## Wednesday, February 22, 2012

### 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.

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.

1. Yeah gday.
Nice idea. I've hardly used geogebra - if you want to write a couple of hundred words about how to make geogebra draw this graph (from a table of values? or as a continuous function?) I'd be mighty grateful - and promise to try it out.
Cheers,
@bewdyrooster

1. Ooh - a rich treasure awaits! Actually - this is a good start. Fire up GeoGebra. Go to the input bar at the bottom left, type in "ln(x)" and press return. You should see the graph. Now explore moving around, zooming in and out. You can right click on an empty area to change scale, or you can hold down the ctrl key which hovering on an axis, then drag. If it would help, I could make a video clip next time I get some hours free.

2. The ln() investigation - A great 'low floor-high ceiling' task!

I think that I'll probably do it with the log() function since I think they'll find it easier to see and understand the connection with exponents.

Thank you for this post.

3. For most kids, younger ones anyway, you could do the same thing with the 1/x button/function.

1. Great observation Kate!

I do note with sadness that many calculators don't seem to have a 1/ x button. Instead they use the shift/function button to access an x^-1 functionality above a proper button. Lots of reasons I think this is unfortunate, most importantly, having such an obvious 1/x button encourages students to use multiplicative inverse as part of thinking through calculations involving division. (Bring back RPN! :-)