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:

(c) I was wondering what a 3D version of this might look like, but it turns out that this exists!

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!

Some Nice Twitter/Bloggy/Desmos Things

I love how creative I am with my blog titles. Meh. I realized I “favorite” tweets on twitter a lot when I want to save them for later, because they are awesome. But as I was looking through them recently, I was like: I should put some of these in a blogpost for others. And so one day if I’m looking for something, I can actually find it by searching my blog (something I do way too often) instead of scrolling forever on twitter.

@rawrdimus shared this applet he made on Desmos for helping kids to understand the idea of a derivative as “slope-iness.” What I like about it is (a) you only get a small line segment instead of the whole tangent line (the whole line would be distracting), and (b) that kids can drag the slider for a and get a sense for what’s happening and how that relates to what’s being plotted, and (c) that kids can then make a prediction where the next point will be (and then drag the slider to see if their thinking was correct.


Related to this is something many people worked on earlier this year based on a tweet I wrote (I wanted a surfer or skier to be travelling on a curve, and the surfboard/ski to be the tangent line)… an updated version of this was posted by @lustomatical…


My friend @pispeak posted a nice calculus puzzle that I enjoyed thinking through and solving: “Found this cool question below online (for a challenge) but got stuck…thoughts? help? @samjshah @calcdave @stoodle #mtbos “The line y=0 is tangent to both x^2 and x^3. But there exists another line tangent to both curves. What is the equation of that line?””

I don’t know this teacher, but I like the idea of doing this. Maybe next year I can make it a goal to do write one positive note to each student. Something heartfelt and genuine. A student met with me before school to talk about a “math exploration” she was going to do, and I loved how into her idea she was. I can totally write so many notes saying good things like that to my kids. Like this teacher, doing stuff like this will make me feel good.


@mikeandallie retweeted a link to a page that explains the unsolvability of the quintic without needing all that abstract algebra. I forgot to dig into this page. But OMG it looks like it’s going to be aweeeeeesome.

@bowmanimal wrote a freaking amazing blogpost about something he did in stats class before winter break. I still am reeling with how awesome it was. The question: “How can we use basic statistics to examine and tell apart writing styles? What do statistics about your own writing say about your style?”  Doesn’t get you excited? Trust me, click on the link and read how he does this. I don’t often come across lessons that I’m desperate to teach, but this is one of them. It also clearly comes from a master curriculum designer.

@dandersod wrote a blogpost ages ago about how to turn a graph into a 3D printed object. I desperately loved it, and had our tech integrator teach me how to do this on our school’s 3D printer.


I wanted to have my precalc kids make mathematical ornaments based on beautiful polar or parametric graphs they tinker around with/discover (maybe have a christMATH tree? haha sorry)… but the timing wasn’t right this year for ornaments (we do polar in the spring). But I still want to make this a reality this year. I hope I remember!!!

@fermatslibrary tweeted out this picture:


I love it because I remember doing something two years ago with my geometry class, arguing that we don’t need cosine and tangent, and that having sine is enough. We showed that we could have done all of trig with just sine. But then we talked about why having cosine and tangent in the mix makes our lives easier. I love this chart because it clearly illustrates what life would be like if we didn’t have multiple trig functions. (On a side note, I wonder what kids would notice and wonder about this chart if they hadn’t ever seen or heard of trig before. Like a middle school kid or a late elementary school kid.)

@mzbat (don’t know who this is) wrote a riff on my fav Carly Rae Jepsen, which I feel often enough:

hey i just met you
and this is crazy
but could this meeting
be an email maybe


POP! Popcorn Optimization Problem

I’m in the middle of optimization in my calculus class now. I had a “long block” (every seven school days, I see a class once for 90 minutes) and for the second half of that long block, I like to do something slightly different. Since I knew my kids hadn’t seen or done the traditional “box optimization problem” in precalculus (since I taught them last year also!), I decided to do that.

This might jog your memory if you don’t know what I’m talking about. You take a piece of paper. You cut out four squares (the same size squares) from the corners. You then fold up the four flaps and tape the box shut. There you go!


You can probably tell that the box’s volume is going to be based on the original paper you start with, and the size of the square you decide to cut out. The question is: what’s the largest volume you can get for this box. 

If you cut out a teeeny tiiiiny square, you’re going to have a very large base for the box, but almost no height. And if you cut out a giant square, you’re going to have a large height but a teeeeeeeny tiiny base. And somewhere between a teeeeeny tiny square and a giant square is going to be the perfect square to cut out which will give you the largest volume.

So the question is: given a specific piece of paper, what size square do you need to cut out to get the maximum volume.

This question has been done to death in middle school classes, in Algebra II classes, in Precalculus classes, and in Calculus classes. And I recognize that this post is just another rehashing of the same old problem. But I remember reading about a teacher who did a variation of this by including popcorn. And I wanted to do the same. No surprise, when I looked it up, it was dear Fawn. But I had such a lovely time in class today watching this unfold that I wanted to share the specific sheet I made up for kids to do this.

