"Because you have to figure out how much in one section and then multiply by the number of sections."
"It's the top layer times how many layers."
So far so good. Next I filled a coffee mug with cubes and asked them to estimate how many cubes in the cup, without touching the cubes inside. 5 minute exercise: count the cubes in the top layer, stack cubes up next to the cup, multiply... we dumped out the cubes and counted. The student who got the closest even thought to take one layer off to account for the thickness of the bottom of the mug. So I figured we were all set for tomorrow's task.
I set out 12 cylinders - cans, jars, salt boxes - and grouped them into threes and fours. In each group, I asked them to decide which cylinder was biggest by volume. We had an interesting discussion about how extra height might cancel out extra weight and vice versa. Their thinking seemed to be largely linear:
"This can is just a little bit wider, but the skinny one is way taller."
[Turning the tuna can sideways and measuring it against the soda can] "I am trying to picture how many times this will go into that one."
Me: "How did you guys get the number of cubes in the coffee cup yesterday?"
Them: "Oh, we counted the number of cubes in one layer and multiplied by the height."
Me: "So how could we use that here?" Nothing.
Me: "What were you finding when you counted the number of cubes in the bottom of the cup."
Them: "How many cubes were in one level of cubes."
Me: "And what is that?" Blank faces again.
I am afraid I was not wise enough to abandon ship and plan to circle around the next day. I dragged the thing out another painful five minutes or so, AND THEN abandoned ship and tried to regroup.
There was a disconnect between the cubes on the bottom of the coffee cup and the actual area of the circle at the base. They knew that the area of the circle would be different than the number of cubes because of the spaces between the cubes, so that seemed to prevent them from using one to substitute for the other, which makes perfect sense when I think about it their way.
The next day, I handed out this sheet of paper:
I asked them to find the radius and diameter of one of the circles. What kind of units do we measure that in?
Next I asked how they might measure the circumference of the circle. The formula was rusty from the earlier lesson, and I realized some students almost equated circumference with area, since both "had an inside." I also asked them what kind of units we use to measure circumference. It was almost split between "units" and "square units", with a few "don't know"s. As soon as
Next I asked them to arrange as many cubes as they could in one of the blank circles and count them. We collected numbers and then I asked them to estimate the number of squares in the circle with the grid. Would that be more or less or equal to the number of cubes they just counted? Why? [I realize now I first could have asked them to tell me what they thought the number of cubes told me about the circle, and what units that number represented.]
"The squares will be more because there are no spaces in-between." Pretty much unanimous.
What are we measuring about the circle when we count the squares? "Area." Okay good, this seems to be working. What value for the area does the formula give you? They calculate that and it seems natural to everyone that we got a number that is close.
Did I fix it? The class before lunch seemed to have lots of aha's, the one after lunch not so much. In fact, they way over-counted the partial squares by taking every sliver, no matter the size, as half a square. "We're tired." But when I ask again how to calculate the volume of the cylinders, many more students are on board.
Overall, I am pleased. I think the activity clarified some misconceptions, and this experience is a reminder to me that there can be gaps in the students' understanding that are invisible to me unless I ask about their thinking. This is a lesson that I have traditionally tried to simply illustrate and explain very well, then have them practice the formula. Did it work? When they got started actually measuring and calculating, most of them could do it on their own. Next week we get to see which ones are larger. I hope they still care. Next year it will be smoother.