Exploration (IA) Ideas
Some ideas for a student Exploration (IA):
Idea #1: Building parabolas
Idea #2: Triangle dissection
Idea #3: Choosing with replacement
Idea #4: Pentagonal tessellations
Idea #5: Area of a parabolic segment
Idea #6: Directrix of a parabola
Most students have little or no experience in writing a paper where they need to explore, analyze and fully explain mathematical ideas. They have completed countless hours of doing mathematics - doing homework exercises, showing algebra working on assessments, etc - but they often lack practice in writing about mathematics. I want to give my students guidance that will help them produce a successful Exploration, but I also genuinely hope that they will enjoy writing their Exploration. I think there is a greater chance of both of these occurring if clear and effective support and encouragement is provided to students - especially before they start writing their Exploration.
Introduction / Preparation stage
Students need to be guided and encouraged by their teacher at all stages of completing their Exploration. I believe that this is most crucial in the first two stages - during the introduction / preparation stage and the topic choice stage.
See a discussion of the different stages of completing the mathematics internal assessment task (Exploration) from the point of view of a student and from that of a teacher on the Managing the Exploration page.
There are two simple but effective activities that I think students should do during the intro/prep stage: (1) read short articles that are examples of good writing about a mathematical topic at a suitable level, and (2) read past student Explorations - preferably with some information on how well the Exploration addressed the assessment criteria.
For the intro/prep stage, it's not particularly easy to find mathematical articles that: (i) are not too long and not too advanced, (ii) demonstrate good writing with appropriate mathematical working, explanations and diagrams, and (iii) involve mathematics at the right level (commensurate) for the Analysis & Approaches course (either SL or HL). I have managed to collect a few articles that I think include these three ingredients There are descriptions and links for four such articles in the blue box below.
mathematics articles
good writing about interesting mathematics at a suitable level for upper secondary mathematics
1. Pathways and Barriers to Counting by Peter G. Brown
This is a short 5-page article on interesting identities related to the binomial theorem taken from the online magazine Parabola Incorporating Function published by the School of Mathematics & Statistics, University of New South Wales, Australia. The magazine "publishes articles that can contribute to the teaching and learning of mathematics at the senior secondary school level, in the areas of applied mathematics, mathematical modelling, pure mathematics, statistics and the history of mathematics."
2. Chasing Imaginary Triangles by Ian VanderBurgh & Serge D'Alessio
A short but interesting article that includes right triangles, Pythagorean Theorem, imaginary numbers, quadratic equations & inequalities, semi-perimeter of a triangle, tangent line, hyperbola & Heron's formula. The article is taken from the Canadian Mathematical Society's publication Crux Mathematicorum - which claims to be "an internationally respected source of unique and challenging mathematical problems ... designed primarily for the secondary and undergraduate levels ... and has been referred to as 'the best problem solving journal in the world' ."
3. Rugby & Mathematics: A Surprising Link among Geometry, the Conics, and Calculus by Troy Jones & Steven Jackson
This is a 6-page article investigates the optimal location for kicking a conversion in rugby to maximize the angle at the goalposts. The article was in the Nov. 2001 issue of the Mathematics Teacher magazine published by the National Council of Teachers of Mathematics (NCTM) in North America.
4. Thinking out of the Box ... Problem by Walter Dodge & Steve Viktora
A fascinating 7-page article (intended for teachers but certainly suitable for students to read) that considers the 'classic' optimization problem of maximizing the volume of an open-top box - and extends the problem well beyond the standard treatment. The investigations described in the article are structured around six questions. It appeared in the Nov. 2002 issue of the NCTM's Mathematics Teacher magazine.
Topic Choice stage
I have also written my own articles that are not Explorations but present interesting mathematical topics that I believe could provide teachers and students with ideas appropriate for developing into a successful Exploration. These articles have a title prefaced by IA idea #1, IA idea #2, etc. I have written these - and continue to write more - with both teachers and students in mind. As mentioned, my intention is not for them to be examples of an Exploration but more to present some thoughts about a mathematical topic sufficiently constrained and focused to make it a good 'starting point' for an Exploration.
My first article is titled IA idea # 1 - Building parabolas. It looks at constructing parabolas from linear and non-linear functions, and considers how to go about finding the vertex and axis of symmetry of a rotated or non-standard parabola (non-vertical, non-horizontal). It's certainly takes more effort and thought than finding the vertex and axis of symmetry of a standard parabola. My hope is that this article might contain some ideas for starting and developing a student Exploration.
All of the current IA idea articles can be accessed by clicking on the links in the menu on the left - and all of them have been designate for filtered student access, so you can give access to these articles to students through the site's student access system.
Also, see my list of 200 Exploration ideas/topics.