Reviewing The Properties of Multiplication

Reviewing The Properties of Multiplication

Review Properties of Operations

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)=cRepresent the assumption as
an equation
(−1) ⋅ a ⋅ (−1) ⋅ b=(−1) ⋅ c_
(−1) ⋅ (−1) ⋅ ab=(−1) ⋅ c_
(−1)(−1) ⋅ (−1)⋅ ab =(−1)(−1) ⋅ c_
1 ⋅ (−1) ⋅ ab=1 ⋅ c_
(−1) ⋅ ab=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

Opening

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

Explain and Justify Proofs

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 thata + 0 =a . You can rewrite this equation asa + ( −a ) = 0 .

A Negative Times a Negative

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 thata + 0 =a . You can rewrite this equation asa + ( −a ) = 0 .

Prepare a Presentation

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

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

Work Time

Multiply Integers

Calculate:

(−6) ⋅ 9

8 ⋅ (−7)

(−9) ⋅ (−7) ⋅ 2

(−7) ⋅ (−5)

(−7) ⋅ 5

7 ⋅ (−5)

Rules for Multiplying

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

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 …