# Tiny Game Re: Euler’s Number

I’m teaching Algebra 2 this year and the other teacher and I decided that we should introduce e to our kids. The reason it’s challenging is that it’s hard to motivate in any real way. You can do compound interest, but that doesn’t do much for you in terms of highlighting how important the number is. [1] I asked on Twitter for some help, and I got a ton of amazing responses (read them all here). My mind was blown. This year, though, I didn’t have time to execute my plan that I outlined at the bottom of that post. So here’s what I did:

1. The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form $(1+\frac{x}{n})^n$, but in a super concrete way so kids can see the polynomial for different n-values. It shows why the x-intercept travels more and more left as you increase n, so when you finally (in the class discussion) talk about what happens when n goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when $x=1$. And they see how this polynomial gets to look more and more like an exponential function as you increase the value of n. If you want to introduce e, this is one fantastic way to do it.
2. A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)

Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

Finally there is a class winner.

So I then went up against them.

And when we both said our numbers, I said: e.

The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

We sit down and I show them on my laptop how this works:

The red graph is my score, for any student number chosen ($e^x$).
The blue graph is the student score, for any student number chosen ($x^e$).

Clearly I will always win, except for if my opponent picks e.

I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

3. After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but e pops up in all of them!

I re-emphasize e is a number like $\pi$ and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore e in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in e and ln.

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

UPDATE: Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for e.

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!

# Clothesline Math – Logarithm Style

I remember when I first heard about Clothesline Math, I was excited by all the possibilities. And in a few conference sessions with Chris Shore, I saw there was so much more than I had even imagined that one could do with it!

It’s basically a number line, that’s all. But it’s a nice public giant number line which can get kids talking. Today I came back from spring break and before break, students learned about logarithms. However I wanted to have them recall what precisely logarithms were… so I created a quick Clothesline Math activity.

I hung a string in the classroom. I highlighted it in yellow because you can’t really see it in the photo…

I then showed them this slide – explaining the string is a number line…

I then showed them this slide, which explains what they have to do if they get two of the same number. (I brought cute little clothespins, but mini binder clips or paperclips would have worked just as well):

And then I gave them the rules of play:

I handed out the cards and let kids go. It was nice to see they didn’t get tripped up as a class on too many of them, but I got to listen to debates over a few trickier ones, which we collectively resolved at the end.

Here are the cards I handed out: .DOC FORM: 2019-04-01 Clothesline Math – Logarithms

Here is a picture of some of the cards. The two on the left are average level of difficulty. The two in the middle caused my kids to pause… it took them time to think things through (they haven’t learned any log properties yet). The one on the right doesn’t belong on the number one (it is undefined) and the kid who got that card immediately knew that. Huzzah!

Here’s a picture of the numberline at the end.

And… that’s it!

I was excited to try it out as a quick review activity. And it worked perfectly for that!

(Other things of note: Mary Bourassa made a clothesline math for log properties and shares that here. The author of Give Me A Sine blog does something similar here, but has kids create the cards. I couldn’t find anything with basic log expressions — so I made ’em and am sharing them in this post. Chris Hunter has a nice tarsia puzzle that sticks with basic log expressions here, but I wanted to try out clothesline math so I didn’t use that!) But if anyone has others out there involving logs, I’d love to see them in the comments!)

# Digits

I’m about to start a unit on logarithms. Kids don’t technically know that yet. To prime them, today I gave both my Algebra 2 classes a warm up. I was super nervous about this, because I haven’t seen a crazy amount of endurance from many of my kids when they get stuck on something. And I was going to give them something totally open-ended! And without a calculator allowed! Gasp!

I asked them to do the following. Think about $2^{60}$. It’s going to be a long number when it is all written out. I wanted them to come up with a guesstimate about how many digits there are in the expansion. To scaffold, I asked them for three things:

a) What’s a guess (for the number of digits) that is too low? How do you know? (Can you come up with a larger low estimate?)

b) What’s a guess (for the number of digits) that is too high? How do you know? (Can you come up with a smaller higher estimate?)

c) Based on your work and your intuition, if you had to make a guess, how many digits are in the expansion of $2^{60}$?

Honestly, it was one of the best things I’ve done recently. Kids were showing grit and so much flexibility in their thinking! I had to correct a few misconceptions and nudge a little here and there, but it was all on them how they wanted to go about this. It was beautiful. (At one point, a kid said they wanted to give up, but I came back around a few minutes later and they were rapidly making progress and hadn’t given up.)

