Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.
Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y or f(x) given x and vice versa, not reading graphs, and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:
- identify the changing and unchanging quantities;
- determine the effect of the change of one quantity over the others;
- describe the properties of the relationship; and,
- think of ways for describing and representing these relationships.
These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.
Sample introductory activity:
What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?
Click here or the image above to go to dynamic window for the worksheet.
I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.