# Ideas for high school pure maths projects

I am thinking of giving my high school students some pure maths projects to do. It is a lot easier to think of some interesting stats projects but not in pure maths. The students' maths background are weak, they might not really understand the relationship between integration and area finding. I am planning of giving a short project that could be done in 1-2 weeks time (for 1-2 pages long) that emphasises on conceptual understanding but if possible exciting and upon completing the projects students should feel more confident tackling more problems.

From my experience when I studied maths back then in high school, I preferred to understand the concepts first before trying to do any problems. But nowadays, I think it is a common phenomenon that students tend to jump straight into doing problems without bothering the purposes, motives, and relationships from one concept to the other. In other words, students are not interested in connecting the dots, they just want to do and pass the tests.

The topics that we are discussing now are: differential and integral calculus of trigonometric and logarithmic functions.

Here are some ideas that I have in mind:
1. Summarise different interpretations of derivatives (for example: geometric interpretation, algebraic interpretation, physical interpretation, etc)
2. Relate the idea of (definite) integration (continuous) and summation (discrete) and give some properties or equations which are very similar in both cases.
3. Summarise common trig identities and prove them. For example, $sin^2\theta+cos^2\theta=1$ can be proved using Pythagoras.

I would really appreciate if anyone could share some other ideas or examples which might be helpful in enhancing students' understanding of concepts, I have a very limited knowledge on the current research, maybe there are some exciting research topics which do not require tons of advanced maths. Many thanks!

• Can basic elementary number theory be handled by them? – Git Gud Mar 30 '14 at 11:55
• @Git Gud: Some really basics stuff yeah I think, like divisibility, prime factorisation, gcd, etc.. But we don't want to go to far to something like Fermat's Little Theorem, Chinese Remainder Theorem etc. But I would like to know some ideas in elementary number theory which might be useful in some other occasions. Thanks! :) – user71346 Mar 30 '14 at 12:03
• You've made your issues clear, but not in the form of a question. Putting it into a question, with a question mark, focuses the post and makes it easier for people to answer well. – user173 Mar 30 '14 at 12:47
• Would something like asking them to provide primality certificates of some given prime numbers be feasible (i.e. I'm not asking about this particular question, but its character)? – dtldarek Mar 30 '14 at 13:29
• ams.org/programs/students/high-school/emp-student-research – David Ebert Jun 4 '14 at 6:14

Joseph Malkevitch, based out of CUNY York College but also a visiting professor at Columbia University Teachers College, has a fair bit on his website about (high school) student research.

Depending on the student's mathematical level, one could point to Malkevitch's pages on Mathematics Research Projects or a couple other pages directed at undergraduate students (1) (2).

Other helpful links and pieces of writing can be found on his main site (click his linked name above).

• Very good, substantive answer. A good standard to set. :) – paul garrett Mar 30 '14 at 23:13

Just a list of some cool things:

• Recurrent relations
There are a lot of different ways to find solutions for recurrent relations. Logistic maps are also very neat way to illustrate what's going on.
• Bernoulli numbers This is what I gave a short presentation on. It may be a bit boring for highschoolers sinec there is not really a cool way to explain or illustrate what you're doing, but maybe for the students who are more interested in pure math.
• Linear algebra
• Fractals
• Complex numbers
Some very basic complex analysis can be suited for high-schoolers (of course, formal proofs for all the theorems that are commonly used is way too much, but maybe some arguments or some informal proofs would be feasible).
• Infinite series
There was an excellent pdf on this, which presented the difference between the different types on convergence for infinite series, together with some convergence tests, all in a very intuitive way very suited for highschoolers, but unfortunately I can't find it now.
• Cardano's formula for solving (some) cubic equations
this can be done geometrically like the quadratic formula, but probably requires guidance.
• Fourier theory
An illustrated approach could be really cool, together wth an explanation of some of the basic concepts of Fourier analysis.
• Calculus
some semi-rigorous proofs using limits and limits of summations can probably made by high school students and can be very interesting.
• Trigonometric identities
like cos and sin addition and multiplication formulas
• Platonian solids
There is a very nice proof that there are only 5 platonian solids, and there are some nice properties for 3-dimensional solids as well (F + V = E + 2), and some basic linear algebra for calculating areas, volumes and angles would be appropriate.
• Ramsey theory
This can be explained quite nicely by using the analogy of people in a group who are either friends or not. Also, the proofs have a nice graphical representation. Graph theory in general has some really cool topics for high school students, so there are definitely more topics here.
• Programming an AI for a simple game
It's always fun to play a game against a computer, and it's impressive when you can program a computer in such a way that it performs better in a game then most humans can.
• Maps and projections
Things like the Mercator projection and stuff. I don't know the specifics, but I know a guy who did this as his high school project.
• Non-euclidian geometry I don't know too much about this, so this may be too hard, but this is a really cool demonstration of how axiomatic systems work and a fun mind-experiment that, suprisingly, can be embedded in a picture.
• Game theory
Game theory provides a lot of interesting question. For example, I was just playing that 2048 game, and I found myself repeating: right, down, right, down, ... until i got stuck, then I did left or up, and started over again. Provided you have the probabilities of the new blocks appearing (else you could estimate them or just create your own variant in which the probabilities are known) you can, for example, try to analyze the average score. Then you could also try to find a pattern that ends the game as quickly as possible, or one that ends the game with the lowest average score (and are these patterns the same?). The nice thing is that even when a mathematical analysis is not feasible, you can just let your computer play for a day and come up with a very good estimate.