At first, kids didn’t know where to start. I told them they were going to get time to work on this, so they could take on strategies that might take a while. (Normally, we start class with something short and quick. I wanted to indicate this wasn’t that.) Initially, I gave 7 minutes, but since so many kids were on a roll, I expanded it to 14 or 20 minutes. I honestly don’t remember how long.

What I adored is that this problem was definitely in their wheelhouse. Most groups were gung ho, and just started writing stuff down — and eventually (sometimes with a little encouragement/prompting from me), they came up with SUPER awesome solutions. Seriously, things I had never thought of.

The main two approaches I saw were:

1. Kids noticing that $2^{10}=1024$. Which is close to $10^3$. So $2^{60}=(2^{10})^6 \approx (10^3)^6=(10^{18})$. So that puts us at around 19 digits.
2. Kids noticing this pattern:

So after going up about every 3 exponents, we add an additional digit to the number. (I say about 3 because all groups who did this method saw that a few times, you’d get 4 exponents in a row which keep the same number of digits instead of 3. But it was usually 3.)Assuming the number of digits increases after going up every 3 exponents, that means that exponent 12 has 4 digits, exponent 15 has 5 digits, exponent 18 has 6 digits, exponent 21 has 7 digits… etc. So exponent 60 has 20 digits.

So that puts us around 20 digits (or maybe a little lower because of those occasional 4 exponents in a row).

That’s about all I wanted to share. I was a little out of my comfort zone because I didn’t know if they would all just throw their hands up and give up. But they didn’t, and instead did some phenomenal thinking.

I just realized… you might want to see how this relates to logarithms. It turns out that the number of digits is equal to doing the following: take the log of the number, and then take the floor function of that result, and then you add one. I won’t spoil it by explaining why, though. See if you can figure it out!

# Problem Solving with Trig

So I’m at #TMC17 and Rachel Kernodle nerdsniped me. Or rather, I asked to be nerdsniped. Her session is at a time when there were a lot of other amazing sessions I wanted to go to, so I wanted to know if hers was one where I could hear about it and get the gist of things instead of attending. After some internal debate, she said that since it involved working on a problem, and then using that problem solving to frame the session, the answer was maaaaybe not. But then she thought: maybe I can try the problem on you and see how it goes. As long as you’re willing to put in the time to problem solve. Of course I said yes.

First, you can see her session description, which then framed how I approached the problem:

And then this is what she gave me (but it was hand drawn):

From the session description, I knew I had to find the ratio of the side lengths, so I could find exact trig values for angles other than 30, 60, 90, 45.

Rachel also gave me a “hint page” which she told me to look at when I was stuck (and to time how long it took me before I opened it). Let’s just say I’m extremely stubborn, and so as long as I think I have the capability to solve something and I am not completely stuck, I knew I wasn’t going to open it. Turns out my stubbornness paid off, and I ended up solving it.

In this post, I wanted to write a little bit about my experience with the problem. Because now when I look at that triangle, I have an duh, there’s an obvious approach to use here and everything I know points at that obvious approach. And the answer feels really obvious too. It is funny that I’m almost embarrassed to post this because there are going to be people who see it right away, and I worry (irrationally) (math pun) that they are going to judge me for not seeing it as quickly as they did. Even though I know being good at math has nothing to do with speed. And that it was important to go through the steps I did!

It took me over an hour to solve this problem. I had to do a lot of play and make a lot of random leaps before I stumbled across the “obvious approach.”  And I needed to do that in order for me to mine it for lots of things. It was true problem solving. And I know I really deeply understand this because at first the problem looked flummoxing and interesting, and now it looks obvious and somewhat trite. That’s my metric of how I know I deeply understand something. There are still certain things that I teach that I don’t deeply understand: like how the cross product of two 3D vectors yields a third vector perpendicular to the original two. I have done the math, but it’s non-obvious to me why the crazy way we compute cross products give us something perpendicular.(When I only understand something by doing brute algebra, I rarely feel like I get it.)

I’m going to try to outline the messiness that was my thought process in this triangle problem, to show/archive the messiness that is problem solving.

1. The first thing I noticed was 36 and 36 sum to 72. So I was like: obviously put two of those figures together, and just play around. Something nice will happen. I remember when seeing the problem that approach felt immediate, obvious, and would lead to the solution. I was like yes! I have an inroad! This is going to rock, and I’m going to solve it quickly! And I’ll even impress Rachel!

