Multidisciplinary problem

I am looking for ideas for an activity for high school students, which involves plane geometry and another field, such as algebra, series, etc...

For example, in junior high there is a nice activity in which students are investigating the idea of pi, by taking rounded shapes, such as plates, measuring the perimeter and radius of each one of them, then setting an XY Cartesian axes, putting points from the measures and calculating the slope, which is pi (if we take 2r on the x axis).

I am looking for similar activity, has to be geometry, but slightly harder, for high school students. It has to involve algebra or any other field (perhaps apart from Trigonometry because that makes the geometry too easy).

Any ideas will be most appreciated !

• Something like this? – Paracosmiste Mar 26 '18 at 16:07
• Are matrices too advanced for your students? – Peter Taylor Mar 26 '18 at 16:30
• Matrices are fine ! – Jankel Mar 26 '18 at 17:32

Pick's Theorem is a great link between area of polygons and the algebraic skill of writing expressions. Nrich maths has a ready to go investigation of Pick's theorem: https://nrich.maths.org/pickstheorem.

Coordinate geometry and transformations are another area with a huge amount of overlap between geometry and algebra.

This was a problem set in a Spanish newspaper in the summer of 2011. I've stripped away some fluff about t-shirt designs: you can add your own "real world" motivation if desired.

Take two straight lines meeting at a point. Let's call that point the origin $O$, so the lines are $O + sA$ and $O + tB$. Now draw a zigzag starting at $O$. The first segment runs from $O$ to $O + A$, the second segment runs from $O + A$ to $O + t_2 B$ for some $t_2$; the third segment runs from $O + t_2 B$ to $O + s_3 A$ for some $s_3$; etc. We add two more constraints: each segment has the same length, and the 20th segment runs perpendicular to $A$. What is the angle between the two lines which form the boundary?

There's an extremely elegant solution using only basic geometry. But an alternative approach goes via the matrix power $$\left[ \begin{matrix}-1 & 1 + \cos\frac{\pi}{2n} \\ -2 & 1 + 2\cos\frac{\pi}{2n}\end{matrix} \right] ^n$$ (where $n=10$ for the stated problem), so the basic geometry solution ends up giving a nice identity for the matrix power.

I don't have a specific project in mind but you might give some thought to Diophantine equations, i.e. equations whose solutions have to be integers. This connects coordinate geometry and number theory. (For example, number theory concepts are used to show when the equation (a/b)x + (c/d)y = 1 has integer solutions and when it doesn't.)

If you want something a little lighter, easier, consider the derivation of pi with experimental probability. Kind of fun to draw the grid and do the experiment. And it doesn't involve math, they don't know yet.

You can get more or less into the statistics of if if you need more math content: https://www.statslife.org.uk/the-statistics-dictionary/1976-statistics-bootcamp-estimating-pi-with-r-and-buffon-s-needle

Or you can just make it an experimental physical activity with a 9th grade algebra equation as the math content. https://www.sciencefriday.com/articles/estimate-pi-by-dropping-sticks/

I wouldn't use the computer simulator as I feel like, we are physical monkeys and get more from dropping sticks than looking at a screen. But it is an option if you want.

Wikipedia has a decent article on the concept (Buffon's Needle), but they go off into a bunch of minutia about using needles that are shorter or longer than the grid spacing. I think if you use an equal spacing, the thing is more intuitive and the pi value pops out clearly.

I have some calculus students who are working on a mini "research project" with me.

The idea is the following: when we define a tangent line, we take two points $(a,f(a))$ and $(a+h,f(a+h))$. We find the unique line passing through these points, then we take a limit as $h \to 0$.

We are investigating what happens if you take the three points $P= (a,f(a))$, $Q= (a+h,f(a+h))$, $R=(a-h,f(a-h))$, find the unique circle passing through all three points, and then take a limit as $h \to 0$. It took most of the semester but we have it!

Even the task of finding the equation of the circle probably the hardest algebra and geometry problem these students have ever done:

1. Juggling terms like $f(a+h)-f(a-h)$ and interpreting these things geometrically really stretches their understanding of graphs of functions.
2. Figuring out the geometry of the situation: I can find the center of the circle by finding the intersection of the perpendicular bisectors of the two segments $PQ$ and $PR$.
3. Translating that geometric statement into an algebra problem. Understanding how similar triangles show that the perpendicular bisectors have opposite reciprocal slope, finding the midpoint, and using point slope form.
4. Solving the resulting system of equations (which has a lot of variables!) for the center of the circle.
5. Finally using the center and one of the three points to get the equation of the circle.

This would be a fine stopping point for your students!

To find the limit of the resulting expressions as $h\to 0$ you need to do some careful work with Taylor expansions. For instance, you need to expand the terms $f(a+h) \approx f(a)+f'(a)h+\frac{1}{2} f''(a) h^2$. In the end everything works out, and you get a "tangent circle"! Of course, this is already well known as the "osculating circle", but this is a pretty novel and fun way to go about generating it.

Here are two desmos links, one to the 3 point approximation, and one to the actual osculating circle that you get from taking a limit.

https://www.desmos.com/calculator/eehupqcbtg

https://www.desmos.com/calculator/dhzsnvcifz

I have a couple of suggestions:

1) The natural extension to your junior high exercise is to use areas instead of circumference to estimate pi. This can be done the same way Archimedes is said to have experimented with it - by making circles with various diameters out of the same material and weighing them. It is a great exercise in proportion.

2) Euclid's Book II of the Elements is mostly geometric proofs that have direct algebraic equivalents, eg a(a+b); a(b+c+d); (a+b)^2; (a+b)(a-b); (a-b)^2 etc. It is both fun and challenging to explore algebraic expansions visually through geometry.