Two Boxes in a Circle
The March MathsJam had an interesting geometry puzzle.
In the course of the evening, we noodled on this for a while and initially made no progress. So I decided to model it in geogebra to confirm whether there was a unique solution. What I found visually was that there appeared to be one when the smaller box was tilted 45 degrees. And that was in fact not hard to confirm because the triangle DFA inscribes the diameter in that case, along with the triangle FGA. You can then solve with the Pythagorean theorem and verify that they give the same result in this configuration.
\(\text{the diameter: } AF^2 = AD^2 + DF^2 = 16 + (4 + 2\sqrt{2})^2 = 40 + 16 \sqrt{2}\) \(\text{and also the diameter: } AF^2 = FG^2 + AG^2 = 4 + (2 + 4 \sqrt{2})^2 = 40 + 16 \sqrt{2}\)
But this isn’t entirely satisfying because it basically presumes that answer and confirms it works and it doesn’t settle whether this is the only answer.
New Beginnings
I’m back
It’s been a long 3 years. In the course of Covid and life changes, I took a hiatus from my the old blog site and in the interim it broke down. But the itch to post math related posts has been growing along with the desire to migrate off of blogger and onto to a more open source platform.
Two Hinged Triangle Geometry Walk Through
Setup
It’s time for another geometry walkthrough motivated this week by this interesting problem from James Tanton.
“Two isosceles right triangles are hinged at corners as shown. Line segment connecting midpoints of their hypotenuses is used as the hypotenuse of yet another isosceles right triangle.
Prove A, B, C lie on a straight line. Can anything be said about where B sits on segment AC?”
Trig identity Three Ways (at least)
This post starts with reading elsewhere about someone struggling with the following trig identity given a triangle with three angles x, y, z show:
Mastodon: The Wild Wild Neolithic West
Its been a long while since my last social media post: how-i-use-twitter and everything is all of a sudden in huge flux. With all the turmoil on Twitter I’ve been exploring Mastodon on the math focused server a friend runs: mathstodon.xyz. However, a new platform means starting over again
- You have to rebuild your network of follows and followers. This is huge and discovering people has made my previous two attempts at using Mastodon unsatisfying. But this time is different due to the chaos at Twitter. Large enough groups of people I know have migrated that I could start with a core group of folks and participate in enough conversations to have fun while finding new people.
Why things are vital this time - The extraordinary growth of the network
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Peculiarities. Mastodon isn’t twitter and has a few quirks that you have to get used. No quote tweets due to a fear of harassment (which seems overly paranoid to me - since you’re just a screenshot away from the same effect) and a general stance that makes discoverability harder. You have to use hashtags since full text search doesn’t exist across the fediverse. And sometimes the other instance’s data is an extra hop away from your server . For instance, you’re surfing a profile on another server and sometime have to click to it to get full info. Crucially, since each instance only stores the posts that users on it follows - there is a **deep effect on search **even for hashtags. You can only see what you know about or someone else on your instance knows about. As a consequence, the larger the server grows the more useful it becomes if you’re interested in finding things . There’s also an overly precious stance on content warnings that doesn’t fit my theory of action. But I can live with that.
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Usability Once you have enough people in your network Mastodon is quite usable despite the large discoverability issues. Can it be your only microblogging platform? That remains an open question for me. The overall network is much smaller than twitter. I don’t need most of the twitterverse though just the parts I read. And on that front - the missing piece is probably government and media accounts. For now there are bridge sites like birdsite.wilde.cloud that will publish tweets to toots. But they aren’t quite realtime or completely reliable. But the network is growing very rapidly (See above) so the situation could very easily shift in the upcoming months.
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Trust. This is a huge general issue. I went with mathstodon because I knew the admins and could implicitly assume they would operate in good faith. But how are millions of people going to make that leap? I feel like there needs to be some type of vetting process or change in structure for many of the instances to answer this question. Perhaps there will be multitudes of small sites where everyone personally knows someone involved - perhaps companies will enter the space and you’ll pay for some additional guarantees of stability/security? This also remains to be seen and I expect new developments as growth continues. But that’s the key as well - the network is growing despite this issue so its not an adoption blocker yet.