That appraoch didn’t work. Nothing popped out. I saw 54s and 18s and 144s pop out. But those weren’t angles that helped me. But I did then realize something nice… 36 is a tenth of 360! So I was going to use a circle somehow in this solution. Obviously!

2. So I drew this:

and I was like, I have something here! But after looking around, I was getting less. You can see I was trying to draw in some other lines lightly and play around — I thought maybe creating other triangles within these triangles would work. But nothing seemed to pop out. At one point, I thought I had possibly created an equilateral triangle in this (even though I saw one of the angles was 72! I was clearly desperate!). I started to get dejected at this point. I knew the circle had something to do with it…
3. But seeing that 54s and 18s and 36s and 72s kept appearing, I thought maybe algebraically I should play around with the numbers (adding in 180 also, since I can draw a straight line wherever) to see if algebraically I could get a 30, 60, or 45. I tried adding and subtracting numbers from the set {18, 36, 54, 72, 180} looking for 30, 60, or 45. I figured if I could somehow do that, then I could find a diagram that would have angles I could get side relationships from. And then like a domino effect, I could get others. I don’t know. But after like 2 seconds, I got bored with this and didn’t see it as very efficient. My intuition was strongly saying I was going in the wrong direction. So I stopped:

4. At this point, I was pretty dejected. I was slightly losing interest in the problem, thinking it was too hard for me. I tried to “force” a 60 degree angle in a diagram of that original blasted triangle. Hope! And then hope dashed!

5. Damnit! I know the circle had something to do with it. It is just too nice to abandon the circle! Maybe…

At first I drew all ten vertices for a 10-gon. I started connecting them in different ways. I thought I could exploit the chord-chord theorem in geometry, but that wasn’t good. I tried in that second diagram to extract part of the circle diagram and investigate it more. And the third was just more of the same. At one point, I was like $e^{i\theta}=\cos\theta+i\sin\theta$ and was thinking I could somehow think of this as a problem on the complex plane, where each vertex was $e^{ni\pi/5}$ and then look at the real parts for the x-coordinate and the imaginary parts for the y-coordinate. Clearly my mind was whirring, and I was going anywhere and everywhere. I actually thought maybe this complex plane thing seems ugly but it will be so elegant. But then I realized I didn’t know where to go if I labeled each of the points on the complex plane. Done and done and doneAt this point I put the problem away. Nothing was working.

6. But after a minute, I couldn’t let it go! I wanted to solve it!!! So I went back. I thought I was getting too complicated, so I went simple.

Nope. Didn’t help. But for some reason, this diagram and looking at the 72 reminded me of something I hadn’t thought of before. This is the leap that helped me get to the answer. And I can’t quite explain why this diagram sparked this leap. Which sucks because this is that moment that led to the rest of the problem for me! But I immediately remembered something about 72s and pentagons. And it hit me.

7. So I drew what this connection was. My brain was whirring, and I was somewhere good…

I remembered the 72 degree angle appeared in a star. And this star was related to a pentagon. And that the pentagon had something about the golden ratio tied up in it. So I knew that maybe the golden ratio was involved in the answer. And where does the golden ratio appear? When there are similar triangles and proportions. I had my new approach and my inroad that I thought would work. Two triangles next to each other failed. Circles failed. But star/pentagon might work!

8. So I looked at the original triangle and tried to figure out where I could find a similar triangle. And so I drew one line and created a similar triangle. I labeled the two legs as having length “1.”

Initially, I was thinking I could do something with the law of sines. Because if you think about it, this is the ASS case — where you have that 36 degrees (circled), the side I labeled 1 (circled), and the other side I labeled y (circled). But you note that last side could be in two different places, which is why there are two ys circled. I still think there is something fun that I could do with this. But as I was doing this, I realized I was making things more complicated.

I knew that the golden ratio came out of a proportion. So I abandoned the law of sines for the proportion. I simply set up a proportion with the two similar triangles. I first found “?” by doing $1/y=y/?$. So ? was $y^2$This was exciting. I knew the golden ratio came out of solving a quadratic. Yeeeeee! At this point, my excitement was growing because I was fairly confident I was almost at the solution.

