Good Math Problems

A record of conference takeaways — TCM 2024

On Thursday, after school, I hopped in an Uber to the airport. I was flying to a conference, the “Teaching Contemporary Mathematics” conference (TCM) held at the North Carolina School of Science and Mathematics. I think I’ve been at least twice over my career, maybe three times, and always found it to be a really solid conference. The big sell for me is that it’s primarily high school math-focused, and most of the sessions are given by actual math teachers about their own teaching practice. And more importantly, it’s felt like forever since I just got to geek out with other math teachers. A fellow teacher from my school and I arrived pretty late and just passed out, with dreams of conference day on Friday.

Although I haven’t blogged in forever, I wanted to at least outline a few things tonight that I can return to. But it’s late and I’m tired, so I’m only going to do a few tidbits.

There were two sessions that got the mathematical side of my brain whirring.

Ryan P gave a talk on “A Rainbow of Random Digits,” where he went from 1D to 2D to 3D to think about a interesting problem. I thought what he showed was beautiful, especially for the 1D and 2D which I could see using with students in a sort of ongoing independent study/investigation. For the 1D question, he asked us if we had two dowels, of length 1 and length 2, and we split the larger dowel into two pieces, what is the probability that the (now three) dowels could form a triangle. A nice fun introductory question with various approaches to answering it. For the 2D question, he asked as a warm up: if you had two numbers chosen randomly between 0 and 1, (a) what is the probability that the sum is greater than the product, and (b) what is the probability that the sum of between 5 and 6? Lovely scaffolding, and the first question requires a little bit of calculus, which is fun. And then the full 2D question (which can be answered using some calculus, but I think it’s even more beautiful without it): if you have two random numbers between 0 and 1, what is the probability that the first non-zero digit of their ratio is a 1? a 2? 3?… 9? Totally fascinating 2D probability space, with lots of “triangle slivers.” To whet your appetite, here it is:

And totally not my intution, but the probability of getting a 1 (33.3%) is greater than the probability of getting a 2 (14.9%) which is greater than the probability of getting a 3 (10.2%), etc. The calculation involves some fun infinite geometric series. All of this was new to me. I also have no idea why getting a 1 digit is more likely than getting a 2 digit which is more likely than getting a 3 digit, etc. My intuition–which was way off–made me think all digits would be equally likely. Now honestly, I don’t know how teachers can actually build something like this problem into their normal classroom practice, but I do know that I’d love to work with a student to get them from the statement of the question to the answer in a lovely set of independent investigations, and some well-thought-out hints to guide. (And maybe chatgpt to write some simple code to do an initial simulation.) My friend and colleague suggested that maybe the distribution might be related to Benford’s law.

The second session that got the mathematical side of my brain whirring was by Bryan S. He had first learned about Conway’s “Rational Tangles” a few years ago, and wanted to present it to us. Wonderfully, I had first learned about these from Conway himself when I was in high school attending Mathcamp (and Conway was a guest lecturer). Conway was an electric speaker — and this one lecture of his imprinted itself on my mind. Now skip forward to this year. I had students work on “Explore Math!” projects and one worked on knot theory. I mentioned in my feedback to one student I had a really cool knot theory-adjacent thing I learned and I could show her. She responded saying “yes, please!” Of course it’s been years since I learned Rational Tangles. And it’s like the universe said “Oh, let me bring Bryan S. to you to remind you about all the nitty gritty of it.” And he was fabulous — a marvelous instructor who somehow managed to convey the excitement, weirdness, inquiry, all in a single short session. The crux of the setup is that you have two ropes held by four people:

There are two moves: T(wist) which has the front right person and back right person switch positions, where the front right person brings their rope under the back right person’s rope, and R(otate) where the four people just rotate clockwise. It turns out that by doing seqences of moves like this, such as TTRTRTTRT (etc.), you can get a pretty tangled tangle in the middle of the two ropes. One question — the main one we talked about — is if you can do a series of Ts and Rs to “undo” the tangle and get back into the original position of just two untangled ropes. Amazingly, a few Mathcampers created a digital version of this twisting and rotating and it took me about 20 minutes today to find it even though I knew it existed and I had played with it before. Here it is!

