Happy weekend, dear readers. My math classes are currently studying surface area and volume of solids and we came across the conundrum of cylinder volume. How oddly counter-intuitive is it, that a short, seamingly smaller cylinder, can actually hold more capacity than a taller, skinnier cylinder? Hence the video above, in which I try a classic conservation of quantity experiment out on Zombie Teacher's 4-year old son Ethan. (who is not currently a zombie)
Piaget would tell you that the conservation of continuous quantity is a developmental skill. By the time we are six years old, most of us have some understanding that equal amounts of liquid or solid don't change quantity simply by putting them in taller or fatter containers. But it's quite amusing to try out the experiment on younger children! And occasionally on older kids and grown-ups, just to see how spatially advanced they are.
For example, these two cylinders do NOT have the same volume. But can you guess which one holds more liquid? Think it's the taller one? You're WRONG! We are studying the math in class right now and it's as simple as stacking coins. Prism volume is equivalent to the area of its base, (big B) multiplied by its height. In the case of a cylinder, the area of the bottom "coin" or base, is equal to volume of a 1-coin cylinder (except in units cubed instead of squared). The "height" of the cylinder then becomes how many coins are stacked; thus, increasing the volume of the first "layer" by its height/layer factor.
Area of the shorter, "horizontally gifted" cylinder, then, is 7 x 7 x 7 pi = 343 pi
Area of the taller, "vertically gifted" cylinder, then, is 18 x 4 x 4 x pi = 288 pi
I like to show Ethan's conservation video after having taught the formula, and then to challenge my students to go home and try the experiment with family members (hopefully younger ones). AS LONG AS they make an attempt to explain not only the purpose of the experiment, but the math behind it, to the subject of their experiment. This forces them to internalize the formula in an effort to explain it to someone else. Because we all know the best way to really learn material is to have to teach it yourself!!
|Yes, those are match box cars playing with the solids.|
Other cool tricks that kids can try, or you can demo before the class, include the pyramid/prism and cone/cylinder volume conundrum. Did you know that there is a proportional relationship between the volume of a cylinder, and the volume of a cone? and a sphere too? There is a similar relationship between square- or rectangular- based prisms and pyramids of equal base sizes. It just doesn't visually make sense.
I like to survey my class first, either with fingers or on paper, to write down how many times they think the pyramid solids will fit into their prism counterparts. I tell them they can give me "half a knuckle" if they think it needs a decimal answer (it doesn't). Then I show them either with rice, or on this absolutely fantastic interactive web page from CMP2. Make sure you have the sound on because it has fantastic sound effects! The little sink "fills" the solids, and the drain "empties" them out.
First, fill the cone and pour it into the cylinder. It's not full. Do it again. It's still not full. Do it again. It's full! The cone fills the cylinder 3 times. Hence, the pi x radius squared x height formula is proportionally true for a cone; it's just 1/3 the answer.
Try filling the cone and dumping it into the sphere. The cone fills the sphere TWICE!
And the creepiest, coolest one of all, is to fill the cone and sphere, and empty them both into the cylinder to fill it perfectly! The cone fills up all the gaps between the round parts of the sphere, like melting ice cream from the cone back into the tub.
The 1/3 relationship also works with a rectangular- or square-based pyramid and it's equal-based prism counter-part, LxWxH (x1/3 for the pyramid). And you do NOT need to spend any money at all to use the online app. You just need either a computer bay for students to try it themselves, or a Smart board to demo it in front of the class.
I am lucky enough to have both... so I show it on the Smart board and with the rice first, and then give the kids a chance to try it themselves. You'd think it was a preschool party, with how many 7th graders try to swarm the rice "sensory" table to play with my solids sets. And oh, how nasty the floor gets. I get down on my hands and knees after school to scrape up as much of the rice mess as I can, so the custodians don't report me for destruction of the carpet! And I feed them a lot of cookies at Christmas time :o)
Try it out. Play. Experiment. See!? Math is fun.