Then I labeled the part of the leg that wasn’t ? as $1-y^2$ (since the whole leg length was 1). Finally I looked at the third triangle in the diagram that wasn’t similar to the original triangle. It was isosceles and has legs of $y$ and $1-y^2$ so I set them equal and solved and not-quite-the-golden-ratio came out! (There was a mistake I made where I set $y^2=1-y^2$ and got $y=\sqrt{2}/2$. But I then found it and rewrote the equation $y=1-y^2$. This was the most depressing part of it. Because I couldn’t find my error because I was so tired. I went through my work multiple times and nothing. But taking some time away and then looking with fresh eyes, it was like: doh!)

And so that was the end. I found if the original triangle had leg lengths of 1, the base was going to have a length of $\sqrt{5}/2-1/2$.

I was so proud. I was on cloud nine. I was telling everyone! SO COOL!!!

It probably took me in total 90 minutes or so from start to finish. So many false starts at the beginning, and one depressing transcription error that I couldn’t find.

The point of this post isn’t to teach someone the solution to the problem. I could have written something much easier. (See we can draw this auxiliary line to create similar triangles. We use proportions since we have similar triangles. Then exploit the new isosceles triangle by setting the leg lengths equal to each other.) But that’s whitewashing all that went into the problem. It’s like a math paper or a science paper. It is a distillation of so freaking much. It was to capture what it’s like to not know something, and how my brain worked in trying to get to figure something out. To show what’s behind a solution.

# Graham’s Number

TL;DR: If you have an extra 45-60 minute class and want to expose your 9th/10th/11th/12th graders to a mindblowingly huge number and show them a bit about modern mathematics, this might be an option!

In one of my precalculus classes, a few kids wanted to learn about infinity after I mentioned that there were different kinds of infinity. So, like a fool, I promised them that I would try to build a 30 minute or so lesson about infinity into our curriculum.

As I started to try to draft it — the initial idea was to get some pretty concrete thinkers to really understand Cantor’s diagonalization argument — I decided to build up to the idea of infinity by first talking about super crazy large numbers. And that’s where my plan got totally derailed. Stupid brain. At the end of two hours, I had a lesson on a crazy large number, and nothing on infinity. You know, when that “warm up” question takes the whole class? That’s like what happened here… But obvi I was stoked to actually try it out in the classroom.

In this post, I’m going to show you what the lesson was, and how I went through it, with some advice for you in case you want to try it. I could see this working for any level of kid in high school. Now to be clear, to do this right, you probably need more than 30 minutes. In total, I took 35 minutes one day, and 20 minutes the next day. Was it worth it? Since one of my goals as a math teacher is to try to build in gaspable moments and have kids expand their understand of what math is (outside of a traditional high school curriculum): yes. Yes, yes, yes. Kids were engaged, there were a few mouths slightly agape at times. Now is it one of my favorite things I’ve created and am I going to use it every year because I can’t imagine not doing it? Nah.

We started with a prompt I stole from @calcdave ages ago when doing limits in calculus.

Kids started writing lots of 9s. Some started using multiplication. Others exponentiation. Quite a few of them, strangely, used scientific notation. But I suppose that made sense because that’s when they’d seen large numbers, like Avagadros number! I told them they could use any mathematical operations they wanted. After a few minutes, I also kinda mentioned that they know a pretty powerful math operation from the start of the school year (when we did combinatorics). So a few kids threw in some factorial symbols. Then I had kids share strategies.

Then I returned to the idea of factorials and asked kids to remind me what $5!$ was. Then I wrote $5!!$. And we talked about what that meant ($120!$). And then $5!!!$ etc. FYI: this idea of repeating an operation is important as we move on, so I wouldn’t skip it! They’ll see it again in when they watch the video (see below). While doing this, I had kids enter $5!$ on their calculator. And then try to enter $120!$. Their calculators give an error.

Yup, that number is super big.

Then I introduced the goal for the lesson: to understand a super huge number. Not just any super huge number, but a particular one that is crazy big — but actually was used in a real mathematical proof. And to understand what was being proved.

Lights go off, and we watch the following video on Graham’s number. Actually, wait, before starting I mention that I don’t totally follow everything in the video, and it’s okay if they don’t also… The real goal is to understand the enormity of Graham’s number!

I do not show the beginning part of the video (the first 15) because that’s the point of the lesson that happens after the video. While watching this, kids start feeling like “okay, it’s pretty big” and by the end, they’re like “WHOOOOOOAH!”

