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?


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.

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?

Addendum 1

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.


Addendum 2

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).



addendum 3

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.