**Approach 1: **Since we were in vector land, a few kids solved it like this. They found the vector from P to Q, and then set the magnitude equal to 5 and did the algebra.

**Approach 2: **Similarly, some students just used the distance formula that they had memorized.

I like that this kid expanded out and then later eventually factored. Because that was slightly different than how the student in approach 1 solved it (leaving as is, and then taking the square root of both sides.

**Approach 3: **One student found the equation of a circle of radius 5 around the point . Then they realized that they were looking for the solution to a system of equations for the circle and the line . So they substituted into the equation of the circle and solved!

**Approach 4: **Most students took a geometric/visual approach. They drew the point and the line . Then they drew these two triangles (seeing that the vertical distance from the point to the line was 3 and the diagonal distance from the point to the line was 5, since we want a distance of 5 away from . Then they used 3-4-5 right triangles to get the horizontal distance.

All of these were lovely. I enjoyed seeing them all together and drawing some connections among them. Most kids were awed when they saw Approach 3. And since so many students didn’t take a straight up algebraic approach, they were like “ooooh” when they saw Approach 1/2. I supposed what I most like about this is that it really highlights how circles, distance, and vectors are all essentially tied together. I mean kids *should* know that circles and distance are fundamentally related (but of course they don’t always remember that). But this problem connects those two concepts with something new: vectors. And that the magnitude of vectors simply being an equation involving a circle, secretly.

***

Actually, while I’m writing this, I might as well share this other problem that had a couple of approaches. This was the basic question:

**Approach 1: **Most students took this approach. They drew the vector, and then drew a smaller vector with unit length, and then used similarity to find this new vector (with a scale factor of .

Related to this were students who simply saw the scalar that was multiplied by the vector to be a “scale factor” that stretches/shrinks the vector by a particular factor. But why this works is because of this similar triangles argument.

**Approach 2: **A bunch of students used trig. They first found “the angle” and then realized that angle put on a unit circle would work! The fact that so many students saw the problem this way made me happy. I then asked what if we wanted the vector to have length 2 or 3 (instead of unit length), and they were able to answer it. We also talked about one huge deficit of this approach: you lose exactitude since they approximated the angle with their calculator. Even if they didn’t round, they wouldn’t “see” the square root of 5 pop out, when they would with the similarity argument.

**Approach 3:** Okay, so strangely this year, none of my students used this approach. But it is related to the similar triangles approach, and in years past, I’ve had students come up with it. So I showed it to them so they could see another approach. It’s an algebraic approach to find the scale factor.

Fin!

]]>It’s not a coincidence. And in fact, a circle also has a nice property: the derivative of the area of a circle () gives you the circumference of the circle (). So yesterday I decided I wanted to come up with a short investigation that at least exposes my kids to this idea.

After working for around 90 minutes this morning, I ended up with a packet, and these things on my desk which I’m going to use for illustrations (blocks, dumdums, and tape):

**UPDATE:** I was shopping yesterday and found these gems. YAAAS!

I’ll post the packet I whipped up below. It goes through the standard argument, so in that way it’s nothing special. But in the past when I taught the course, I used to just kinda stand up at the board and give a 5 minute explanation. But I wasn’t sure who was really grokking it, and I was doing too much handwaving.

The big picture trajectory:

*At the start of class, but way before doing this activity, I’m going to have kids recall what a derivative is graphically (the slope of a tangent line), and then how we approximated it before we used limits (the slope between two points close to each other). And from that, I’m going to remind kids of the formal definition of the derivative:

* I may also start class with this problem, suggested to me on twitter by Joey Kelly (@joeykelly89). It’s a classic problem that was featured on xkcd, but oh so unintuitive and surprising!:

*Way later in class, I will transition to this activity. The first idea is to get kids to see the connection between the volume of a sphere and the surface area of a sphere. And then again for the area of a circle and the circumference of a circle.

*Then I try to get kids to understand what’s going on with the sphere first… followed by the circle.

*Then I show kids the “better explained” explanation. Why? Because at this point, kids are spending a lot of time thinking about the algebra, and I’m afraid they might have lost the bigger picture. The algebra focuses on one “shell” of the sphere, or one “ring” of the circle. But how does it all fit together? [@calcdave sent me this video, which I’d seen before but forgot about, which has the same argument… this is where the licorice wheels above come into play.]

