## Are You Utilizing GeoGebra?

An Example Of Using A GeoGebra File To Impart Conceptual Understanding

(The original GeoGebra file used for the dynamic image above was created by Melina O’Brien, Moss Vale High School, NSW, course participant, September 2011)

## You don't know what you don't know!

The saying ‘You don’t know what you don’t know’ applies to many aspects of life but especially to the use of technology. In regards to teaching, there can often be an incredible tool with game-changing potential available for our use, if only we knew about it. For mathematics teachers, GeoGebra is one such tool.

## How many of the teachers in your department do not yet use GeoGebra on a regular basis - and across all math classes?

Most mathematics teachers will have heard of GeoGebra. However, it appears many math teachers are unaware of the extent of GeoGebra’s power, simplicity and applicability to numerous aspects of middle and high school mathematics. I propose that the number of teachers who utilise GeoGebra (or similar) on a regular basis, even simply projected onto a board, to be less than 50% – and that’s 50% of those who have access to a computer and data projector!

## GeoGebra is mostly promoted to senior maths teachers

This message is directed to the everyday math teacher here, or the school leaders who influence them - with my feet planted firmly on the ground. However, the message is equally relevant to the ‘pure math’ types. My sense is that because GeoGebra is an exceptional tool for higher levels of mathematics, most of the files developed and most of the articles written in support of GeoGebra have been by teachers of higher mathematics. We rarely see files and articles written about GeoGebra as an approach for middle and junior high school. Therefore, teachers of these younger grades can be excused for assuming GeoGebra is for the teachers of higher mathematics!

Hopefully, those who read this and my other GeoGebra articles will begin to see that GeoGebra has applications across all high-school year levels and across many topics beyond graphing and geometry.

Hopefully, those who read this and my other GeoGebra articles will begin to see that GeoGebra has applications across all high-school year levels and across many topics beyond graphing and geometry.

## A solid foundation for understanding

To support the argument (that GeoGebra ought to be a key tool in every math teacher's toolkit) take a 30-second look at the dynamic image (gif) at the top of this article. It features a very simple GeoGebra file designed to help convey the principle behind reading a bearing from two points. The file is not meant to replace the need to ‘crunch numbers’ around bearings. However, showing such a file before tackling this principle will engage students, provide a stimulus for questions and provide a solid foundation for understanding. Alternatively, showing the file after covering the work will provide students with a visual summary.

## A paradigm shift

The power of using cleverly-designed GeoGebra files in this way is underestimated by many mathematics teachers. Such a file demonstrated as part of a series of quality files provides a paradigm shift in teaching when compared to using an approach devoid of dynamic demonstrations. The staggering fact is that GeoGebra can have this much impact even when ONLY using the plug-in-the-data-projector-and-show-the-file method. Setting up student-led investigations, which require more extensive planning and a non-typical approach for teachers, offer even greater benefits. (Make sure you read the Pythagoras investigation article to see how powerful GeoGebra can be in leading a user into new-to-them mathematical discoveries).

The gif below was created from a visually simple file containing some added novelty. It was created by past participant Anne Wolkowitsch and is a classic example of a teacher’s creativity being unleashed through GeoGebra. For the purpose of the gif I added some visuals – the dotted lines and the “Ouch” - during the video editing process.

The gif below was created from a visually simple file containing some added novelty. It was created by past participant Anne Wolkowitsch and is a classic example of a teacher’s creativity being unleashed through GeoGebra. For the purpose of the gif I added some visuals – the dotted lines and the “Ouch” - during the video editing process.

(The original GeoGebra file used for the dynamic image above was created by Anne Wolkowitsch, The French School, NSW, course participant December 2013)

The file is clearly ideal as an introduction to the notion of gradient. The original file had the title and dynamic calculations showing at all times because the file was intended to be used for pre-calculus. For the purposes of this article, I modified the file so that it became an example of a file to support the teacher - using a conceptual approach - to help bring understanding to junior high students. To achieve this, I simply added checkboxes to hide both the calculations and the title, preferring to start the file with the gradient dynamically displayed as ’Surfboard Number’.

## Using the file as part of a conceptual approach

Here’s one suggestion for embracing a conceptual approach to teaching gradient with this file.

Note that the above gif could be appropriately displayed to a beginning calculus class. However, know that the gif by itself has little use for a middle-school class especially when introducing gradient. The gifs, remember, have been designed for this article to help convey the message that GeoGebra is a tool that all mathematics teachers could be using.

