Help my pick my capstone project

For my masters I need to do a capstone project this term. I pick a subject and then develop a series of lectures that are intended to be an ‘enrichment topic’, and likely targetted at upper year HS students.

Here’s the outline:
Students create a mini-course on a single mathematical enrichment topic. The course created
will be roughly three one-hour lessons that include both teaching and mathematical problem
• An “enrichment topic” means a topic that is outside of the union of the curriculum documents
(including courses that are not mathematics) for a generic jurisdiction. For example, teaching
calculus to Grade 10 students would not count as an enrichment topic.
• The mini-course can be aimed at any level from Grade 9 to Grade 12 (or college for college
• The mini-course can be approached from a number of different angles, including an historical
angle, a contest preparation angle, etc.

And here’s some possible ideas:

  1. Applications of … (eg. Logarithmic Functions,
    Complex Numbers, etc.)
  2. Approximating Functions
  3. Arithmetic, Geometric and Harmonic Means
  4. Binary Numbers
  5. Cauchy-Schwarz Inequality
  6. Ceva’s Theorem
  7. Circle Geometry
  8. Chinese Remainder Theorem
  9. Compass and Straightedge Constructions
  10. Congruence Classes
  11. Conic Sections
  12. Continued Fractions
  13. Cryptography
  14. Cubic Equations and Polynomials of Higher
  15. Diophantine Equations
  16. Euclidean Algorithm
  17. Euler Phi Function
  18. Euler’s Formula
  19. Factoring Polynomials
  20. Fermat’s Last Theorem
  21. Fermat’s Little Theorem
  22. Game Theory
  23. Generating Functions
  24. Geometric Inequalities
  25. Graph Theory
  26. Graphing Polar Curves
  27. Group Theory
  28. History of … (eg. Trigonometry,
    Indian Mathematics, Fibonacci)
  29. Infinite Series
  30. Invariants
  31. Inversion
  32. Limits of Sequences
  33. Linear Programming
  34. Magic Squares
  35. Markov Chains
  36. Mathematical Logic
  37. Mathematics of Gambling
  38. Matrix Theory
  39. Mersenne and Fermat Numbers
  40. Nine Point Circle
  41. Open Problems (eg. Goldbach’s Conjecture,
    Do any odd perfect numbers exist?)
  42. Partitions
  43. Pell’s Equation
  44. Pencils of Planes
  45. Pick’s Theorem
  46. Platonic Solids and Polyhedra
  47. Polar Coordinates
  48. Primality Testing
  49. Properties of the Pedal Triangle
  50. Ptolemy’s Inequality
  51. Pythagorean Triples
  52. Quadratic Residues
  53. Random Walks
  54. Regular Expressions
  55. Set Theory
  56. Spherical Trigonometry
  57. Tessellations

So I have it narrowed down to two ideas.

  1. Golden Ratio
    Pros: I think it’s a cool enrichment topic and I’m personally interested in it.
    Cons: I don’t know squat about the golden ratio. So in addition to developing the course, I have to learn and understand something about it. Like right now I couldn’t even do an outline of what I’d be teaching. And I’m concerned that the math behind it may be too much for a HS enrichment course. So basically, unsure if this is too much for me to handle over the summer.
  2. Mathematics of loans
    Pros: Easy for me to do, zero learning, The opposite of 1), I’d likely be done this early since I’d be just drafting an outline, screwing around in latex and spending some time building out some interesting exercises like ‘calculate PV/FV’ and ‘build an amortization schedule’.
    Cons: Less certain this is a viable ‘enrichment’ topic.

Looking for your opinion on which way I should go. And if you choose 1) would welcome guidance on what area of the golden triangle I should focus on (I did some reading on this over the weekend, and holy crap that ratio is everywhere).


I suppose I could do #28 on the list, graphing polar curves. I plan on getting a tattoo once I’m done the degree and was thinking of getting a unit circle, which would tie in nicely. But, a golden ratio tattoo opens up some interesting possibilities as well.

Even before I had finished reading, Interest Theory had come to mind.

It’s math. It’s algebraic. It’s interesting (I think) It is outside the scope of anything in a HS curriculum. And since virtually every student borrows money from a bank at some point, it is a GD useful tool that could actually be life altering for some students.

I’ll just throw out there that I had a lot of fun introducing a class of high school students to the basics of actuarial math by having them calculate auto insurance rates given some very simple data, etc.