*Finally, I problematize what they’ve learned. I have them mistakenly make a conjecture that the derivative of the volume of a cube is going to be the surface area of the cube, and the derivative of the area of a square is going to be the perimeter of the square. But quickly kids will see that isn’t quite true. So they have to tease out what’s happening.

My document/investigation [docx version to download/edit]:

My solutions:

I haven’t taught this yet. So it could be a complete disaster. I don’t have a sense of timing. I don’t know how much of this is me and how much will be them. I am just hoping tomorrow isn’t a disaster! Fingers crossed!

]]>**At the very end, you’ll see a problem that nerdsniped me for a good while. It hit a sweet spot for me. **

***

First off, though, something about me! Kara Newhouse contacted me and Joel Bezaire about how we use reading in our math classes, and then she wrote an article about it for KQED Mind/Shift. I didn’t quite know what it was, but I figured why not answer a few questions!

Even though I hadn’t heard of Mind/Shift before, it apparently is something that gets around. Because my friend Julie told me someone sent it to her, and my sister saw it in the NAIS newsletter. And even Steve Strogatz tweeted it out (with some links to my blog). So that’s random and awesome. If you want more information about how I organized the book club, here is my post-mortem after the first year of doing it. Now I am not teaching the course I had the book club in, but what’s awesome is that I have a few kids who wanted to read math books with me… so informally I meet with them for 40 minutes each 7 school days, and over donuts, we talk. First we read *Flatland*, and now we’re reading *The Man Who Knew Infinity*. And I’m loving it! So don’t think you need to do it *in* a class. I’ve read books and discussed them with kids one-on-one!

Okay, now that’s over! Onto the other less self-aggrandizing things!

***

I love Richard Feynman. I don’t think it would be too strong a statement to say that I would be a different person today if I hadn’t had been introduced to him when I was young, when my father gave me a copy of *What Do You Care What Other People Think?* And I saw this, and I immediately wanted to get it printed on a business card to hand to kids at the start of the year.

***

Now that we’re talking about Feynman, James Propp wrote a powerful piece about “genius” which problematizes Feynman. I already knew Feynman was self-fashioning himself in the way he presented himself to the public-at-large (and his contemporaries). But this article goes further, in a reflection connecting to a powerful piece by Moon Duchin about the sexual politics of genius. He notes:

But Duchin makes me ask, for whom does Feynman’s advice work well? Who in our culture is forgiven for putting aside personal relationships in the name of single-minded pursuit of truth? Who is permitted to be a joker? And who in our culture is steered, from an early age, toward an excruciating attunement to what other people think?

I highly recommend reading James Propp’s piece.

***

Steve Phelps (@giohio) shared this desmos applet which plots lines normal to a curve. You can change the curve!

It reminds me of my *family of curves* project (post 1, post 2, post 3). I wonder if I couldn’t have kids come up with a way to get any perpedicular line to a curve in calculus, and then have them play around with this applet to generate beautiful designs!

***

Nanette Johnson tweeted out a powerful slide from a talk she was at given by @danluevanos. I don’t need the rest of the talk. I get it.

I often feel like a crappy teacher. Right now, I’m on day 3 of Spring Break, and since it started, I’ve been contemplating how crappy I feel about my teaching. I know I’m not a bad teacher, but … maybe I am? I don’t know. But yes, these two questions screamed at me. Because they are part of something I need to recommit myself to: *focus on the positive* and *take the positive and multiply it*. Because I always focus on questions #3 and #4, and rarely give myself time to think about #1 and #2.

***

Mark Kaercher is using Desmos to do warmups.

Here’s the first link in his tweet: https://www.desmos.com/calculator/xxxmahtp91

Here’s the second link in his tweet: https://teacher.desmos.com/activitybuilder/custom/5a770219c1b9e208ca83895c

I love this idea of doing warmups using Activity Builder. Must remember for next year!

***

Patrick Honner wrote a great article in Quanta magazine: “How Math (And Vaccines) Keep You Safe From The Flu”. I’m just mad I didn’t send this article to my calculus kids after we did our *point of inf(l)ection *activity (adapted lightly from Bowman Dickson @bowmanimal).

***

This tweet from Kara Imm just made me so happy. I always believe that formalism and stuff should come *after* something has been explored (whenever possibe). And second graders were absolutely doing that! Sixogon! Navada! So awesome!

