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

The second could be notated

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.

Typo at the start: it’s 16 sets of 3 which would be (4*4)*3 for those of us who see in columns, or 12 sets of 4, (4*3)*4.

For me I see it as 4 sets of 12 at first, I think because of the coloring, and then as 3 rows of 16 next, which for me is 3 * (4*4).

At the bottom I see the 3rd and 4th picture by rearranging. I wish I could see the second one that way too, but rotating a grid of dots is easy and rotating the bunnies while keeping the ears up is just a bit beyond my immediate visualization, so I see the 1st and 2nd picture by multiplying.

I love all the uses of these peeps — as long as we’re not actually eating them, it’s all good.

Typo fixed.

This is what I want to see from you.

My son, who is in Kindergarten, went through your Peep photos. Here are a couple of things he said.

1) “They are the same because purple has 8, 12, 8, 12 and pink has 8, 12, 8, 12.” He was looking at the photo that had four complete boxes. He was counting two by 4s, but skipped 4 for some reason. I guess he sees them as them same because of the same pattern, almost like a 1-1 correspondence.

2)”The purple is more because it is taller and they ate less.” This was for the picture that had two rows of 4 pink and 3 rows of 3 purple.

3) “There are 12.” This is for the photo that had four boxes stacked. I thought this was interesting because I would have said 12(4)=48. But we really don’t know what is in the boxes below. So, when I asked how many peep are there, he seemed to only count they ones he saw, where I assumed the pattern continued.

For most of those photos he had to count. However, after he mentioned, “they ate less” I started to ask who ate less. That seemed more visual for him. I guess because of the smaller number.

From my 3 1/4 year old…

Which has more?

Picture 1: Liam: Pink. Me: Why? Liam: Because I like pink. Picture 2: Liam: Purple. Me: Why do you think? Liam: Because I like purple. Picture 3: Liam: Purple. Me: Why? Liam: Because I like purple. Picture 4: Liam: Pink. Me: Why? Liam: Because I like pink.

No need to overthink this one… Since he is 3 1/4… But that of course doesn’t stop us…

He is actually incorrect on all of them.

Or correct on all, but one, if he is thinking about which has more empty spaces.

What I love about this activity is that is can appeal to any age. It’s rare to find interesting ways for Algebra students to think about the distributive property as anything other than a manipulative trick. The area model is first used in Grade 3 or 4 to show the distributive property. The same area model can (and should) stretch into multiplying binomials and factoring and finally into completing the square. However, this basic numerical property of addition over multiplication loses its magic in algebra and students learn the “rainbow method” or the “FOIL method” using algebraic symbols and completely forget from whence this idea came. Can you tell I’m on a mission?

For some funny reason, I have this huge feeling to sit down and write a paper..(but then it passed) haha. Got me thinking again about the properties, which I really haven´t thought about. Thanks again for making me think even when I am not in your class anymore..Miss those classes a lot!

I agree with Elaine. This is an activity that I can use with my fifth graders when learning about the distributive property. In the beginning of the year, We explore other ways to compute multi-digit multiplication. We discuss how expressing the multiplication of 85 x 34 can be done by writing 85 as (80 + 5) and 34 as (30 + 4) so that we could multiply in this manner: 80 x 30 + 80 x 4 + 5 x 30 + 5 x 4. There was so much push back from many who had done the lattice method. Last month I reintroduced distributive property for solving multiplication with mixed numbers. Another push back and not fun. Since then, I’ve been asking my self if there is any other way for my students to learned distributive property in a way that makes more sense for them, less abstract. These Peeps are great for teaching distributive property. the use of distributive property happens so natural here!!

Iztchel,

You said “We discuss how expressing the multiplication of 85 x 34 can be done by writing 85 as (80 + 5) and 34 as (30 + 4) so that we could multiply in this manner: 80 x 30 + 80 x 4 + 5 x 30 + 5 x 4.” Are you doing this only symbolically as you describe here? Or are you showing it as a area model, a rectangle with the top dimension divided into 80 and 5 and the side dimension divided into 30 and 4? The result is that the large rectangle is divided into four sub-areas, each representing the 4 products you mentioned (80 x 30, 80 x 4, 5 x 30, and 5 x 4). This is the model that I find causes the aha! moments and also is the model that nicely transitions into multiplying mixed numbers, multiplying (x + 5)(x + 4) and then factoring x^2 + 9x + 20, and finally completing the square. You may be doing it this way, but it was not clear from your comment.

I feel your pain on the “push back” from those who have learned the lattice method. While is works, it is difficult to see WHY it works. I haven’t found a way to use the lattice method as a conceptual springboard to multiplying mixed numbers, binomials, factoring, and competing the square as well as a springboard to long division and algebraic division, although there may be someone who has figured this out. The area model is consistent and, in my experience, easily understood by students once they play around with it a bit. The lattice method is a shortcut. Eventually students need to use shortcuts, but not at the expense of understanding the conceptual underpinnings of multiplication and division.

Oh!! Yes that’s what I meant. Thanks!! Area model is the way to go. I never thought about it with fractions. Thanks!! That is really an aha!! Moment.

This is so cool! What a great use of Peeps!

I’ve noticed (in doing things like this with pictures or blocks — but not yet with Peeps!) that it’s difficult for students to describe an amount as 4*3 groups of 4 even when “12 groups of 4” is no problem and even when 4 groups of 3*4 is ok. Why is that? What else is it related to?

By the way, I’m posting a little more about this on the Mathematics Teaching Community at

https://mathematicsteachingcommunity.math.uga.edu

and linking back to here.

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