Discovering the Distributive Property of Multiplication

Distributive Property of Multiplication…oh, my!
Yes, captain.  Shields up and ready to fire on the Distributive Property of Multiplication.

Root canal without Novacaine.  Pins under my fingernails.  Sitting through an insurance seminar.  Those all seem preferable to teaching the Distributive Property of Multiplication (DPM) to third graders.  Seriously, didn’t my generation learn that sometime in MIDDLE SCHOOL?  I know, stop with the whining and just teach it.

Here’s the rub.  I have never had to teach this property!  I met with my team and we talked about how to teach this.  Should we break it down into steps?  Should we do only hands on first?  Should we use the Go Math book?  We decided that we had to start with the concrete.  Get the students to use manipulatives to build arrays that could then be broken apart and make the connection to the DPM.  So here’s what I did.

I presented the problem/question then highlighted key words.
First I introduced the problem:

How can we break apart the array 3 x 7 to make it easier to solve?

If you have read my previous posts about Lesson Study and bansho, then you know I always start with a question that begins the lesson.  We read the question together, we underlined important key words (break apart, easier).  I explained to the students that I would give them the bag of foam tiles and a blank paper.  They were to use the tiles to build the array first and the paper was to record their solutions with drawing and words and numbers. 

I began to walk around the room interested in how the students interpreted “break apart.”  Some students literally scattered their tiles around leaving no semblance of an array. Some students broke apart the array by putting the tiles into equal groups (hey at least they were equal groups!). Some students broke apart the array into 2 smaller arrays.  I saw that one student was already recording his solution so I went to see what he had done.  He had broken the array into 2 smaller arrays, counted the tiles in each array, then added.  Not bad!  I had the student go up to the board and copy his solution and then explain to the class what he had done. This was an excellent opportunity for me to step in after his explanation (which by the way I wish I had recorded because he used the correct vocabulary and his explanation was worthy of any common core question!) to talk about what it means to “break apart.”

Student used tiles to find a solution, then recorded answer then shared it with classmates.
I like to tell stories, so I told the students this story about my backyard.  In my backyard, I had to move a lot of bricks from one location to the other.  There were more than 30 bricks.  I said that there was no way I could move all the bricks at once.  So I decided to move about half of them at a time using a hand dolly.  This way I divided or divvied up the work to make the task easier.  I kept referring back to this idea of breaking up a hard task into easier tasks.

I guided the students to understand that breaking apart the array meant that after you broke it apart, you still had to have an array!  Not groups, no piles, not unequal shapes.  So once that was cleared up, I gave them another array to make and break apart:  3 x 8.
Another student found a solution for 3 x 8.
This time, the students understood that they had to end up with 2 smaller arrays made out of the bigger array but keeping the same amount of rows or columns.  Most of the students I observed broke apart the array into 2 arrays of 3 x 4.  Then they added the two arrays (12 + 12).  I didn’t see any multiplication sentences yet, so I had to make sure I encouraged using multiplication sentences when labeling.  However, I did find one student who had written multiplication sentences so I picked the student to come up and record the solution while the others continued to work.

After the student explained what she had done, I then asked the students:

Where is the best place to break up an array?

I guided the conversation by asking them which factors were easiest to multiply.  They came up with 0, 1, 2, 5 and 10.  Great!  I kept guiding the conversation by saying pointing out that you’re not going to break up an array by 0 or 1. Since we are working with large arrays, I pointed out that 5 would be a better choice.  Once that was established, I asked the students to turn their paper over and take notes.  A very quick lecture followed introducing them to the DPM.  We brainstormed synonyms for the word distribute and we talked about WHY they had to learn this property. 

From there, I gave them another array to break apart:  4 x 9.   Most of them began by breaking apart the array at 4 x 5 and 4 x 4.  From there I saw them write their multiplication sentences and added the partial products to get the total product.  Success! For now at least. 

I did quickly explain to them how their 2 multiplication sentences needed to be combined into one multiplication sentence 4 x (5 + 4) so they could see where we were going with this whole process.  That’s tomorrow’s lesson.  Stay tuned.
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