Associative Property

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Students will be able to prove that multiplying numbers in any order will produce the same product (associative property), with models.

Big Idea

Children need to understand why the mathematical properties work and how they are applied. In this lesson, we explore modeling and "proving" the associative property.

Teacher Background Information

2 minutes

The teaching of the properties is a tricky one for some teachers.  It is important to remember that the properties help students build their number sense, use that sense to efficiently solve problems, and to make sense of the math world.  They do not have to "know and memorize" the names of the properties, but the structures and patterns are useful and necessary.

When you are working today and, and each day, present the properties as helpful patterns to be used in the right moments. 

Mini-Lesson & Active Engagement

15 minutes

To begin this lesson, I gather everyone in the community center to share this math problem:

3 + 5 + 2 = S

I ask the students to first solve for "S" in their heads, and then share with their partner the value of "S" and the strategy they used to solve the problem.  Many students add left-to-right in order, while a few see that they can add 3 and 2 to get 5 and then add that 5 to the 5 in the equation 5 to get the sum of 10.  

Next, I ask 3 students to stand in a group, then 5 others in a separate group, and 2 in another group. I tell them to "act out" my instructions.  

Using my arms like parenthesis, I put them above the group of 3 and 5 while saying, 3 + 5 equals 8. At this point, the groups shuffle together.  Then I call out, 8 plus 2 equals 10.  They all gather together and I ask the class what happened to our groups.  (They become one large group).

Next, I repeat the same situation, but start with 5 + 2, and continue through the same problem. Finally, I start by adding the 3 to the 2 in my first step and go on to complete the problem.  The goal is that students are able to see that all 3 situations produce the same sum. Make sure to explicitly talk this through, so that you hear students reason that because the group amounts never change, they are merely "joined" in different order, the sum does not change. This may not be as obvious to all of your students, so check for understanding before you proceed and if needed, do this with manipulative a few times.

If you have a math vocabulary board, this is when you add the card "associative property" to it and have the students explain that joining numbers in any order does not change the end value. (A tip about math vocabulary boards/walls - they work best if there are visuals as well as numbers. These can be examples or pictures.)

I then put the equation 4 x 2 x 5 =  on the board and tell the students that the associative property is the same for multiplication. This understanding is immediately assessed because I next ask them to solve the multiplication problem in more than one way, working with a partner. 

Finally, I model how to draw out (4 groups of 2) 5 times using arrays. 

Independent Practice

20 minutes

For independent practice, I ask the students to work on proving the associative property for our multiplication equation, creating one more drawing for the situation, and then to reorganize the factors and draw their own models.  

Be careful to watch for students that are not modeling correctly.  I ran into several situations where the students were just drawing arrays for the totals because they knew the product. The purpose of this lesson is for them to understand why the property works, not to find a product and make a random drawing. 

This student has several ways of modeling the equations and is getting close to understanding why it works.  My goal is to get her to where she can verbalize why the products are the same. 

In this video, I listen to the student's strategy and just point her to the next step, which was to group different factors together and create a model to prove it. 


Close and Sharing

10 minutes

To close the session, I remind the students that in addition and multiplication, it doesn't matter what order we group the numbers in, because in the end all the groups will be together. 

I ask them to swap journals with a partner and look for what they each did similarly and differently. I then allow them a few moments to revise and add to their journals. Revision is a meaningful activity for students for a few reasons. One is that it demonstrates your commitment to "making sense" of the work (rather than just getting the right answer). Another reason it is powerful is that it is entirely student-centric. The student is in charge of their learning, and as they rethink reasoning they are deepening understanding and transfer.