Do you remember your early encounters with the animal kingdom? So many wonderful different animals - it may even have been a bit overwhelming at first. But very quickly we learnt to group the animals into a scheme that made sense to us. In mathematics we have a similar extravaganza of different 'animals', which can be overwhelming for students to make sense of. Enter the idea of The Function Zoo - first introduced to me by Mary Barnes in her amazing Investigating Change books.
Here is how I worked the idea into a Year 11 class, several lessons into the Functions topic:
|A look at the different species of animals ....|
|... and how we might organise them.|
Students worked in groups of four, using large sheets of butcher paper to sketch their ideas. There were at least two laptops per group and the students had just enough GeoGebra skills to be able to turn algebraic expressions into graphs.
The results were incredible: great conversations between students about functions. With GeoGebra on hand, I was able to encourage students to explore their questions, rather than give them answers, and even ask them more questions if they were ready for it.
Twenty minutes later I quietly threw this slide on the screen but otherwise said nothing:
The groups noticed it soon enough - and went wild. Seeing a few more functions they knew but had forgotten gave them new energy to keep going. Others asked each other questions, trying to work out the graphs they didn't recognise. Most recognised the last graph from our "explore your calculator" game. We then debriefed as a class, and explored why the idea of the Function Zoo is helpful and interesting. Apart from the obvious benefit of being able to organise our thinking, the real benefit comes in being able to make connections - as I suggested in these slides:
As often happens in student exploration activities, the class produced something unexpected, a gift from them to extend the lesson idea. One group drew the absolute value of a quadratic function - a blend of two of our function families. We decided this new function was like the cross-species breeding you sometimes see on display at the zoo : the lion bred with a tiger to make a liger.
|Absolute value of a quadratic function : a "liger" in our function zoo.|
Liger drawing: St Hilare (1772-188). Function by GeoGebra.
A fun and powerful idea - allowing students to see that even quite unusual functions can be seen as blend of function attributes they already know how to work with.
- A graphing tool makes a huge difference to the success of this lesson. Without it, students would spend a very long time plotting to explore their ideas. There is time for careful plotting later - this lesson is about seeing the bigger picture.
- I found the group structure allowed for a high degree of differentiation - I could customise leading questions for each group, depending where they were up to on the functions journey.
- I can't stress enough the value of developing students' GeoGebra skills (or other computer graphing application) when doing mathematics at this level. I sneak some GeoGebra learning into every lesson - even if it's just the class watching me do a quick check of an equation or a graph. Show them one small GeoGebra idea per lesson and by the end of term they will know the product well - especially if they are using GeoGebra at home as part of their study.
- Why am I such a GeoGebra fanboy? Most importantly because all my students can download a copy to use home. GeoGebra is free and runs on Windows and Macintosh and it doesn't need an internet connection to run.