Reversing the Meat-a-Morphosis machine : inverse functions

A short followup to the post Two Ideas for Introducing Functions.

Yesterday I discovered another payoff from the hilariously gruesome Meat-a-Morphosis video: a powerful and memorable analogy for Inverse Functions. Imagine if we ran the machine backwards - if we put chicken nuggets in and the original chickens came out? My students thought this was hilarious.

Inverse Functions: What if you put in nuggets and out came chickens?
Image adapted from original at Meat-a-Morphosis ( Patty Hill &  Michael Word)

Even more fun: for students still coming to terms with f(x) notation, faced with this:

doesn't this convey the same idea even more clearly?

Put a chicken into the nuggetiser, then into the de-nuggetiser and you get back your chicken
(OK , maybe with  few ruffled feathers and a lot of squawking!)

Who would have thought there was so much laughter in a lesson on functions? Thanks to the Meat-a-Morphosis team at Kealing Middle School, Austin Texas for a wondeful teaching resource.

Who's afraid of the error monster?

Maybe it's built into our very survival instincts : if something is wrong, it's uncomfortable - so run away and hide from it. We see it in class every day, every hour - students (and teachers!) running away from errors. No-one wants to be wrong, or even worse, be seen to be wrong. And yet, when it comes to learning, "errors" are valuable tools. I see it as one my most important roles as a teacher to convince students not to be afraid of errors - on the contrary, to look for them, appreciate them and share them.

When I talk about errors with my students I introduce them to the Error Monster:

Image from
http://conservationbytes.com/2009/10/21/sleuthing-the-chinese-green-slime-monster/ 
(by CJA Bradshaw?)

We discuss ways in which this scary monster is in fact a good friend - how every error we make, or another person makes, is a valuable gift to our learning.  I encourage students to face their monster head on - whenever they do an assessment and see their mistakes, to run towards their monster and embrace it:

Adapted from CJA Bradshaw's slime monster.

Now our monster becomes an object of fun and affection - helping overcome embarrassment and disappointment at making mistakes - and allowing us to instead focus on resolving those errors.

Teacher notes:

• I like to use the analogy of a blind person using a walking cane. How could they see where to go, if they didn't make "mistakes"? It's only by having the cane bump into things the person can see where to go. Errors help guide us on our learning path.
• You have to walk the walk : be happy - and not embarrassed - to face your own errors in class. I highlight to my students my specific weaknesses when doing algebra : I know (and they know) I make silly errors with signs and expansions - so I laugh at my monster and then keep a careful eye out for him. I make a show in front of the class of checking for my common errors. Hopefully over time I will get better at these!

A visit to the Function Zoo

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.

The challenge:

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.

Download lesson slides & annotations: (Google Drive)  PDF  PowerPoint

Teaching Notes:

• 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.