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

Naturally, I bought some a few months back.

I asked Tabitha (6 years old) and Griffin (8 years old at the time) 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 why she thought that. I thought she might be mistaking side length for area. That is, maybe she was paying attention to the lengths of the sides of the two crackers rather than to the amount of cracker. So I asked about it.

She pointed to the uncovered part of the BIG Cheez-It and argued that this wasn’t enough to make another full regular Cheez-It. So she *was* paying attention to the amount of cracker.

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

I imagined that this chipmunk would be nibbling the crackers next door and that our conversation would be at an end.

I was wrong.

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.

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

## So what do we learn?

Notice the differences between the children’s strategies. Tabitha, the six-year old, worked with the crackers. She put one cracker on top of the other and tried to picture whether the leftover space made up a whole cracker. She was very concrete in her thinking.

Griffin, the eight-year old, worked with representations of the crackers. He traced and cut out squares of paper which he could manipulate with more precision than the actual crackers.

The two children reached similar conclusions.

Neither child used tools to calculate areas.

Knowing whether one cracker is twice as big as the other does not require measuring how big either cracker is.

All of this is very typical for young children. Younger children tend to work with the actual things they are comparing. They are what we call *concrete thinkers*. Older children begin to work with representations of the things (e.g. Griffin’s cut outs). They are more likely to be *abstract thinkers*.

## Starting the conversation

Investigate advertising claims. Have a healthy, skeptical attitude towards these claims, and encourage your children to wonder about them, too.

Be forewarned, though! You may create critical thinkers who question your authority, too.

And you may end up spending a LOT of time trying to figure out whether Double Stuf Oreos are really doubly stuffed.

For older kids, there are a lot of other things you can do:

Measure the sides and compute areas.

Take the ratio of the side lengths and square the ratio (easy to do from the picture, and it does seem to be a 2:1 ratio).

Put the small cracker on top at a 45º angle, so the corners are at the midpoints—Use the folded-paper trick of folding the corners to the center to demonstrate that the area is half.

Count all the crackers in each box. If the weights are the same, there should be half as many big ones.

Weigh individual crackers (if you have a $10 centigram scale), and plot a histogram of their weights. Compute the mean and standard deviation of the weights.

These are all lovely demonstrations, for sure,

gasstation! Thanks for sharing. An important difference between these and the ones I documented in the post iswho thought them up. As much as possible, I want to start the conversation by asking my children to do the intellectual work of creating strategies. I try to save my own ideas for the end of the conversation—extensions to build on their own thinking and support future innovation.I agree that it is better to start with the kids’ ideas. Sometimes it is worth nudging them with something they did not think of, though. Most of the ones I listed here would be for slightly older kids, anyway.

Could you elaborate on the comment – “Knowing whether one cracker is twice as big as the other does not require measuring how big either cracker is.” I’ve read the post through a few times and don’t quite understand what you mean.

Sorry about that,

mjlawler. That is pretty cryptic, isn’t it?One of the goals of this website is to help parents (and teachers) see how much mathematics there is to play with

outside of textbooks. Many people are made nervous by textbook math, and so they may shy away from doing math with kids, or worry that they have to do it like a textbook would. This is a shame, and I want to change it.A standard American textbook approach to this comparison would ask students to find (or estimate) the area of the small cracker, the area of the large cracker, and to compare the resulting computations. (Notice that this is suggestion number 1 in

gasstationwithoutpumps‘s list of extensions above.)So out come the rulers and the computations.

Tabitha is 6 years old. She can’t do that. But she can still do some math.

We can do math without computing, and that this makes all sorts of fun and interesting ideas accessible to children who don’t know how to multiply .

So we don’t need to know the

areasof these crackers to check whether one is twice as big as the other.That is what I was trying to say.

Thanks for asking!

I realize the odds are quite low, but in the one in a zillion chance that you haven’t seen Fawn Nguyen’s picture frame project, I think you’ll like it a lot. Her approach to a problem that probably is in every algebra book is similar to what you are doing here:

http://fawnnguyen.com/2013/04/10/20130410.aspx

Not one in a zillion at all,

mjlawler. Those are my cutouts at the bottom of her post. Fawn rocks.Always looking for real-world applications of math investigation. This is a good one. I also like exploring probability with colored candy. THe “double” stuff oreo investigation is a funny one. Some years back i was in Oakland, and a kid up in Berkeley ran a study of the parking meters downtown. She discovered a discrepancy between the rates offered by different machines. Something to consider in the future may be how to explicitly help parents initiate and continue productive mathematical conversations with their kids.

Thanks for the comment,

Richasaurus! I would love to hear about your probability tasks. That is such a difficult topic because right/wrong are so abstract. How—exactly—is a 30% chance different from a 50% chance, and how can I help children (or adults) to experience that difference in a way that theyknow it? This is a problem I have been working on for some time, with very limited success. Any help is appreciated!I also welcome your thoughts about how

to explicitly help parents initiate and continue productive mathematical conversations with their kids. That is precisely the thing I am trying to figure out how to do. I tend to err strongly on the side ofshowingovertelling, but I try to do both. Feedback on places where more telling would be helpful is greatly appreciated. Comments are good, and you can hit me up on the About/Contact page too.