Unexpected Binomial Theorem Connection
I saw this post and immediately thought: “What an odd connection to Pascal’s Triangle. What’s going on?”
Due to time constraints I didn’t get a chance to look into this for several weeks and while its not quite as geometric as I hoped the connection is fairly natural.
First just as in the triangle you need an initial row of values from which the relation will then generate all the rest of the rows. In this case its the triangle with a single median. Stewart’s Theorem is helpful here for finding the initial equation. What’s novel is while I’ve done this before I’ve never rearranged in this fashion.
$ a^2 \cdot m + c^2 \cdot m = 2m \cdot (m^2 + b^2) $
Properly fitting Problems
What motivated this essay was two posts I recently saw. The first started with the idea that no student should take more than five minutes on a problem. If stuck it was unlikely to be productive and they should just move on. In case, there was any confusion this expanded to the idea that investigations or inquiry based learning was ineffective. Secondly, was a note from an acquaintance about how he was about to start a class for pre-service teachers about the pedagogy of problems in the classroom which piqued my interest about which problems he would be using. Together these readings motivated me to think again about “Where do mathematical problems fit?”
To start though you have to step back and define the problem. The classic “word problems” of elementary school are generally a misnomer. Students typically (but there are exceptions) know all the operational procedures to solve them and are being asked to practice translating from natural language into mathematical statements. Instead, I’m fond of the dichotomy between an exercise versus a problem. An exercise is a mathematical task that one knows how to do or has seen demonstrated and is aimed at practicing a skill or procedure in the service of mastery. On the other hand, a problem is a mathematical task where the solution is unknown at the start and one must reason how to solve it along the way. Importantly, this distinction is relative to the practitioner. What may be an exercise for me, can be a problem for someone else or vice versa depending on what prior knowledge we each have. That also means problems exist at every grade level but because they reflect where the solver is, have almost infinitely different and evolving forms. For me, problems are also central to learning mathematics. Skills like basic arithmetic are important in of themselves but also are only stepping stones and tools towards the service of mathematical exploration and understanding. This creates some difficult tensions though. Especially when beginning what is the proper balance between mastering skills and thinking about unsolved questions? Channeling the first author, the answer would be initially its mostly useless to try problems, the learner doesn’t have enough prior knowledge and domain experience. Problems just waste time and more seriously are ineffective for knowledge acquisition. In other words, even when one solves one, the process is not well synthesized. There’s usually some discussion of how experts approach problems differently but not much discussion of how or when one reaches that state (other than usually the impression its postsecondary at least)
On the other hand, there are plenty of Inquiry Based Learning advocates that are all in with problems on a daily basis. Its easy to find a class vignette with everyone standing around whiteboards working on some communal task that will take a class period. All of which seems delightful but then the sneaking suspicions enter about the amount of mathematical ground being covered or if enough practice is occurring for growth to occur. After seeing yet another task involving rearranging the digits 1-10 to fit into some equations I often end up thinking how its very easy to idle in the domain of pre-algebra and never really get anywhere.
Instead, I inhabit a middle ground. Time is limited but its important to constantly encounter problems through school in the service of developing patterns of thinking. At the same time, there needs to be a large amount of exercises along the way or you flounder without ever mastering concepts confidently enough to utilize them later. That’s a very tricky space given the constraints. So I think you always have to be strategic in what you choose not only in how to to balance the hours between longer problems and practice exercises but what forms each take. To keep to the flow, the problems should riff on the concepts being worked on in an exercise or require one to practice a skill in the service of the solution so you are getting double duty out of it. They should also advance the arc of the subject being explored. That’s constraining in the sense there are plenty of interesting non-curricular tasks one could do but only a subset align well with the overall goals for a class. Like many artforms though within those limits there still is tremendous opportunity for creativity and interesting experiences.
