A nice proof for the Law of Cosines

We’ve been working on the Law of Cosines in my precalculus classes. And I am having them prove it by scaffolding up from specific triangles to more general triangles. And then with the most general triangles, students consider acute, obtuse, and right triangles.

Kids tend to struggle a bit on the first triangle, but as soon as they realize they need to draw an altitude, they see all that opens up with right triangles and are good. After that, for the rest of the concrete ones, they tend to breeze through. The place where they first stumble again is when they get to the fifth triangle, the one with the angle \beta. They get to L^2=(5\sin\beta)^2+(4-5\cos\beta)^2 but then don’t go any further. But since I know I want them to get to the law of cosines, I tell them to expand and look for something nice. Sometimes I’ll give them the answer (L^2=41-40\cos\beta) and then say: work your work until it looks like this, with one trig function in it. From that point on, kids are in the zone.

For years, I used to teach this by giving kids waaaay too much information.


And I kinda told them what to do… Meh. I was jumping way ahead to get to the formula. We weren’t savoring the thinking to get to the formula. Now we are.

That being said, I ran across something quite beautiful. A stunning proof of the Law of Cosines (at least for acute triangles) on the site trigonography.


I love it because it looks like a proof for the Pythagorean theorem.  Which is nice because  the Law of Cosines is essentially a more generalized version of the Pythagoren theorem.

The area of the bottom square (the green one) is clearly the area of the top two squares (the red and blue ones) minus two green areas. Ummm…. c^2=a^2+b^2-2ab\cos(C) anyone? You see the c^2 and the a^2 and b^2, but what you also see is that you’re taking out some area (the green bits). [1]

When introducing it to my class, I showed them this image:


and said it was a proof of c^2=a^2+b^2-2ab\cos(C).

And I just said to observe. To make statements based on what they see visually…Anything and everything. And if students could, see if they could make connections to the equation (but without writing anything down). After a short while of observations, I opened this geogebra applet and played around. I showed them what happened when we made angle C a right angle.


They saw the green rectangles disappear, and how this would be a proof of the Pythagorean theorem if the blue areas and the pink areas were equal to each other. And then I squiggled and smushed the triangle about and eventually kids conjectured that the blue rectangle areas were equal, the pink rectangle areas were equal, and the green rectangle areas were equal. I told them that was true, and they were going to prove that. But before doing that, I asked them: if this was true, do you see a connection between this diagram and the law of cosines?

And kids eventually got there. They saw this argument, essentially…


…and they realized that the green rectangles were probably the thing that was being subtracted out in the law of cosines!

At this point, I gave my kids a blank paper copy of the diagram, and groups work on proving that the blue rectangle areas were equal, the pink rectangle areas were equal, and the green rectangle areas were equal. They had seen all these right triangles before,  when they were looking at the diagram and making observations, so this went pretty quickly for most of them.

I love this proof of the law of cosines! Of course when I went online, I saw so many other beautiful proofs (look here, and the links at the bottom, for some). Troll the internet and be amazed! They are so elegant! This “scaling up” one might be my favorite. And here’s one that David Butler sent me (that is on the site I linked to above). And I remember proving the Pythagorean Theorem in geometry using the crossed chord theorem, and now the same argument here can be used for the law of cosines.


[1] To be clear, this diagram only works for acute triangles. I haven’t yet modified the argument to work for obtuse triangles.


Some very cool things about the Law of Sines

So yeah, I’m teaching the law of sines and cosines, and I’m finding some awesome things. What’s totally ridiculous is that after I introduced some of it to my class, I looked back at my stuff from last year and apparently I had done the same thing. Like… I don’t remember it at all.

In any case, when I teach the Law of Sines, I tend to have kids derive it by finding the area of a triangle in three different ways.


We set these three different ways to get the area of a triangle equal to each other, and divide by \frac{1}{2}abc to get the Law of Sines:



Now, to be clear, there are some subtleties that have to be addressed here. Like for example, this argument clearly works for acute triangles, but what about an obtuse triangle or right triangle, like:


It turns out with just a tiiiiny little bit of extra work, we can show that the Law of Sines holds. (Here’s a fun little applet you can play with for this… one important thing that can help you for the obtuse triangle proof is that \sin(\theta)=\sin(180-\theta).)

