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!