One approach is to begin with the formal definitions: Relations are pairs of numbers, and functions are sets of numbers such that for every input there is a single output. However, most high school students are not capable of abstract thinking. It is essential to begin with concrete examples, and then, afterwards, point out the abstract principle.

 

I began thinking about this problem when I noticed that the students were restless during the explanation. They were not restless because it was the afternoon. When a different topic was introduced later during the class, they were not restless. Instead, it may be fair to assume that their restlessness may have been due to the inability to comprehend the abstraction as it was introduced.

 

Here is one way to introduce the topic, starting with concrete items rather than abstract concepts. Write down three pairs of numbers, and show the points on a graph. Then on a second graph write down the same three pairs of numbers, with a fourth point on the same vertical line as the second point. Call on students, one at a time, by name. For the first point, if x = …, what is y? Do the same with the second graph. Here a student, say, John, will not be able to answer where y is, because there are two points. Explain that this relation is not a function, because John (look at him) could not answer where y is. Then write the formal definitions, referring to the examples.

 

This has the advantage of making immediately clear the vertical line test. With the book’s approach, the vertical line test seems to be one of the mysteries of mathematics, true because the teacher said so. We want them to understand it because it is obviously true

 

This approach is better than the approach in the textbook, which involves drawing two rectangles, one for the x and the other for the y, with arrows going from one to the other. Once they understand the concept of relations the rectangles can be used, but to use them in the initial explanations may not be helpful, as they may not fully understand what the rectangles are all about. It may be better to use points on graphs, which is more meaningful to them.

 

Indeed, this approach is frequently used by mathematicians to learn new ideas. Looking at specific situations, the mathematician generalizes to a basic principle. Here too, the students look at the specific points on the graphs, and use it to understand the principles of relations and functions.

 

Another example of relations is age of trees (number of rings) vs. the size. We see that since the same number of rings may have different sizes, size is not a function of age. This is a good example, and is given in the book. However, when explaining it we have to write everything down explicitly on the board:

 

Input:     age (number of rings)

Output:  size (width of trunk)

 

We cannot write size(age), and so size is not a function.

 

Incidentally, when making this statement, the teacher can write down size(age), and so introduce the function terminology f(x), which would be introduced later.