## Instructor 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.

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