***

So Anna Blinstein asked about higher level mathematics and this happened. I learned that arithmetic with complex numbers is akin to arithmetic with polynomials mod x^2+1. WHAAA?!

Of course I took out pencil and paper and had some fun with this. Blows my mind.

***

Steve Strogatz tweeted out this interesting article by Maria A. Vitulli about Writing Women in Mathematics in Wikipedia. The abstract is here:

***

When it comes to polynomial division, I’ve seen the connection to standard division (where x=10). But I think I need to exploit this more in my teaching, especially to make the polynomial remainder theorem seem obvious to kids (and not like magic, which they often feel, even when we’ve figured it out). Erick Lee tweeted a perfect reminder:

***

Joe Cossette tweeted out a neat idea — a stop motion photography race between two figures. (Click here for original tweet so you can watch the video.) I wonder if this can be adapted to calculus when we talk about rectilinear motion. Regardless, I could see it be interesting to give a physical understanding to various functions. Especially when comparing them (like exponential versus quadratic). “Which will eventually win?” Plotting versus is one thing, but *seeing* them in a race is another.

***

David Butler read my recent post about The Law of Cosines and shared his own post which talks about how the Law of Cosines doesn’t actually need *cosine* in it. Worth seeing! Trig without trig!

***

I leave you with a problem from Abram Jopp that nerdsniped me!

More constraints: You can’t use domain restrictions. You can use compound inequalities, but desmos only allows simple ones (e.g. 2<x<3 or x<y<x+2) and not complicated ones. A good number of tweeps got obsessed. There were many different proposed solutions, but I am proud of mine. I don’t think anyone else’s was quite like it.

In the process of working on this problem, David Butler and Suzanne Von Oy reminded me of this beautiful relationship: min(A,B)=1/2(A+B)-1/2|A-B| and max(A,B)=1/2(A+B)+1/2(A-B). And so I wanted to illustrate that with this Desmos graph. I definitely want to remember this gem when I teach Advanced Algebra II at some point when we’re exploring the power of absolute value. It’s so awesome.

]]>

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 . They get to 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 () 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…. anyone? You see the and the and , 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 .

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.

]]>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 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 .)

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 . 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: , where is the radius of the circumscribed circle. I asked them to prove it algebraically, and they did:

. But we know that . So let’s manipulate the rea equation to get an in it.

.

Now we have .

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.)

]]>He was on the way to making a neat Pascal’s Triangle argument. Look at that 70. That’s :

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 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? . To be clear, this is just 1. The word is {BBBB}.

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

And when we concatenate them, we are going to have eight letter words. But we know . So this is simply . 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? . 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? . To be clear, this is just 4. The words are {AAAB, AABA, ABAA, BAAA}.

And when we concatenate them, we are going to have eight letter words. But we know . So this is simply . 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? . 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? . 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 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 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 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

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 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 ways to create eight letter words with four As and four Bs.

]]>***

*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!

]]>

And then… they got stuck.

You see, I showed them two alternative forms for the double angle formula for cosine ( and ). 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: ).

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.

]]>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 of 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 was 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 2.3*n* times [where *n* is the number of pixels]. With a computer, this can go quickly.)

Thoughts/Questions:

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

(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.

Thoughts/Questions:

(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 how 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).

Thoughts/Questions:

(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, which 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.

]]>The hard part is: if we have a function , we can approximate the derivative at a particular point by doing the following.

Find two points close to each other, like and .

Find the slope between those two points: .

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 on any level. They seem disconnected!

But I’m on my way there. I’m following things in this way:

Check out this thing I whipped up after school today. The diagram on top does and the diagram on the bottom does . 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

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!

]]>@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

]]>

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

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).

]]>

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 (http://bit.ly/TMC18sessug). 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

]]>In one of my Advanced Precalculus classes last week, I saw a group of three students successfully figure something out. To celebrate, one group member taught his group how to do a *three person handshake* which was elaborate and awesomesauce.

Yeah, it moved me. Those are the things that get me.

Because what it showed me, in that moment, was how solid a group that was. They came to a *collective* understanding. They were having fun together — being wonderfully silly. And they were celebrating their success. It was a sign that the group had gone beyond being three people working together; they had created some sort of synergy. It was a lovely instantiation of that synergy.

Sadly for them, two days later, I changed groups. (My groups stay with each other for 6-7 weeks, usually.)

