Connecting the Open Array to the Distributive Property

Read the following vignette¹ to view how students used an array model to represent their  thinking.

 

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This learner inquiry is designed to help students explore why the distributive property works.  Consider the multiplication problem 16 x 16: What are all of the different ways we could break this multiplication problem up?  How are the parts connected to the whole?  The students in Anne Denney’s third grade have been working with several depictions of arrays and discussing how the numbers could be spilt.  Now they are moving on to try it with larger numbers.  Let’s eavesdrop…

 

Josey decides to try to multiply 16 x 16.  She splits each 16 into 10 and 6.  Then she multiplies 10 x 10 and 6 x 6 and declares, ‘There, the answer is 136.” 

 

Cleopatra, who is working next to her, says “That can’t be right...10 x 16 = 160 —- 136 is too little.”

 

Josey ponders this insight and agrees, although she is puzzled.  “But what did I do wrong,” she asks.  Neither girl can figure out the problem so they bring their work to Anne. 

 

Why didn't’ my way work,” Josey asks Anne, perplexed.

 

Anne calls the class together and asks Josey to share what she did.  Why didn’t this work, Anne asks.  She split the 16;’s up correctly, didn't she?  10 + 6 is 17=6. 

 

I tried 16 x 16 too, Cleopatra offers.  10 x 16 = 160, and 6 x 16 = 96, so we know that the answer is probably 256.  She shows the class how she added 16 + 16 + 16 + 16 + 16 + 16  by pairing 16’s and adding 32 + 32+ 32.

 

Josey agrees.  Yea, we know that 136 is too little, so we think you can’t just split the two numbers.

 

Who wants to investigate this?  Anne asks the class.  “Can you split both numbers when you multiply, and when do you know that you have done all of the pieces?”

 

Several children decide to join Josey and Cleopatra and Anne suggests that they use graph paper and make a picture of what they are doing.   They set off with their materials.

 

Josey says, “Lets start with 12 x 12, because we know that one.”  She outlines a 12 x 12 square on the graph paper.  I’ll split the 12 into 10 and 2.  Here’s the 10 x 10 (she outlines a 10x10 square inside the 12 x 12 as shown below), and here’s 2 x 10. 

So, what is left, Anne asks?

 

2 x 12 Josey concludes, confidently finishing the array.

 

There, I did all the parts.  It is 144.

 

Now I’ll try 16 x 16 again.  She draws an outline of a 16 x 16 array and outlines the 10 x 10 within it.  And here is a 10 x 6, and a 6 x 16.  she continues drawing in the parts.

 

But you had a 6 x 6 before, didn’t you? Anne inquires.  Where is that?

 

AT first Josey looks perplexed, but then she sees it.  Oh, here it is, and draws it in.  It is this little square.  So I left out these other parts before! She explains as she points to the other two 6 x 10 arrays. 

 

Is there a little piece like that in the 12 x 12? You have four arrays here, (in the 16 x 16), and four numbers when you split each number, two 10’s and two 6’s.

 

Oh yea, a 2 x 2, Josey beams, proud of herself  You can do it by splitting both numbers.

 

Because Josey owns the question, she persisted. Because her dilemma was intriguing and meaningful to her classmates, they joined her.  Often the best question comes from the child and teachers just need to listen intently and trust.  Josey is engaged in real mathematics; she has raised an important question: Can both numbers be spilt when multiplying?  As she sets to work to prove it is not possible, she proves that it is.  Historically, some of the best mathematics has been invented this way.

 

 

 

¹Taken from Young Mathematicians at Work: Constructing Multiplication and Division , by Catherine Twomey Fosnot (pp. 43-44).

Vignette