Another session I went to was Chris B’s Estimathon! I participated in my first estimathon at the Park City Math Institute many years ago (and again, a couple years ago). I hated it both times, for a few years. First, I hate estimating. I love thinking and calculating — but Fermi problems? I get really annoyed because I feel I rarely have the adjacent information that can unlock the problem. Second, the other people (both on my team and on all the other teams) were very, very competitive. I prefer a cooperative board game over a competitive board game, and in this, I think I feel similarly. That being said, I really enjoyed doing the estimathon with Chris and our other math conference participants. It was fun because the other people on my team were chill about it, and also let me sort of work alone at times when I got obsessed with a problem. Here are two example questions we were tackling:

Yes, I went to #5 because you can calculate that. Here’s what the scoring sheet looked like:

So you get to guess a minimum value and a maximum value for the range that the answer is in. And your score is ceiling(maximum/minimum). In other words, take the maximum value, divide it by the minimum value, and round up. So min: 2000, max: 3000 would yield a score of 2, and min: 2000 max: 4000 would also yield a score of 2…. but min: 2000, max: 4001 would yield a score of 3. Your goal is to get the lowest score. At the end, Chris gave us the absolute best idea (which I think he got from his colleague Emily). You have kids find out numbers for something that they are passionate about or would be an expert in. It could be “the number of pokemon” or “how many grandchildren does Sameer’s father have?” And then later in the year, you could create an estimathon out of these numbers — where kids have to see what they know about each other and their passions. I love this as a way for people to get to know each other.

There were two additional sessions that I attended, which were about students and math, and I loved both. First was by Jenny W, Lauren B, and Kevin J, and reminded me of the “5 practices” (https://www.nctm.org/Store/Products/5-Practices-for-Orchestrating-Productive-Mathematics-Discussions,-2nd-edition-(Download)/). It’s about using Desmos to highlight and discuss student thinking and to uplift student brilliance. Although I’ve seen many, many presentions on the 5 practices over the year (especially at PCMI), this was a great reminder of a lot of the things I don’t know, and teacher moves I’ve stopped flexing.

The second was by Lauren B called “I am, We are, You are.” It highlighted a few things to me. First, there’s a gap between the demographics of people who teach mathematics (and their identities) and the population of students who learn math (and their identities). She posed a question (of which she thinks the answer is yes): “Is there a way to expand identity in a math class?” I think this is a great question to chew on — and not easy. Especially if you take away classes like statistics or data science from the mix. We played with a super engaging Desmos activity which gave us choice on which data sets to plot against each other, and the fun part was guessing what the scatter plot would look like before we saw it. And she had a quote from Rochelle Gutierrez (who I’ve met before briefly!) which I couldn’t copy down quickly enough, but went something like “Do I have to be a better you in this classroom, or can I be a better me?” This is a student asking the question — essentially saying “Do I have to mold myself to be a miniature version of you, the teacher, to succeed in this room we’re in together?” I also thought that had a lot to chew on… in terms of what we expect from students and the culture that we build together.

Okay, it’s now 9pm and I’ve been at this for way longer than I intended, so to sleep I go.

***

It’s the next day and the conference has finished and I wanted to archive — briefly — the remaining sessions.

First, Hollylynne L. gave the keynote talk “Data Science is Everywhere and For Everyone.” Some takeaways for me are that there are a few large organizations (like NCTM, NCSM, ASA, etc.) that are collaborating to create a united data science position. It’s drafted, and about to be adopted, and has four guiding principles: (1) data science is contextual and multidisciplinary, (2) data science is an investigative process, (3) data science understanding and experiences are for everyone, and (4) data science educators must develop and practice ethical uses of data. The presenter shared her experience ethnographically observing data scientists for 9 months and what traits they exhibited. Then she shared with us that she feels like data science needs to be presented to kids with larger data sets — in terms both of cases but also in number of attributes shared (e.g. not just a survey with one or two questions, but a survey that has a ton of questions!). As an expert on CODAP, she shared how to fluently use it to show data, ask questions, and then interrogate the data. That part was inspiring, and in only a few minutes showcased the power of CODAP (codap.concord.org). Her talk also got me thinking about how our department has over the years shoved all of data science/statistics to our AT Statistics courses as we were making room for everything else we need to teach. Lastly, at the end, she shared a resource I want to follow up on called InSTEP. It is a free online site [https://instepwithdata.org/public/] that is designed to get teachers ready to teach data science and statistics, and it sounds like you learn lots of content, pedagogical moves, and you learn to use various tech tools but primarily CODAP (which is what I want to learn). So yay!