- Project and manipulate the file (with ’Surfboard Number’ displayed) to the whole class before mentioning the word ‘gradient’.

- Note that the word ‘Number’ has been deliberately used in preference to Slope or Gradient. This is because we want students to think about what the Surfboard Number means as the surfboard moves. It won’t take students long to relate this number to the position of the surfboard!

- Once students begin to see the relationship – horizontal means zero, steep means bigger number, up means positive, down means negative – ask students to come up with a name for the number. ‘Slope’ tends to be a common suggestion.

- Now ask the really interesting question: “How is the surfboard number calculated?” If the class is capable you could, at this point, ask them to explore, preferably in pairs, on some graph paper.

- The following clue could be offered: “Every point has an x coordinate and a y coordinate”. Granted this is an obscure clue. But the aim here isn’t that they will discover the gradient formula - some may discover it or get close (or they might simply look it up in their textbook) - the main aim is to have them explore and to engage the students.

- Allow students a few minutes to ‘wrack their brains’ with this challenge. If the students are engaged they will be eager when you say “Would you like to see how the surfboard number is generated?”

- If you possess a performer’s streak then you will likely, immediately prior to clicking the checkbox to reveal the calculations, deliver a 'fanfare' by blowing your imaginary trumpet and uttering several exclamations of the type “Are you ready for this … really, are you? … wait for it … “, etc.

- Now, reveal the calculations. The gif image, created for this article and not for the classroom, shows the calculations being revealed 5 sec from the start. However, in our example, when using the actual GeoGebra file, it would be best to demonstrate the file for 5+ minutes prior to revealing the calculations to students.

- Manipulate the file stopping at points similar to those demonstrated by the gif. It is essential when teaching conceptually, to be a facilitator rather than a ‘teacher’. In other words, try to lead students to see the relationships between the numbers and the graph rather than immediately showing them. Note that the added annotations highlight some of these relationships and point to the sorts of clues and explanations the conceptually-oriented teacher would attempt to lead students into discovering.

- The final step is to show the title. However, given we are addressing a junior high school class, the title in the gif is probably best left hidden. Or you might want to modify the title.

- If using the file for a calculus class who already know the workings of the gradient formula then everything could be displayed from the beginning as there is little new to establish other than to clarify student understanding about the gradient of a function at a point.

Note that the above gif could be appropriately displayed to a beginning calculus class. However, know that the gif by itself has little use for a middle-school class especially when introducing gradient. The gifs, remember, have been designed for this article to help convey the message that GeoGebra is a tool that all mathematics teachers could be using.

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## Download the modified surfer file

The modified version of Anne's file - the one used for the gif - is here.

The principles underpinning the surfboard gradient file can be applied to numerous aspects of mathematics, and especially to assist the conceptual understanding of middle and junior high mathematics. For more examples of dynamic images of GeoGebra files check out this page of files created by participants of the 'GeoGebra Proficiency' course.

The principles underpinning the surfboard gradient file can be applied to numerous aspects of mathematics, and especially to assist the conceptual understanding of middle and junior high mathematics. For more examples of dynamic images of GeoGebra files check out this page of files created by participants of the 'GeoGebra Proficiency' course.

## Increase your ability to impart conceptual understanding

Are you one of the few who utilises GeoGebra extensively across all your math classes and across numerous topics? If not then hopefully this article has inspired you to embark on a journey of becoming proficient with GeoGebra.

## Learning GeoGebra solo is frustrating and ** s l o w** !

Becoming a proficient user of GeoGebra on your own (or without an expert-guided learning journey) can be a slow and painful process. This is, in part, because GeoGebra can do so much more beyond high school mathematics and so therefore many of the freely available resources have no relevance to high school mathematics. However, it is also a slow 'solo' process because when seeking to learn a new aspect of GeoGebra, it is often impossible to know what to search for - you simply have no idea what search phrase to use! I could share numerous examples of this from my own experience. This is why a 'one-stop shop' to enable you to become proficient GeoGebra user (for all high school classes and across multiple topics) is an ideal path forward.

## Your turn ...

Having heard the argument for Geogebra to be utilised proficiently and widely by mathematics teachers, what do you think? I'd love your honest comments, for or against. (Your email address will not be required)