It's funny how a little bit of a change to a lesson can make a humongous difference.
Last year (and a couple of years prior), I reviewed volume formulas with my students by putting them into groups and rotating a bunch of prism- or cylinder-shaped containers around. (Largest container was a plastic bucket. Most were tupperware.) They had to measure the objects and to calculate the approximate volume, in cubic centimeters.
This year, I added a new piece. I wanted the kids to be able to visualize that their volume measurements are correct (or incorrect), without me giving them my answers. So, I collected a bunch of liter water bottles and filled them up in advance with water. At the end of the kids measuring/calculating all of the volumes, I said to them that we could certainly verify the volumes with cubic centimeter blocks, but it would require thousands of them per container, and it just doesn't scale. So, instead, we were going to use water. I held up a liter bottle and held up a plastic 10cm-by-10cm-by-10cm container and asked them to vote on which one they thought looked bigger. Once we got outside (Ooh! So warm and sunny!), a kid volunteered to do the demo where they poured the water carefully from the bottle into the cube. Gee whiz! They're exactly the same! To emphasize that this means that a bottle of 1 liter water is equivalent in volume to 1000 cubic centimeters, I showed them using a math manipulative item how you can neatly fit 1000 cm^3 cubes snugly inside the cube container, the same way that 1 liter of water had filled the same container to the brim.
Now that we knew that 1 liter = 1000 cm^3, we started to fill up some of the containers they had measured with liters of water. I kept asking kids what values they had gotten for each container, and volunteers kept pouring in more water to see if it would overflow. (We had some measurement instruments, obviously, to obtain increments smaller than 1 liter.) It was super neat. Every step of the way I kept exclaiming to the kids, "Remember that this means that you're adding in another ______ of those little yellow cubic centimeters!" (Kids were getting secretly competitive, obviously, when other groups' answers were getting eliminated and when the containers turned out to hold about as much as their own answers.)
It was super cool! We collectively marveled in the end at how even a relatively small container can hold a couple of liters of water -- proving that they can fit thousands of those little yellow cm^3 cubes!!
Afterwards, we went back up to the classroom. Our next task was to figure out how we can predict the height of the water inside a new prism-shaped container, once you transfer it from an old container that was filled to the top. I let the groups struggle with this for a while on their own, and most of them figured out one way of doing it (with some guiding questions, mostly). At the board, I had the kids explain their ways of doing it, and we saw that both ways -- finding % of volume taken up in new container, then multiplying it by the total height of the new container; or setting up V = l*w*h with volume of water and solving for h -- arrived at the same answers using very different geometric understanding. I made the kids show me work both ways, using their own measurements/numbers, to verify that they hadn't made an arithmetic error somewhere and that they did indeed understand both methods, and I heard kids say, "Tsssssss..." when they were finished. It's a noise that Salvadoran children make to indicate that something is unexpectedly cool, and that noise made me smile.
And of course, we did the actual experimentation. I filled Container A with water to the top, transfered it over to Container B, and announced to the class that the height rose to about 6cm. Kids got to see for themselves that their calculations brought them to the right ballpark of predictions!
How fun! Now we're ready for big boy conversions!
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