Introducing the concept of function

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 or shown to the students is how to set up and represent the relationships in tables, graphs, and equation. 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 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: Read more of this post

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