Tag Archives: associative property

Math Peeps at Play

I have test driven these photographs and questions with a 6-year old, an 8-year old, a 43-year old and a classroom full of 19—40 year olds. Good conversations were had with all populations. I turn them over to you. Use them for the forces of good, not evil.

Associative and distributive properties

How many Peeps in this picture?


Do you see 4 boxes of 12?

Or do you see 12 sets of 4?

The first could be notated 4\cdot\left(3\cdot4\right)

The second could be notated \left(4\cdot3\right)\cdot4

That these two are equal is an instance of the associative property of multiplication.

There are, of course, other ways to view these guys, and to notate how you see them. The mathematics doesn’t live in the Peeps, it lives in the interactions we have around the Peeps.

Careful discussion and notation will demonstrate the associative property and/or the distributive property in each of the pictures below.

Which is more?

In each of the following images, are there more purple Peeps or more pink Peeps? Of special importance is this question: How can you know without counting?

Here is the full set of Peeps photos for easy downloading (.zip).

In the meantime, these guys are still hanging around my office. Got any other arrangements you’d like to see?

I know, for instance, that I wish I had a fifth box so I wouldn’t have repeated 4’s in that first picture.


Beyond the Textbook wrap up

What does this have to do with mathematics?

I had a question at the beginning of the day on Thursday, which I shared through Twitter.

The question got louder in my head as the day progressed. From my perspective, a tremendous amount of time was being invested in designing the platform for a mathematics textbook-of-the-future while not very much evidence was being presented that any of our work reflected knowledge of mathematics for teaching.

My worry continued to deepen that we were designing a better platform for delivering Khan Academy content.

Considering that my critique of Khan Academy has nothing whatsoever to do with the platform, and everything to do with the pedagogical content knowledge of the instructional designer, this was fast becoming a problem.

So I sought out some sympathetic ears in a lull in activity. I hit Frank Noschese and Chris Harbeck with a vulgar version of this question: What in the world does this have to do with mathematics?

Angela Maiers took me up on this question by arguing that, essentially, Mathematics has nothing to do with this, and that’s the way it should be.

In the end, it turns out that the two of us had very similar concerns. An example helped to bridge the gap. That example follows.

At heart, multiplication is about same-sized groups. Whether you write five groups of three as 5×3, 3×5, 5(3) or some other way, multiplication structure is about some number of same-sized groups.


We can use multiplication to count the water bottles in this photograph because they are arranged in an array—rows and columns.

But many children do not count things this way.

We can know this by observing children as they count. It is quite common for children to count an array by circling around the outside, or even in a seemingly haphazard order. Even very skilled counters may not notice the unique structure of an array.

A common counting sequence for a child who does not use the rows-and-columns structure of an array

A common counting sequence for a child who does not use the rows-and-columns structure of an array

If they do not notice this structure, they cannot use it.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the commutative property of multiplication. One view of the array below is as five groups of three. The other is as three groups of five. The array makes those groups for you, and it suggests that a groups of b will always be the same as b groups of a.


The array support the general argument that ab=ba for all whole numbers.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the associative property of multiplication. One view of the collection of shoes below is as four groups of three. A different view is as four groups of six.


How many shoes? (Credit to my student, Marissa Brown, for the photo. She submitted it for a class assignment.)

If you see four rows of three, then we can express the total number of shoes as (4•3)•2. If you see four rows of six, we can express the total number of shoes as 4•(3•2). Of course these are equal—each of them correctly counts the number of shoes on the shoe rack.

Therefore, (4•3)•2=4•(3•2).

And again, the deep connection between (1) multiplication, and (2) the structure of rows and columns suggests a more general argument.

There was nothing special about 4 rows, nor about 3 pairs, nor about the fact that these were pairs. Anytime we have A groups of B groups of C, we can compute either (A•B)•C or A•(B•C).

That is the associative property of multiplication.

So What?

But what can we use this property for? What good is it?

For one thing, it’s good for mental math.

Quick: what is 6×60?

If you are like most of us, you unconsciously multiplied 6•6, then by 10. You used the fact that 6•(6•10)=(6•6)•10. You used the associative property of multiplication.

And Javier, in an IMAP video, uses it to figure 5•12. Go there and watch for it.

Did you catch his implicit use of the associative property?

He knows that:


Or dig this. What is 35×16?

Use the associative property twice:


This is about number sense; it’s about the numerical relationships that form the heart of mathematics.

But it’s also about the inner working of paper and pencil computation. Let’s say you want to multiply 35×16 by the standard American algorithm. Then you would, at some point, say to yourself “3 times 6 is 18”. But that 3 doesn’t mean 3. It means 3 tens. The fact that you can treat it as a 3 is due to the associative property of multiplication.

Division, by contrast, is not associative. (a÷b)÷c is not the same as a÷(b÷c). This explains why we do not operate digit by digit in the standard long division algorithm.

There is much, much more.

Contrast with what Sal Khan has to say about the associative property of multiplication.

Khan knows this property. But he does not know (1) that an array is an important representation that can help to establish this property, (2) that children need to be taught to see the multiplication structure of an array, (3) that—at 1:55 in the video—he is using the associative property to do the computation 12•30.

Et cetera, and on and on.

This video demonstrates my concern perfectly. Too much attention to delivery method (exercises! badges! energy points! sympathetic narrator!) and not enough attention to mathematics, not enough attention to how people learn mathematics.

Bringing it home

And—to be frank—if Discovery Education doesn’t have someone paying extremely careful attention to all of this throughout their beyond-the-textbook writing process, they’re not going to produce something that will have an impact on mathematics teaching and learning in this country.

But if they do? Perhaps the sky is the limit.

I have been through a brainstorming/prototyping process before that was very much like Thursday’s session. That other one didn’t have the same attention to the possibilities of electronic student materials that this one did. If Discovery can get both parts of this right, they could create some exciting stuff.

I believe they want to do that. I really hope they can.