# 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.

### 9 responses to “Incommensurate Cheez-Its”

1. I really enjoy your “talking math with your kids” posts because I also like to take advantage of opportunities to “talk math.” (See my Dinosaur Math post: http://k12mathpassion.com/2013/05/30/dinosaur-math/). Plus, I think relating math to everyday events is just good practice for making content comprehensible (not to mention, fun) for students.

I particularly like the comparison of Tabitha’s and Griffin’s approaches, reflecting the different maturity levels in their mathematical thinking. Nice!

2. Just last week we discovered that regular Cheez-its are basically a square inch, so we were going to build lots of area practices with them….

3. Christopher

One square inch strikes me as a reasonable estimate of the area of a standard Cheez-It. I shall need to investigate this further.

I dig the Dinosaur Math post, galenaylor. I especially like that you seize the opportunity to follow up on something the student/child is interested in and develop it further. That really is at the heart of what I hope to help parents of young children get better at.

I was thinking about your follow up question there, and it strikes me that I don’t really know how long 65 million seconds is. That’s kind of lovely. I am also reminded that someone (was it xkcd?) blew my mind recently by pointing out in a web comic that Stegosaurus and Tyrannosaurus were separated from each other by more time than separates us from T. Rex. That’s how long the dinosaurs were doing their thing.

As you observe, it is listening that makes these conversations possible.

4. Cool. I’m going to get some of these for my kids. They love “yummy math.”

5. I wanted your picture to show the regular cracker at 45˚ to the big one, so that I could see if the corners matched up to the midpoints. That’s the way I generate squares of half the area.

6. mrdardy

Christopher – Slightly off topic, but I wanted to share a conversation with my soon to be four year old daughter from this past weekend, We were on a long car drive and she was asking how far we were from our hotel. I replied that we were twenty minutes away. Later in the pool she was jumping to me from the pool steps and commanding me to back up some. I asked her how far I should go and she told me to be five minutes away. I said “Do you mean five feet away?” and she replied, firmly, that she meant for me to be five minutes away. I am wrestling with whether I think this is just charming and (semi) clever on her part or whether I need to start answering her pleas in the car with distances. Curious to hear some ideas on this.

• Christopher

This is fabulous, mrdardy! I’ll work up a post on the matter soon. Short answer is that you’ll do no harm either way. Just keep talking, and keep her talking!

7. thanks for the post. I am going to try this as an eighth grade lesson. I posted some ideas on my blog and welcome any feedback. I think it could be a great way to discover irrational numbers