Multiplication – Division Connection: Cousins.

Multiplication and Division are cousins.

Did you know that?
They’re totally related. That’s what we figured out today in math –
during a totally mind-blowing lesson
that I’m going to share with you.

We’ve been working with arrays and how they represent multiplication.
I’ve tried to cultivate this language: combining the same amount over and over.
It’s starting to click.

So we worked on this foldable.
–If you’ve read my Math Notebooks: Pink? post, you know why it’s obnoxiously pink.
It’s cause I’m such a good sport.


First we did the left side (about five days ago).
We wrote problems that required multiplication on the outside flap.
On the inside, we drew a sketch of what was happening in the problem.
Then we drew the array.

That was just the left side. The first problem reads: “Mary has 8 bags of candy. She has 10 pieces in each bag. How many pieces of candy does Mary have in all?”

Not exactly rocket science, but pretty challenging for my little guys.
It was only two months ago that I amazed them all with my mathematical magic: regrouping.
Anyway, we drew the actions of the problem (equal groups being combined)
and sketched an array with 8 rows and 10 in each row, for a total of 80.

So today,
I blew their minds.
We’ve been working with division for a few days.
We have already identified that an array can represent multiplication AND division.
The kids said they were totally down with this idea.
They got it.
No problem.

I KNEW they were faking it.

So today, I handed out the foldables again.
On the right side, we wrote this problem:
“Mary has 80 pieces of candy. She is putting an equal amount of candy in each of 8 bags. How many pieces of candy will be in each bag?”

You get it, right?
Same problem?
Same objects?
Same amounts?
Different angle?
Instead of combining over and over, we’re taking away over an over. What you may call…division!

But no one seemed to notice that it was the same idea.
They WERE faking it!
Buncha fakers.
So we sketched the actions of problem inside the the foldable.
(a total being shared equally into groups)
Then we drew the array.
80 total, 8 rows, 10 in each row.
Then I said,
“So,  open your flaps and look at the arrays……
Notice anything?”

And Randy went, “OH!”
but it was an “OH” that is impossible to capture in text.
It was like, “Oh, the universe makes sense to me now!” in only two letters: O.H.

And it should make sense now.
Because it was the same array for both problems.
Because multiplication and division are inverse operations, or, as we say in third grade,
opposites.

We wrote on the back of our foldable.
I don’t know how to teach without having kids write about their thinking. This was kind of a fast one, just to record some basic reflections:

So the kids had kind of an epiphany
and now the world is a logical place.
Multiplication and division are related. They are inverse operations, representing groups, an amount in each group, and a total. The difference is…
which parts to you have?
and
which part are you trying to find?

So that’s why they’re cousins.

Question: How do you blow your kids’ minds? What are they the most surprised to learn?

On a different level, I recently added a couple of very basic math games to my TPT and Teacher’s Notebook stores.
They’re made for Kinder, mostly: Number matching. Each pack includes cards with numbers 0 through 20 in word form, number form, tallies, and pictures for students to match.

There are three different styles:
Rainy Day Number Matching TPT and Teacher’s Notebook

 and Have a Heart Number Matching for Valentine’s Day TPT and Teacher’s Notebook

 Lucky Number Matching for St. Patrick’s Day TPT and Teacher’s Notebook

Couldn’t help myself.
KPM Doodles is SOOOO cute.