Applying the Distributive Property

Students work in partners on the Think About It problem.  After 3 minutes of work time, I have students share out the expressions they've come up with.  There are a few possibilities here, but the most common expressions from kids will be:  10 x 65 and 10 x 60 + 10 x 5.

I focus the conversation around 10 x 60 + 10 x 5.  I'll underline the 10 x 60 in one color, and ten the 10 x 5 in another color.  I ask students what the two underlined expressions have in common.  Students will identify that each expression has 10 as a factor.  I'll ask if 10 (60 + 5), then, is an equivalent expression.  We'll refer back to the picture, too, to talk about what the (60 + 5) represents.

To start Example 1 in the Intro to New Material section, I ask students to articulate what 4(5+1) means.  I'm looking for someone to name that it is 4 groups of 5 + 1.  The first equivalent expression, then, will be what we get when we expand the expression out:  5 + 1 + 5 + 1 + 5 + 1 + 5 + 1+ 5 + 1.

For the second expression, I'll ask students how many 5s there are in the expanded form, and how many 1s there are.  We'll write 4(5) + 4(1) as the second expression.

Finally, the third expression will be a simplified version of the second:  20 + 4.

For the Example 1, I don't use the words 'distributive property.'  I want students to understand conceptually what is happening when we distribute a coefficient first.

For Example 2, I have students work with their partners to create three equivalent expressions, starting with repeated addition.  Students should write: n + 2 + n + 2 + n + 2 = 3(n) + 3(2). Students should also simplify to 3n + 6 by combining like terms.

To prove that 3(n+2) = 3n+6, I have students use substitution.  I'll break the class into thirds, and have each third substitute in a different value for n.

I'll then tell students that although we went through multiple steps to create the equivalent expressions 3(n + 2) = 3n + 6, we can apply the distributive property to create these expressions much more easily.  We multiply the coefficient by both terms inside the parentheses. 

In this lesson, I have students independently complete some Quick Practice problems, before we move into partner practice.  This gives me a chance to check for student understanding.  After each problem, I'll ask students:

• What is the coefficient?  What are the factors?
• How did you create an equivalent expression?
• Why are both expressions equivalent?

Students work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group.  I'm looking for:

• Are students correctly distributing the coefficient?
• Are students accurately rewriting the expression in application of the distributive property?
• Are students correctly recognizing the distributive property when used to create an equivalent expression?

I'm asking:

• What is the coefficient?  What are the factors?
• How did you create an equivalent expression?
• Why are both expressions equivalent?

After 10 minutes of partner work time, students independently complete the Check for Understanding problem.  I pull one student's work to display on the board.  The class critiques the work, giving both positive and critical feedback. 

Students work on the Independent Practice problem set. 

In the first problem, using the distributive property with the expression (n + 4)r  may at first seem tricky to kids.  I let them struggle a bit here and work to figure it out.  Some students will apply the commutative property, and rewrite the expression as r(n+4) before applying the distributive property.  Other students may apply the distributive property twice - first the n, and then the 4.  Either method is fine!

The proof on Problem 3 may also be difficult, because students need to substitute in for two variables.  As I circulate, I make sure to check this problem.  It is important to me that students master the distributive property.  It's just as important to me that students can reason about their work and prove their claims using examples and non-examples.

After independent work time, I bring the class back together for a conversation about Problem_6. First, I have students share their expressions with their partners.  Then, I have students share out the expressions with the class (we'll come up with more than the two the problem asks for).  I'll have a students summarize why the expressions are equivalent, and how the distributive property was applied.  

Students then independently complete the Exit Ticket to close the lesson.