Quadratics: the unit that I don't proud of

[This post is part of MTBoS mentoring program [2]. This is post number #4 and is related to “My lesson” topic [3].]

Quadratics is the first unit I have to teach in ESPA 4 [1]. This is a course of adults education in Balearic Islands (Spain). What the students should know to do at the end of this unit are:

1. Solving second grade equations
2. Representing quadratic functions and getting the vertex, their orientation and the cut points with the axes.
3. Solving real problems (in particular modelize situations with quadratic functions)

How have I faced the unit until now

I have split the unit in three parts according with these aims. So there are three parts, which seem unconnected:

• In the first part, essentially, we solve 2nd-grade equations
• In the second part, we use 2nd-grade equations just for finding cut points with the axes
• And for solving problems we use 2nd-grade equations or finding the vertex of the parabola

Solving 2nd-grade equations

For introducing this topic, I put this problem¹:

From this point, we make problems of finding the dimensions of a square with a fixed area.

After that, I put the analogous problem with a rectangle and we see that there are infinite solutions. So we have to restrict the problem: perhaps the height of the rectangle could be something related to the width. For example, the height could be 3 times the width. Also we trait the triangles:

But we restrict our dependency to number times a dimension. In this step all the 2nd-grade equations are of the form \(ax^2 = b\).

Then, we make a small change: the dependency of one dimension is a number plus the other dimension:

With this kind of problems, we get complete equations (of the form \(ax^2 +bx + c = 0\)). After that we practice solving 2nd-grade equations. Just equations, no problems.

Representing quadratic functions

In this part, I remember the cartesian plane and how we could read or write coordinates and points to/from it. And inmediately I teach how to represent quadratic functions and we just “resolve” exercises of representing them:

Using quadratic functions and 2nd-grade equations in the reality

It is supposed that this section represents the real applications to all of the previous stuff. But I can’t achieve what is pretended. A sample of my problems is this:

(in the last two problems of this sample, the students have just to apply a formula)

I have never been able to find optimization problems which are real, easy and interesting.

What I have to improve

• The introduction to 2nd-grade equation must be shorter. We usually spend two or three weeks.
• I need better models for introducing the second grade equations.
• I need to attach the lesson to the reality, overall the quadric functions. I have been thinking about it many times. The real application of quadratic functions is the parabolic shot [4,5]. But its governing equations are very much complicated for my students. So I’ve been stucked.

References