So… yeah. That’s a pretty traditional way to teach the Law of Sines. But did you know that the ratio that pops up with the Law of Sines has a geometric interpretation?

Like, look at this triangle. And look at the ratio of (side length)/(the sine of the angle opposite the side).


That 10.36 has a geometric meaning. Ready for it? READY? I don’t think you are, but I’m going to show it to you anyway…


Dang! HOLY MOTHERFATHER! Yuppers. That triangle has one circle that can perfectly circumscribe it. And twice the radius of that circumscribed circle is that ratio!!! Don’t believe me? Ok, I know you do, but play with this applet I made to see it happen! Maybe try to create a right triangle and see if that reminds you of something you learned in geometry?

Now this year I told my kids that \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}=2R. And I sent ’em up to the whiteboards and asked them to prove it. I gave some hints. Like, for example, the inscribed angle theorem. But eventually kids got it!


This is actually another proof of the Law of Sines! (To be clear, you will also have to make an argument for an obtuse triangle, which requires a tiny bit of modification. You have to see a central angle and one of the angles of the triangle are the same because they both subtend the same arc. And a right triangle.)

So I had a follow up after this… I asked kids to prove that the area of any triangle is: \frac{abc}{4R}, where R is the radius of the circumscribed circle. I asked them to prove it algebraically, and they did:

\text{Area}=\frac{1}{2}ab\sin(C). But we know that \frac{c}{\sin(C}=2R. So let’s manipulate the rea equation to get an R in it.


Now we have \text{Area}=\frac{1}{2}abc\frac{1}{2R}=\frac{abc}{4R}.

I asked kids to show this algebraically. They did it in various ways (all correct), similar to the argument above. However I had a student present me with a stunning geometric argument that proved this area formula. I honestly don’t know if I would have been able to come up with it. It was so stunning I had to take a photo of it. I leave this as an exercise for the reader. MWAHAHAHA.

(All of this Law of Sine stuff was inspired by this webpage.)

A beautiful combinatorics argument

Today a teacher in my department was struggling to understand algebraically and conceptually why:


He was on the way to making a neat Pascal’s Triangle argument. Look at that 70. That’s \binom{8}{4}:


He started working backwards, and saw that 70=35+35.

But each of those 35s came from 15 and 20. So 70=(15+20)+(20+15).

And then going backwards more, we see the 15 comes from 5 and 10, and the 20 comes from 10 and 10. So 70=((5+10)+(10+10))+((10+10)+(10+5)).

And then going backwards once more, we see the 5 comes from 1 and 4, and the 10 comes from 4 and 6. So 70=(((1+4)+(4+6))+((4+6)+(4+6)))+(((6+4)+(6+4))+((6+4)+(4+1)))

In other words, 70=1*1+4*4+6*6+4*4+1*1.

By the time we make our way down from the 1-4-6-4-1 row to 70, we see that:

The first one in the 1-4-6-4-1 row just is added once when making the 70.
The first four in the 1-4-6-4-1 row is added four times when making the 70.
The six in the 1-4-6-4-1 row is added six times when making the 70.
The second four in the 1-4-6-4-1 row is added four times when making the 70.
The second one in the 1-4-6-4-1 is added just once when making the 70.

In other words: 70=1^2+4^2+6^2+4^2+1^2.

The other teacher and I realized we could generalize this. But we were left unsatisfied. It was a Pascal’s Triangle argument, but I wanted to see the answer with an understanding of combinations. I wanted something even more conceptual. So my friend and I started thinking, and he had an awesome insight. And I want to record it here so I don’t lose it! It made me so happy — little mathematical endorphins exploding in my head!

Let’s assume we have a set of 2n letters, where n letters are A and n letters are B.

Blergity blerg, let’s just keep things concrete, and have 8 letters, where 4 are As and 4 are Bs. (We can generalize later, but I want to just see this happen!) Given 4As and 4Bs, there are \binom{8}{4} ways to arrange them to make different 8 letter words [1]. Great! That was the easy part.

Now we are going to construct a whole bunch of different sets of 8 letter words, in a particular way (using AAAABBBB), so that when we add up all those sets, we’re going to get all possible 8 letter arrangements of AAAABBBB.

How are we going to do this? We are going to create special of 4 letter words and concatenate them together to make 8 letter words. 

Set 1: We are going to create a 4 letter word with 0As (and thus by default, 4Bs) and a 4 letter word with 4As (and thus by default, 0Bs).

