Quadratics is the first unit I have to teach in ESPA 4 . 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 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.
 Cultura de les Illes Balears, C. d’Educació i. 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. 2009. 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. A new exploration! 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.
 Explore the MTBoS. Week 4 of the 2016 blogging initiative! 2016. https://exploremtbos.wordpress.com/2016/01/31/week-4-of-the-2016-blogging-initiative/.
 Meyer, D. Will it hit the hoop? 2010. http://blog.mrmeyer.com/2010/wcydwt-will-it-hit-the-hoop/.
 Teaching Channel. To the Moon! 2015. https://www.teachingchannel.org/videos/paper-rocket-lesson-plan.
All the problems here are translations from their original ones in catalan.↩