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Quadratics: the unit that I don’t proud of

Authors
Xavier Bordoy
Date of creation

Abstract

I explain why Quadratics is the unit that I don’t proud of

[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:

  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:

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

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.


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