using the properties of operations for addition or multiplication to prove that if the values for
*a*and*b*are positive, then both (−*b*) and −(*ab*) are negative.

HANDOUT: Explaining and Justifying Proofs

## Hint:

- The
*addition property of equality*states that you can add any quantity to both sides of an equation without changing the equation. - The
*multiplication property of equality*states that you can multiply both sides of an equation by any quantity without changing the equation. - The
*addition property of zero*states that*a*+ 0 =*a*. You can rewrite this equation as*a*+ ( −*a*) = 0 .

# A Negative Times a Negative

# Mathematical Practices

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

As they work through the proofs, students must justify each statement and understand how one statement follows from the next.

# Answers

−b + b | = | 0 | Additive identity property of 0 |
---|---|---|---|

−a ⋅ (−b + b) | = | (−a) ⋅ 0 | Multiplication property of equality |

−a ⋅ (−b + b) | = | 0 | Property of multiplication by 0 |

(−a) ⋅ (−b) + (−a) ⋅ b | = | 0 | Distributive property |

(−a) ⋅ (−b) + (−a) ⋅ b + ab | = | ab | Addition property of equality |

(−a) ⋅ (−b) + [−(ab)] + ab | = | ab | The rule you just proved: a negative number times a positive number equals a negative number |

(−a) ⋅ (−b) + 0 | = | ab | Definition of additive inverses |

(−a) ⋅ (−b) | = | ab | Additive identity property of 0 |

## Work Time

# A Negative Times a Negative

The rule “a negative times a negative equals a positive” can be represented by the equation (−*a*) ⋅ (−*b*) = *ab*. The following proof shows that this equation is true.

- Work on the table and justify each step using the properties of operations for addition or multiplication to prove that if the values for
*a*and*b*are positive, then both (−*a*) and (−*b*) are negative and*ab*is positive.

HANDOUT: Multiplying a Negative by a Negative

## Hint:

- The
*addition property of equality*states that you can add any quantity to both sides of an equation without changing the equation. - The
*multiplication property of equality*states that you can multiply both sides of an equation by any quantity without changing the equation. - The
*addition property of zero*states that*a*+ 0 =*a*. You can rewrite this equation as*a*+ ( −*a*) = 0 .

# Prepare a Presentation

# Preparing for Ways of Thinking

Select students who give clear, understandable justifications for the steps in each proof to present in Ways of Thinking. Also look for students who give different ways of summarizing the rules for multiplying positive and negative numbers.

If any students solved the Challenge Problem, select solutions to be shared.

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

As they prepare their presentations, students must communicate their reasoning precisely, using correct mathematical language.

# Challenge Problem

## Answer

- 6 ⋅ (−$32.50) = −$195

This tells you how much money was deducted from Mia’s balance for the gym membership after six months of withdrawals (because the number is negative, it indicates a loss of money).

## Work Time

# Prepare a Presentation

Show how you justified each step in the two proofs.

- Summarize the rules for multiplying positive and negative numbers, and give examples of each rule.

# Challenge Problem

Each month Mia’s bank automatically deducts $32.50 from her account to pay for Mia’s gym membership.

- Find the value of 6 ⋅ (−$32.50) and explain what your answer tells you about this situation.

# Make Connections

# Mathematics

Have students share presentations for the proof justifications, rule summaries, and the Challenge Problem.

Make sure students understand that from now on, they can simply use the rules to find products. They do not need to show all the steps in the proofs. You might write a few multiplication problems on the board, and choose volunteers to come to the board find the product.

Here are some examples:

$-\frac{5}{6}\cdot -\frac{2}{3}$

56 ⋅ (−0.01)

$-\frac{3}{8}\cdot \frac{4}{5}$

−0.5 ⋅ (−2.4)

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ explanations and justifications for the proofs.

## Hint:

As your classmates present, ask questions such as:

- What happened in that step?
- Can you explain how that property justifies the step?
- Can you give another example of that rule?

# Multiply Integers

# Lesson Guide

Have students work individually on the problems, applying the rules for multiplying positive and negative numbers.

# Answers

(−6) ⋅ 9 = −54

8(−7) = −56

(−9)(−7) ⋅ 2 = 126

(−7)(−5) = 35

(−7)(5) = −35

(7)(−5) = −35

## Work Time

# Multiply Integers

Calculate:

(−6) ⋅ 9

8 ⋅ (−7)

(−9) ⋅ (−7) ⋅ 2

(−7) ⋅ (−5)

(−7) ⋅ 5

7 ⋅ (−5)

# Rules for Multiplying

# A Possible Summary

We can write proofs for mathematical rules by using properties. Once we have proved the rules, we can just use the rule without going through every step of a proof.

SWD: Clearly summarize the lesson, and write down all salient information. Make sure students are recording Summary of Math notes in their notebook.

When multiplying positive and negative numbers, there are two rules that we have proved:

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

## Formative Assessment

# Summary of the Math: Rules for Multiplying

Write a summary about the rules for multiplying positive and negative integers.

## Hint:

Check your summary.

- Do you include all four rules?
- Do you explain how the properties were used to prove the rules for multiplying positive and negative integers?

# 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 still they find confusing about the properties of operations.

ELL: Before students start to write their reflection, allow some additional time for ELLs to discuss their ideas with a partner, to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in their preferred language if they wish and to use a dictionary.

## 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.

**One thing that confuses me about the properties of operations is …**