# Tag Archives: similarity

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?

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.

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.

## Tree problem, redesigned

So I work for this curriculum project.

Back in the old days (late 1990’s) we had a problem that had interesting data, but impossible to teach. Basically seventh-grade students were supposed to write sentences comparing the popularity of various activities among various demographics based on this table of data from the Statistical Abstract (which document, by the way, you need to spend some time with-there’s some great stuff in there for math teachers of all stripes):

Nice task. But where it sat in the curriculum, it was too much. Too much data, too much variation in size of groups, too much everything. It crashed and burned for me and many others every year.

So then we revised and scaled things back. The present version of a problem that uses real data to get seventh-grade students making comparison statements involves this table of data:

Much cleaner and more manageable.

Now we’re revising again. I’m looking at the problem not through the lens of whether it’s manageable in class (which this one is), but through the lens of whether the comparisons it generates are as engaging as they could be for kids (I think they’re not).

What would make it better?

1. Having a reason to compare
2. Making the comparisons visual

### Why compare?

It’s unclear why I would compare the Giant Sequoia to the White Oak, except to notice that even a big oak tree pales in comparison to a Giant Sequoia.

But what if I’m dealing with bonsai trees? In Karate Kid, Daniel-san calls them baby trees. Are bonsais really miniature versions of real trees? If so, how can we describe the similarities? And if not, how can we ferret that out?

### make the comparison visual

We’re comparing heights and widths (essentially) of trees. I want to see that, not just have it be numbers in a table.

Here was my first attempt at this:

In both cases, that’s a 1 foot by 5 foot rectangle.

In both cases, that’s a maple tree (different kinds, but still…)

And we ask students to compare heights, spread and trunk-circumference of the bonsai and full-sized trees. Then there’s another pair-this one two Ponderosa Pines. Long story short: height/spread ratio consistent within species, but the trunk is out of whack.

Personally, I am more interested in comparing the bonsai to the real than I was in comparing two different types of trees in a table. It’s a better task, I think. But the visual isn’t quite right.

So here’s a second try:

This one is more provocative. Not as good for making estimates of measurements, but better for seeing the comparisons in question.

So we launch with something like this one. And provide something like the initial pairing while students work. It’s on a photocopy so they can write all over it.

And then we follow up with that table of trees and ask kids to give reasonable measurements of bonsai trees for the other tree types.

Now…these images are scammed from various places on the Internet. These particular pictures are only in use for classroom trials. The real deal will require new photos. So I turn to the web for a bit of guidance…

Since the new pictures will be from scratch in the spring/summer, what modifications would make the images more compelling?