# Tag Archives: squares

## Incommensurate Cheez-Its

There are now BIG Cheez-Its (U.S. only, it appears). The package claims that they are “Twice the size!” of regular Cheez-Its.

On seeing this claim, I thought for sure that we were gonna have a We mean four times, but say twice sort of a situation on our hands. So I bought some.

And then I asked Tabitha (6 years old) and Griffin (8 years old) what they thought. I started with Tabitha when Griffin wasn’t around so I could get her pure thoughts.

She put one cracker on top of the other and proclaimed, “No”.

I wanted to know the source of that. I thought she might be making the classic linear v. area error (i.e. interpreting twice to mean twice the side length). So I asked.

She pointed to the uncovered part of the BIG Cheez-It and argued that this didn’t constitute another full regular Cheez-It. Score one point for argumentation, but minus one for spatial visualization.

A few minutes later, it was Griffin’s turn. He ran like a chipmunk with his two crackers into the dining room. Experiment over, right?

Nope.

He was in search of paper and a pen. He carefully traced each cracker, cut out the uncovered part of the BIG one and attempted to partition and reassemble this remainder on top of a tracing of the regular cracker, which it did not completely cover.

Sadly the cut outs are lost forever.

His conclusion: BIG Cheez-Its are almost but not quite twice the size of the regular Cheez-Its.

Volume perhaps?

If the crackers are twice as big, but the mass of one serving is constant, and if one serving of regular Cheez-Its consists of 27 crackers, how many crackers should be in one serving of BIG Cheez-Its?

There are 14.

If the area of a BIG Cheez-It is about twice the area of a regular Cheez-It (as Griffin confirmed), then the side lengths should be in a ratio of approx. 7:5 (a reasonable estimate of the square root of 2).

Notice the progression in the children’s strategies. The six-year old worked with the crackers. The eight-year old worked with representations of the crackers. Similar conclusions were reached; the child who worked with representations could manipulate those representations in order to achieve a greater degree of accuracy, and to investigate hypotheses that the child working concretely could not.

Neither child used tools to calculate areas.

## Triscuit squares

After chatting with the Mathalicious crew about a lesson on square roots and irrational numbers, I was inspired to talk math with Tabitha (5 years old).

Me: [With a box of Triscuits at the dining room table] Tabitha! Come here, please! I want to talk to you about something.

Tabitha: I know what this is gonna be about.

Me: What?

T: Math.

Me: Right. I want to know whether you can arrange those in a square.

I hand her four Triscuits.

She quickly forms them into a square.

T: Done. Now can I eat them?

Me: Not yet. Can you do it with these?

I give her nine Triscuits and scramble them up.

She is again successful.

Me: How many more are in this square than in the last one you made?

T: I am really tempted to eat them.

Me: Right. But how many more are there this time?

T: There were 4 before. And now there are 9.

Me: Yes. So how many more in the big one?

Some elaborate Triscuit shuffling goes on, lasting about a minute.

T: Five. Wanna know how I did it?

Do you see the beauty of doing this on a regular basis? Children learn discourse patterns through exposure. Not only can she explain her thinking, she expects to do so.

Me: Yes.

T: I took 4 away, then there were five left.

Me: Nice. One more.

T: I really want to eat these.

Me: I know. Soon. Can you make a square with these?

I give her 7 Triscuits.

She moves them around. She is not especially systematic in the order she places them.

She ends with this arrangement.

T: If you took this one [i.e. the one in the upper left] away, you’d have a square.

Me: Is that a square?

T: Oh! No. It’s a rectangle.

You do it.

Me: I can’t. See I can do a square with 1 Triscuit.

T: Of course.

Me: Then I can do 4 like you did, and 9 like you did. Four had two Triscuits on a side. Nine has three Triscuits on a side…

But she can no longer hear me over the sound in her head of the crunching of Triscuits.