Now time for the lesson… My aim? To have kids understand what problem Ronald Graham was trying to understand when he came up with his huge number. What’s awesome is that this is a problem my precalculus kids could really grok. But I think geometry kids onwards could get the ideas! (On the way, we learned a bit about graph theory, higher dimensional cubes, and even got to remember a bit about combinations! But that combinations part is optional!)

I handed out colored pencils (each student needed two different colors… ideally blue and red, but it doesn’t really matter). And I set them loose on this question below.

It’s pretty easy to get, so we share a few different answers publicly when kids have had time to try it out. The pressure point for this problem is actually reading that statement and figure out what they’re being asked to do. When working in groups, they almost always get it through talking with each other!

One caveat… While doing this, kids might be confused whether the following diagram “works” or if the blue triangle I noted counts as a real triangle or not:

It doesn’t count as a real triangle since the three vertices of the triangle aren’t three of the original four points given. During class I actually made it a point to find a kid who had this diagram and use that diagram to have a whole class conversation about what counts as a “red triangle” or “blue triangle.”. Making sure kids understand what they’re doing with this question will make the next question go more smoothy!

Now… what we are about to do is super fun. I have kids work on the extension question. They understand the task (because of the previous one). They go to work. I mention it is slightly more challenging.

As they work, kids will raise their hand and ask, with trepidation, if they “got it.” I first look to make sure they connected all the points with lines. (If they didn’t, I explain that every pair of points needs to be connected with a colored line.) Then I look carefully for a red or blue triangle. Sometimes I get visibly super excited as I look, saying “I think you may have gotten it! I think you may… oh… sad!” and then I dash their hopes by pointing out the red or blue triangle I found. (So here’s the kicker: it’s impossible to draw all the line segments without creating a red or blue triangle… so I know in advance that kids are not going to get it… but they don’t know this.) After I find one (or sometimes two!) red or blue triangles, I say “maybe you want to start over, or maybe you want to start modifying your diagram to get rid of the red/blue triangle!” Then they continue working and I go to other students.

(It’s actually nice when students try to modify their drawings, because they see that each time they try to fix one thing, another problem pops up. They being to *see* that something is amiss!)

This takes 7-8 minutes. And you really have to let it play out. You have to ham it up. You have to pretend that there is a solution, and kids are inching towards it. You have to run from kid to kid, when they think they have a solution. It felt in both classes like a mini-contest.

Then, after I see things start to lag, I stop ’em. And then I say: “this is how you can win money from your parents. Because doing this task is impossible [cue groans… let ’em subside…] So you can bet ’em a dollar and say that they can have up to 10 minutes.!That it takes great ingenuity to be successful! What they don’t know is… you’re going to get that dollar! Now we aren’t going to prove that they will always fail, but it has been proven. When you have six or more dots, and you’re coloring all lines between them with one of two colors, you are FORCED to get a red or blue triangle.” [1]

Now we go up a dimension and change things slightly. Again, this is a tough thing to read and understand so I have kids read the new problem aloud. And then say we are going to parse individual parts of it to help us understand it.

And then… class was over. I think at this point we had spent 35 minutes all together. So that night I asked kids to draw all the line segments in the cube, and then answer the following few questions:

These questions help kids understand what the new problem is saying. In essence, we’re looking to see if we can color the lines connecting the eight points of a cube so that we don’t get any “red Xs” or “blue Xs” for “any four points in a plane.” Just like we were avoiding forming “red triangles” and “blue triangles” before when drawing our lines, we’re now trying to avoid forming “red Xs” and “blue Xs”:

So the next day, we go over these questions, and I ask how this new question we’re working on is similar to and different from the old question we were working with. (We also talk about how we can use combinatorics to decide the number of line segments we’d be paining! Like for the cube, it was $_8C_2$ and for the six points it was $_6C_2$ etc. But this was just a neat connection.) And then I said that unlike the previous day where they were asked to do the drawings, I was going to not subject them to the complicated torture of painting all these 28 lines! (I made a quick geogebra applet to show all these lines!) Instead I was going to show them some examples:

It’s funny, but it took kids a long while to find the “red X” in the left hand image. Almost each class had students first point out four points that didn’t form a red X, but was close. But more important was the right hand figure. No matter how hard you look, you will not find a red X or blue X. Conclusion: we can paint these line segments to avoid creating a red X or blue X. Similar to before, when we had four points, we could paint the line segments to avoid having a red triangle or blue triangle!

