As our semester in geometry drew to a close, we investigated solids. Rather than dispense formulas (
Go, Kate!), we tried to follow the
Van Hiele levels. Play and touch to get some understanding and visual recognition, sorting to start thinking about characteristics, and summarizing findings in definition-like descriptions. Then we started thinking about measures. Surface area is so natural, especially combined with the idea of a net. But what about volume. It's very interesting to have students sort the power solids by volume. The sphere and hemisphere are very subtle. I find it's also very common for students (college math majors) to be unable to remember formulae. "Isn't there one with a 4/3?"
We filled them with water, and with no instructions from me, they immediately set out to try and verify their conjectured order. (It's not as messy as you'd think. The top of the solids container makes a pretty good tray.) Two methods come up: adopting a unit, and measuring each of the shapes in terms of the smallest, and filling one and trying to pour it into the next. The different methods lead to noticing different things. It seems like the groups that adopt a unit notice more numerical relationships from the data, and the groups that directly compare notice more of the geometric properties of the solids themselves. ("Where does the water go?")
Usually from this data, you can suggest the idea of comparing solids with similar relationships. Cone, Sphere, Hemisphere, Cylinder; Triangular Prism-Pyramid, Cube-Square Pyramid, Cylinder-Cone; Square Prism-Rectangular Prism-Cube, Small Triangular Prism-Large Triangular Prism or Hexagonal Prism. Brilliantly designed little set. (Although it does get into the experimental error that Dan Meyer cautioned about (was excited by?) in his
TEDx talk.) We compare the exterior of the solids and the water compares the interior. Significantly different for the smallest objects. Students have suggested measuring then by immersion, but we have yet to try it.
So this brings us to the boundary of informal and formal argument/reasoning. How can we relate the volume of the prism and pyramid. I do like models that fit together, but then that's just one example. Of course, then, I tried to model it in geogebra.
Webpage or
geogebra file.
It didn't help most of the students.
So I tried again:
Webpage or
geogebra file.
This was helpful. Or far more helpful, anyway.
Both sketches make use of
Cavalieri's Principle to show equivalent volume. We got at that in class by doing some block building, where each student had the same number of blocks per level. This we extended into understanding the volume of a generalized cylinder.
Resources: One of my favorite resources for this kind of classical problem is David Joyce's Java implementation of
Euclid's Elements.
Book XII is the one you need for these problems, especially
Proposition 7 and
10. Our department's java wiz
David Austin is the one who connected us to those. David A's visualization work is literally inspiring, and worth checking out.
We finished this all by building with polydron tiles a plethora of polyhedra. I love how, left to their own devices, students invent regular polyhedra, antiprisms, various truncations, and completely original solids. (Unfortunately these are pretty expensive, but they are durable and usable by kids as young as 2nd grade. Cf.
ETA Cuisinaire.) I set them the challenge of building a polyhedron with volume between 1 and 2 liters as a fancy new container for a boutique. The need for actual measurement and estimation as well as decomposition and formula use makes this quite a challenging problem.
Extension: as I was thinking about this and looking for resources, I came across Archimedes' proof that a sphere is 2/3 of the circumscribed cylinder. Famously, this is the relationship that Archimedes wanted put on his tomb. I took the translation from the
Archimedes' Palimpsest that was posted at
Cut the Knot (an invaluable geometry site), added some clarifying comments and made it into a
handout with an accompanying
geogebra sketch. The sketch isn't really for visualization, but allows the reader to experimentally test some of Archimedes' unjustified claims. (All correct, though. Were it today, the justification of the steps would be left as an exercise for the reader. Pretty good exercise.)
