The key to mastering calculus seems to be gaining a good understanding of rates of change, how this relates to the idea of a function and then seeing how we can use the tools of algebra and geometry to develop the gradient function. Now as exciting as it is for some of us to play with a quadratic or a cubic function, I recently discovered, quite by chance, a very personal and highly engaging way to explore rates of change of a function.
I was exploring the introductory concepts with a Paul*, a teenager just starting on the calculus road, when he made the connection that he was experiencing a very dynamic change process: his height had started shooting up in the last few years and very soon he expected to be nearly 2m tall. He knew he was experiencing a "growth spurt", growing at a faster rate than when he was younger. Just as I was wondering how we could use this connection in our exploration, Paul told me his parents had been marking his height on the kitchen wall for the last 10 years. Wow! This was exciting - some real data we could plot and explore. Paul measured the markings off the kitchen wall and made a table showing his height at different ages. And here's what we were able to do with that data in GeoGebra:
In the process of this exploration we uncovered many ideas about slope, functions and the use of modelling, each time applying them in a context Paul had a profound personal interest in - we even named our polynomial the "Paul function" in his honour. I'm certain he will never forget the idea of rates of change, the gradient function or the power and fun of modelling a function based on data points.
Do your students have a wall somewhere with their heights measured over the last 15 years? If so, I highly recommend working this into your calculus activities.
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| This data is all about ME! |
I was exploring the introductory concepts with a Paul*, a teenager just starting on the calculus road, when he made the connection that he was experiencing a very dynamic change process: his height had started shooting up in the last few years and very soon he expected to be nearly 2m tall. He knew he was experiencing a "growth spurt", growing at a faster rate than when he was younger. Just as I was wondering how we could use this connection in our exploration, Paul told me his parents had been marking his height on the kitchen wall for the last 10 years. Wow! This was exciting - some real data we could plot and explore. Paul measured the markings off the kitchen wall and made a table showing his height at different ages. And here's what we were able to do with that data in GeoGebra:
In the process of this exploration we uncovered many ideas about slope, functions and the use of modelling, each time applying them in a context Paul had a profound personal interest in - we even named our polynomial the "Paul function" in his honour. I'm certain he will never forget the idea of rates of change, the gradient function or the power and fun of modelling a function based on data points.
Do your students have a wall somewhere with their heights measured over the last 15 years? If so, I highly recommend working this into your calculus activities.
I'm a deeply indebted to the work of Mary Barnes, in particular her 1999 series "Investigating Change", on teaching and learning calculus, still available at Curriculum Press. Google Books previews are available. Thanks also to a new Google+ friend and teacher Steve Phelps for showing me how to use the GeoGebra FitPoly function. Using GeoGebra to build a gradient trace function comes from the original GeoGebra documentation by Marcus Hohernwater, however I have found in practice it can take students some time to understand what is going on. Priscilla Allan's YouTube demo shows a good way to use colour to make it clearer and I have extended her idea to actually show the trace point moving along the x-axis, prior to adding in the gradient value.
And special thanks to Paul (*not his real name of course) and his parents for allowing me to share this story of our ongoing exploration of mathematics.
