Intent

This activity offers students opportunities to find several rules that fit a single data point.

Mathematics

An infinite number of functions can be found to fit a single data point. Their behavior—aside from what happens at that point—will vary widely. This activity gives students some experience with the diversity of functions at their disposal to address such a situation. As part of the discussion, they will be introduced to function notation.

Progression

Students will work on the activity individually and then share their results with their groups and the whole class.

Approximate Time

5 minutes for introduction

25 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Doing the Activity

If time permits, present an example of a single data point for which students are to write a rule. For example, for the pair In = 1, Out = 2, three possible rules are y = x + 1, y = x² + 1, and y = 3x -1.

Discussing and Debriefing the Activity

For each of the two questions, you might compile a list of all the rules that students created to fit the given number pair and then plot the number pairs for at least three of those rules.

Then ask, What do your graphs from Question 1c have in common? It may be obvious, but be sure students realize that any correct graph must go through the point (2, 5). Because their rules were created to fit the In-Out pair (2, 5), the graphs for those rules will also include the point (2, 5).

In preparation for later work in this unit, it is important that students see both linear and nonlinear examples for each case.

For a nonlinear rule, try to draw out the rule y=xy=x size 12{y= sqrt {x} } {} for Question 2, as a square-root function will be needed for the final pendulum analysis. (Students will become aware of square-root functions in other ways before that time, so there is no need to push this topic if it doesn’t come easily.)

Now tell students that they will learn another way of expressing a function, called function notation. Explain that, with function notation, we give each function a name consisting of a single letter. For instance, suppose that one of the rules found for Question 1 was y = 2x + 1 and that a student found the additional ordered pairs for this rule shown below.

Ask for a letter of the alphabet, and tell students that we could use that letter as the name of this function. For this discussion, we will use the letter b.

Tell students that they can think of the table as “the b rule” and of the output as the result of “doing b” to the input. Explain that we write “doing b” to something, say x, as b(x), and that the standard way to read this expression is “b of x.”Some alternative, and perhaps clearer, ways to read this include “the b-value of x” and “b applied to x.” You may want to use such phrasing initially, but students should gradually get used to the standard terminology.

Illustrate this new notation with examples. For instance, in the first row of the table, 2 is the In and 5 is the Out. Ask how this might be expressed using b. Help students see that they can express this using the equation b(2) = 5. It is standard terminology to refer to 5 as “the value of the function b at x = 2” as well as calling it simply “b of 2.”

Ask how this notation could be used to write the general equation for the rule. Bring out that, instead of writing y = 2x + 1, the equation for the function can be written as b(x) = 2x + 1.

Have students practice this notation and language. Ask, for example, What is b(0)? b(7)?

Use similar examples to develop the insight that any letter can be used as the In to define the function. For instance, the equation b(t) = 2t + 1 defines the same function as the equation b(x) = 2x + 1.

Let students make up some letter combinations to stand for other functions that fit the given data pairs. For example, they might write h(x) = x + 3 or f(x) = 5 for Question 1, and c(x) = x - 2 or g(x)=xg(x)=x size 12{g \( x \) = sqrt {x} } {} for Question 2.

This new way to represent functions symbolically is a good occasion to review the four approaches to functions that students have seen, especially as a lead-in to connecting functions to the unit problem.

Ask, Can you think of a way to express the goal of the unit in terms of function language and notation? If necessary, have students express the goal in words first. They have seen that the period of a pendulum is determined primarily by the length of the pendulum, so they may say that they want to figure out what they can do computationally when they know the length in order to find the corresponding period.

If necessary, ask them to suggest variables to represent a pendulum’s length and periodand use them to express the goal using function notation. For example, if they choose L and P to represent the length and the period, they might express the unit goal like this.

Goal: To find an equation for a function f for which P = f(L).

Ask more specifically what they want to know about this function in terms of the exact situation in Poe’s story. If needed, remind them of their interest in the 30-foot pendulum. Help them to see that the goal for the specific situation can be expressed like this.

Goal: To findf(30), where f is the function that gives the period of a pendulum in terms of its length, which is 30 feet.

Students should see that if they have an equation or rule for f(L), finding f(30) is just a matter of substituting 30 for L.

Key Questions

What are all the rules you found for each number pair?

What do your graphs from Question 1c have in common?

How can you express the goal of the unit in terms of function notation?