My next session was by Reed H and was an invigorating conversation on Standards Based Grading. He presented a “post mortem” of him implementing SBG in his precalculus classroom for the first time, sharing why he made the switch but also the tradeoffs that occurred. Although our school is moving in a different direction, I was still curious to see how the SBG conversations were going — and it reminded me how much I liked SBG even though it took me 4 years until I had refined it to the point where I could run it fluidly in my standard calculus classes. Reed’s own observations, and the conversations we had as small and large groups, also reminded me of my own path to SBG, and how I now know there is no single flavor of SBG that is going to work for all, because its success is dependent on so many cultural and institutional factors. And there is no magic bullet that is going to make it suddenly easy.

My last session was by Verónica Z and Doru H and was on Linear Programming and Other Means of Optimization. The presenters shared three ways to do optimization without calculus. First, the standard linear programming. The second was something called the “simplex algorithm.” Honestly, I got very confused at this point, because it isn’t in any way intuitive and I think that part of the presentation was meant for people who knew the method. So I ended up stopping my notes and writing “very confused” on my page. (I did find this example that went through the algorithm that I’m curious to read though, to get the crux of the algorithm, but don’t think it will explain “why” it works.) Finally, we learned the TOPSIS algorithm which was just invented in 1991 (“Technique for Order of Preference by Similarity to Ideal Solution”). New math! And the presenter, Verónica, did a really cool job of showing us how to make a decision of which new phone to buy (out of a choice of three) if we were looking at two variables: picture quality and battery life. It’s such a simple algorithm that — at the highest level — has you develop two new “fake” phones that exist — the best phone and the worst phone — one with the best of the traits of the three phones that exist and one with the worst of the traits of the three phones that exist. We plot all three phones and the best “fake” phone and worst “fake” phone. And then we find the “distance” from each of the three phones to the best and worst “fake” phones, and use those distances to rank the phones. Details are in my notes, but I loved learning new math in the universe, and new math to me!

Lastly, the conference was raffling off math art, and although I didn’t win, a new friend did, and she saw how much I coveted them and offered me hers. I demurred and then eventually accepted.

***

Personal Note: Speakers for the most part shared their slides with attendees. So I’ve downloaded the sessions I went to and saved them on my google drive. But I don’t know if they are officially public, so I’m just linking to them here for my own easy access.

Hypercubes and more! Three problems you may enjoy working on!

One way I start my Advanced Precalculus classes is by having them thinking about n-dimensional cubes. We get there by first exploring the “Painted Block Problem“.

First I have kids look at one of these blocks (I think I give a 5x5x5 block) and have them notice and wonder. Eventually kids wonder what’s on the inside, why different parts are painted different colors, etc. And after some drama, we open up the block, and kids see the new color inside. The question they then are tasked with are how many mini-cubes of each color exist in an n x n x n block.

To be clear, green blocks have 0 exposed faces, yellow have 1 exposed face, blue has 2 exposed faces, and pink has 3 exposed faces.

Something that kids eventually stumble upon as they are working on this problem: in a cube, you have 8 vertices (pink), 12 line segments (blue), 6 faces (yellow), and 1 cube (green).

After they solve this problem, I introduce the idea of 0-dimensional cubes (a point), 1-dimensional cubes (a line segment), 2-dimensional cubes (a square), 3-dimensional cubes (a cube), and 4-dimensional cubes (a tesseract)… and how to draw them:

Now let’s look at a 2-dimensional cube (a square). We can see it’s made up of 4 points (0-dimensional cubes), 4 line segments (1-dimensional cubes), and 1 square (2-dimensional cube).

This snip from Wikipedia shows that fact, along with how many smaller dimensional cubes make up a higher dimensional cube.

Finally, here are three questions that I think are fun to ponder:

First, if you look at this chart, there is a really striking pattern…

If you take any number in the chart, double it, and add the number to the left of the number, you’ll get the number in the row below.

