Month: March 2018

More Things I Want To Highlight From Twitter

As I mentioned before, I often see neat stuff on Twitter and want to remember it, but I just “favorite it” and then forget it. Sometimes I remember to go back and look for it, but it’s arduous. So I decided if I get the time, I’ll blog about some of my favorite things which will help me remember them better.

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

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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!

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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.

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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.

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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!

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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.

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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!

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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).

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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!

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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.

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

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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:

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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 $x^2$ versus $e^x$ is one thing, but seeing them in a race is another.

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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!

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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.

A nice proof for the Law of Cosines

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Some very cool things about the Law of Sines

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

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

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

$\frac{\frac{1}{2}ba\sin(C)}{\frac{1}{2}abc}=\frac{\frac{1}{2}cb\sin(A)}{\frac{1}{2}abc}=\frac{\frac{1}{2}ac\sin(B)}{\frac{1}{2}abc}$

$\frac{\sin(C)}{c}=\frac{\sin(A)}{a}=\frac{\sin(B)}{b}$

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

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

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

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

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

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

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

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

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

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

$\text{Area}=\frac{1}{2}abc\frac{\sin(C)}{c}$ .

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

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

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

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now

Month: March 2018

More Things I Want To Highlight From Twitter

A nice proof for the Law of Cosines

Some very cool things about the Law of Sines