Teaching Notes for Patterns in Polynomials
When we developed this project, we were really torn between two possible project statements. We considered just defining the Chebyshev polynomials recursively and then asking kids to see what properties the polynomials had. The appeal of this approach was that it did't require any trigonometry
a topic we expect some kids, and teachers to react to negatively. On the other hand, trigonometry seems to be the easiest (but probably not the only) way to derive many properties of the Chebyshev polynomials.
As an alternative to the Teaching Notes you see here, you
could start with the recursive definition from the Results
section.
A "phase" in the following outline could very well equate to two or three days, particularly if your classes are less than an hour long.
Phase 1
Don't distribute the problem statement immediately. We presume that your students will be familiar with polynomials and how to manipulate them. However, a phrase like "a polynomial in 
" might not be familiar. Explain that they will be exploring an interesting "family" of polynomials that will be described with this terminology.
Start with the example that serves as preface to the first Warm-up Problem: to show that 
is a polynomial in 
. You'll probably find that many students will get only as far as 
. You'll need to clarify that you want y only in terms of x. If no one comes up with the solution, provide the hint that 
and ask what to fill in. Finish the example as a whole class and then go on to the rest of Warm-up problem 1. The last two parts, d and e, will probably require a digression to review of the identities for 
and 
. If you've recently covered these identities you might not need to do Warm-up Problems 2 and 3.
After the class has completed the first few Warm-up Problems, distribute the project statement. The definition of 
is crucial, so make sure that everyone understands how to find 
. Assign Question 1 in the project statement before the next phase.
Phase 2
Collect results from Question 1.. You might want to go one step further and derive the formula for 
as a class, to review the process. See if any observations can be made from this preliminary information. Discuss how you might proceed if you wanted a polynomial that is much further down the list, such as 
. If nobody has brought it up at this stage, suggest the possibility of using recursion to generate the polynomials. Point out how the identity in Question 2. relates three consecutive 
. Use the recurrence formula to generate a few more 
.
You should now have enough polynomials to start looking for patterns in the coefficients. Give your class time to look for patterns. See if they can prove some conjectures, or at least verify them by extending the list of polynomials.
Phase 3
Use graphing technology to examine the graphs of the Chebyshev polynomials. Start with the standard window, which is normally 
and 
. Your students should notice that zooming in will be necessary to highlight the interesting parts of the graphs. Eventually, you'll want to zoom to 
and 
. Ask for observations on the graphs, including why the range is 
when we restrict the domain to 
.
To bring closure to the project, ask your students to write up a report that summarizes the properties of the Chebyshev polynomials that they have learned. The report should start with their own a definiton of the Chebyshev polynomials, trigonometric or recursive, whichever they are most comfortable with; and a table of the first few polynomials in the sequence. Some of your students might also want to try to find out something about the mathematician Pafnuty Lvovich Chebyshev. He's famous for several scientific accomplishments
so famous that he is on a postage stamp!