How many ways can we create 4 letter words with 0As? \binom{4}{0}. To be clear, this is just 1. The word is {BBBB}.

How many ways can we create 4 letter words with 4As? \binom{4}{4}. To be clear, this is just 1. The word is {AAAA}.

And when we concatenate them, we are going to have \binom{4}{0}\cdot\binom{4}{4} eight letter words. But we know \binom{4}{0}=\binom{4}{4}. So this is simply \binom{4}{0}\cdot\binom{4}{0}. And this is just 1, because the only eight letter word possible is {BBBBAAAA}.

This is a degenerate case, so it’s hard to really see what’s going on here. So let’s move on.

Set 2: We are going to create a 4 letter word with 1A (and thus by default, 3Bs) and a 4 letter word with 3As (and thus by default, 1Bs).

How many ways can we create 4 letter words with 1As? \binom{4}{1}. To be clear, this is just 4. The words are {ABBB, BABB, BBAB, BBBA}.

How many ways can we create 4 letter words with 3As? \binom{4}{3}. To be clear, this is just 4. The words are {AAAB, AABA, ABAA, BAAA}.

And when we concatenate them, we are going to have \binom{4}{1}\cdot\binom{4}{3} eight letter words. But we know \binom{4}{1}=\binom{4}{3}. So this is simply \binom{4}{1}\cdot\binom{4}{1}. And this is 16 eight letter words. (Each of the first four letter words can be paired with each of the second four letter words… so this is merely 4*4. Just to be clear, I’ll list the first few eight letter words out: ABBBAAAB, ABBBAABA, ABBBABAA, ABBBBAAA, BABBAAAB, BABBAABA, …

Set 3: We are going to create a 4 letter word with 2As (and thus by default, 2Bs) and a 4 letter word with 2As (and thus by default, 2Bs).

How many ways can we create 4 letter words with 2As? \binom{4}{2}. To be clear, this is just 6. The words are {AABB, ABAB, ABBA, BAAB, BABA, BBAA}.

How many ways can we create 4 letter words with 2As? \binom{4}{2}. To be clear, this is just 6. The words are {AABB, ABAB, ABBA, BAAB, BABA, BBAA}.

And when we concatenate them, we are going to have \binom{4}{2}\cdot\binom{4}{2} eight letter words. And this is 36. (Each of the first four letter words can be paired with each of the second four letter words… so 6*6 eight letter words.)

Set 4: We are going to create a 4 letter word with 3As (and thus by default, 1B) and a 4 letter word with 1As (and thus by default, 3Bs). By the same logic as above, we are going to end up with \binom{4}{3}\cdot\binom{4}{3} eight letter words. This is just 4*4 eight letter words.

Set 5: We are going to create a 4 letter word with 4As (and thus by default, 0Bs) and a 4 letter word with 0As (and thus by default, 1B). By the same logic as above, we are going to end up with \binom{4}{4}\cdot\binom{4}{4} eight letter words. This is just 1*1 eight letter words.


Now look at all the different eight letter words created by this process, from Set 1, Set 2, Set 3, Set 4, and Set 5. We have captured every single possible eight letter word with four As and four Bs. Let’s check a few random words:

AABABBAB… okay this is in Set 4.
BAABAABB… okay this is in Set 3.
BBBABAAA… okay this is in Set 2.

Cool! I only have to look at the first four letters to decide which set it is going to be in!

But look at what we’ve done. We’ve shown that we can get all eight letter words in these five sets… so the number of eight letter words is:


If we simply write the squares out…


But we saw at the very start that the number of eight letter words is simply \binom{8}{4}

So the two are equal.

All the hard work is done, so I leave it as an exercise to the reader to generalize.

P.S. I take no credit for this amazingly wonderful letter rearrangement solution. I just bore witness as my friend figured it out, and I got giddier and giddier. I love it because it’s abstract, but still understandable to me. But it’s close to my threshhold of abstraction!


[1] If you don’t quite see this, imagine 8 blank slots.

___ ___ ___ ___ ___ ___ ___ ___

You choose four of them to put the As into. There are \binom{8}{4} ways to choose four of these slots. Put As into those four. By default the rest of the slots must be filled with Bs — they are forced! So there are \binom{8}{4} ways to create eight letter words with four As and four Bs.

