Good Math Problems

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:

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

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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!

    pic1

    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:
    pic2
    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:

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  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!

    pic3
  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.

    pic8

    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…  
    pic9

    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.”
    pic10.PNG

    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^2This 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.

 

 

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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.

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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.

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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.pic3

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:

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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.

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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.

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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:

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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”:

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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:

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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!

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

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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.

Waiters, Waiters, everywhere…

Today I was nerdsniped in the math office. My department head and two colleagues were working on this problem. I don’t know where it came from. But golly, did I enjoy it!

Imagine you have a row of waiters all facing forward. Each waiter has a beautiful silver platter that they are carrying. They have to choose: will they hold it directly in front of them, or on their left side, or on their right side? Here’s a diagram showing the three options (I imagine I’m looking down on the waiter.)

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Okay, so there is one constraint. Remember the waiters are all standing in a row. So you can’t have the platters crash into each other. So here’s an example of an OK way the waiters could hold their platters, and then a NOT OK way the waiters could hold their platters.

w2w3

So here’s the question… If you have n waiters standing in a row, how many different ways could they hold their platters?

I am not going to post the answer here, because I like to nerdsnipe! But if you want to check your answer, for 20 waiters, I calculate 267,914,296 different positions!

I bet you will have a lot of fun with this problem. One person in our office came up with many pages of work, and had a very complex approach which yielded some deep insights. She was super psyched about the intricate superstructure she was building. Another person got to review solving a particular type of thingie using matrices (I want to keep things vague so I’m going to use the word thingie to avoid giving anything away). I and another person had the same approach that led to a quick and elegant solution, but left me with rich conceptual questions to pursue. And as I started doing that, I realized that I had accidentally stumbled on the complex approach that the first person had taken.

Spiral Challenge

Megan Schmidt is obsessed with spirals. Her obsession got me hooked — for hours — on a math problem. I thought it would take maybe an hour or two, but I’m still at it and I’ve probably been working four or five hours.

I’ve been having so much fun with it.

Here’s the problem. Look at the spiral below…

obsessed.png

We see that 1 is located at (0,0).
We see that 2 is located at (0,1).
We see that 8 is located at (-1,0).

If we continue this spiral in this manner, can you come up with a formula for the coordinates of the kth number?

So what I want to know is if we consider the number 2016, can we come up with a way to precisely define where it is? What about 820526487?

One easy way around this is to write a computer program that just brute forces our way through it. So here’s the constraint: I want a closed formula for the x-coordinate and y-coordinate. That means no recursion! No if/then statements! Just an equation that relies on k only.

You know one of the most frustrating things? Going down a path and feeling good about it, even though it is pretty complicated. And then having a new insight on how to attack the problem (which *just* happened to me now as I typed up the problem and look at the image I created for this post) [1]. And realizing that approach might yield it’s secrets so much easier!

In any case, I thought I’d share the problem because it’s given me so much enjoyment thus far. If you do get obsessed and solve it, please feel free to put your answer in the comments. I have a feeling there are a variety of valid solutions which look very different but yield the same answer.

[1] What this reminds me is how slight changes in representations can lead to new insights! Before I was using this image that Megan sent me:

megan

UPDATE!: I solved it!

If you want to see that I did solve it, check out this Geogebra sheet. It won’t give away how I solved it (unless you download it, look at how I defined each cell, and then reverse engineered it).

https://www.geogebra.org/m/cXhwYh3P

So yeah… 2016 is at (-22,13), and 820526487 is at (-14322, 4784).

I am so proud of myself! I came up with a closed form solution!!!

I am going to put a “jump” below here, and then show what my solution is, and write a little about it. So only read below the jump (meaning: after this) if you want some spoilers.

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A New Insight on the Famous Painted Block Problem

There is a famous, well-known problem in the world of “rich math tasks” that involves taking an nnn cube and painting the outside of it. Then you break apart the large cube into unit cubes (see image below cribbed from here for n=2 and n=3):

cubes

Notice that some of the unit cubes have 3 painted faces, some have 2 painted faces, some have 1 painted face, and some have 0 painted faces.

The standard question is: For an nnn cube, how many of the unit cubes have 3 painted faces, 2 painted faces, 1 painted face, and 0 painted faces.

[In case you aren’t sure what I mean, for a 3 x 3 x 3 cube, there are 8 unit cubes with 3 painted faces, 12 unit cubes with 2 painted faces, 6 unit cubes with 1 painted face, and 1 unit cube with 0 painted faces.]

Earlier this year, I worked with a middle school student on this question. It was great fun, and so many insights were had. This problem comes highly recommended!

Today we had some in house professional development, and a colleague/teacher shared the problem with us, but he presented an insight I had never seen before that was lovely and mindblowing.

Spoiler alert: I’m about to give some of the fun away. So only jump below / keep reading if you’re okay with some some spoilers.

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Clock Puzzle

In our last department meeting, one teacher presented a puzzle/problem for us to figure out.

At 3:00, the hour and minute hands on a clock form a right angle. What is the next time that happens?

clock

The presenting teacher had a pretty darn elegant solution. But I enjoyed working it out using brute force. (That’s pretty much my go-to.) I’m going to type my solution down below the jump.

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Stuffing Sacks

Matt Enlow (math teacher in MA) posted a fascinating problem online today, one he thinks of when storing all those plastic bags from the grocery store. You shove them so they all lie in a single bag, and throw that bag under the sink. Here’s the question: how many different ways can you store these bags?

For 1 bag, there is only 1 way.
For 2 bags, there is still only 1 way.
For 3 bags, there are 2 ways.

Here is a picture for clarification:
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Can you figure out how many ways for 6 bags? 13 bags?

You are now officially nerdsniped.

A number of people had trouble calculating 4 bags correctly, so I’ll post the number of ways 4 bags could be stored after the jump at the bottom, so you can at least see if you’re starting off correctly…

Additional Information: Matt and I figured the solution to this problem together on twitter. It was an interesting thing. We didn’t really “collaborate,” but we both refined some of our initial data (for 5 bags, he undercounted, and I overcounted). It seemed we were both thinking of similar things — one idea in particular which I’m not going to mention, which was the key for our solution. What blew my mind was that at the exact time Matt was tweeting me his approach that he thought led to the solution, I looked at my paper and I had the exact same thing (written down in a slightly different way). I sent him a picture of my paper and he sent me a picture of his paper, and I literally laughed out loud. We both calculated how many arrangements for 6 bags, and got the same answer. Huzzah! I will say I am fairly confident in our solution, based on some additional internet research I did after.

Obviously I’m being purposefully vague so I don’t give anything away. But have fun being nerdsniped!

Update late in the evening: It might just be Matt and my solution is wrong. In fact, I’m now more and more convinced it is. Our method works for 1, 2, 3, 4, 5, and 6 bags, but may break down at 7. It’s like this problem — deceptive! I’m fairly convinced our solution is not right, based on more things I’ve seen on the internet. But it is kinda exciting and depressing at the same time. Is there an error? Can we fix the error, if there is? WHAT WILL HAPPEN?!

The number of ways 4 bags can be stored is… (after the jump)

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