But inspired by this group, I changed what I did when I had kids sit with their new group members (in all four of my classes). They all said “Hi!” but then I dramatically and mysteriously had them hush up, and I showed them the first 11 seconds of this video.

They were entranced. So I asked if they wanted to see it again. They did, so I showed them the first 11 seconds again. They all thought they were going to learn that handshake. Fools! FOOLS!

Instead, I told them about how amazing it was to see this precalculus group develop their own handshake. I shared with them what that handshake meant to me, an outside observer… what it said about their group to me.

And then… I gave kids 3 minutes to develop their own group handshake together. The only thing I said was that the handshake had to involve everyone from the group. (Of course, this took 4 minutes, but saying you are giving them 3 minutes gets them working together very quickly.)

Now I’ll be honest. I thought this could go either way. I thought kids might be hesitant to do something corny/dorky like this, and it would be a huge flop. *But in all four classes, every group did it*. [1] And they were doing SUPER COOL THINGS including pounding the table, incorporating fistbumps, incorporating dance moves, and creating beautifully symmetric hand formations. It was super fun to watch. And some kids wanted to share their handshakes publicly, so those who were comfortable (and that was most of them) demonstrated their handshakes for the rest of the class.

What is going to happen with this?

I don’t know. Maybe nothing. At the very least, it was a great quick way to get kids working as a team on something when switching groups. And if the group can use it when celebrating a collective success, it will make visible and public what fun and friendly groupwork can look like. And that might just inspire other groups to do the same. I like an atmosphere where kids are propping each other up, patting each other on the back, and see themselves working as a team. And the more structures that I can develop that promote this [2], the better.

[1] Okay, one of my calculus classes was a little less enthusiastic as the others, but they all did it too! I didn’t get the same JUMP RIGHT IN feeling in all groups I got from the other classes. Some groups had it, but not all.

[2] Like the hotel bells

]]>This post is going to share the talk. If you scroll to the bottom, you’ll get access to the slides and the handout.

*Abstract*: Don’t like the way the textbook approaches a concept but are intimidated by creating your own content? Bowman and Sam both write their own content from scratch. We’ll share the simple lesson-design tricks we use to write investigations that lead to vibrant discussions and a-ha moments. You will leave ready and excited to write your own content!

* Hack #1: Old Problem, New Problem
The Important takeaway:* This is the simplest of all the hacks. You might already do this naturally, and textbooks sometimes have questions that switch what students are traditionally given and what they are asked to find. If you’re hankering to see if students have gotten what they’re doing conceptually, mix things up. Just look at a problem and see if you can’t refurbish it by maybe giving them some information and “the answer” and asking them for some other piece of information that they traditionally are given. When you do this, kids will think harder, talk a

My favorite slides (one content, one funny):

Relevant blogposts:

- Give students right triangles and have them associate the correct trigonometry equation that corresponds with those right triangles: http://bit.ly/NCTMSamTrig
- Come up with the equation for a parabola given a focus and directrix, and the backwards question: http://bit.ly/NCTMSamParabola
- Give students definite integrals and signed areas but missing the function, and see what functions they can draw: http://bit.ly/NCTMSamIntegral
- Play Rational Function Headbandz with students, where students have a rational function (or trig! or logarithm! or whatever!) on their forehead so they can’t see it, but they ask each other yes and no questions to determine the equation of the graph: http://bit.ly/NCTMSamHeadbandz
- Students use protractors to attack forwards and backward questions on inverse trigonometry on the unit circle: http://bit.ly/NCTMSamInverseTrig
- Instead of giving students visual patterns and ask them to come up with the sequence, why not have them come up with their own visual pattern using blocks?: http://bit.ly/NCTMSamBlocks
- Mathematical Iron Chef using group-sized student whiteboards: http://bit.ly/NCTMBowmanIronChef

* Hack #2: Thinking Before Mathing
The Important takeaway: *Too often, mathematical notation and premature abstractness get in the way of student thinking instead of being the tool for efficiency and communication that it is for those of us that already understand the concept. Let students play around with ideas in their heads, with their own framing, and own vocabulary, before you develop abstract structures. Let them do it their own, inefficient way before you show a better, more efficient, “correct” mathematical way – the right way won’t stick unless they’ve created something in their brain to stick it to!

