Saturday, October 13, 2012

Polynomial stories

What's not to like to about polynomials? They look amazing - and they are just great fun to play with - especially if you have dynamic graphing software to explore their shapes. Here are a few teaching ideas I developed over the last few weeks.

First and foremost we need a character : meet Polly the Amazonian parrot. There's a reason she is from the Amazon... you'll see soon.

Amazona agilis by Jacques Barraband(1767-1809)

Polly featured throughout my lessons - my favourite was the zero polynomial $P(x) = 0$

Reminds me of a Monty Python sketch ....

and later when we looked at taking the second derivative, then the third, fourth, and fifth derivatives - the disappearing Polly:

The Disappearing Polynomial

I'm a big believer in having 'characters' to help teach mathematics - I think they act as 'mental anchor points' to help link related concepts, and then make high level linkages across topics more visible - as in the example above of the disappearing polynomial.

Then we need to explore the properties of polynomials. If we have a quartic polynomial, how does its graph change if we change the roots? I got a good response to my students with this homework exploration tool - I used it for a 'flipped' lesson:
Lesson 02 FLIPPED Graphs of Polynomials

Early on in the presentation of polynomials, I think it's a good idea to show some of the interesting applications:

Using polynomials for modelling. I used this example of a photograph of the Amazon river, loaded into GeoGebra, fitted to a polynomial using the FitPoly function.  Why would we want to do this? I suggested in this case, having an equation for the Amazon River could help us model water flow - perhaps helped by working out the gradient function:

Amazon River – photo from NASA. 
Curve fitting using GeoGebra FitPoly[] function.

Using polynomials to approximate other functions: A good time I think to introduce the Taylor Series:

No need to go into deep explanations - just show what is possible with polynomials. I returned to this idea in the next topic when showing higher derivatives, and have another visit planned when we do complex numbers to help demonstrate the famous Euler Formula.

And finally, after demonstrating the closure of polynomial operations for addition, subtraction and multiplication, students may find it interesting to learn about the role polynomials play in many encryption systems.

I looked hard for an online paper suitable for advanced high school students - the best I have found so far is Christopher Cooper's notes for his "Languages and Machines" course at Macquarie University.

More GeoGebra HowTo Sheets

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