Visualizing Volume is Tricky!

This was one of those lessons that I think I gained a lot more than my 6th graders did. It was meant as a one-period activity, but I kept going off on different tangents and brought the students along for the ride.

Over a month ago Andrew Stadel tweeted me a picture that he took of a William Sonoma display of their cupcake mixes. I was at our local mall last Sunday and saw a similar display. We both thought about buying the mixes to make cupcakes for our kids, but it was $15 a can, and we'd need three, so the poor teachers said no can do.

I projected the images above and asked the kids to give me a guess of how many dozens or how many individual cupcakes can the large container make when the small [real] container can make 1 dozen or 12 cupcakes.

Their guesses were all over the place, ranging from 47 to 994 cupcakes. (We were very careful whether the submitted guesses were in "dozens"or in "individual cupcakes.")

So I did the only thing I knew. I replicated the two cans so the kids could see them physically in the room instead of just on a still photo. Granted the large "can" made from butcher paper was pretty awful.

But, before I asked for another guess at the number of cupcakes, it occurred to me that I wanted to know if kids were better at guessing one and two-dimensional items.

Question 1: How many times taller is one segment than the other?

The two segments below are proportional to the heights of the two cans.

Their estimates:

Question 2: How many times longer is one circumference than the other?

The two circles are proportional to the cans' tops/bottoms.

Their estimates:

Question 3: How many times larger is the area of one circle than the other?

The two circles are proportional to the cans' tops/bottoms.

Their estimates:

Then, I let the kids — row by row — come up to get a visual check at the paper replicas of the cans. But they may not manipulate the models because I was more interested in seeing the difference between their guesses of the still photo and the physical models.

Their estimates:

If I wrote this post as Dan's 3-Act Lesson, then it was time for Act 2: figure out the volumes of the two cans using their measurements. (Yes, I carry a measuring tape wherever I go now.)

I tried to show the kids equivalent measures whenever possible. We worked a lot with centimeter cubes this year, so this was a rare time that we measured in cubic inches.

And here are some estimates from grown-ups who only saw the left image at the top of post:

(Got more guesses after I printed this: 1,000, 182, 600, 576.)

The straight lines [heights] seemed easiest to estimate. It got a little bit tougher when these lines bend into circles [circumferences] — and there was a large number of over-estimates here. I thought area estimates were pretty good, average of all 33 student guesses was 28.3 (calculated was 26).

The volume estimates, from kids and adults, remained well under the calculated numbers. I don't know what to make of all this. But I kept wondering: Are boys or girls better at making volume estimates? (From my small sample of 33 students, the girls were closer.) How about science teachers? The 3D models helped overall; and I bet if I let the kids do everything but measure the cans, their numbers would be closer. Interestingly, at one point I'd placed the smaller can inside the larger can, and kids who stood nearby kinda gasped. One said, "Oh, you can fit a lot inside."

Thank you, once again, to Andrew for sending me that tweet that started all this! He did a wonderful presentation of 3-Act Lessons to his staff.