# Discovering relations between graphs and algebraic formulas

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.

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/.