[2018-01-31 Popcorn Activity .docx version to download]

Teacher Moves / Outline

This activity requires students knowing and using the quadratic formula. My kids (standard level calculus) are pretty weak with algebra, so I started the class with a “do now” that had kids use the QF. So I recommend that.

Show kids the popcorn. (I had two different flavors.) Show your excitement about the activity. (I was genuinely excited!) Get this psyched. Hand out the worksheet but nothing else.

Put a three minute timer on the board. Explain the problem. Show kids a piece of cardstock with 4 squares drawn on it. Show kids a second piece of cardstock with those same four squares cut out and the flaps folded up so it looks like a box (but untaped, so you can unfold it too). Tell them the volume they create is the amount of popcorn they are going to get. And that you aren’t going to overfill their boxes — just to the brim. Tell them they have 3 minutes to work with their partner to come up with the best size square they want to cut out. And they are not allowed to do any calculations. Just visual estimation. 

At that point, give cardstock, ruler, scissors, and tape to kids. Do not let kids start until you press “GO” on the timer. Then… GO!

After three minutes, my kids were done. They measured the side length of the square they cut out and recorded it on the worksheet. They then cut and taped. They weren’t allowed to get their popcorn until they did one more thing… some math…


It was super important to me that kids didn’t measure anything, except for the side length of the square, to do these problems. Why? Because this is where I want kids to recognize the side length of the square is the height (that was obvious to all my kids), but also that when calculating the length and width, they were going to be doing 216-2x and 279-2x (where x was their side length). Only a few kids didn’t get the 2bit (they only subtracted x), but I sent them back to their seats to rethink their length and width and they immediately got it. It was actually awesome to hear their OOOOOOHHHHH moment. But yeah, no measuring. They have to use their brainzzz to come up with the length and width with what they are given!

Only after checking their volume with me, and I said it was correct, could they fill their boxes with popcorn.

As an aside, when writing this activity, I had to decide what level of scaffolding I wanted to give for this. I decided not much. So I didn’t include any diagrams. (Well, I did put two on the very last page of the worksheet in case a kid needed some additional help. Turns out no one did.) I also initially wrote the worksheet to be in inches, but then changed to centimeters, and then after thinking a bit more, I changed to millimeters. Why? So kids don’t have to deal with fractions (inches) or decimals (centimeters), and we could keep our eye on the prize. It also made the volume huge — and so kids would have to do a little work to get the correct window when graphing.

At this point, I sent them back to their seats with popcorn in their box to then solve the general case. Close to the end of class, I posted the different volumes students got by estimation (it was a tiny class today… kids were absent or at sports).


Overall, I spent about 35 minutes on this in class. One pair finished completely. All the others are at the place where they are in the middle of the calculus work (close to being done).

Submit a Proposal for TMC18! You, yes you!

I’m on the organizing committee for Twitter Math Camp. If you don’t know what it is, you should know it’s the greatest professional development I’ve been to. And I have gone to it every year since it started. So check out the website ( The key aspect is that it’s a grassroots conference that was started by math teachers for math teachers (and related educators). By people who were passionate about their classrooms and just wanted to get together with each other. Here are some fun pictures from TMC last year:


Below I have an invitation to submit a proposal to talk at TMC this summer. There are three options: a short 30 minute session, a regular 60 minute session, and leading a  6 hour multi-day session. If you want to come to TMC and haven’t considered giving a talk, I want you to take a moment and think “well… if I did put myself out there, this is what I would talk about… this is what I know.” If you’re a first year teacher, it could be a session called “If I could do it over” and talk about what you learned, to help other early career teachers. If you’re a math coach, it could be about how to wrangle your more challenging teachers and getting them on your side. If you’re an experienced teacher, it could be about how you design your quadratics unit or how you bring outside speakers to the classroom or … I’m just asking you to consider leading a session.

We in the online math teaching community and at TMC believe that everyone has things of value to share, and we can all learn from each other. TMC is a welcoming place, and if you’re scared of presenting, you’ll know that you’ll be doing it at a small conference to a small and friendly audience (anywhere from 5 to 20 people, usually). It’s a place to just put yourself out there! I personally am terrified of public speaking, but it was at TMC that I first put myself out there, and it turned out to be so much fun to design and implement my sessions, and just a lot less scary than I thought. And I did it with someone else, which made it more fun! So yeah, I’d love for you think about it. Think about what you know, think about what you have to say, think about what you’re strong at… and if you think you don’t have anything, I’d argue you’re being too hard on yourself. We all have things of value to share. And we all can learn from each other.

Now without further ado…


We are starting to gear up for TMC18, which will be at St. Ignatius High School in Cleveland, OH  (map is here) from July 19-22, 2018. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are. 

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it! 

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past nearly everyone who submitted on time was accepted, however, we cannot guarantee that will be the case. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 19 and 48 one hour sessions that will be either Thursday, July 19, Friday, July 20, or Saturday, July 21). That means we are looking for somewhere around 70 sessions for TMC18. We are requesting that if you are applying to speak for a 30 or 60 minute session that there are no more than 2 speakers and if you are applying for a morning session that there are no more than 3 speakers.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 15, 2018 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, David Sabol, Sam Shah, and Glenn Waddell