It seems like much of the focus in mathematics in High Schools (beyond standardized testing, that is) is on problem solving. However, without good problems to solve, it is difficult to engage in any real problem solving. The What-If-Not strategy for creating and posing problems fills the need for good problems. I first stumbled upon the What-If-Not problem posing strategy in the book The Art of Problem Posing (2004) by Stephen Brown and Walter Marion. This book prompts teachers to change their thinking about good problems from that of a scavenger (you need to search for and find good problems for your students) to that of a creator (you and your students create and solve your own problems). Over the next few blog posts, we will put the What-If-Not problem posing strategy to work by exploring the traditional Parabola Construction. These blog posts will approximate the trajectory I followed with some of my high school Geometry classes.
Consider the parabola construction shown below (or play through the construction at http://www.geogebratube.org/student/m232937).
The construction begins with a line BC that serves as our Directrix and a point that serves as the Focus. Next, a point A is placed on line BC, and a line perpendicular to line BC is constructed at A. Then, the perpendicular bisector of point A and the Focus is constructed. Finally, the intersection point P of the perpendicular and the perpendicular bisector is constructed. The Locus Tool is used to construct the parabola.
The proof that this construction leads to a parabola goes something like this: Point P is equidistant from the Focus and point A because it is on the perpendicular bisector. The distance from point P to the directrix is measured along the perpendicular to the Directrix. So, the distance from the Focus to point P is equal to the distance from point P to the Directrix.
Now onto our question: How do we create new problems from this construction using the What-If-Not strategy?
The What-If-Not problem posing strategy applied to the parabola construction works like this: Take one of the characteristics of the construction, change it, and pose a new problem in the form of a question. Using the strategy in this way, here are a four questions we will explore – one at a time – over the next few blog posts:
- What would happen if the perpendicular bisector was not a bisector, but a perpendicular trisector (or some other ratio)?
- What would happen if the perpendicular bisector was a bisector, but was not perpendicular?
- What would happen if the perpendicular line constructed to the Directrix was not perpendicular?
- What would happen if the Directrix was not a line, but was a circle?
For each question, we will do our best to answer the question “What is figure is produced by this change in construction?” as well as to provide proofs for each.
In the meantime, I would love to see some of your What-If-Not questions for the parabola construction! Leave them in the comments.