# Exploration (IA) Ideas

Most secondary students have very little or no experience in formal writing tasks in which they discuss and analyze mathematical ideas. They have completed countless hours of * doing mathematics* - e.g. doing homework exercises, showing algebra working on assessments, etc - but they often lack significant practice in

*. I truly wish my students will derive some pleasure from completing the internal assessment (IA) requirement for Maths SL & HL - the*

**writing about mathematics***Exploration**. And I think that there is a greater chance of this occurring if clear and effective support and encouragement is provided to the students. [ * when the word "

*Exploration*" is capitalized and italicized it refers to the required IA task for Maths SL & HL ]

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 - the **introduction / preparation** stage, and the **topic choice** stage.

If you have not already done so, see my Exploration SL - Student Guide and Exploration HL - Student Guide . Each is a 10-page detailed guide for students (PDF file in which you can input your own deadlines).

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 for Maths SL and/or Maths HL, 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 satisfy the three ingredients of (i) not too long and not too short (roughly 6-12 pages; like an *Exploration*), (ii) demonstrate a good quality of writing with appropriate mathematical working, explanations and diagrams embedded in the writing, and (iii) that the level of mathematics is at the right level (commensurate) for Maths SL and/or HL - not too basic and not too advanced. I have managed to collect a few articles that I think include these three ingredients and there are link to them below. I am also in the process of writing my own articles that are not Explorations but present interesting (at least, I find them interesting) mathematical topics that I believe could provide teachers and students with ideas appropriate for developing into a good student *Exploration*. I am calling these articles of mine, ** Exploration Starting Points**. I'm writing them with both teachers and students in mind. As mentioned, my intention is

__not__for them to be examples of an Exploration (maybe somewhat) but more to present some thoughts about a mathematical topic sufficiently constrained and focused to make it a good 'starting point' for an

*Exploration*.

Take a look at my first ** Exploration Starting Point**, "Building Parabolas" - which 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 8-page paper might contain some ideas for starting and developing a student Exploration.

#### Exploration - Introduction / Preparation Stage

mathematical articles:

**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.