So now we’re ready to understand the problem Graham was working on. So I introduce the idea of higher dimensional cubes — created by “dragging and connection.” I don’t take forever with this, but kids generally accept it, with a bit of heeing and hawing. More than not believing that it’s possible, kids seem more enthralled about the process of creating higher dimensional cubes by dragging!

And then… like that… we can tie it all together with a little reading:

And… that’s the end! At this point, kids have been exposed to an incomprehensibly large number. And kids have learned a bit more about the context in which this number arose. Now some kid might want to know why we care about higher dimensional cubes with connecting lines painted red/blue. Legit. I did give a bit of a brush off answer, talking about how we all have cell phones, and they are all connected, so if we drew it, we’d have a complex network. And analyzing complex networks is a whole branch of math (graph theory). But that’s pretty much all I had!

In case it’s helpful: the document/handout I used: 2017-04-04 Super Large Numbers (Long Block).

[1] I like framing this in terms of tricking their parents. We’ve been doing that a bunch this year. And although I understand some teachers’ hesitation about lying to their students about math, I think if you frame things well, don’t do it all the time, it can be fine. I don’t think any student felt like I was playing a joke on them or that they couldn’t trust me as their math teacher because of it.

# Getting familiar with the Unit Circle

In our standard precalculus class, we’ve spent 4 days “getting ready” for trigonometry. Which sounds crazy, until you see what awesome thing we’ve done. But I’ll blog about that later. Right now I want to share what I created to help kids start learning the unit circle.

Here were the hurdles:

1. We are introducing radians for the first time this year. So they’re super unfamiliar.
2. The unit circle feels overwhelming.
3. Although I am familiar with the special angles in degrees and radians, kids aren’t. So I know when I hear 210 degrees that’s “special” but kids don’t know that yet.

Here is what we have done:

1. Filled in a “blank unit circle” using knowledge of 30-60-90 degree triangles, 45-45-90 degree triangles, and reflections of 1st quadrant points to get the points in the other quadrants.

In this post, I’m going to do here is to share what I’m going to be doing to help kids learn the unit circle.

Phase I: Get confident with angles

I am going to talk about these like pizza. And to start, focus on radians.

I’m going to remind kids that $\pi$ radians is a half rotation about the circle. Then we can see that each pizza pie slice is $\frac{\pi}{2}$,$\frac{\pi}{4}$, $\frac{\pi}{3}$, and$\frac{\pi}{6}$ radians. [The “top” half of the pizza is divided into two, four, three, or six pieces! And the top half is $\pi$ radians!]

Then I’m going to work on the “easy-ish” angles by pointing at various places on the unit circle and have kids figure out the angle. I am going to have kids not only state the angle in radians, but also explain how they found it. For each angle, I will ask for a few different ways one could determine the angle measure. Then I’m going to repeat the same thing with the “easy-ish” angles, except I am going to do it in degrees.

And then… you guessed it… I’m going to do the same exercise but with the “harder-ish” angles. Start with radians. Then again with degrees. Always justifying/explaining their thinking.

Finally, I am going to let them practice for 5-8 minutes using this Geogebra applet I made. The goal here? To focus on getting kids familiar with the important special angles. Not only what the values are of these angles, but also to get them to start finding good ways to “see” where these angles are.

Phase II: Start Visualizing Side Lengths — utilizing short/long

Next comes getting kids to quickly figure out the coordinates of these special angles.

We’ve already been working on special right triangles, so I think this should be fine. And then…

Kids are asked to visualize the side lengths/coordinates based on the drawing. So, for example, for the first problem, kids will see that the angle is $\frac{4}{3}\pi$. They hopefully would have mastered that from the previous exercise. They also will see that if they would draw the reference triangle, the x-leg is shorter than the y-leg, so they know the x-coordinate must be $\frac{1}{2}$ (but negative), and the y-coordinate must be $\frac{\sqrt{3}}{2}$ (but negative).

After practicing with this for these four problems, kids are going to practice some more using this second geogebra applet I created.

Phase III: Putting It All Together

It’s now time to take the training wheels off. No longer do I give the picture to help visualize things. Now, I give the angle. This is more like what kids are going to be seeing. They need to know $\sin(315^o)$ and $\cos(3\pi/4)$. No one is going to be giving them nice pictures!