Double Angle Formulae

I posted this on my Adv. Precalculus google classroom site. I don’t know if I’ll get any responses, but I loved the problem, so I thought I’d share it here.


I mentioned in class that I had stumbled across a beautiful different proof for the double angle formulae for sine and cosine, and I would post it to the classroom. But instead of *giving* you the proof, I thought I’d share it as an (optional) challenge. Can you use this diagram to derive the formulae? You are going to have to remember a tiiiiny bit of geometry! I already included one bit (the 2*theta) using the “inscribed angle theorem.”

If you do solve it, please share it with me! If you attempt it but get stuck, feel free to show me and I can nudge you along!



Below this fold, I’m posting an image of my solutions! But I say to get maximal enjoyment, you don’t look further, take out a piece of paper, and take a stab at this!


What felt like Forever… It was maybe 20 minutes?

On Friday in one of my Advanced Precalculus classes, kids were working on figuring out the double angle formulas for sine and cosine. They got \sin(2\theta)=2\sin(\theta)\cos(\theta) and \cos(2\theta)=\cos^2(\theta)-\sin^2(\theta).

And then… they got stuck.

You see, I showed them two alternative forms for the double angle formula for cosine (\cos(2\theta)=1-2\sin^2(\theta) and \cos(2\theta)=2\cos^2(\theta)-1). I showed them these forms. And I said: figure out where they came from.

All groups in a few minutes were on yellow cups (“our progress is slowing down, but we’re not totally stuck yet”). I didn’t want to give anything away, but I didn’t have any group have a solid insight that I could have them share with others. I let things remain a bit more, no luck, so then I said: “this looks related to something we’ve seen before… a trig identity… maybe that will be helpful. Bring in something you know to open up the problem for you.” Eventually kids realized they needed to bring in some outside information (namely: \sin^2(\theta)+\cos^2(\theta)=1).

I was sure that was going to be enough. Totally certain. But after another 5 minutes of watching them struggle, I wasn’t so sure. I didn’t want to give anything more away, but I had to because we had to move forward. But what more could I give without giving the whole show away? Since many groups were trying some crazy stuff, I said: “this is a simple one or two step thing…” Why? I just wanted them to take fresh eyes and see what they could do thinking simply. They kept on saying I was trying to trick them, but I told them it wasn’t a trick!

And then, in the span of the next five minutes, all my groups got it.

But what was more interesting was that we had three different ways to do it. As kids moved on to the next set of questions (and I breathed a sigh of relief that they figured this out), I reflected on how awesome it was that they persevered and then came up with different approaches. So while they worked, I put up the three different approaches.


And with a few minutes to go at the end of class, I had everyone put everything away and I just pointed out the embarrassment of riches they came up with. And it was great to hear the audible reactions when kids who had one way saw the other ways and say things like “ooooh, I never would have thought of that!” or “that’s so clever!”

I had (have?) so many mixed feelings when I saw how difficult this question was for my kids. And I was hyperconscious about how much time we had to spend on this. But the ending made me feel like it was time well-spent.

Alone with “Starry Night”


So I wasn’t actually alone with Van Gogh’s Starry Night. But I went to MoMA this morning and got to tour the museum with other math teachers before the museum opened. Our sherpa? George Hart, mathematical artist. A few months ago, I got an email from two different teacher friends letting me know about this opportunity to take a master class on Geometric Sculpture put together by the Academy for Teachers. What an opportunity indeed!

I show up at 8:30 am and me and a gaggle of math teachers (a gaggle is eighteen, right?) are raring to go. We have fancy namecards and everything. (Note to self: at the book club I’m hosting in a bit over a month, create fancy namecards.)


Beforehand, we were assigned a tiny bit of homework. We were asked to go onto the Bridges website (it’s an international annual math-art conference, organized by our sherpa), look at submitted papers for their conference proceedings, select three papers, and then read and reflect on them.

