Remember that classic maximum area problem? Here’s a version of it: *Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose*.

This problem is usually given as an application problem and is solved algebraically. For example if *x* is one of the two equal sides to be fenced then the area is the function *f*(*x*) = *x* (18-2*x*). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.

Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:

- Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
- If you were Pam, what garden size will you choose? Why?
- What do the coordinates of P represent? How about the path of P, what information can we get from it?
- As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
- What equation of function will run through the path of P? Type it in the input bar to check.
- What does the tip of the graph tell you about the area of the garden?

Feel free to use the comments sections for other questions and suggestions for teaching this topic. How to teach the derivative function without really trying is a good sequel to this lesson. More lessons in Math Lessons in Mathematics for Teaching.