# 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
solving.
• 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
teachers).
• 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:

Summary
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
Degree
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
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).

Summary

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.

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…

1 Like

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.
1 Like

IMO, this is the most useless piece of information I have ever received in my life.

Because Present Value.

2 Likes

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.

2 Likes

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.

1 Like

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.

1 Like

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.

1 Like

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.

1 Like

https://www.amazon.com/Books-Peter-Neuwirth/s?rh=n%3A283155%2Cp_27%3APeter+Neuwirth

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.

1 Like

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

1 Like

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 amazon.ca, and it’s fREAKIN \$40 and shipped from australia. What the heck.

Graphing polar curves was #26 on the original list.