Monday, October 26, 2015

Halloween Geometry

Link to PSet

Since Halloween's nearly here, I thought a problem set with some spooky shapes would be fun. There's no answer key for this one because these are more like exercises than real problems: We walk through two fairly simple compass and straightedge constructions to make some Halloween-y moons and ghosts that have some special properties. If you end up making either of one of these (or both) feel free to send them my way and I'll post some student images here on the site!

A practical note: I suggest in the PS that you may want some graph paper for Exercise 2 in particular. If you don't feel like buying yourself a whole pad, you can find some patterns to print out a sheet at a time here. This site also has hexagonal graph paper and some other fun options we may use in later problem sets.

We're really only introducing two new topics here, but they may lead you to some fun ways to play around with Euclidean constructions. Once you know how to bisect lines and arcs, there's a lot of neat stuff you can do to play around making symmetrical shapes of your own design. Those shapes may also be a fun starting point for making your own tessellating shapes, per Exercise 2. I don't really talk about tesselations in depth here (I figured this could be more of a "follow-the-steps" kind of problem set), but we'll almost certainly return to them in future assignments and talk about their properties in more detail. For now, I'd say that trying to work out why the ghost works (especially how his final zig-zag needs to be drawn) is a good way to build some intuitions about tesselations.

If you like tesselations (and who doesn't?), you may want to try playing around with the two shapes below, which are called the "Kite" and the "Dart." These are prototiles for an aperiodic tiling of the plane, which means that try as you might, you won't be able to come up with a way to make a repeating pattern with these.

If you want to find out more about aperiodic tiling, go check out the wikipedia page for Penrose Tiling, which will point you to more fun ways to cover the plane. Happy Halloween, and happy constructing!

Sunday, October 18, 2015

Cookie Cutting in HW5!

Link to PSet
Link to Answer Key.

In this assignment, I've decided to go back to some problems that are more geometric and encourage kids to play with making (and breaking) shapes. You may find that you want either a compass or a small circle stencil for some of these problems...I found a pretty handy stencil for about $2, and they're are fun things to play with that may come up again in future problem sets. If you're really stuck, however, coins of different sizes would probably do the trick.

My main goal in these problems was to give kids an excuse to experiment. To that end you'll find two worksheets worth of "cookies" to chop up in whatever way you like, and I'd definitely say that trying out different kinds of shapes (especially concave vs. convex shapes) is a great way to extend what's going on here.

Finally, if you want to read more about cutting up pastries for the sake of mathematics, Martin Gardner's article about cutting up a real donut is quite funny (especially the reader mail from people who tried to do it and found that real donut-cutting is heavily constrained by the crumbliness of actual donut matter).

Friday, October 9, 2015

Alternate Q2 and Q3 for HW4

Link to revised PSet

Last time I mentioned that the last two questions of HW4 seemed too hard (or at least too tedious), so I've revised those to be a little simpler. We're still asking you to think about how to count up the mismatched sets that Boofernaut can wear, but the number of colors is reduced from 5 to 4, which ought to make this easier to think through.

The last question is now one about rearranging a set of 3 colors into distinct orders, and I decided to actually give the answer but challenge students to fill in the possibilities. In general, we've found this is a decent strategy for making problems easier to attack: The uncertainty of knowing whether or not you're done is a big stumbling block sometimes.

Somewhat unrelated to the homework, but I just finished Siobhain Roberts' book "Genius at Play" about John Conway, who is an incredible mathematician who you can find lurking around the biographies of other incredible mathematicians and in lots of material about recreational mathematics. His own biography is fantastic and includes lots of great descriptions of games, puzzles, and pure mathematics.

Saturday, October 3, 2015

Notes from the field, HW3 and HW4 suggestions

Now that we've had a chance to work through HW3 and part of HW4 in our house, I wanted to share a few things we ran into that might be stumbling blocks for other kids.

1) In the answer key to HW3, I suggest that kids might try to figure out the pattern for how many socks Grufftina needs to pull out depending on the number of colors. This part is pretty straightforward: She needs to grab a sock for each color in the drawer and then one extra.

The last part of this question suggests that maybe we could come up with a rule even if we don't know the actual number. In the text, I wrote this down as Grufftina having "n" colors in the drawer, and I was looking for kids to come up with the idea that whatever "n" is, "n+1" socks will do the job here. This turned out to be trickier than I thought it might be, mostly because my daughter was trying to sort out what "n" might stand for (maybe "nine?" maybe "ninety?")

My wife had an excellent way of breaking through this logjam, though, that I want to share here: She said that the sock monsters had some words for numbers that weren't the same as ours, so Grufftina might tell you that she has "Blorg" different colors in the drawer, but we don't know what "Blorg" means. Can you still tell her how many to pull out?

This turned out to totally do the trick, so definitely feel free to cross out my very boring "n" and do something like this instead if it helps.

2) OK, this one we haven't really run into yet, but as I was writing the answer key to HW4 it struck me that Q2 and Q3 are probably just a little too hard because of how many colors you need to keep track of. I was trying to get away from needing to talk about how you might multiply the numbers together to get to the right answer, but with 5 colors and 3 feet to think about, the tables/diagrams just get kind of big and probably annoying for young kids to deal with.

My suggestion here is that reducing the number of colors to 4 probably makes this all MUCH easier to think about. In particular, for Q2, it makes it easy to do "leave one out" reasoning about the number of different mismatched 3-sock sets with all colors different: If there are 4 colors and 3 feet that have to be all different, there's always one color left out! So leaving out each color in turn tells you that there are 4 different mismatched sets (as long as we don't care about what foot the sock goes on).

