Mathematical discoveries through exploring GeoGebra - A Pythagorean discovery with GeoGebra. Conceptual Understanding.

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.

GeoGebra Can Open Doors To New Discoveries

For the record, I am no geek. I’m a feet-on-the-ground math person, fascinated by the craft of teaching mathematics. For many years I’ve been driven by the question “How do we stop boring kids to death with mathematics?” Let’s face it, at least 70% of people today over the age of 12 will state they 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, I see that the sum of the two smaller areas is equal to the larger area is true.”

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 squares and only squares on the three sides of the triangle.

Mathematical discoveries through exploring GeoGebra - A Pythagorean discovery with GeoGebra. Conceptual Understanding.

Geogebra inspires mathematical discoveries!

So Began the Pondering

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

  • 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 was truly investigating mathematics!)
  • I discovered that the principle worked for equilateral triangles.
  • Well then, what about regular pentagons?
    • Yes, it works for regular pentagons.
  • Wait… maybe I can use a slider with ‘n’ varying from three sides, 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).

Mathematical discoveries through exploring GeoGebra - A Pythagorean discovery with GeoGebra. Conceptual Understanding.

Geogebra inspires mathematical discoveries!

GeoGebra Causes Curiosity

Now I’m not for a minute thinking I discovered anything new here—I’m sure I’m not the first! Anyone wanting to create some (real) mathematics out of this is welcome to do so. The point of the 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.

It was GeoGebra which armed me with the ability to pursue the ‘what if’ question. It was GeoGebra that caused the ‘what if’ question to arise in the first place. 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 to inquire, to play, and then to discover.

If Me…Then Why Not For Students?

My point is this: If I can discover something mathematically new simply by investigating with GeoGebra then what’s stopping students benefitting from the same experience? 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. 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 utilize GeoGebra on a regular basis—across multiple year levels and topics?

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

In ‘Developing a Conceptual Understanding of Maths 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.

Learn Implement Share

Learn Implement Share provides an engaging path for teachers to become proficient with GeoGebra. Ultimately, professional learning is best when undertaken in TEAMs. This allows school leaders to better guide the pedagogy of their staff.

Here’s what one teacher had to say after completing the online guided journey:

“[The course] allowed me to achieve what I had set out to do, and that was 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 my 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? We’d love your thoughts below!

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