The question I ask my kids to answer: why?

Second, without thinking in this recursive way, can you come up with an explicit formula for how many k-dimensional cubes are in a d-dimensional cube? (So if k=1 and d=4, you should get 32.)

(I have only asked my kids this question once, but we had tons of scaffolding. I remember giving it to them the year I devised and solved the problem, and I wanted my kids to have the same gargantuan a-hah moment I had… A couple groups got it, but I realized looking back that the class time we needed to spend on it wasn’t worth the payoff.)

Third, if you add up all the numbers in a row of the chart, you see powers of three. WHAT?!?! ZOMG! Why!?!

Semicircle Puzzle

Matt Enlow posted an interesting geometry puzzle on twitter (tweet here), and I think the thing that got me intrigued was his initial challenge: “I can’t tell how hard this problem I just made up is.” Not knowing if there is an elegant/easy/obvious solution or not got me hooked.

I’m going to try to outline my approach/solution, because I sometimes like deconstructing my thinking to see how I actually think/learn… so from this point on… SPOILERS.

Some things that stood out to me… First, it looked like there was initially a single circle in a square, and the circle got cut in half and then it started sliding. So I initially drew the full circle in the square (before sliding), I drew the diagram shown, and then I drew the two semicircles in a rectangle after they fully “slid”… I saw the cut circle “in motion” — but after a short while I didn’t see how that would help me.

Then I drew the image and solved the problem and felt proud about it. But then I realized I drew the picture wrong. I circled the wrong part in my diagram, so you can see. I had the “slice” hit the corner of the rectangles, and then I was able to use similar triangles to come up with a solution.

I was proud but for some reason, probably because Matt’s initial tweet suggested to me that it would be harder than this, something was nagging me about it. So I went back and quickly saw my error. But I have always found that taking a wrong approach can help eliminate pathways to a solution, but might also help me see possible tools to use in a solution. And in fact, this idea of using that “cut line” and similar triangles was important in my pathway to the end.

So when I went back to the drawing board, I wanted to really see how this diagram worked… Some things were fixed (the 12 by 19 rectangle, the fact that the semicircles sort of “slid,” and importantly, the fact that the semi-circles were tangent to the rectangle at two places). So I decided to build this diagram in geogebra (with only one of the semi-circles), and as I built it, I saw that everything hinged on the movable point “G.”

I made the line where the semi-circles touched movable, based on the location of point G. Play around with moving point G here on this web-based geogebra page, and try to get it so the semi-circle on the bottom is tangent to the right and bottom side of the rectangle!

So to me, everything hinged on location of point G, or in other words, the distance from A to G (which is the same as the distance from H to C). We are looking for the location of point G which makes the semi-circle perfectly tangent to right and bottom sides of the rectangle. So to me, those appeared to me as “keys” to the problem. [1]

Sooooo I drew my diagram, and importantly labeled the distance from point A to point G with a variable, a. And then I labeled lots of things in my diagram in terms of that variable and the radius of the semi-circle, r.

I had two variables, so I needed two independent equations. And here is something nice… because I initially went down a wrong path earlier with my mis-drawing, I had already gotten similar triangles in my head! So I got one equation from that.

I hunted and hunted, and found another equation I could get… using the Pythagorean Theorem!

So now I had two equations and two variables.

… and since I knew this was going to be a beast to solve, I just used Desmos, and got that the solution is a=1.5 and r=7.5.

I did a little of the algebraic gymnastics to try to work this out by hand, but it was pretty uninteresting to me and I was pretty convinced that if I really wanted to, I could. To me, getting the equations was the interesting part, and the rest felt like pencil-pushing. So I stopped there. It was nice that the geogebra applet I created seemed to confirm my answer for me:

So that was my process to solving this mathematical puzzle. Who knows – I could also be totally wrong! I’m left thinking of the following:

(1) Is there a more elegant way to come up with the answer? Because the answer is so nice (a diameter of 15?!?!) but it comes out of such an ugly set of equations, I bet there is a nicer way. In other words, is there a better “conceptual” approach that gives a stronger insight into the geometric nature of the setup?
(2) How did Matt come up with this puzzle? How did he come up with the 12 and 19, so that the answer worked out so neatly to a diameter of 15 (radius of 7.5)? Based on my playing around with this puzzle, I wouldn’t have expected a nice answer — so that shocked me. I would have anticipated nice side lengths and an ugly diameter, or ugly side lengths and a nice diameter.