My Paper Choices and Thoughts

1. Prime Portraits, Zachary Abel

This mathematician was able to construct portraits using the digits of prime numbers. The digit 0 was black and the digit 9 was white, and the other digits were various shades pic1of gray. The digits of a number were put in order in a rectangular array (e.g. 222555777 would be put into 3×3 array, where 222 is the top row, 555 is the middle row and 777 is the bottom row) and an image results. For most numbers, the image will look like noise. But this author was able to use prime numbers put into a rectangular array to create images of Mersenne, Optimus Prime, Sophie Germain (using Sophie Germain primes), Gauss (using Gaussian primes), and others. I wpic2.pngas blown away. This intersection of math and art doesn’t quite fall neatly into any of the categories that George provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct that art. What makes it interesting is that the math version of these portraits feel unbelievable. Senses of awe and wonder and curiosity filled me when seeing the portraits for the first time because how could it be? It was like a magic trick, because nature couldn’t have embedded those portraits into those numbers. And before reading the paper on how these were constructed, I had a nice few moments thinking to myself how this could have been done.

(If you’re curious, the answer is to start backwards. First take an image, pixelate it, and then turn those pixels into a number. Take that number and check if it’s prime on a computer. If it isn’t prime (which is likely), slightly alter the image by the colors by +1% or -1% (some imperceptible noise), repixelate it, and turn those pixels into a number. And again, check if that number is prime on a computer. If it isn’t, do this again. It turns out that you’re going to need to do this about 2.3n times [where n is the number of pixels]. With a computer, this can go quickly.)



(a) Math: I recall faintly from college classes that the distribution of primes is related to the natural logarithm. Which explains why the 2.3n comes from something involving a natural log. But what is this relationship precisely, and how does it yield the 2.3n?

(b) Content: I think prime numbers are very rarely taught in high school math in a meaningful way. Number theory is ignored for the “race to calculus.” However there is so much beauty and investigation in this ignored branch of math. Where could I fit in conversations of prime numbers in an existing high school curriculum? Could ideas from this paper be used to captivate student interest (by letting them choose their own image), while showcasing what various types of prime numbers are?

(c) Extension: Are there other things that we teach that have visualizations that look impossible/unbelievable, but actually are possible? Can we exploit that in our teaching? I’m thinking that often numbers in combinatorics are crazy huge and defy imagination… Perhaps a visualization of the answer to some simple combinatorial problem?

(d) In order to fully appreciate this work, the viewer needs to have an understanding of prime numbers. Without that understanding, this is just a pixelated image with some numbers superimposed. All wonderment of these pieces is lost!

2. Modular Origami Halftoning: Theme and Variations (Zhifu Xiao, Robert Bosch, Craig Kaplan, Robert Lang)

I chose my articles on different days, and I didn’t even notice that this article is very similar to the first article! I chose it because I love the idea of a gigantic public art project in a school (I tried once and failed to make a giant cellular automata that students filled in). But this article basically shows how to fold orgami paper (white on one side, colored on the other side) in five different ways to make squares where all of the square is colored, ¼ of the square is colored, ½ the square is colored, ¾ of the square is colored, and none of the square is colored. A number of each of these origami pieces are constructed.

Then an image is converted to grayscale and scaled down to the number of origami pieces you want to use. Then the image is scaled-down image is pixelated with “origami piece” size pixels, and each pixel is given a number based on brightness [0, ¼, ½, ¾, 1].

Then this origami image can be created by putting these five different origami pieces in the correct order based on the brightness of the pixelated image!


Just like with the previous paper, this intersection of math and art doesn’t quite fall neatly into any of the categories that George Hart provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct a variation of that art.


(a) Math Classroom: I really love the idea of having kids take an image with a particular area (w by h) and figure out how to “scale down” the image to use a particular number of origami pieces. It is an interesting question that will also involve square roots! It seems like a great Algebra I or Algebra II question.

(b) Extensions: How could this project be extended to the third dimension? 3D “halftone” origami balloons? Unlike a photograph which can be easily pixelated, can we find a way to easily pixelate the “outside”/”visible part” of a 3D object and create a balloon version of this? Similarly

(c) This is not just a low-fidelity copy of an existing piece of art. If we took a random non-professional Instagram photograph, we might call it “pretty but not art.” But if someone made this Instagram photograph out of origami sheets, we would be more likely to call it art. But why? Just one thought, but there is something about the intentionality of the artist (and the craftsmanship that goes into creating the origami piece) that isn’t in the original photograph. It also is likely to evoke something different in a viewer – a viewer will instantly wonder “how was that done” when seeing the origami piece (so the art piece evokes process) while a random photograph might not do the same (they just pressed a button on their phone and got a cool photo).