So this is what they’re tasked with:

I have a strong feeling that breaking down the unit circle in this way is going to make all the difference in the world. Fingers crossed!

If you want the file I created for my kids, here you go (.docx2017-02-xx Basic Trigonometry #2.docx:  , PDF: 2017-02-08-basic-trigonometry-2)!

# Visualizing Standard Deviation

A few days I got an email from someone (Jeremy Jones) who wanted me to look at their video on standard deviation. And then today, I was working with Mattie Baker at a coffeeshop. He was thinking about exactly the same thing — how to get standard deviation to make some sort of conceptual sense to his kids. He said they get that it’s a measure of spread, but he was wondering how to get them to see how it differs from the range of a data set (which also is a measure of spread).

Of course I was hitting a wall with my own work, so I started thinking about this. While watching Jeremy Jones’s video, I started thinking of what was happening graphically/visually with standard deviation.And I had an insight I never really had before.

So I made an applet to show others this insight! I link to the applet below, but first, the idea…

Let’s say we had the numbers 6, 7, 7, 7, 11. What is the standard deviation?

First I calculate the mean and plot/graph all five numbers. Then I create “squares” from the numbers to the mean:

The area of those squares is a visual representation of how far each point is from the mean.[1] So the total areas of all those five rainbow squares is a measure of how far the entire data set is from the mean.

Let’s add the area of all those squares together to create a massive square.

As I said, this total area is a measure of how far the entire data set is from the mean. How spread out the data is from the mean.

Now we are going to equalize this. We’re going to create five equal smaller squares which have an area that matches the big square.

We’re, in essence, “equalizing” the five rainbow colored squares so they are all equal. The side length of one of these small, blue, equal squares is the standard deviation of the data set. So instead of having five small rainbow colored squares with different measures from the mean, the five equal blue squares are like the average square distance from the mean. Instead of having five different numbers to represent how spread out the data is from the mean, this equalizing process lets us have a single average number. That’s the standard deviation.

I’m not totally clear on everything, but this visualization and typing this out has really help me grok standard deviation better than I had before.

I created a geogebra applet. You can either drag the red points up and down (for the five points in the data set), or manually enter the five numbers.

https://www.geogebra.org/m/EatncEg2

My recommendation is something like this:

1. {4, 4, 4, 4, 4}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
2. {8, 8, 8, 8, 8}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
3. Set the five numbers to {2, 4, 4, 4, 6} and look at what you see. What is the standard deviation?
4. Consider the number {5, 7, 7, 7, 9}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #3. Then set the five numbers to {5, 7, 7, 7, 9} and look what you see. What is the standard deviation? Were you right?
5. Consider the numbers {3, 7, 7, 7, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #4. Explain your thinking. Then set the five numbers to {3, 7, 7, 7, 11} and look at what you see. What is the standard deviation? Were you right?
6. Consider the numbers {3, 6, 7, 8, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #5. Explain your thinking. Then set the five numbers to {3, 6, 7, 8, 11} and look at what you see. What is the standard deviation? Were you right?
7. What do you think the standard deviation of {4, 8, 8, 8, 12} be? Why? Check your answer with the applet.
8. Can you come up with a different data set which matches the standard deviation in #6? Explain how you know it will work.
9. Set the five numbers to {4, 4, 4, 4, 4}. Initially there are no squares visible. The standard deviation is 0. Now drag one of the numbers (red dots in the applet) up. Describe what the squares look like when they appear? Eventually drag that number to 15. What do you notice about the standard deviation? Use your understanding of what happened to describe how a single outlier in a data set can affect the standard deviation

Okay, I literally just whipped the applet up in 35 minutes, and only spent the last 15 minutes coming up with these scaffolded questions. I’m sure it could be better. But I enjoyed thinking through this! It has helped me get a geometric/visual sense of standard deviation.

Now time to eat dinner!!!

Update: a few people have pointed out that the n in the denominator of the standard deviation formula should be n-1. However that would be for the standard deviation formula if you’re taking a sample of a population. This post is if you have an entire population and you’re figuring out the standard deviation for it.

[1] One might ask why square the distance to the mean, instead of taking the straight up distance to the mean (so the absolute value of each number minus the mean). The answer gets a bit involved I think, but the short answer to my understanding is: the square function is “nice” and easy to work with, while an absolute value function is “not nice” because of the cusp.