Finally: If you like puzzles like this, you might want to google “Sangaku” and look at the twitter feed of Catriona Agg.

[1] At this point, I had a small detour where I briefly tried to work this problem on a coordinate plane, where I was finding the intersection of the two lines to find the location of the center of the circle, point I, based on the coordinates of G… but when I realized that once I had the intersection point, I’d still have to find find the right coordinates for G to make the circle tangent to the edges, I realized that would be annoying. So I abandoned the coordinate plane work, though I could always return to it if I needed.

Two Problems that Got Me To Think

Here are two problems that have gotten me to think a lot.

The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem: We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability that your entire group was the exact same if you were in a 4-person group?

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I thoroughly enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

Second Problem:

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to fully exploit this structure in my thinking.)

I hope you enjoy thinking about these!

Archiving some gems from Twitter (April 2019)

I have seen a lot of great stuff on twitter lately, and I’ve missed a lot too, I’m sure. I wanted to just archive some of the things that I’ve saved so they don’t disappear! I also think it might be a benefit for someone who reads this who isn’t on twitter or missed some of these tweets. But that’s just a side benefit. I’m writing this for me!!!

***

Desmos writes interesting job descriptions when they have openings. When someone pointed that out to them, they mentioned that this article on reducing unconscious bias helped informed how they write their job descriptions. It’s pretty great and I highly recommend it if you’re hiring. I have thought a lot about “fit” in the past few years when doing hiring, but it’s tricky to think about it well. I have come to recognize that someone entering our department needs to be open and willing to collaborate and compromise, but also have sympathetic pedagogical beliefs with what our department values (and can’t compromise on those). One way I have tried to avoid it is thinking about these things:

cultural fit.png

But also I have found it harder to balance these thoughts, which I admittedly have a lot:

cultural fit 2.png

Not quite those things, but similar thoughts that get at my own personal views on the what persona/personality traits make an effective teacher. Which I tend to think mirror my own traits. But that’s only because I have these traits because I think they make an effective teacher. But I have worked with enough amazing teachers to know that amazing teachers come in all personas! Just like amazing students don’t all have to have the same personas. But this type of bias is something I am trying to be super cognizant about when on hiring committees.

***

I saved this just because I like the question and wanted to work on it. And I can see all kinds of extensions. A formula for n circles? What about spheres? I’m guessing (without working on this problem yet) that this is a classic “low entry point, high ceiling” type problem.

***

I just really liked this quotation, and I need to think about the ways that students can see themselves in the mathematics they do. It is part of a larger thing I want to do which is “humanize math” — but I’m not very good at making it a core part of what I do in the classroom. Small bits here and there humanize and expand what kids think about math, but I’m not there yet. I want to one year leave the classroom and know that kids have looked in the mirror and saw something. (It kind of reminds me in a super literal way of how Elissa Miller put a mirror in her classroom, and I think on the bottom she wrote “mathematician.”)

***

Okay, I love this so much. If you’ve never seen it before, it a great trick. You have someone pick any number between 1 and 63 secretly. They just point to the cards that number is on. In about three seconds, I can tell you your number.

I actually made a set of these cards where the numbers are more jumbled up, so kids don’t see a pattern to it. I do put the powers of 2 in one of the four corners though to make things easier for me. Oh wait, have I said too much?

If you don’t know this trick, or how or why it works, I’m sure you can google it. But I’m going to recommend the awesome book “Math Girls Talk About Integers” (there are a lot of great “Math Girls” books out there, so make sure you get the Integer one.

Not only is the book awesome (and great for kids to read), but it breaks down this trick so well. *Shivers with joy*

***

I was excited with Karen Uhlenbeck won this year’s Abel Prize, the first woman to win it ever! I had my kids read this article in the NYTimes about it, and write down three notes about the article. We started the next class with a “popcorn sharing” of what people wrote down. (I also said that although I liked the article, it was a bit dense and thought it could have been written more lucidly.) One thing that came up in both classes I did this in was what a “minimal surface” was — so I told kids it is a surface with minimal area.

