Ideas from Classroom Teachers for
Functions: Polynomial, Rational, and Radical
A good review of what has been taught in year 1 should be included at the beginning of this section. If possible, use data that have been collected and saved earlier as examples. This could be student, teacher, or class data.
On defining function as a rule: Use iterations to show input and output just a flowchart of the input - output for a specific number of iterations.
When reviewing linear and quadratic functions: A review of 1st and 2nd degree polynomial topics would be time well spent here. Then expand the lesson to include polynomials of higher degree.
A connection between piece-wise and absolute value functions should be emphasized.
Function notation should be emphasized at this point, and a connection between graphical and algebraic representations should be made. Also end behavior of even- and odd-degree polynomials needs to be addressed informally (without the use of limit notation).
Concerning linear factors: More advanced factoring techniques, such as the difference and sum of cubes, and factoring by grouping, could be addressed at this stage. Also the Rational Zeros Theorem could be addressed, especially for those with access to graphing calculators.
When finding the vertical and horizontal asymptotes, students should be able to use a discovery approach to determine the asymptotes. Graphing calculators would be useful here. This may reduce the time necessary for this section.
End behavior and local behavior are important ideas that should be developed informally (without limit notation) at this time.
Under radical functions: Square root and cube root should be the emphasis here with an introduction to other root functions and especially their relationship to the square and cube root functions (domains, ranges, end behaviors, growth rates).
On inverse functions: Emphasis should be placed on illustrating inverses in numerical form (switch x and y in ordered pairs), algebraic form (switch x and y in the equation), and graphical form (reflection over y = x line). Also the definition of inverses relative to composition should be emphasized to illustrate why some functions are inverses (y = x^3 and y = x^(1/3)) and why some are not (y = x^2 and y = x^(1/2)), even though powers and roots are treated as inverse operations when solving equations.
For this chapter, an alternative assessment could be used here with some journaling, especially with the graphing calculator activities in finding asymptotes.
For the B’‘ group, range and domain need to be stressed several times. Also, I would spend more time on higher degree and absolute value functions and less time on piece-wise functions.
Pizza Pi: Work Force (http://www.hsor.org/modules.cfm?name=Pizza_Pi) does a good job of connecting tables, graphs, and symbolic representation for lines, parabolas, and exponential functions. It also has a good linear programming problem that can be used with this topic.
On the ODE website, search for IMS lessons that address these standards; there are several on roots and solutions in particular.
For sources of functions, datasets are available online or elsewhere. For example, see Exploring Data (http://exploringdata.cqu.edu.au).
Many activities will generate data that are represented by various functions, e.g., Spaghetti and Pennies, Stacking Cups, Pendulum (in Navigating Through Algebra, NCTM), and using a CBL with Cooling and Warming Models.
See the exploration activity in Algebra in a Technological World (NCTM).
Return to Functions: Polynomial, Rational, and Radical