Auto insurance costs are something high school juniors and seniors are interested in for some reason…

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If you’re doing polar graphing you should get a tattoo of a cardioid curve. On your chest.

I’ll second that interest theory could be both a good life skill and a viable enrichment topic.

Many adults struggle with the math behind mortgages, car loans, the power of compound interest,…I’m sure with your knowledge you can come up with some interesting math problems using interest.

Miss me with that P&C stuff because I know less about auto insurance actuarial-donking than I do about the golden ration :). It’s actually an interesting idea, but once I’m done mocking the whole P&C industry I have no idea where I’d even start.
Auto insurance pricing, how do you even.

So folks are leaning towards interest theory. My spouse is about the same, wants me to have free time this summer.

OK, presuming that’s the case, where/how do I make this fun? Start with the basics of compound interest/PV/FV. Then:

  • some exercises on how shocking to see how much interest you pay over the course of a mortgage. Maybe look at total cost of financing between two different cars.
  • revive that old exercise about how one person starts saving at age 20 and stops at 27, the second person starts at 27 and saves til 65, and they both have the same amount saved at 65.
  • something something credit card debt. geez, both my kids had to get a proper ass-kicking over this.
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IMO, this is the most useless piece of information I have ever received in my life.

Because Present Value.


That’s a good one, imo.

Also, I would include a very brief overview of utility theory—WhoTH cares if you’re a millionaire when you retire if your roof is leaking now and the furnace doesn’t heat the house and you’ve had ramen noodles for your last twelve meals.


Fair enough. But I’m not going to start teaching HS kids about the benefits of leveraging lol.
Maybe I could do some work around increasing caution on variable rate mortgages. People are taking an absolute kicking on this right now. Rates jump up, people are starting to have to sell their houses because they can’t afford the payments and IIRC, in Canada they push out the amortization period to a max of 40 yaers and then start increasing payments. And when that becomes unaffordable, time to sell the house.

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Man, I love utility theory even if I’m not overly expert on it.

Good stuff. I’ll have to think on how I can introduce that.

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Besides some of the stuff you mentioned, other possible topics:

Perpetuities are a geometric series

Annuities are a finite series.

Car lease with optional balloon payment at end vs loan.

Keep it simple and do the Interest Theory one. The Golden Ratio gives me some hippie vibes also.

Don’t let these actuarials talk you up on the complexity - people are much, much dumber than smart people think they are when they start trying to teach something.

Edit: This post was not directly aimed at ArthurItas lol.

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Also, binary arithmetic could be a good candidate, just to use the following joke:

There are 10 kinds of people. Those that understand binary numbers and those that don’t.

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Read this, take some examples or make your own.

It takes the idea of approaching your whole financial life as the present value of future expenses and revenue.
I can’t find my copy, so I probably lent it out.
Highly valuable (IMO) to upper level high school students.
Could easily turn into a discussion on Utility Theory, as some students will choose to purchase (or buy on loan) something while others don’t.

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Ooh, yes, especially if you teach them the number four using fingers…or, using two hands, 68.

Bond questions are also good ones for interest theory

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Geez, I’m looking at taht book on amazon. On ‘.com’, it’s like 10 bucks, with a bunch available used for like $3. But, shipping from the US. Head over to, and it’s fREAKIN $40 and shipped from australia. What the heck.

Graphing polar curves was #26 on the original list.

Shall I add you to the innumeracy thread?

I find the “wow, that’s a lot of interest” discussion innumerate. You pay a mortgage on a house because you cannot afford it all at once. One should consider interest as the “rent of money” and the payment of principal as an investment in a house (with an unknown rate of return). then discuss financial leverage.

Same goes for a car, so, “lease or own” decision exercise. It always makes me laugh a bit when someone says “never lease a car” without all of the inputs required to make such a decision. I mean, if someone offered to lease you a $40,000 car for $10/month, why wouldn’t you (after checking all of the conditions, of course)?

Can also see what the Present Value is when someone chooses to save by going farther to the cheap gas station every time for a lifetime. (Spoiler: I don’t think it’s a lot, relative to other items.)
Or, the current (heh-heh) discussion of EV versus ICE, or whether to buy Solar Panels or continue to pay for electricity (what inputs are required? is the big thing here so practical application of math).