I've taken some subjects that I found interesting myself, and some others from a course where everybody had to give a presentation on a mathematical subject. Some of these subjects may be too hard or maybe too specialized to write a 100-page paper about, but I'll let you be the judge of that.

I've mainly focussed on things that have either a nice geometric representation (because a picture usually makes things more interesting and easier to understand) or some nice things that are not to hard to understand but still are considered interesting and/or nontrivial. I think all but the first three have elements that allow for a nice graphical explanation.

That being said, the students probably require a considerable amount of guidance. Most of the material they'll find is not suited for them, so the teacher either needs to have some material that is appropriate, or help out in another way that allows the students to do the research relatively independent.

There are some very nice questions in art gallery theorems that can be explored via paper-and-pencil experimentation. Below, placing guards at the three blue vertices visually guard the entire interior of this polygonal "art gallery":
(Wikipedia image)

• Note that this is one of the theorems mentioned in my link above. The specific site is york.cuny.edu/~malk/high-school-research/… and includes as recommended reading: O'Rourke, Joseph, Art Gallery Theorems and Algorithms, Oxford U. Press, Oxford, 1987. – Benjamin Dickman Mar 31 '14 at 18:42
• My Art Gallery book is online at this link. – Joseph O'Rourke Jun 4 '14 at 20:03

Here are a few projects done at my high school:

1) Given a quadratic in the form $y=ax^2+bx+c$, determine what changing $a$, $b$ and $c$ does to the turning point (vertex) of the parabola and prove your answer. Most students are able to figure out $a$ and $c$ relatively quickly, and they are good practice for $b$, which turns out to be much harder.

2) Prove that 7 congruent circles form 12 points of tangency as shown in the diagram below. Challenge students to prove this at least two ways using separate branches of maths.

3) Taxicab Geometry. Resources and questions abound online. This is one example of a good guiding worksheet. I haven't done Taxicab Geometry with my students, but I think it's an interesting way to challenge students to rethink all of their basic geometry definitions and axioms.

4) Have students revisit every maths theorem and formula used in high school. Have them summarise the theorem in their own words and give two examples of how it is used. (This project was given to the "bottom set" of students, with the feeling that they needed more practice, whereas the top students were more prepared for an extension.)

• For (2) it looks like 12 points of tangency, how do you count 18? – JTP - Apologise to Monica Jun 4 '14 at 2:59
• @JoeTaxpayer Oops. You're right. – David Ebert Jun 4 '14 at 6:15

Solving quadratic and cubic equations. At least for cubics, this really does require that your students have seen complex numbers.

• Solving the quadratic by completing the square is simple, clever, and you may not have ever shown them the proof in class.

• For the ambitious, Cardano's method for solving the cubic is quite straight forward, based on the lemma that for any two numbers $S$ and $P$, there are two (possibly complex) numbers whose sum is $S$ and whose product is $P$.

Staying within the realm of algebra, some basic things about complex numbers could make for a decent two-page project. Students could prove the basic geometrical facts about complex numbers:

• Multiplication correponds to multiplying the lengths and adding the angles.