My favorite slides (one content, one funny):

Relevant blogposts:

- Bowman’s blogpost on the fold and cut problem: http://bit.ly/NCTMBowmanFoldCut
- Sam’s blogpost on the fold and cut problem: http://bit.ly/NCTMSamFoldCut
- Fawn Nguyen’s Visual Patters: http://visualpatterns.org
- Kids conceptualize how solids in calculus are formed before learning technical notation and abstract vocabulary: http://bit.ly/NCTMBowmanVolume
- Have kids “notice and wonder” with a single polar equation to generate lots of curiosity about conic sections: http://bit.ly/NCTMSamConicSections
- Kids are asked to find the center of rotation of a figure, and playing with patty paper first makes it more puzzle-like and fun!: http://bit.ly/NCTMSamPattyPaper
- Dan Meyer: If math is the Aspiring, then how do you create the headache? http://bit.lt/NCTMDanAspirin

* Hack #3: Make Math Magical Again
The Important takeaway: *This hack takes some time, but it is worth it. You are trying to build up a moment of surprise and curiosity for kids – something that will make them want to learn more. (It’s like watching a magic trick. You’re in awe, but you desperately know how the trick was performed because magic isn’t real.) You have to think about something you find interesting and really dig deep to figure out for yourself why it is interesting. That takes some thinking! But once you find the answer, I’ve found it often points directly to a way to get kids to appreciate that thing. Often times, I’ve found that having kids explore uninteresting things is powerful because it gives context for the interesting outcome (e.g. appreciating that the complex solutions to polynomials when plotted aren’t that interesting, but solutions to x

My favorite slides (one content, one funny):

Relevant Blogposts:

- A way to motivate trigonometric identities which will make them feel a bit more interesting/magical: http://bit.ly/NCTMSamTrigIdentities
- A way to get kids to be surprised that the perpendicular bisectors of a triangle always meet at a single point: http://bit.ly/NCTMSamPerpBisectors
- A mistake and observation that a student made can be turned into a fascinating lesson: http://bit.ly/NCTMSamMistake
- The moment Sam vowed to change his teaching so that he put a focus on the artistic and creative aspects of mathematics – because he loves the surprise and beauty and wants kids to get that surprise and beauty: http://bit.ly/NCTMSamCreativity
- A simple way to make the joy/magic students feel public and contagious by using a simple $5 hotel bell: http://bit.ly/NCTMSamBell
- Helping students understand the idea of limits by hiding the value of a function at first: http://bit.ly/NCTMBowmanLimit

* Hack #4: Toss ‘Em An Anchor
The Important takeaway: * Math instruction doesn’t always need to go from skill to practice to application. Instead, application to some interesting context, whether that be abstract or “real world” can actually drive student learning, and help them learn the more mundane skills and contexts. Great anchors are both natural to the mathematical context, and sticky – tangible, novel, memorable, easy to refer back to.

My favorite slides (one content, one funny):

Relevant Blogposts:

- Introducing Inflection with Infection: http://bit.ly/NCTMBowmanInflection
- Modeling Memory with Calculus: http://bit.ly/NCTMBowmanMemory
- Using Income Tax to review piecewise functions: http://bit.ly/NCTMBowmanIncomeTax
- Use inflated balloons to introduce the idea of related rates to make the idea of two rates being connected in different ways “sticky”: http://bit.ly/NCTMSamBalloons
- A series of posts about “pseudocontext” in math textbooks from Dan Meyer: http://bit.ly/NCTMDanPseudo
- An anchor problem for Riemann Sums involving estimating the area of a weirdly shaped object: http://bit.ly/NCTMBowmanRiemann
- When introducing arithmetic series, have every kid in the class fistbump with each other in the most efficient way possible… Kids will constantly be referring to “the fistbump problem” for the rest of the unit:
- http://bit.ly/NCTMSamFistbumpsTwo posts about the concept of letting kids come up with the idea of “Squareness,” basically, inventing a measure for how square a quadrilateral is, one from Avery Pickford, one from Kate Nowak: http://bit.ly/NCTMSquareness1, http://bit.ly/NCTMSquareness2

**Photos of Me and Bowman Presenting:**

**A photo of Bowman, me, and my colleague who came to support me!**

**Some Tweets about the Presentation:**

Click to view slideshow.

**Resources: **

Slides (with one taken out…):

]]>