Discovering relations between graphs and algebraic formulas

blog, en, mtbos, mtbos-week1, functions, formulas, graphics, lines, curves

This post is part of MTBoS mentoring program [2]. This is post number #1 and is related to “One Good Thing” topic [3].

One of the aims of my official curriculum [1] is to discover regularities and patterns. Another one is to plot afine functions and knowing that the formula \(y=ax+b\) is associated with (straight) lines and other types of functions correspond to curves. Since two courses ago, I have combined these aims letting students discover by themselves this relation. This is the chronicle of how I have done this year (this week).

Preliminaries

Before threating the problem of differentiating curves or lines from the algebraic formula, we need several preliminaries. The first is to know the cartesian coordinates. In the first classes, I explained what is the cartesian plane, what are the axes and I gave some terms like origin and quadrants. Then I gave exercises of reading coordinates from points and to draw points giving their coordinates (see the picture above). Since this point, we will identify the points and their coordinates.

Discovering the relation

After assuring everyone knows how to read and write points to the cartesian plane, I started to represent functions. I put a simple formula in blackboard (like \(y=2x+10\)) and asked what are the points which satisfy the equation. Then we plotted these points and we got the representation of the formula. We represented several functions: for example \(y=3x-2\), \(y=x^2\), \(y=60/x\) or \(y=\sqrt{x}\), calculating the tables of values of \(x\) and \(y\). We saw the impossibility of knowing exactly what the graph of the function is exactly, because we represented only a finite number of points but the graph itself has infinite number of those. So we had to guess the form of the graph corresponding to one formula.

This is a workout exercise. After that, students know that there are a lot of possible graphs which could generate a formula and they are self-confident representing functions. Then, I asked them:

  • “Is there a relation between those graphs and the formulas?”
  • “Is there a way of knowing what kind of graph is generated by a formula, or not?”

For knowing that, I asked students to list a bunch of functions (we put in the blackboard) and then I asked them to conjecture their rule (their relation between graphs and formula). The rule could be simple or complex as they want. They choose. If they have not guessed any rule, then I suggest to represent as many function as they need for getting the guess. And if they got the rule, I ask them to prove or falsifying. Falsifying is always possible: we just need two graphs and two formulas which contradicts the rule. But we learnt that it’s impossible to prove the rule with their tools. And even we knew that there are rules more general than others and rules which implies other ones.

This week, the rules are:

  • “A function of the form \(y = x^{even}\) has a ‘U’ graph”
  • “A function of the form \(y=\text{number divided by }x\) has two curves”
  • \(\sqrt{\text{something}}\) is a curve”
  • “If the functions contains \(x^{\text{2 or greater}}\) then, it is a curve because when \(x\) grows the change of \(y\) is greater”

Next day, we checked these rules and I informed which “proved” rules are really true or false (read mathematically proved). See graph above (in catalan). Sorry for the quality.

(full-size image)

After that, I asked them if there is a rule to know if a functions gets a line or curve. And we deduced that we get a line when the formula is \(y=ax+b\).

Sequel

After that, because it’s needed only two points for defining one line, then when we would represent a function of the form \(y=ax+b\) (that we have learned that is a line), we will need to calculate just two points in our table of values.

Valuing the experience

I love this experience because it’s very exciting when students deduce their own rules and try to prove or disprove them. We feel like a really mathematicians. I strongly recommend you to apply in your class. Obviously you could use Geogebra or other interactive tools instead of blackboard.

References

[1] 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.

[2] Explore the MTBoS. A new exploration! 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

[3] Explore the MTBoS. Week 1 of the 2016 blogging initiative! 2016. https://exploremtbos.wordpress.com/2016/01/10/week-1-of-the-2016-blogging-initiative/.

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