University of Arizona

Math 407, Fall 2006, Functions Module

Goals

In the functions module we continued our focus on the structure and meaning of algebraic expressions, but applied them to the concept of function. Functions are to algebra what variables are to arithmetic. Just as in algebra, variables are used to represent numbers, in the study of functions we use letters to represent entire functional relationships. Our goal in this section was to sort out the relationship between these levels of abstraction, and develop ways of thinking about functions that would help high school students see the difference betweent hem.

Syllabus

Week 1: Extended analysis

Extended analysis of a high school mathematics problem (see sidebar). Here is the whiteboard showing the various solutions.


Week 2: Definition of function

Definition of a function, different ways in which functions arise. Slide showing the list of function candidates considered in Tuesday's class. From class discussion we generated the following list:

  • Given sets A, B a function assigns at most one element in B to an element in A, making ordered pairs (a,b)
  • A function is defined as an equation in one variable where each input produces a unique output. f(x) = x^2+6
  • a function is a rule or set of mathematical operations that is performed on an input and consequently produces a specific output
  • a function is a system of components affecting an operation.

For Thursdays's class we refined the first definition (see sidebar).


Week 3: Study of high school texts

Study of the definition of a function as presented in various high school textbooks. Group presentations on the questions

  • In what ways does the text define and use the concept of function?
  • How does the textbook support meaningful understandings of the concept of function?
  • In what ways does it allow the student to avoid meaningful understandings of the concept of function?

Week 4: Analysis of functions through algebraic representations

The importance of proof: The comparison of growth rates of power functions and exponential functions: preliminary explorations on a mathematical justification that exponentials grow faster than powers. Assignment 2 and Hints

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Week 5: Analysis of functions through functional equations

The importance of definitions: definition of logarithm, deriving the properties of logarithms from the properties of exponents.



Car A sets out going at 50 mph. Starting 3 hours later, car B tries to catch up. If car B goes at 75 mph, when does B catch up with A?

  • Solve the problem with the constants replaced by parameters: Car A going at vA mph, car B starting th hours later and going at vB mph.
  • Think of the solution as a function of the parameters. How does it depend on the head-start time, th?
  • How does it depend on the velocities? Can you rewrite the expression for the function to bring out these dependencies more clearly?
From Rethinking Secondary Mathematics, part of the TEXTEAMS materials of The Charles A. Dana Center, The University of Texas at Austin.

Defining functions

Look at the first definition we came up with in class.

  • Think about how to modify this definition to say what a function is, rather than what it does. You can use wording from the other two definitions, but bear in mind that we want to make clear that in the definition of a function it doesn't matter how you get the output, just what the output is.
  • Also, look at the function candidates we considered and think about what you need to add to each one to make it a function according to your definition.