A few months back, I got an email from a local suburban elementary school. They had been given a bit of money to “give all of our fourth graders a unique math experience,” and they were seeking advice.

My first thought was, “Send them all to New York to visit the Museum of Math!” but this was off by a couple orders of magnitude.

As the conversation continued, it became clear that they weren’t seeking advice so much as someone to make it happen. So I said yes.

I am spending three Thursday mornings, and one afternoon, with these fourth graders. Today was day 1.

The theme of the residency is scale. We are playing with small versions of big things and big versions of small things.

A few favorite moments from today:

### Horses

When asked to share a big version of a small thing, one girl said “Horses”. I pressed her to state her meaning. “If you had a map with stables on it, the horses in those stables would be really small, then when you went to the stables, the *actual* horses would be really big.”

Ladies and gentlemen, I give you the big math idea of *inverse!*

I thought *the horses on the map are small versions of the big real-life horses.* But she was very clear that her experience was *small horses on the map, then see the big ones*. The small-to-big relationship isn’t just the opposite of the big-to-small one; it is its own relationship. These two relationships are inverses—each existing on its own, but with a special connection to each other.

### Which One Doesn’t Belong?

I cooked up a little *Which One Doesn’t Belong? *set in preparation for our work.

*Which One Doesn’t Belong? *never disappoints. (Student/home version and Teacher Guide coming this summer from Stenhouse, by the way!)

We noticed all the things I had hoped for, and more. And then afterwards a girl came up to me to make her case that we weren’t being totally precise about our description of the upper-right image. If—as we claimed—the shape in the upper right is composed of four of the upper-left triangles, then the big triangle wasn’t exactly the same size as the one in the lower left because the triangles have outlines which are not infinitely thin.

### Composing triangles

I brought in many small laser cut triangles of these seven types:

I gave them time to play with these triangles. One student said she knew what we were going to do with them. So I asked her what that was, and she replied that we were going to see which ones could fit together to make other ones. This was not the plan, but was behavior I was eager to encourage.

She asserted that the pink and the black make the red.

This was a detour worth five minutes, so we took it. Arguments were presented pro and con. The major pro argument was based on the *close enough* principle. Con arguments were of two flavors: (1) put the red underneath and you’ll see some red peeking out from underneath, and (2) the long side on the pink plus black shape is not straight, while it is on the red one.

### Composing similar triangles

The main question I wanted to get to—remember that our focus is *scale*—was *Which of the triangles in our set will do what the upper-right shape in our Which One Doesn’t Belong? set does?* Which of our triangles can you make into a larger version?

**Spoiler Alert!**

All triangles do this. But these fourth-graders don’t know that. And because they don’t know that, they got to feel a little thrill of success when they found one that did.

And of course they produced some evidence that the relationship we’re investigating is a challenging one.

This is what we had on the document camera at the end of one of three sessions this morning.

HOLD THE PHONE! LET’S LOOK AT ONE OF THESE CLOSE UP!

Do you see? All the others use four triangles to make the bigger version, and this one can too. But this can scale up to make a bigger version that uses only two of the original!

Of course there is a part of my math-major brain that knows this about isosceles right triangles, but it’s a wonderful wonderful thing to have pop up unexpectedly in the middle of fourth-grade math play.

Overall, a delightful morning of math. We got to only a small fraction of what I’ve got chambered so we’ll pick up where we left off next week. I’m hoping I can get them to build one of these.

Either way, I am thankful for the opportunity to play math with this group of kids. They are creative, enthusiastic, curious, and delightful. Their teachers have been very welcoming and open to the intellectual chaos I began to unleash today.

### Addendum

I chose a set of triangles that would have interesting variety and some discoverable properties.

**Purple**: 3-4-5

**Pink**: Isosceles obtuse

**White**: Isosceles right

**Red**: 30-60-90

**Light blue**: One-eighth of a regular octagon

**Black**: Equilateral

**Dark blue**: One-fifth of a regular pentagon

I also made some yellow obtuse scalene triangles, but they are missing so they didn’t make the trip. Within these classes, these triangles are all congruent. Each class has at least one side that is one inch long.

What a fortunate group of 4th grade students!

There’s a math museum in New York?! Field trip!

Yes there is. It’s lovely. You should go.

Hey Christopher,

I’m an education student and will soon get my first chance to teach math to a group of fourth graders.

During my studies I focus on exploratory learning and your post gave me plenty ideas.

I am planning to let the student explore different shapes and their characteristics (parallelism, angels) by letting them cut paper strips and then asking them to build different forms.

I was wondering:

How much instruction did you give to your students? Did you try to lead them in a certain direction or did they mention important points by themselves?

Thanks in advance!