## Instructor Overview

Students use the distributive property to rewrite and solve multiplication problems. Then they apply addition and multiplication properties to simplify numerical expressions.

# Key Concepts

The distributive property is stated in terms of addition: *a*(*b* + *c*) = *ab* + *ac*, for all numbers *a*, *b*, and *c*. However, it can be extended to subtraction as well: *a*(*b* − *c*) = *ab* − *ac*, for all numbers *a*, *b*, and *c*. Here is a proof. (We have combined some steps.)

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

= a(b + (−c)) | Subtracting is adding the opposite. |

= a(b) + a(−c) | Apply the distributive property. |

= ab + a(−1 ⋅ c) | Apply the property of multiplication by −1. |

= ab + −1(ac) | Apply the associative and commutative properties of multiplication. |

= ab + −(ac) | Apply the property of multiplication by −1. |

= ab − ac | Add the opposite is subtracting. |

We can use the distributive property to make some multiplication problems easier to solve. For example, by rewriting $1.85 as $2.00 − $0.15 and applying the distributive property, we can change 6($1.85) to a problem that is easy to solve mentally.

6($1.85) | = | 6($2−$0.15) |
---|---|---|

= | 6($2) − 6($0.15) | |

= | $12 − $0.90 | |

= | $11.10 |

One common error students make when simplifying expressions is to simply remove the parentheses when a sum or difference is subtracted. For example, students may rewrite 10 − (6 + 9) as 10 − 6 + 9. In fact, 10 − (6 + 9) = 10 − 6 − 9. To see why, remember that that subtraction is equivalent to adding the opposite, 10 − (6 + 9) = 10 + [−(6 + 9)]. Applying the property of multiplication by −1, this is 10 + (−1)(6 + 9). Using the distributive property, we get 10 + (−6) + (−9) = 10 − 6 − 9.

# Goals and Learning Objectives

- Apply addition and multiplication properties to simplify numerical expressions.

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