## A Pythagoras Investigation Using GeoGebra

How GeoGebra Can Open Doors To New Discoveries

## I made a mathematical discovery thanks to GeoGebra!

GeoGebra is an awesome mathematical tool! It even allowed me to discover something new. Well, at least new to me! For an insight to my ‘new discovery’, take a peek at the dynamic image (gif) above.

For the record, I am no geek. I’m a feet-on-the-ground math teacher, fascinated by the craft of teaching mathematics. For many years I’ve been driven by the question “How do I stop boring kids to death when teaching mathematics?” Let’s face it, 70+% of people today over the age of 12 will state they are or were bored by mathematics at school. But I digress ...

## A Pythagoras investigation using GeoGebra

One day, I was playing around with GeoGebra and wanted to create a file which visually demonstrated the workings of Pythagoras’ Theorem. I had junior high school students in mind. I wanted the file to cause students to respond with “OK, now I see that the sum of the two smaller areas is equal to the larger area is true." I was not consciously aware of the fact that I was undertaking a Pythagoras investigation!
The first file I created was the file dynamically represented below. NOTE: This is different to the one above – in accordance with the theorem it contained only squares on the three sides of the triangle. |

And so through my Pythagoras investigation I began pondering ...

I was happy with the file. However, not long after creating it I began pondering the following:

The file I ended up with is represented below as well as by the one at the top of this article (two slightly different aspects of the same file).

- I wonder what would happen if, rather than having squares on the three sides, I created equilateral triangles?

- As I set about this task I had no idea what the outcome would be. (Heck, I'm truly investigating mathematics here!)

- I discovered that the principle worked for equilateral triangles.

- Well then, what about other regular pentagons?

- Yes, it works for regular pentagons - at least the ones I tested.

- Wait … maybe I can use a slider with ’n’ varying from three sides and in intervals of one and set the polygon to ’n’ sides. Wow, what might happen then?

- “Well look at that … this Pythagorean principle (as it relates to sums of areas) seems to apply to all sets of regular polygons on the 3 sides of a right-angled triangle.”

The file I ended up with is represented below as well as by the one at the top of this article (two slightly different aspects of the same file).

## GeoGebra causes curiosity

Now I’m not for a minute thinking that my Pythagoras investigation allowed me to discover anything new here - I'm sure I'm not the first to discover my discovery! (And anyone wanting to create some (real) mathematics out of this is welcome to do so).

The point of this article is not about my ‘discovery’ nor whether the discovery is or isn’t unique or of value. The point of the article is that GeoGebra, alone, caused me to be curious. Without the use of GeoGebra, I would never have been inspired to pursue the ‘what if’ question. Without the use of GeoGebra, I would never have embarked on this Pythagoras investigation!

Prior to this discovery I certainly had not been walking the streets of Sydney pondering the nature of the areas of regular polygons sitting on the sides of right-angled triangles! It was the act of using GeoGebra that caused me, a non-geeky, not overly mathematically gifted bloke, to inquire, to play, and then discover.

The point of this article is not about my ‘discovery’ nor whether the discovery is or isn’t unique or of value. The point of the article is that GeoGebra, alone, caused me to be curious. Without the use of GeoGebra, I would never have been inspired to pursue the ‘what if’ question. Without the use of GeoGebra, I would never have embarked on this Pythagoras investigation!

Prior to this discovery I certainly had not been walking the streets of Sydney pondering the nature of the areas of regular polygons sitting on the sides of right-angled triangles! It was the act of using GeoGebra that caused me, a non-geeky, not overly mathematically gifted bloke, to inquire, to play, and then discover.

## If me ... then why not students?

My point is this: If I can discover something mathematically new (to me) simply by investigating with GeoGebra then what's stopping students from gaining the same experience, especially given students have a lot more maths to discover? In the classroom students would need to be guided by a well-planned investigation - some tips are provided here - but this is all very doable. And wouldn't it be great to have our students doing some 'real' mathematics?

## The important question

This leads me to an important question: Why isn’t GeoGebra (or any similar tool) an integral part of every mathematics teacher’s toolkit? Let me rephrase that:

Why doesn’t every middle and high school mathematics teacher who has access to a computer and data projector utilise GeoGebra on a regular basis - across multiple year levels and topics?

The article 'Are you utilising GeoGebra?' contains a detailed example of using a file to impart conceptual understanding (with downloadable file). The article also argues the case for wider adoption of GeoGebra in maths classrooms globally.

In 'Developing conceptual understanding with GeoGebra' I show that simply projecting quality files and allowing higher-order questioning to occur requires no change in pedagogy from the teacher yet can powerfully improve conceptual understanding.

Here’s what one teacher had to say after 'becoming proficient' with GeoGebra:

In 'Developing conceptual understanding with GeoGebra' I show that simply projecting quality files and allowing higher-order questioning to occur requires no change in pedagogy from the teacher yet can powerfully improve conceptual understanding.

Here’s what one teacher had to say after 'becoming proficient' with GeoGebra:

I had set out to increase my confidence with my use of GeoGebra. Richard set the course out in such a way that not only did you learn the mechanisms of GeoGebra, but it allowed you to think of other uses for the same applications. The reactions of the students at school to the (files) have∞∞ been great and it has given them more insight into some concepts that they always find difficult. Susan Hoy, 26/05/2015

Do your students make genuine mathematical discoveries? Can you see how well-constructed GeoGebra investigations can help? Would you use a version of this Pythagoras investigation? We'd love your thoughts below! (Your email address will not be required)