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?
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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.
* 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.