Example:
I’m going to draw from my own experience doing math circle activities where I’m very purposely fairly random about activities because we exist outside the curriculum. But I often run into things that would work really well in a classroom setting paired with a learning goal. Recently we tried out determining the number of rectangles that could be fit in one larger one.
There are 60 sub rectangles in the above 3x5
The process for exploring this was fairly messy and time consuming and involved a lot of counting strategies and looking for patterns. But the ultimate solution has deep connections to combinatorics and would work really well embedded in a unit on combinations, pascal’s triangle (or both). If doing so you’d want to combine it with simpler exercises on calculating combinations and using their properties along with other simpler problems that built towards being able to tackle this one.
Creating, integrating and curating these tasks is the crux of the instructor “problem”. Its easy to go a little astray and end up with a problem that falls flat and no one can solve or is too easy or fails to make the crucial connection or is boring. The chance for failure is fairly large. But practice helps and keeping one’s eye on this ultimate goal and reflecting on how each attempt at problem integration into a sessions plays out aids in more effectively using them. And to return to my central philosophy, problems are too essential to Mathematics to not keep tilting at figuring out how to properly use them.
Carnival of Mathematics 203
Graphic scores by John De Cesare (1890–1972).
Welcome to the 203rd Carnival. For all the other carnivals future and past, visit The Aperiodical where you can also submit future posts. This is my fifth go around hosting a carnival and the first time I’ve done so during March. So hold onto your hats and prepare to be inundated with Pi Day material. For those jaded souls among you, there will be plenty of interesting math and no digit reciting contests. (If you’re in the other camp I found this Pi Day Digit Song quite amusing this year)
Before we really get going via wikipedia.org here are few facts about the number 203. (We’re getting higher up the ordinals and sadly the list of references is getting shorter)
203 is the seventh Bell number, giving the number of partitions of a set of size 6, 203 different triangles can be made from three rods with integer lengths of at most 12, and 203 integer squares (not necessarily of unit size) can be found in a staircase-shaped polyomino formed by stacks of unit squares of heights ranging from 1 to 12.
15-75-90 on the quarter circle
I just saw a nice visual proof of the ratios for the 15-75-90 on the internet via @ilarrosac. This one works via symmetry and the Pythagorean theorem.
Two Circumcircles Walkthrough
I really enjoyed working through this problem from Stanley Rabinowitz I saw recently on Twitter. This was one of those problems where I circled around it and after finding the first solution realized I could improve it a lot and simplify the algebra fairly drastically.
One first impression, I saw two things. First there was power of the point at D which could be used to give expressions for DQ and AD. On the other hand AP looked like it would be hardest segment to get a purchase on.
My second thought was around the midpoints and filling in the similar triangle DEF looking important as well finding all the congruent and parallel segments it created.
I started also filling in the cycle quadrilaterals as well at this point but didn’t immediately see to do with all of this.
Next I fired up geogebra and modelled a few thing. One I was curious about the circumcenters and if they could help. That didn’t immediately feel promising. The alignment in particular didn’t give me anything. However, I did notice the centroids completely aligned as expected but I hadn’t though about that yet. (Others online went from there to homothety)
At this point I went back to a combination of a few of my initial experiments with the cyclic quadrilaterals and similar triangles and started just working out the ratios.
You can see the basic algebra I worked out with the similar triangles and power of the point around the 2 intersections solved the proof. But it wasn’t entirely satisfying. Yes the calculation for the two parts were equal but why? It didn’t really feel like it was elucidating the structure. At this point I could see that the 2x scaling for a and b above was key but I wondering what would happen if I started from that point. What I found below was I could drop all the ratios and just rely on the cyclic quads.
But I kept thinking about the problem some more and had my final conceptual breakthrough. Above I ended up proving GH = 2EO at which point I could just do segment addition and subtraction but what did that imply? Everything I kept looking at was 2x scaled. So what about the whole cyclic quads and their diagonals? As soon as I adjusted my mental model the final version fell out which is almost completely algebra free and focuses directly on scaling throughout.