Good Mathematics Projects for High School Students

by Mary Parker, Professor of Mathematics, Austin Community College

If I had a niece or nephew who will later take calculus and who has now completed Algebra II or is currently enrolled in Trigonometry or Elementary Analysis, here are some of the ideas I'd suggest. Your student's high school math teacher will undoubtedly have more good ideas.

Notice that most colleges have just recently increased their attention to letters of recommendation from high school teachers during the admissions process. A student who shows the initiative and perseverance to do an out-of-class project will give the teacher more evidence from which she/he can write a really good recommendation.

1. Analytic Geometry:
   A thorough investigation of various properties of parabolas, circles, ellipses, and hyperbolas gives students opportunities to explore geometry and algebra together and see how understanding one aspect enhances their understanding of the other aspect. In the sixties and before, this was the major "precalculus" course, because it was believed to enhance the students' mathematical maturity. I believe it did that. Students can find any number of analytic geometry books at the right level and can pick and choose among the topics that look interesting to them.
2. Using graphing computer software:
   Most high school students have been exposed to graphing calculators and can use them to graph functions of the form is equal to some formula containing the variable x. However there are lots of other types of graphing (polar coordinate system, parametric equations, functions defined implicitly, functions of two variables) that students in Calculus II or Calculus III will need to use. Most students arrive in calculus with very little facility with any of these. That made more sense ten years ago when all graphing was done by hand and students had to understand quite a lot of the algebra before they could even produce a graph. However, some graphing calculators and virtually all mathematical computer software (including lots of public-domain programs) make graphing of all sorts very easy.
   I'd download an appropriate software package (see our description of free software), and find a calculus book (in some used book store so that it wasn't too expensive), and have the student explore which types of graphs that software program will do and read the corresponding sections of the text. Skip all the part that seems to have to do with taking derivatives or finding integrals - that's the calculus part.

These are some other areas that I think would be interesting, although I haven't developed them very much yet. The latter three in the list are not particularly relevant to later calculus courses, but might be interesting for some students.

• Explore and learn to do some things in a computer algebra system, like Mathematica, Maple, Derive, or Scientific Notebook. (See Marcus McGuff's web page describing software for discussions of capabilities and prices.) Many of the larger universities with substantial money are having all calculus students and above do some work with one of these systems. If a student has learned one, she is at a great advantage in learning another, so don't feel that it is important to find the "right" one.
• Explore fractals.
• Read selections from several appropriate books and write a paper explaining the difference between the concepts of chaos and randomness.
• Investigate linear regression: Learn how to interpret it, when it is appropriate to use it, how to do it by hand and how to do it in some computer program. Find these discussions in an introductory statistics textbook. I recommend any of the books that we (and UT) have been using in our statistics courses for the last several years, by David Moore. There are a number of public domain little statistics packages available and most spreadsheet programs, certainly Excel, will do the basic calculations. They won't necessarily do as many diagnostics as would be useful. Critique the software package you find based on what you learn from the textbook.

Go back to the outline of Mary Parker's "Where should I start in college-credit mathematics?"

This page was prepared by Mary Parker, 512-223-4846, mparker@austincc.edu.