# Circles with Geogebra

I used Geogebra to introduce my Year 12 further maths class to circles. I wanted the students to see the algebraic equation of a circle in action as certain parameters were changed. This was my first lesson with the students using Geogebra so before we began the circles work I gave them a quick tour of the main commands and its set up. I still use geogebra.org/classic as I find it easier to navigate.

I designed the following sheet to guide students – they were to work individually (at first) on the computers whilst writing answers on the sheet.

The tweaks to the general equation of a circle surprised students and generated some rich discussion that was beyond the scope of this unit of work. For instance, we discussed other conic sections such as hyperbolas and ellipses. This felt natural and because the students had generated these shapes themselves and they were eager to find out more and try to explain their thinking to me and their peers.

The students enjoyed playing with the sliders and the trace function. They created some pretty pictures by tracing different conic sections with variables sliding at different speeds. Given it was their first lesson using Geogebra I thought this was worthwhile as they were experimenting and having fun. One girl said she “hated computers until about 10 minutes ago” which was nice to hear.

On reflection, I would certainly tighten the questions so that students’ answers were more precise. For instance, when investigating equations such as k(x-a)2 + (y-b) 2 = r2 the results depend very much on the value of k, yet students did not categorise the different outcomes for the different ranges of k such as k<1, k=1, k>1 etc. Their thinking is not fully developed to do this independently yet, and I didn’t leave them with enough space to do so anyway! I would also make it more explicit for the students the significance of a, b and r by getting them to describe what they represent. Here’s a sample of a students’ sheet mid-lesson:

The lesson went well. The students had a positive first encounter with Geogebra, investigated the general equation of a circle, and played around with some variations. However, I do wonder whether this would have worked better as the second lesson on circles so that students were more confident to explore the concepts. I’m aware that I just gave them the general equation of a circle immediately without proving it until our second lesson. So some tweaks to consider – I will make the improvements and try to record my findings in another blogpost.

May 2019 Update: I have refined the worksheet and included specific cases for students to explore. This is because most students did not consider and explore the critical values of k in the equation on the worksheet. I have also made the role of a, b and r more explicit so students hopefully remember it better.

The update worksheet is here – feel free to use/edit  Circles with Geogebra Unit 1

Comments, questions or suggestions more than welcomed

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