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.
Extended analysis of a high school mathematics problem (see sidebar). Here is the whiteboard showing the various solutions.
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:
For Thursdays's class we refined the first definition (see sidebar).
Study of the definition of a function as presented in various high school textbooks. Group presentations on the questions
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
.The importance of definitions: definition of logarithm, deriving the properties of logarithms from the properties of exponents.
Look at the first definition we came up with in class.