Distributive Property

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SWBAT express a sum of two whole number with a common factor as a multiple of a sum of two whole numbers with no common factor (e.g., 36 + 8 = 4(9 + 2) ).

Big Idea

Using the distributive property to express a sum of two whole numbers as a multiple of a sum of two whole numbers with no common factor can make numerical reasoning easier.

Think About It

7 minutes

Students work in partners on the Think About It problem.  After 2-3 minutes of work time, I ask students to share out the expressions they've come up with.

Here is what I'm looking for:



I frame the remainder of the lesson by letting students know that we'll be working to create expressions like 4(9+2).  In this lesson, we'll use the Distributive Property to express the sum of two numbers as a multiple of a sum of two whole numbers with no common factor.

Intro to New Material

15 minutes

To start the Intro to New Material section, I ask students to jot down on their papers the Greatest Common Factor (GCF) of 36 and 8.  I'll then ask students to identify the expression that would result if we factor the GCF from the sum, 36 + 8. I'll say, "What expression would we write using the Distributive Property?"

Once students have identified that the correct expression would be 4(9 + 2), I'll ask students to check our work by distributing the 4 back into the terms in the parentheses.  I want my students to build the habit of checking their work by getting back to our original expression. 

I'll then tell students that we can say say that 4(9 + 2) shows the sum 36 + 8 as a multiple of the sum of two numbers (9 + 2) with no common factor.  I ask students to help me fill in the key ideas in the notes:

  • We can express the sum of two whole numbers with a common factor as a multiple of two whole numbers with no common factor using the distributive property.
  • We can say that 4(9 + 2) is a multiple of (9 + 2) because a multiple is the product of a number and a whole number.  4 is a whole number.

It may seem strange for students to think about 4(9 + 2) as a product of a number and a whole number, because they see two numbers in (9 + 2), but they should think of (9 + 2) as one quantity.  The parentheses around (9 + 2) help them to see that.

I work with students to complete Problems C and D.

Partner Practice

15 minutes

For about 15 minutes, my students will work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group.  I am looking for: 

  • Are students explaining their thinking to their partner?
  • Are students using the vocabulary of factor, GCF, and multiple to describe their work?
  • Are students checking by distributing the number they factored out? 

I'm asking:

  • Explain how you determined your answer.
  • How do we know that 2(9 + 4) is a multiple of (9 + 4)?
  • How do we know that (9 + 4) have no common factors?
  • If 15 was the GCF of 15 and 45, how did you write the expression with the distributive property?

After 10 minutes of partner work time, I'll ask students to independently complete the Check for Understanding problem.  I circulate around the room to check their responses.  An exemplar response might be:

I know that 16 and 60 have a greatest common factor of 4.  4 x 4 = 16 and 4 x 15 = 60.  I can express the sum of 16 and 60 as a multiple of 4 and the sum of 4 and 15. 

4(4 + 15)

Evidence that students checked their expression by distributing the 4 to the terms inside the parentheses should also be visible.

Independent Practice

20 minutes

I'll now ask students to work on the Independent Practice problem set.  I'll again encourage students to check their work by distributing.  Students should use the fill-in notes from the INM to help them use the vocab to describe the expression. 

The steps students should follow to solve these problems:

  1. Find the GCF of both numbers.  Write this number on the outside of the parentheses.
  2. Factor out the GCF from both numbers.  Write both numbers that are left with an addition sign inside the parentheses.
  3. Check your work by redistributing the GCF and making sure that the expression you get matches the original expression. 

Closing and Exit Ticket

8 minutes

After independent work time, I bring the class back together to discuss Problem_5.  I'll start by having partners turn to one another to talk.  One partner shares what Norbert is correct about.  The other partner shares what Norbert is confused about.  I then Cold Call a student to share how Norbert should express 48 + 18 as a multiple of the sum of two whole numbers with no common factors.

My students will then work independently on the Exit Ticket to close the lesson.