## Friday, 3 July 2015

### y = mx + c

I was first introduced to GeoGebra in 2011 during my teaching training at the University of Leeds. I remember being shown how to use sliders to manipulate the x-coefficient and constant term in a linear equation that was controlling a graph. I was struck by the power of this demonstration for teaching links between functions and graphs. The graphical response of manipulating the m and c values was immediately apparent and with careful questioning a deep understanding of the mathematics at play could be obtained by a student in a matter of minutes. For a student to explore these relationships using a pencil and paper would take hours of graph plotting and even with a graphical calculator at their disposal the relationships would take much longer to explore and become clear. The session was only short but the snapshot of GeoGebra I’d seen was impressive.
Unfortunately I didn’t endeavor to actually use GeoGebra during my training year; I was too busy planning three part lessons, filling in paper work and writing essays. Indeed over three years passed before I saw it in use again, not in a school classroom but back in the same seminar room at the University of Leeds. This time, whilst enrolled on the excellent TAM course run by the MEI, I was shown how GeoGebra can be employed to help students obtain a deeper understanding of calculus (this will be the subject of a blog post in the near future).

I think there are several barriers aside from being very busy that meant I did not embrace GeoGebra sooner;
1.       Whilst I was impressed by what I had seen of GeoGebra whist training I was only shown its use in a fairly limited capacity – I was unaware of its amazing potential to aid in understanding a vast array of concepts linking number, geometry, algebra, statistics and calculus.

2.       I did not come across any teachers using GeoGebra at the schools where I trained or have worked and so I had little inspiration or support to develop its use in my teaching.

3.       Although there are a lot of excellent tutorials on the web on how to use GeoGebra and a plethora of GeoGebra worksheets available via GeoGebraTube, few authors have attempted to explain how they actually integrate GeoGebra into their day-to-day practice as teachers and use it as they cover different topics across the maths curriculum.

At the beginning of this year I took some time to really get my teeth into developing and using GeoGebra resources. Now, whilst I wouldn’t call myself an expert; if I can visualise a way to represent a mathematical concept - I can usually make it happen using GeoGebra and the result is generally far more eloquent, informative and engaging than I could achieve with a whiteboard and a pen.  My motivation for writing this blog is to allow you to look over my shoulder in the classroom, to share some ways in which I use GeoGebra and hopefully help some readers to overcome the three barriers that I have discussed. I welcome any comments, ideas, corrections or questions that you have relating to any of my posts or resources.
The first applet I will share here is a very simple GeoGebra sketch based on the one I was first shown that I have eluded in this post. Thanks for reading.

Some suggested questions you might ask are:
What effect does changing c have on the graph?

What can you say about the line if m is negative?

What will happen to the line if m is set to zero?

Can the line ever be vertical and if so how would you write down its equation?
Can you think of a way to find out where the line crosses the y-axis from the equation?

What about where it crosses the x-axis?

If you know the x-value of a coordinate point on the line how could you find its y-value? e.g. The point (3,A) lies on the line y = 3x + 10; find A.

If you know the y-value of a coordinate point on the line how could you find its x-value? e.g. (B,5) lies on the line y =2x - 15, find B.

How could you check if a particular coordinate point lies on the line using just its equation? e.g. Does the point (2,7) lie on the line y = 6x - 5?

If you were given two points how could you find the gradient of the line going through them? e.g. (5,2) and (3,1).

Could you then find the lines' equation?