[This post is part of MTBoS mentoring program (Explore the MTBoS 2015). This is post number #4 and is related to “My lesson” topic (Explore the MTBoS 2016).]

Quadratics is the first unit I have to teach in ESPA 4 (Conselleria d’Educació i Cultura de les Illes Balears 2009). 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:

- Solving second grade equations
- Representing quadratic functions and getting the vertex, their orientation and the cut points with the axes.
- 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 problem^{1}:

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 (Teaching Channel 2015; Meyer 2010). But its governing equations are very much complicated for my students. So I’ve been stucked.

# References

Conselleria d’Educació i Cultura de les Illes Balears. 2009. “Ordre de La Consellera d’Educació I Cultura de 22 de Juliol de 2009, Per La Qual S’estableix El Currículum de L’educació Secundària Per a Persones Adultes Que Condueix a L’obtenció Del Títol de Graduat En Educació Secundària Obligatòria a Les Illes Balears.” http://www.caib.es/eboibfront/pdf/VisPdf?action=VisHistoric&p_any=2009&p_numero=117&p_finpag=55&p_inipag=4&idDocument=629435&lang=ca.

Explore the MTBoS. 2015. “A New Exploration!” https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

———. 2016. “Week 4 of the 2016 Blogging Initiative!” https://exploremtbos.wordpress.com/2016/01/31/week-4-of-the-2016-blogging-initiative/.

Meyer, Dan. 2010. “Will It Hit the Hoop?” http://blog.mrmeyer.com/2010/wcydwt-will-it-hit-the-hoop/.

Teaching Channel. 2015. “To the Moon!” https://www.teachingchannel.org/videos/paper-rocket-lesson-plan.

All the problems here are translations from their original ones in catalan.↩