3. A Pattern Tracing System for Generating Paper Sliceform Artwork, Yongquan Lu and Erik Demaine

I chose this paper because of the beautiful sliceform image on the first and last page. I had only seen them once before, but forgot what they were called! I wanted to learn howpic4 to make them. In this paper, the authors share that most existing sliceforms are created in separate pieces (e.g. the image on the first page, a bunch of hexagons created separately) and then pieced together afterwards. The authors wanted to instead thread the paper slices together so they could create the same intricate patterns—but with the paper slices interconnected. So instead of individual hexagons placed together, a giant connected sliceform was created (e.g. the image on the last page). The authors came up with a way to do this for designed created in polygonal tiles, like in many Islamic star patterns, and then created a program to “print” the strips of paper needed – with red lines indicating where folds are, and blue notches indicating where cuts need to be made so the paper slices can be fit into each other.

They accomplished this in two steps. First, they came up with a way to notate the internal structure of a paper slice within one polygon. One notation captured lengths (where slices of paper intersected other slices of paper and where slices of paper needed to be bent/folded), and another notation (not provided) recorded angles that needed to be folded. The second step was more tricky. An algorithm was created that looked at the edge of a polygon (where a paper strip initially ended), and looked to see if it could be extended into another polygon. In that way, one strip could start in one polygon and then enter another, and then another, etc. This is the threading that the authors wanted to get. The authors created a three-step algorithm for deciding if a paper strip could enter another polygon at all, and if there were multiple possible paths for this strip to take, which one it should choose.

After doing all of this, the authors then created a program that could take in an image, calculate out the different strips of paper needed to create the sliceform, and with the notation they created, print out the appropriate slice (see image on page 370 for an example).


(a) There were two big things I didn’t totally understand when reading this paper. First, how were angles recorded/notated? Second, where did the 3-step algorithm for extending paper slices come from? How do we know if we follow it that all segments in the figure will be created by the paper slices, and no segment will be repeated?

(b) Besides just being “cool,” is there an application to this in a high school math class? What higher level research does this connect up to? (Just like origami was simply beautiful but then it also was exploited to create new and interesting questions for mathematicians, what does this bring up for us?)

Note: When I went to research these, it turns out that Lu and Demaine created a website to help amateurs out: https://www.sliceformstudio.com/app.html

(c) I was wondering what a 3D version of this might look like, but it turns out that this exists! https://www.sliceformstudio.com/gallery.html

Back to the Master Class

After getting coffee and pastries, and introducing ourselves to each other in small groups, we all were taken on a tour of MoMA, where George led us to certain pieces to spoke to him as he looked at them through mathematical lenses. There was one sculpture in particular that George stopped us at — a sculpture he remembered seeing as a kid visiting MoMA — that I would have walked right by. It was a figure cast in bronze (?), that had a lightness and movement despite it’s medium. To me, it screamed that it was a figure in tension. Rooms later, I was still thinking about how it was a collection of oppositions, form and formlessness, fluidity and stability. For George, describing what drew him to it was ineffable.


Here are more photos of George taking us around.


The whole walkthrough, George kept on saying “I’m not an art historian, but this is what I see in terms of my perspective as a mathematician…” which was just what I needed to hear. I know so little about art history and contemporary art, but hearing that let me feel a bit more “free” in looking at something and thinking about it with my own lens, instead of me passively waiting to hear what the piece is “supposed” to convey or what philosophical/conceptual trend it is a part of. In general, I feel ill-equipped to make statements/ judgments about art in museums that go beyond “I like this” or “I didn’t really like this.” But listening to George talk about what he sees as a mathematician and mathematical artist was liberating. Because I can see mathematical ideas/principles (intentional and unintentional) in some of the art too! This walk and talk reminded me a lot of what I imagine Ron Lancaster’s math walk around MoMA would be like!

And as the title of this blogpost suggested, there was something so special and magical about being able to have the run of the museum before the general public was let in. And a random fun tidbit: I also learned that there is no simple mathematical equation for an egg. I (of course) had to google that when I got home, and came up with this webpage.

We Become Card Sculptors

We get back to the room that was our home base, and some people share out interesting things from the articles they read. I was going to share mine, but I noticed that even though the ratio of men to women was low, more men were taking up airtime than women proportionally. So I kept my hand down.

