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How to maximize an area without calculus - using high school math to solve new problems #SoME4

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Tags: algebrageometryoptimizationproofstrigonometrytriangleprecalculus

About a year ago, my friend had a question about triangles that came up at work: it boils down to maximizing the area of a right triangle given a hypotenuse length. Initially I found a pretty basic solution with calculus, but I got interested in solving the problem with more basic methods, which led to a bit of creativity having to make up for the lack of the stronger tools. In this video, I'm presenting proofs that I discovered using tools I would understand to align with high school classes in algebra, basic geometry, and trigonometry. It felt appropriate for me to make that my entry for my first Summer of Math Exposition, because it reminds me a lot of something that Grant says about important facts in math: because important math facts are important because they're related to so many other things, if you take some random thing and dig deep enough you will inevitably find something important. In this case, we end up touching some really fundamental ideas like the Pythagorean theorem, FOIL, Thales' theorem, the isosceles triangle theorem, the triangle angle sum theorem, SOHCAHTOA, and the double-angle identity for sine. Something I'm going for in this video is for the content of these proofs to be a vehicle for starting students down a path of understanding math in a more holistic way, interfacing with it more as a language of truth than a memorized methodology, and seeing where the creativity and problem-solving of math comes into play, as well as how seemingly disparate topics end up getting connected. My hope is that because that overarching idea is more abstract than there being a specific piece of math which is elaborated on throughout the video, it might be able to be adapted into a number of different settings - the proofs are arranged from least knowledge required to most knowledge required partially so that a teacher in a geometry (read: pre-trigonometry) class might be more able to present just the first two proofs, if they feel that would be appropriate. I also think this video could be well-suited to provide a class activity, with the students stopping and trying to solve the problem when the video indicates: I think this problem is actually in a really good spot in terms of difficulty to be attempted by classrooms with different degrees of collaboration across the spectrum of skill going from algebra up to precalculus. I didn't get to say so in the video (ran myself out of production time to add this to the end) but I did just want to say thanks to everyone running the event, Grant in particular, for the opportunity and the inspiration to give this a shot. I think I'm in an interesting spot where the video is more or less what I wanted to make, but I'm aware that there are some spots where it's rough around the edges, and it'll be interesting to see how people react to it, whether people like the core thrust of the idea enough to make up for its shortcomings, and where some areas are that I might be able to improve as a presenter for any future projects like this, or just in my work as a tutor. I'm excited to be getting some real feedback on this after its time in production. Thanks for reading, and I hope you have a nice rest of your day!

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7 Overall score*

24 Rank

11 Votes

10 Comments

Comments

A clear and understandable presentation of three elementary proofs. However, if I were a student, I probably will become tired during the second second proof. To keep the students focused, there probably should be a bit more entertainment. Maybe, the meta perspective at the end of the presentation should have been placed in between the proofs to keep the students focused and curious about the next proof.

7.5

I really liked lots of different parts of this video: the real-world story and application, the way you talk about a “decent guess” vs knowledge, your patience in getting people to try the problem first, your historical notes about dodgy mathematical attributions, and comments about problem-solving for new learners.

This style is different to what we usually see: instead of trying to teach new maths concepts, there’s a lot of benefit to revising maths ideas but connecting them to new and interesting problems. I’m very inspired!

8.3

I really really enjoyed this video! It is definitely accessible for high school students and I love that it demonstrates how you might go about solving a problem which is not a “standardized” which you would normally learn in a classroom.

The geometric example is very well known, but the algebra and trigonometry ones are quite good at explaining.

7.6

This is a good exercise for optimisation, nice animation. The idea of proving it in several ways is useful. Thank you. Maybe this video is a bit long for students, I understand you wanted to be complete.

Should have started directly with the ice example, not the abstract question.

You spent quite a lot of time explaining Pythagoras. You should have given it more space on screen, probably even removing the ice box completely.

Lastly, the “algebraic solution” is very much geometric.

I really like the animations and I can see clear inspirations from larger channels. maybe you could explain the calculus and the multivariable calculus solutions at the end as a bonus couple? :)

The author did a nice job of showing three different approaches to solving the problem.

This video is not bad. I like the animation. The problem is fairly basic, but it is a nice example of a real world problem that can be solved many ways. I think the biggest problem with the video is that the video keeps getting sidetracked onto tangent problems like proving the Pythagorean Theorem or Thales Theorem. You should focus on the main problem. You have to assume that student know these things.

Solid clear explanation of the problem and it’s solutions. What would happen if you slid the partition to the opposite corner? (Note: Audio level really low, needs another 20 db.)