Exploring pi with GeoGebra
The case for all middle & high school maths teachers becoming proficient users of GeoGebra
Richard Andrew, 10th March 2017
Transform your math teaching with GeoGebra.
Conceptual understanding
The article Procedural knowledge vs conceptual knowledge in mathematics education sheds some light on the conceptual approach to teaching mathematics. In the article I state that mathematical understanding is fostered through the use of activities and strategies specifically designed to engineer ‘aha moments’ to occur for students and that through using a conceptual approach, metacognition and other higher-order thinking processes are encouraged.
GeoGebra is an incredibly powerful and easy to use tool which mathematics teachers can use to help impart conceptual understanding to students. This article is another in a series to promote the idea that (all) middle and high school mathematics teachers ought to be regularly utilising GeoGebra. As I argue in 'Let GeoGebra transform your mathematics teaching' there appears to be an unnecessarily high portion of math teachers who do not regularly utilise GeoGebra (or similar dynamic geometry software). I also argue that the main reasons for the lack of utilisation of this amazing tool is a lack of knowledge by math teachers of the power of GeoGebra and its applicability to so many aspects of mathematics.
GeoGebra is an incredibly powerful and easy to use tool which mathematics teachers can use to help impart conceptual understanding to students. This article is another in a series to promote the idea that (all) middle and high school mathematics teachers ought to be regularly utilising GeoGebra. As I argue in 'Let GeoGebra transform your mathematics teaching' there appears to be an unnecessarily high portion of math teachers who do not regularly utilise GeoGebra (or similar dynamic geometry software). I also argue that the main reasons for the lack of utilisation of this amazing tool is a lack of knowledge by math teachers of the power of GeoGebra and its applicability to so many aspects of mathematics.
Exploring pi with GeoGebra
Let’s consider one powerful, yet simple, application of GeoGebra, namely to highlight the relationships between pi and the circle.
Check out the dynamic image (gif) at the top of this article for 30-seconds or so. As you can see the gif features a GeoGebra file with a circle, its circumference and its diameter displayed dynamically. And yes, this is deliberately a very simple file. Yet when some dynamic text is added to calculate the quotient, we end up with … pi!
Long term users of GeoGebra (or similar) will likely be thinking “So what?” After all, they have, for many years, been immersed in GeoGebra files more complex and impressive than this one. Yet this file is an excellent example of the power and simplicity of GeoGebra. How else can you encapsulate, quickly and easily, the derivation of pi using circumference and diameter, and with a visual representation? The file could be shown as a summary after students have been manually exploring circles in order to derive pi. Alternatively, it could be shown at the beginning of the topic to have students consider any relationships which might be occurring as the circle changes size. The file could also be modified and used for a student-led investigation.
Now consider the dynamic image below. Again, spend 30 seconds or so observing the gif.
Check out the dynamic image (gif) at the top of this article for 30-seconds or so. As you can see the gif features a GeoGebra file with a circle, its circumference and its diameter displayed dynamically. And yes, this is deliberately a very simple file. Yet when some dynamic text is added to calculate the quotient, we end up with … pi!
Long term users of GeoGebra (or similar) will likely be thinking “So what?” After all, they have, for many years, been immersed in GeoGebra files more complex and impressive than this one. Yet this file is an excellent example of the power and simplicity of GeoGebra. How else can you encapsulate, quickly and easily, the derivation of pi using circumference and diameter, and with a visual representation? The file could be shown as a summary after students have been manually exploring circles in order to derive pi. Alternatively, it could be shown at the beginning of the topic to have students consider any relationships which might be occurring as the circle changes size. The file could also be modified and used for a student-led investigation.
Now consider the dynamic image below. Again, spend 30 seconds or so observing the gif.
A teacher-led, guided investigation
Now that students have seen the relationship between diameter and circumference, the file used as the basis of the gif above could be used as a teacher-led, guided investigation with the following question posed: “Can we find a relationship between area and one of the other quantities?” Note that the file uses only three such explorations because I was limited by the size of the gif permitted by this page. But you could include more explorations or create several files for this exploration. In addition, the file could be pre-made or, if you are confident, you could create the dynamic quotients on the fly as students suggest possible relationships.
Note also that dividing area by the square of the radius will not be an intuitive relationship for students to suggest and may require considerable probing of students to have them suggest this. As long as students are able to create dynamic formulas they could be set free on a search to discover pi within the area relationship.
All this is a great deal of fun – very engaging for students and very much about exploring mathematical systems. (Isn’t that what math education is supposed to be about?) And when used as a demonstration tool it is very extremely easy to manage.
Another exceptional file – the Surf Board file – created by December 2013 participant Anne Wolkowitsch - can be found half way down the article 'Are you utilising GeoGebra?' The file has great potential for fostering conceptual understanding and is accompanied by a suggested approach for demonstrating it. The file is included for download.
The engaging, long-term, online pathway to becoming proficient with GeoGebra is ideally undertaken as a TEAM. Information about the GeoGebra pathway can be found on this site.
Note also that dividing area by the square of the radius will not be an intuitive relationship for students to suggest and may require considerable probing of students to have them suggest this. As long as students are able to create dynamic formulas they could be set free on a search to discover pi within the area relationship.
All this is a great deal of fun – very engaging for students and very much about exploring mathematical systems. (Isn’t that what math education is supposed to be about?) And when used as a demonstration tool it is very extremely easy to manage.
Another exceptional file – the Surf Board file – created by December 2013 participant Anne Wolkowitsch - can be found half way down the article 'Are you utilising GeoGebra?' The file has great potential for fostering conceptual understanding and is accompanied by a suggested approach for demonstrating it. The file is included for download.
The engaging, long-term, online pathway to becoming proficient with GeoGebra is ideally undertaken as a TEAM. Information about the GeoGebra pathway can be found on this site.
Here’s what one teacher had to say after completing the course
I found the course extremely helpful and relevant to my teaching. I was aiming to improve my knowledge of GeoGebra and to be able to create my own files. The course equipped me for this and I feel much more confident in using GeoGebra in my
classes and have done so in a number of classes and topic areas.
Sally De Maria, 26/05/201
What other topics come to mind that GeoGebra could be applied to (other than graphing and geometry)? Do you utilise GeoGebra widely? Are you tempted to use it more? We'd love your thoughts below! (Your email address will not be required)