George gives us a set of 13 cards with notches in them. We only needed 12 but you know how we math teachers really like prime numbers… (Okay, that wasn’t the reason for the 13th card, but I want to pretend it was.) We were asked to crease them like so:


And then… we were asked to put them together somehow, into a freestanding sculpture. No glue, scissors, tape, etc. We were given a hint that you can start with three cards. So I figured we needed to create 4 sets of objects that each take three cards. So with my desk partner we made this:


This was the core object we needed to build the final thing together. It was interesting how it took different pairs different amounts of time to get these three things together. Without instructions, it was a logical guessing game, but it felt so good once we hit upon it.

Then came the tough part. Putting these four building blocks together. That took a long time and some frustration, but the good kind. It was one of those problems that you know is within your grasp, and you know that you can come out on the other side successfully, but you don’t quite know how much time and how much angst the journey will cause you. It’s that sweet spot in problem-solving that I love so much. And lo and behold:


Many people got it! I would post a picture of mine, but all my photos look terrible. You can’t see or appreciate the symmetry and freestanding nature of this beast. But it was a moment of such pride when we got the last card to slide in the last notch! (And of course when my partner and I tried building hers after finishing mine, it went much faster and we had a better sense of things.)

Oh yeah, this card sculpture is isomorphic to a cube. I was blown away by that. It was hard for me to see at first, but realized that to get my kids to see it, I would give ’em purple circular stickers to have them put on the “corners” and blue circular stickers to have them put on the “faces” and green circular stickers to have them put on the “edges.” It would help me not only count the different things (maybe put the numbers 1-8 on the purple stickers, 1-6 on the blue stickers, and 1-12 on the green stickers?), but also “see” how they are in relationship to each other. (And George told the class he liked the suggestion and would think about trying it out!) George asked the class what the “fold angle” is for each card (what angle the card was bent at in the sculpture). I loved the question because it’s so obvious when you look at the sculpture from just the perfect angle! (The answer: 60 degrees.)

We See Art and We Build More Art

Lunch was delivered from Dig Inn, and we ate and briefly chatted. And then George took us on a picture tour of his sculptures and their construction. Some choice quotes:

“Kids need to have an emotional connection to math.”

“Math and art are both about creating new things.”

Finally, we ended our day building our own mathematical sculpture. We had 60 pieces of wood that we set up in trios. And we combined those to create a hanging sculpture.


What’s neat is that this hanging sculpture is going to travel to all the schools of the teachers who were at this session for two week periods. It will come to us disassembled and we’re going to get a group of kids (or teachers!) each to build it up and hang it. And then after two weeks, send it on! I love the idea of this same set of 60 pieces being in the hands of young elementary school kids and my eleventh-grade kids.

Takeaways and Random Thoughts

I have recently been into math art. Last year, I helped organize a math-art exhibit in our school’s gallery. I get excited when kids make math-art for their math explorations that I assign in my precalculus class. (In fact, years ago I had two kids make some sculptures and now I know they came from directions George provided on his website.) For me, it isn’t about “art” per se, but about seeing math as more expansive than kids might initially think, and seeing math as a creative and emotional endeavor. That’s why this resonated with me.

At the start of the year, I had intentions of starting a math-art club. Because my mother was sick and I was not taking on any new responsibilities, I decided to put that idea on hold. But now I’m feeling more excited about trying this out. To do this, I want to create 5 pieces on my own based on things I have found online. Things that will kids to say “oooooooooh.” Heck, things that will get me to say “ooooooooh.” (Like the origami image I saw in the second paper I wrote about above.) And then show them to students and get a core group of 4-5 who want to just build stuff with me on a regular basis. Maybe as a stress reliever.

What can we make? Who knows! Maybe stuff out of office supplies? Maybe some of the zillion awesome project ideas that George and his partner Elizabeth have put together. Maybe something inspired by the awesome tweets with hashtag #mathart that I’ve been following (and sites like John Golden’s). Maybe something on geogebra or desmos? Maybe something else? The idea of a large visible public sculpture appeals to me. One that random people walking by can add to also appeals to me. (I tried last year to get a giant cellular automaton poster going at my school, with two students in the art club, but it didn’t quite work as planned.)

Maybe this happens. Maybe it doesn’t. I hope I can muster the energy to start thinking this summer and making this a reality next year.

