Aiding a Confused Parent with Arrays

Found on Twitter with the caption: “kids can't learn 6x8= 48 anymore its now 6x8=(5+1)x8 Why the confusing work”

Dear Confused Parent,

The picture you posted comes from a Powerpoint made by EngageNY to help teachers teach Grade 3, Module 3, Lesson 10.  Lesson 10’s objective is to “use the distributive property as a strategy to multiply and divide.”  So the question “How does topic C use the array model to move learning forward?”  is for teachers to consider as they think about the lesson.  The answer EngageNY gives is: “The array model helps students understand division as both a quantity divided into equal groups, as well as an unknown factor problem.”  Which provides important background, but does not directly answer your question.  

To get back to the first part of your statement, “kids can't learn 6x8 = 48 anymore.”  In fact, the Common Core State Standards state that by the end of third grade students should “know from memory all products of two one-digit numbers.”  So they can and should learn 6x8 = 48.  At the same time, they are learning why 6x8 = 48 and how to figure out that 6x8 = 48.   

That finally leads us to your question, “Why the confusing work?”  The answer boils down to two main reasons, 1) the distributive property and 2) helping students figure out why 6x8 is 48, based in facts they already know and might be able to do in their head.  The distributive property states that a(b+c) = ab + ac.  The property is important for understanding elementary ideas such as how multiplication and addition are related

5 x 3 = 5(1+1+1) = 5 + 5 + 5

to secondary concepts such as multiplying binomials

(x + 4)(x - 5) = x(x - 5) + 4(x - 5) = x2 - 5x + 4x - 20 = x2 - x - 20.  

In the problem you posted, students can break up the array of 48 dots into 
(5 + 1)8 = 5 x 8 + 1 x 8 or 6(5 + 3) = 6 x 5 + 6 x 3 to show that they both equal the same number of dots.  

This is also a strategy for figuring out 6x8 = 48 if students have forgotten the fact.  They may know 6x5 is 30 and 6x3 is 18 and can add to find 48.  Similarly, they may know 5x8 is 40 and 1x8 is 8 which together make 48 again.  An important part of the Common Core is building fluency through practice and conceptual understanding.  So by the end of third grade they should know 
6x8 = 48, but if they don't we should make sure they have the tools to figure it out.

I hope this makes you feel less confused,

Jim Town - Mathematics Specialist at ACOE Core Learning

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