Exploring pi with GeoGebra
The case for all middle & high school maths teachers to become proficient users of GeoGebra
Transform your math teaching with GeoGebra.
Conceptual understanding
The article Teaching Mathematics using an approach that is both conceptual and procedural - Nine keys to the hybrid conceptual-procedural approach 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’ 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 that math teachers can use to help impart conceptual understanding to students. This article, along with others, promotes the idea that (all) middle and high school mathematics teachers ought to be regularly utilizing 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 utilize GeoGebra. I also argue that the main reasons for this is a lack of knowledge of the power of GeoGebra and its applicability to so many aspects of mathematics.
GeoGebra is an incredibly powerful and easy to use tool that math teachers can use to help impart conceptual understanding to students. This article, along with others, promotes the idea that (all) middle and high school mathematics teachers ought to be regularly utilizing 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 utilize GeoGebra. I also argue that the main reasons for this is a lack of knowledge 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 of a simple GeoGebra file. How else can you encapsulate, quickly and easily, the derivation of pi using circumference and diameter 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 looping 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 of a simple GeoGebra file. How else can you encapsulate, quickly and easily, the derivation of pi using circumference and diameter 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 looping gif.
A teacher-led, guided investigation
Now that students have seen the relationship between diameter and circumference, the original GeoGebra file (not the gif) 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. You could include more explorations or create several files for this exploration. In addition, the file could be pre-made or, if you are proficient, 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. (And isn’t this what math education is supposed to be about?) And note that when GeoGebra is 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 halfway 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.
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. (And isn’t this what math education is supposed to be about?) And note that when GeoGebra is 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 halfway 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.
Here’s what one teacher had to say about becoming more proficient with GeoGebra:
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
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What other topics come to mind that GeoGebra could be applied to (other than graphing and geometry)? Do you utilize GeoGebra widely? Are you tempted to use it more? We'd love your thoughts below! (Your email address will not be required)