Proving those facts could be page 1. Page 2 could be using them to prove another geometrical fact about complex numbers:

• There are exactly $n$ $n$-th roots of $1$, and they form the vertices of an $n$-gon centered at $0$. From this you can deduce that there are exactly $n$ $n$-th roots of any number.
• You can use complex numbers to prove certain trig identities.

As a professional mathematician, I certainly appreciate the projects listed above, and this might not be exactly the kind of answer you're looking for. The projects below will definitely help students understand a particular concept (dimension and proof-writing, respectively) but will not directly help them solve any problems.

When I used to teach high school, I would usually try to develop projects that connected mathematics to other disciplines. I would focus especially on the humanities and arts, since students whose interests leaned in those disciplines were often the most challenging to motivate and engage. Two projects with which I had some success:

• In a geometry class, I had students read Flatland and then asked that they come up with a creative project based on what they read. One student turned it into an analysis of women's rights in Victorian England; another painted, on a thin strip of wood, what an inhabitant of Flatland might see; another made the argument that cubist paintings were a 2-d rendering of what a 4-d person might see looking at a 3-d person and then made some paintings in that style.
• Also in the same geometry class, I tried to make an analogy between proof-writing (even though this was only ten years ago, we were still teaching proofs at this school) and editorial writing. What are your givens/the editorial writer's biases and assumptions? What are you trying to prove/the editorial writer trying to say? Can you/the editorial writer justify each step in your argument? I then had the students take an editorial in a local newspaper and attempt to decompose it into a two column proof.

At my daughters high school, last year every student must do a project of student research. They can choose the subject themselves (my daughter choose biology). Some math projects which have been done: -Fractals, programming fractals. -determinants and the eight-queen-problem

• Can you maybe add some details about how deep the projects have been studied? – Markus Klein Mar 31 '14 at 11:33
• surpringly deep! The expectation is that should be used >100hr,and final report about 100 pages. My daughters project was studying birds during winter months in a park nearby our house, detailing which species visited, how many, etc, what they were eating, writing detailed observation protocol. The maths project I found surpringly deep! for the age. – kjetil b halvorsen Mar 31 '14 at 11:38

Continued fractions can produce a vast range of good projects, which you can give at any age from (willing) middle school to (reluctant) high school students. One simple way to motivate the concept is to ask students to discover approximations to pi that were in wide use in ancient world. Sure, everybody knows $\pi \approx 22/7$, but can you do better than that? How do you proceed from $22/7$ to the next approximation? How do you convert continued fractions to "regular" ones, and vice versa? What if you allow not only the fractions like $1/a_n$, but also $2/a_n$, $3/a_n$, etc? E.g., if your first approximation is $\pi \approx 3.14 = 157/50 = 3 + 7/50$, how can you improve it using the continued fractions approach? Has anybody discovered this approximation? Can you produce a reasonably looking approximation that is not listed on Wikipedia?

Ask them to show that the following procedure constructs all rhombuses (rhombi?) of a fixed side length: take two circles of the same radius. Let them be placed to intersect at two distinct points. Now the two centres and the intersection points together form vertices of a rhombus. Now 'pull the circles apart' slightly or push them closer and obtain more of them.

If you understand some French, you can go to the site of Math en Jeans and look at the subjects other people have worked at with high school students.

You might also want to read Euclidlab - http://euclidlab.org/unsolved. They have many interesting projects, which should appeal to high-schoolers (I am a high-schooler myself and am thus speaking from experience''). In fact, I am planning on joining Camp Euclid, which other high-school students might find interesting (it costs a lot, though).

COMAP (The Consortium for Mathematics and Its Applications) publishes a newsletter for its members called Consortium. The current issue of Consortium can be downloaded for free from the web page below (which also includes other free COMAP materials) until the next issue appears, at which time the "current" issue goes behind a firewall.

https://www.comap.com/Free/

Consortium has many articles that can inspire student research (middle school to graduate school) research, In particular, the last couple of issues of Consortium have featured a "new column" that I edit, that discusses research problems specifically designed for (but not limited to) pre-college students. There problems appear under the title: "Student Research Corner." If you use the COMAP search feature you can find the titles of these research problems which are quick starting but all are geometrical questions nearly all involving graph theory and polyhedra.

Maybe something with 3D point groups (not all the details of the group theory, but describing the types of them (mirror plane, rotation, etc.) and classifying some common objects. (E.g. it is interesting that octahedron and cube have same symmetry. also, 2-D tiling patterns (but not 3d space groups...too complicated).