I may come up with an alternate version of the Boofernaut problem that's written this way and replaces Q3 with a question about finding all the different ways to rearrange a mismatched set on 3 feet. Stay tuned for an alternate draft in the next week or so.

If you happen to try either of these problems (or any others) and have thoughts about how to improve or simplify the questions for kids, please let me know! This is all a work in progress, so I'm very much still figuring out how to present some of these fairly tough questions so that young kids can work on them and have fun.

Thursday, October 1, 2015

Mismatching Sock Monsters! - HW4

Link to Pset
Link to Answer Key

Still more socks, but a new set of ideas in this problem set.

With the arrival of Grufftina and Boofinski's siblings, we meet some monsters who don't like matching socks. This (of course) gets us thinking about combinatorics: How many ways can we have a mismatching set? What conditions will we place on sets to be considered different from one another?

Like what we did in HW1, I think experimentation is a key feature of these problems. Actually trying to draw as many sets as you can is a good way to start, especially if you can start thinking about a strategy for how to keep track of what you've done. The core ideas about how to multiply everything together to compute combinations and permutations may be a little tough for some younger kids, but thinking about how to organize the sets of socks they draw may get across the important ideas. Answer key coming soon! UPDATED: The Answer Key for this pset is now available.

Sock Monsters! - HW3

Link to PSet
Link to Answer Key

We're leaving polyominoes behind in favor of socks!

The idea in these questions is to further introduce and extend the idea of proof. Chances are you've heard some version of the sock problem posed in Q1 and by itself it's just a different exercise in working out conditions for some mathematical object to be inevitable (here, a matching pair of socks). With the remaining questions, however, what I wanted to do is start introducing the idea of pursuing more general questions given a specific starting point. We do this by broadening the conditions that constrain the original scenario: What if the monster has more sock colors? What if the monster has more feet? What if both things are true? Besides solving these problems, thinking about other ways to generalize the question is part of what I'm hoping to get kids to do here.

I'm a little hard-pressed to think of a good supporting resource for this, but I will take the chance to mention Paul Erdos' biography "My Brain is Open." There's a bit in there where the author talks about Erdos' quest for ever more general theorems, which is really what this assignment is trying to introduce (very gently). If you haven't read about Erdos, definitely check out this book.

Building Shapes out of Polyominoes - HW2

Link to PSet
Link to Answer Key

Now that we've learned what polyominoes are in HW1, the questions in HW2 ask us to try building some different shapes out of them. We also start talking about how you can actually prove whether or not you can make a particular shape out of specific polyominoes.

The goal for these questions is two-fold: In the early questions, I wanted to encourage experimentation. The value of false starts, dead ends, and general mucking about is a good thing to learn about mathematical thinking. In the later questions, however, I wanted to try and introduce the idea that sometimes you can work something out without having to try lots of experiments. I do this here using the classic "Mutilated Chessboard" problem, which I hope is a clear enough example of a simple proof for kids to get the idea.

If you're still loving the polyominoes, you really need to get a copy of Solomon Golomb's book "Polyominoes: Puzzles, Patterns, Problems and Packings."

Intro to Polyominoes - HW1

Link to PSet
Link to Answer Key

Here is the first proper Bespoke Mathematics Problem Set! In these questions, you'll meet up with some very fun shapes called "polyominoes" which are a mainstay of combinatorial geometry. There's a TON of material about polyominoes out there and if you've played Tetris, you already know what these things are.

This problem set is mostly about teaching kids how we pick a definition for a new class of mathematical objects (establishing the "rules" for polyominoes) and getting them to play within those constraints to make new things. One thing I wanted to emphasize here is that it's okay to try out different rules, but being consistent once you've decided how you want things to work matters a lot.

For a great introduction to polyominoes, I can't think of anything better than the original Martin Gardner article about them. You can find this in the collected volumes of his work (which are unbeatable) published by the AMA. Alternatively, many places online have nice articles about these - try Wolfram or for some pointers.

Constructing a Hexagon

Link to the PSet

This is sort of the beta version of Bespoke Mathematics. Before my daughter wanted math homework in 1st grade, one of her best friends who is a few years older wanted math homework in 1st grade. I've always really liked compass and straightedge constructions, so this is just a short walkthrough of how to construct a hexagon. This is the only Bespoke Math assignment that doesn't have an answer key because it really does take you through the whole process.

If you want more cool stuff to do with constructions like this, I'd recommend checking out Robert Dixon's "Mathographics." You'll find lots of neat stuff there about how to draw an egg with a compass and straightedge, how to do string drawings, and other such things.


Welcome to Bespoke Mathematics! Here's the deal: My 1st-grader tells me shortly after the beginning of the school year that she would really like some more homework. In particular, she'd be very excited if she could finally have some math homework. If the math homework could also be less about counting and adding small numbers together, that would also be good.

The result is what you see here. This is a (growing) collection of short and hopefully fun homework assignments that I thought my daughter would like. Each one has about 3 main questions, which range from the purely fun and speculative to more sophisticated challenges. In each case, I've tried to introduce some fun piece of mathematics that kids are unlikely to see in a standard grade-school curriculum. Each homework assignment has an answer key (often with a few extra questions for fun) and I'll also try to include links to interesting supporting material as I'm able.

Please feel free to share these with anyone who may enjoy them! I'd also love to hear comments, suggestions, or ideas for homework topics. Finally, please feel free to point out any errors in the text, particular difficulties reading my somewhat spindly handwriting, or other issues with the problem sets.