I then showed my kids this short youtube video:

And explained that bubbles, though not “central” to all higher level mathematics, do come up. And then I gave them a question. I’m too lazy to type it out, but watch the first 1 minute and 45 seconds of this video (https://www.youtube.com/watch?v=dAyDi1aa40E) and you’ll see it. Then we talked about some basic solutions. And THEN I revealed the best answer was the answer shown in the video we all watched together.

Of course @toddf9 (Todd Feitelson) used this as inspiration to create his own bubble thingies:

but he also explained how he made them…

and then he EVEN created an awesome desmos activity on this very problem, which I want to archive here for use later: https://teacher.desmos.com/activitybuilder/custom/5cb50bed4dcd045435210d29

(Oh! And Mike Lawler (@mikeandallie) made a mobius strip bubble!)

***

Dylan Kane wrote a nice blogpost about calling on students (and the “popsicle sticks of destiny” — though he doesn’t call them that). My favorite line is this simple question that isn’t about right or wrong:

After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”

It’s so obvious, but even after so many years of teaching, I forget to ask things like this. Or my curriculum isn’t group problem solving based enough for things like this to make sense asking. Or whatever.

***

There’s nothing special about this one… I’ve read it a few places before and it always makes me laugh.

Questions are good. I might have a kid read this at the start of the year and then have a short conversation about why we’re reading it.

It will get at the problematic idea of “obvious,” and when and how learning happens and more importantly when and how learning doesn’t happen.

***

In case you didn’t know, Desmos has a list of all their mathematicians they use when they anonymize in Activity Builder.

https://docs.google.com/document/d/1OY-8dk6vYW1Cags8E6_v3I8YZ-RYROzgsCauW5CZt9w/edit

***

I can imagine putting this picture on a geometry test as a bonus question and asking them why it makes math teachers all angsty… Plus it made me chuckle!

***

I’m so not here yet. Anyone who knows me as a teacher will probably know I’ll probably never get here. I’m such a stickler for making the use of every second of classtime.

***

Crystal Lancour (@lancour28) tweeted out a slide from a session led by Robert Berry (NCTM president) which had this very powerful slide:

Four rights of the learner in the mathematics classroom

The right to be confused and to share their confusions with each other and the teacher
The right to claim a mistake
The right to speak, listen, and be heard
The right to write, do, and represent only what makes sense to you

***

Love the idea of using marbles/paint to draw parabolas (click here to go to the original tweet and watch the video — it’s not a static picture).

***

Bree Pickford-Murray (@btwnthenumbers) gave a talk at NCTM about a team-taught math and humanities course called “Math and Democracy.” Not only did she share her slides (like *right after* the talk) but also she links to her entire curriculum in a google folder. SUPERSTAR!!!

I’ve gone to a few talks about math and gerrymandering (both at MoMATH and NYU) and listened to a number of supreme court oral arguments on these cases. It’s fascinating!

***

I just finished teaching “shape of a graph” in calculus. But I wish I had developed some activities like this, to make it interactive:

***

I’ve literally been preparing to give a talk next month for… months now. And this one stupid tweet summarized the talk. Thanks.

***

I have so many more things I can post, but I’m now tired. So this will be the end.

Double Angle Formulae

I posted this on my Adv. Precalculus google classroom site. I don’t know if I’ll get any responses, but I loved the problem, so I thought I’d share it here.

***

I mentioned in class that I had stumbled across a beautiful different proof for the double angle formulae for sine and cosine, and I would post it to the classroom. But instead of *giving* you the proof, I thought I’d share it as an (optional) challenge. Can you use this diagram to derive the formulae? You are going to have to remember a tiiiiny bit of geometry! I already included one bit (the 2*theta) using the “inscribed angle theorem.”

If you do solve it, please share it with me! If you attempt it but get stuck, feel free to show me and I can nudge you along!

nice

***

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!

(more…)

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:

triangle 2.png

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.

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!
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…
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:
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!
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 done. At this point I put the problem away. Nothing was working.
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.
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!
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.

Continuous Everywhere but Differentiable Nowhere

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