Associative Property With Manipulatives

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Students will be able to model and prove how the associative property in multiplication works.

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.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. 


10 minutes

To review and enhance the lesson from yesterday, I call my students to the community area and ask them to get in "fishbowl" formation. This is when they create a half circle around me with the first row on their bottoms and the second row on their knees.  I ask them to watch me work with one of their peers  to create and solve a multiplication equation using the associative property. 

After I select a volunteer, I show him/her our task, making sure to direct everyone's attention to the directions. 

I ask my partner to roll the three dice and decide what order to write our multiplication problem using all three digits. We rolled a 4, 4, and 5.

Then, I ask him where we should put the parenthesis, knowing that we have to solve that equation first. We decide to put them around a 4 and the 5.

Next, we build the array of that problem.  Then we look to the third digit (the second 4) and see how many times we have to build that same array.   We build those arrays as well, using cubes.  Now we have 4 (4x5) arrays.

Now we can solve for the area of each and add those areas together on paper or a white board.  I chose to use white boards to save on paper. 

Finally, we decide how to move the parenthesis and go through the process again, in order to prove that we get the same product each time. 

Active Engagement

20 minutes

After reviewing the directions and answering questions, I send partnerships off with cubes, dice, and their reflection journals to work in. Their task is to create, build, and prove the products of a three digit multiplication equation in at least two ways.  If they have time, then they can work on a third way.

As I tour the room, I will be looking and listening for the correct building of the original equations, with the students solving the parenthesis first, and for the students to verbally explain how their drawings or models represent the equations. 

This student shows how he and his partner began working and realized an error by asking, "Does this make sense?"

As you look at the student video, and my lesson description, you may well wonder why I am sticking with holding the students accountable in creating models of the associative property of multiplication. There are critical reasons for modeling - it provides repetition and it creates context - both key to remembering. Modeling also allows the student to evaluate their work within a situational context so that they can ask themselves (and others) if it makes sense. My reflection provides some important information on how 3rd grade students grapple with models of the associative property of multiplication.


15 minutes

After about 15 minutes of building and working through their own created equations, I call the students back to the community area with their journals.  We review their work in drawing the equations and using the parenthesis to direct order of operation.  

Then, I hand out a prompt for the partners to respond to.  In my class, we have a rubric to help us write specific and detailed responses.  I remind them to use the rubric and send them off to work.  

The prompt was:

Sarah says the product of 2 x 3 x 4 is less than the product of 4 x 3 x 2.
Is Sarah correct? 

At this point, I allow them to work alone or with their  partner.  Some of my students like and need to be on their own and have quiet to listen to their thoughts when required to write.  I respect this and always allow for it. 

While they are writing, I circulate and read, listen, and watch. I always try to find a way to prompt them to go deeper, explain more clearly, think in a different way, or I challenge their answers - asking for proof. 


15 minutes

After about 8-10 minutes of journaling, I signal the class to silence and ask them to take the 3 sticky notes that I placed on their desks while they worked and travel around to view 3 different journals. While they read and reflect on each journal entry, they are to leave a message to the author, stating whether they agree, disagree, or would like to add something and why.

Following the work tour, the students return to their journals, read the comments, and revise and add if necessary.

If time allows today, we will share our responses with a partner, or at the board with the whole class. There are pros and cons to each of these. Sharing with a partner ensures that each student will be heard and will hear.  Sharing with the whole class allows me to guide the conversation more.