Lesson 19Expanding and Factoring

Learning Goal

Let’s use the distributive property to write expressions in different ways.

Learning Targets

  • I can organize my work when I use the distributive property.

  • I can use the distributive property to rewrite expressions with positive and negative numbers.

  • I understand that factoring and expanding are words used to describe using the distributive property to write equivalent expressions.

Lesson Terms

  • factor (an expression)
  • term

Warm Up: Number Talk: Parentheses

Problem 1

Find the value of each expression mentally.

Activity 1: Factoring and Expanding with Negative Numbers

Problem 1

In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.

Three different multiplication diagrams.

factored

expanded

Are you ready for more?

Problem 1

Expand to create an equivalent expression that uses the fewest number of terms: . If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms?

Lesson Summary

We can use properties of operations in different ways to rewrite expressions and create equivalent expressions. We have already seen that we can use the distributive property to expand an expression, for example . We can also use the distributive property in the other direction and factor an expression, for example .

We can organize the work of using distributive property to rewrite the expression . In this case we know the product and need to find the factors.

The terms of the product go inside:

A multiplication diagram with 2 boxes, 12x and -8 with two dotted boxes above and 1 dotted box on the left.

We look at the expressions and think about a factor they have in common. and each have a factor of . We place the common factor on one side of the large rectangle:

A multiplication diagram with 2 boxes, 12x and -8 with two dotted boxes above and 4 outside on the left.

Now we think: “ times what is ?” “ times what is ?” and write the other factors on the other side of the rectangle:

A multiplication diagram with 2 boxes, 12x and -8 with two dotted boxes above and 4 outside on the left and 3x and -2 above.

So, is equivalent to .