Newton and Descartes channel Dan Meyer

There's a definite pause the first time you show parametric equations to  students well conditioned to Cartesian representations.  I like to imagine Descartes himself staring at the equations pondering : "Why would you do that????"

We're not in Kansas any more!
Descartes: "Why would you do that? It's the same end result!"

Here are three teaching ideas I used this year with my senior mathematics class which may supplement the traditional approach of showing the different representations are functionally equivalent.

1. Extend the function machine idea to show a weird new parametric function machine. Now we have two outputs!  Here are the two function machine images I use for my resources:

Based on a function machine diagram at http://raider.mountunion.edu/ma/MA125/Fall2011/Chapter7/IntroToFunctions.html
I removed text from the original image, then adjusted it to make the parametric machine.

2. Explore the reasons why we might want to use parametric expressions to describe a relationship. 

The best I answer I came up was this (click on the image for a larger view):

Newton and Descartes ponder Dan Meyer's "Will it hit the hoop" lesson.
My students did this activity in a previous lesson, so they got the joke.

In other words, a parametric description of this scenario lends itself to a deeper understanding of the physics of the situation. 

Another reason for using parametric equations is that the maths can be much more interesting - and possibly a lot easier to work with.  Parametrics also give us another way to get a feel for the constraints at work in a locus.  I love this wonderful "move the robot" explanation from James Tanton - and it speaks to my IT background where parameter go in, and things move accordingly!

3. Get a feel for parametrics by controlling the parameter using dynamic geometry software.  I found it really helped my students to build a parametric representation, then adjust the parameter by moving sliders and then seeing points move under their control. Actually touching and moving and parameter reinforces the idea of a point travelling along a path under a constraint.  Here is a resources for students to explore parametric representation of the parabola using GeoGebra:

HowTo Guide: Exploring the Parametric Representation of the Parabola

This guide is part of my collection at GeoGebra HowTo 

TeachMeet East At Kambala

A great way to get inspired: catch up with other teachers at a Teach Meet - come join the movement! Organised by teachers for teachers, Teach Meet seems to be really taking off in Australia.  Thanks to the teachers at Kambala school for hosting us tonight. 

My contribution:

The Function Zoo a Group Exploration Lesson Design v5 Annotated

This lesson is also available for free download at Maths Faculty

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.

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.

Two ideas for introducing functions

Here's two ideas for introducing functions to your class - none of them original, but I used them today and was pleased with just how well they worked.

1. Watch the Meat-a-Morphosis video

This amazing video is a winner with students and teachers. Powerful, simple and clear ideas about functions wrapped up in deliciously gruesome humour. 

Before watching the video: We spent a few minutes exploring the key idea of a function as a 'machine' that maps values of input variables (in the domain) to output variables (to a range), looked at the function notation f(x), and tried a few practice examples using f(x) substitutions.

After the video: We discussed the 'function machine' analogy and reviewed some of the fun examples in the video with their corresponding mathematical analogy.

2. Explore an unknown function on the calculator : the ln() function.

This idea comes directly from Mary Barnes' wonderful "Investigating Change" books*. Let the students know they have their very own "function machine" : their calculator. Ask them if they ever wondered what the ln() button does?

Students have been carrying this function machine with them for years.
So what is that ln() button all about?

Let's investigate! I gave each group of students some sheets of butcher paper and pens, and asked the question: "What does this function do to numbers? What is its domain and range?" - then let them at it, encouraging them to write, sketch, draw on the paper to show their thinking.

The results were astounding. As the work progressed, I dropped some hints to different groups to try different types of values and commented loudly (so other groups could hear!) when I saw group making a nice table or beginning to construct a graph. Some groups discovered logarithmic properties - that ln(100) was double ln(10), one group noticed ln(2) + ln(5) = ln(10), while others had discussions about asymptotes or debated with each other if their calculators were doing the right thing - the numbers seemed so odd and error messages kept coming up. For groups running ahead, I sketched y=x on top of their graph and asked them to draw a reflection. They recognised the resulting graph as an exponential one.

After the activity: I fired up GeoGebra on my board, showing how to graph the function (they groaned, having spent a long time doing it by hand :-) ), then we zoomed in and out to explore the interesting parts, referring to conversations and discoveries made by the class.  Then a good discussion on how to determine the domain and range. My not-so-secret agenda is to convince the students the value of  GeoGebra for this course - coming soon to a lesson near you!

I highly recommend this activity. Don't rush it - it will take at least fifteen minutes. Many great opportunities to develop and practice mathematical investigation skills.

* See http://books.google.com.au/books/about/Investigating_change.html?id=BjOJBR54jkIC for a preview. 

Some teaching thoughts:

• I was surprised how much the activity of exploring ln() on a calculator allowed for differentation through asking different groups different questions. One group finished early, so I gave then the challenge to investigate the hyp() button (hyperbolic trig functions .. hehe!).
• I don't think it's a problem to explore the ln() function a good six to twelve months before we might otherwise look at it. Not knowing about the function is the whole reason the exercise works.
• I think it's a mistake to start with the formal definition of a function that distinguishes between a relation and function. This puts to much focus on the idea of one-to-one mapping, before the deeper idea of the mapping aspect of functions. Start with an interim definition of mapping of a domain to a range - the refinement can come next lesson. This is also Mary Barnes' approach.
• Be ready to explain why we care about functions, as distinct from just working with our usual y = x + 2 expressions. To my thinking, the answer is that functions are themselves distinct mathematical objects - taking us to the next level of abstraction from number -> variable -> function. Equally   importantly, functions are the powerful idea we use for mathematical modelling.