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Discovering the Distributive Property of Multiplication

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

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

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

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

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