Random thought: Based on all the photos that George posted showing him bringing his math art to little kids in public spaces, I wonder if he’s talked to Christopher Danielson who organizes Math-On-A-Stick? Or if he knows Malke Rosenfeld (we had talked about math and dance earlier in the day)? I’m hoping yes to both!

Random note: George said that among his favorite mathematical artists were Helaman Ferguson, Henry Segerman, and John Edmark. Bookmarking those names to check out later.

Random thing: At MoMA in an exhibit about the emergence of computers to help create art was fabric that was created by the artist to hold information in it. What was pointed out to me, 20180208_100531.jpgwhich made me go HOLY COW, is that the punch card idea for the first computers came out of the Jacquard loom. So loom –> computer –> loom. What a clever idea. I wish I knew what information was encoded in the fabric I saw! Additionally, this reminded me of one of the artists we had exhibited at the math-art show I helped organize: the deeply hypnotic and mathematical lace of Veronika Irvine. And that of course got me thinking about this kickstarter that I’m so sad I didn’t know about until after it was done: cellular automata scarves!

Random last thing: totally unrelated to this workshop, last night someone posted on twitter that Seattle’s Center on Contemporary Art is about to open a math-art exhibition, and my friend Edmund Harriss is one of the artists in that show! Along with the work of father-son duo Eric and Martin Demaine who both do amazing paperwork (and amazing mathematics). So awesome. Wish I were there so I could go see it.

There might be light at the end of the Chain Rule tunnel… maybe.

This is going to be a half-formed post. I wanted to get a conceptual way for kids to grok why the chain rule works in calculus. But without doing too much handwaving. And I wanted something visual.

The hard part is: if we have a function g(f(x)), we can approximate the derivative at a particular point by doing the following.

Find  two points close to each other, like (x,g(f(x)) and (x+0.001,g(f(x+0.001)).

Find the slope between those two points: \frac{g(f(x+0.001)-g(f(x))}{(x+0.001)-x}.

There we go. An approximation for the derivative! (We can use limits to write the exact expression for the derivative if we want.)

But that doesn’t help us understand that \frac{d}{dx}[g(f(x)]=g'(f(x))f'(x) on any level. They seem disconnected!

But I’m on my way there. I’m following things in this way: x \rightarrow f \rightarrow g

Check out this thing I whipped up after school today. The diagram on top does x \rightarrow f and the diagram on the bottom does f \rightarrow g. The diagram on the right does both. It shows how two initial inputs (in this case, 3 and 3.001) change as they go through the functions f and g.

At the very bottom, you see the heart of this. It has \frac{\Delta g}{\Delta f}\cdot\frac{\Delta f}{\Delta x}=\frac{\Delta g}{\Delta x}

And then I thought: okay, this is getting me somewhere, but it’s to abstract. So I went more concrete. So I started thinking of something physical. So I went to how maybe someone is heating something up, and in three seconds, the temperature rises dramatically. The temperature measurements are made in Farenheit, but you are a true scientist at heart and want to see how the temperature changed in Celcius.


I love this. I’m proud of this page.

And then of course when I got home, I wanted to see this process visualized, so I hopped on Geogebra and had fun creating this applet (click here or on the image below to go to the applet). These sorts of input-output diagrams going from numberline to numberline are called dynagraphs. You can change the two functions, and you can drag the two initial points on the left around. (The scale of the middle and right bar change automatically with new functions you type! Fancy!)


And of course after doing all this, I remembered watching a video that Jim Fowler made on the chain rule for his online calculus course, and yes, all my thinking is pretty much recapturing his progression.

This, to be clear, is about the fourth idea I’ve had as I’ve been thinking about how to conceptually get at the chain rule for my kids. The other ideas weren’t bad! I just didn’t have time to blog about them, but I also abandoned them because they still felt too tough for my kids. But I think this approach has some promise. It’s definitely not there yet, and I don’t know if I’ll have time to get there this year (so I might have to work on it for next year). But I know to get there I’ll have to focus on making the abstract very tangible, and not have too many logical leaps (so the chain of logic gets lost).

If I’m going to create something I’m proud of, kids are going to have to come out saying “oh, yeah… OBVIOUSLY the chain rule makes sense.” Not “Oh, I guess we did a lot of stuff and it all worked out, so it must be true.”

A blogpost of unformed thoughts, and an applet. Sorry, not sorry. This is my process!