Christmas Lights

aMacHan @ Flickr

Our family put up a Christmas tree this weekend, and it reminded me of one of my favorite problems ever! Students did great work on it, and developed a number of techniques.

On the weekend after Thanksgiving Karen put up a 2 meter tall Christmas tree, 170 cm wide at the base. She put an astounding 1500 lights on the tree. Write a story problem using this information.

One thing you should know about Karen's tree decorating is that she believes that lights should be distributed uniformly throughout the tree, not just on the surface. It's like the tree is full of lights. She started at the base, and put on 750 lights. Then she asked me how many will she need to finish? Now that we know she needed 1500, how far up from the ground did she get with those first 750 lights?

EDIT:
I saw that Abstruse Goose, an funny, edgy cartoon that's hip to math & physics also had some nice Christmas tree problems...

Playing Math

I am, probably obviously, a big proponent of games in math.  I know on some non-research-clinical-double-blind-trial level that when I see students most engaged, in the same way that math engages me, it's as if they're playing.

I walked into a neat building in downtown Holland, MI (the kids were at a science camp at Hope College) that now houses a 5/3 bank.  The facade is beautiful, but I'd never been inside.  It was just as beautiful.  One of the tellers was talking about how they just filmed a scene at this bank for Ed Harris and Jennifer Connelly's coming movie "What's Wrong with Virginia?".  The ceiling has a fantastic pattern (idealized above) with regular hexagons and rhombi.  One of the things I've promised myself I'll think about some day is tesselations of more than one tile, and how to do Escher tesselations.

I sat down and started sketching the pattern trying to think about alterations.  These really boil down to what's the fundamental region and how are you going to move the region about the plane (or what subgroup the tessellation represents if you're more algebraic).  The fundamental region needs to be at least a hexagon and a rhomb.

That led to noticing the obvious translation tessellation, but there's also a rotation tessellation possible.  If you do two rotations and a translation, it becomes really an altered quadrilateral.  (Or, what I actually notice from making this picture is that it's a pentagon with an additional rotation on the side between R1 and T2!)  I also noticed that two of them made a hexagon tiling.  So I thought about making more interesting tilings by dividing up standard tilings into congruent pieces.  Which got me thinking about which shapes can be divided into congruent pieces.  That got me thinking about rep-tiling alterations which is another thing I've promised myself I'll think about someday.  I really got lost in the ceiling for 15-20 min when all I needed to do was make a deposit.

Why don't students play with mathematical objects?  

At this point, take 5 minutes to read this Jonah Lehrer column at science blogs about video games and our interactions with them.  

Video games used to appeal to a limited audience, like math.  More people than in math, because it's easier to get engaged.  The people who got into them could get really into them, like math.  But the Wii has changed things.  A much broader audience of people enjoy playing it.  Why?  Lehrer relates some of the neuroscience, which actually goes all the way back to William James.  (Sometimes I feel like if we had just taken a century to figure out why he, Dewey and Piaget were really saying, things would be better now.)  Someone at Emory's Education Division has assembled some great William James info.  In particular on interest and his talks to teachers (pdf).  He thought a lot about engagement and connections.

"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." - William James (available on a mug)

I think that games help some people see the playful nature of math, but are still not broadly accessible. Is physicality the key?  How do we  broaden the physicality of math?  This is probably a multiple intelligences question.  We played this Prism Power Game, and I shared with the preservice teachers how I had considered having them draw isometric views or just keep data for the game, instead of using blocks.  They unanimously thought it was obvious that the blocks made it more fun.  (I agreed.)

I'm very curious what other teachers do that relates to this idea of physicality, or if you think it is important, too.  Please share your thoughts!

PS>  the game!
Prism Power Game


PPS>  Math at my university actually started in William James College. Those were the days!

PPPS> Had to make a geogebra sketch of this pentagon tiling.  It's pretty cool and flexible. Be fun to Escherize.

Solid Unit

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.)

Net Result

I really enjoy designing nets (2-D plans that fold up into 3-D objects), and I love designing them in dynamic geometry, where you can design all nets. If I ever got the time to do math research again, I can see going in that direction.

I designed these four sketches for my geometry class, which is working on a project to design their own package with a few constraints. Each sketch is available as a dynamic webpage or the geogebra file. Here's 5 nets that were made with the sketches, in a printable pdf format. Geogebra actually has very nice priniting controls, so if you're interested in designing your own, choose that option. Remember you can install it, or run it from your browser at geogebra.org.

General tetrahedron: webpage or geogebra file

Square pyramid: webpage or geogebra file

Convex oblique pentagonal prism: webpage or geogebra file

I was very disappointed that the above sketch, though intuitive, couldn't make concave prisms. This next sketch is the answer, but would be a muddle to try to figure out how it was made. The net is pretty though, for designing solids, and I'm proud of it as work.

General Oblique Pentagonal Prism: webpage or geogebra file

Note that you can use the pentagonal prism nets to make quadrilateral and triangular prism nets by making some of the base vertices collinear.

Let me know what you think, and send me your dynamic geometry design challenges!

PS> I also wrote up a memoir for my class of how I made the pyramids (which really sounds egotistical; reminds me of a Tom Lehrer line about "even the Pharoahs, had to import, Hebrew braseros" Listen at the link. If you do, check out Lobachevsky, a great math song.) Sorry for the ramble - I'm tired. Here's the memoir.