Overview

Time: (35-45 minutes)

Purpose: The objective of this lesson is to have students develop their own definition of function based on the characteristics and behavior of the variables. Too often students become fixated on simplistic, limited features such as the vertical-line test (which is useful only for a subset of Cartesian graphs). By concentrating on the behavior of variables, students engaging in this activity can develop a more meaningful definition on their own, a definition that will help them discover the vertical-line test themselves, in a way that enables them to understand both why it works and under what circumstances it’s useful.

Entrance/Exit Tickets: (optional) The entrance ticket asks students to think about how function is used in non-math contexts. The exit ticket will ask them to define function in terms that relate to their experiences during the lesson, dragging the independent variable and observing the resulting covariation.

Launch: As students arrive, pass out the page 1 worksheet and divide them into pairs, with each pair taking turns on a single computer. The lesson url is at the bottom of the worksheet; if you have a class webpage, you can provide the link there.

Objectives

Students will:

become familiar with independent and dependent variables in a geometric context,
identify an independent variableas a variable that con be moved around, and that controls the motion of some other variable,
identify a dependent variable as a variable that con't be moved directly, but instead responds to the motion of another variable on which it depends.
drag an independent variable while observing the motion of the corresponding dependent variables,
notice that different functions involve different relative movements in the two variables,
experience the motion of variables in a more direct and visual way than is typically the case with algebraic variables,
describe the relative motion (covariation) of independent and dependent variables (attending to both speed and direction),
distinguish functions from relations that are not functions,
locate fixed points of functions,
observe that some functions have multiple fixed points, some have a single fixed point, and some have no fixed point at all, and
write a definition of function in their own words.

Warmup

Time: (10–15 minutes)

Sketch Page 1: Pass out the worksheets and project page 1 of the sketch. Invite a student to drag point `a.` Ask other students to describe the result; they will observe that `a` and `y` are related. Ask students what they think will happen if `y` is dragged, and then ask a student to actually drag `y` to see what happens. Discuss the independent/dependent relationship between the two points, noting that `a → y`. Tell students to fill in the independent and dependent columns of the first row on their worksheets.

If students have already been introduced to the function concept in algebra, ask them to discuss with each other how the variables `a` and `y` are similar to algebraic variables and how they’re different. Students may observe that they can substitute a number for the value of independent variable a in an equation like `y = 2a + 3` and calculate a result for the dependent variable `y`. Ask students, “If `a` is a variable in `y = 2a + 3, `how can you vary it?” Encourage students to realize that they can vary the independent variable `a` in either type of function and see what effect the change has on the dependent variable y.

Ask another student to continue dragging `a` to help classmates fill in the third column for this first geometric function. Does `y` move faster than `a,` slower, or at the same speed? Does `y` move in the same direction as `a,` or a different direction? (Look carefully!). Are there any places (fixed points) where you can make `a` come together with `y`?

Tell students to work with their partners to investigate the behavior of any other functions they can find, and fill in a worksheet row for each function they find. Circulate as students work, encouraging them to explain their observations clearly, asking clarifying questions when appropriate, and prompting them to pay close attention to the behavior they observe. As you circulate make a mental note of students you might call on later to describe what they noticed, or what they wondered, to the rest of the class. As students finish Q1 and Q2 on the worksheet they should go on to page 2 of the sketch and begin filling in Q3.

Sketch Page 2: When most of the class has written answers for worksheet questions 1 and 2, tell the class to move on to page 2 of the sketch. Project this page of the sketch, draw students’ attention to the Q3 table in the worksheet, and tell students "Now we'll fill in the first row of the table together." Ask one student to try dragging points until she finds an independent variable, and then ask another student how to fill in the first cell of the table to identify the independent and dependent variables. As students fill in this cell, describe it using appropriate mathematical language: tell them they can read `p -> q` as "the function that takes `p` to `q`," or (less formally) as "`p` goes to `q`." Have a second student take over dragging duties, and ask a third to describe the relative speed and direction of the variables. Ask a fourth student to drag in a way that reveals the presence and location of zero, one, or multiple fixed points. After filling in the first row of the table, tell students to continue working in pairs, switching roles on each row of the table while they fill in their observations of the remaining functions.

Group Discussion: Shortly after the first groups have completed their tables, begin the class discussion, assuring groups that haven't yet finished their table that they can skip the last question or two. With all computer lids closed, spend three to five minutes in the group discussion, giving students a chance to describe what they noticed and asking them what they wondered.

Examples and Non-Examples

Time: (20-30 minutes)

Sketch Pages 1–2: Pass out the second worksheet (if you didn't print the worksheet two-sided) and tell students to use pages 1 and 2 of the sketch and to record their answers as Q4 and Q5 on the worksheet. Remind them that for Q4 one partner should drive the computer while her partner records their answers and that they should switch roles every time they go to a new sketch page. Circulate as students work, looking for issues to bring up with the entire class.

Group Checkin: (only if necessary) If there's an important issue or problem that needs the attention of the entire class, the time when students are moving from Q5 to Q6 is the best time to do so. Keep such a checkin short so as to save time for a concluding discussion after students have encountered several interesting behaviors and issues on later pages of the sketch.

Group Discussion: Reserve at least 5 minutes, and preferably more, for the wrap-up discussion described below.

Wrap Up

Concluding Group Discussion

Students are likely to have a number of interesting things that they wonder about. Some of the non-functions have more than just two multiple values, and others act like functions over much of their domain. Some dependent variables don't move continuously, and some don't move at all. Some students may point out that when an independent variable has its dependent variable in two different locations (values) at the same time, why can't we just say that it's two different functions that just happen to be connected to the same independent variable?

At this stage students need serious answers to these questions, but the answers should be non-technical, and should bring the class's attention back to the task at hand: defining what a function is. The common thread to approaching such questions should be, "How does this question inform our effort to write a clear and useful definition of function? The hard cases help us to test our definition, but any changes we make to the definition should be governed by simplicity and clarity.

(The efforts of mathematicians to define function have been intense over the years, and have ultimately led to a definition that's based in set theory and of little use to secondary students. This mathematical definition was arrived at to accommodate some very badly-behaved functions—sometimes referred to as "pathological" functions. Though this set-theory definition appears in many secondary math textbooks, it is harder to understand, and less useful, than the definitions your students are likely to arrive at.)

Below are some of the questions that are raised by later pages of the sketch, and that can inform students' efforts to arrive at the best definition that they can.

Pages 5 and 6 contain non-functions that appear to be functions for much of their domains.
Page 7 contains a function that's both discontinuous and periodic; students may find either of these characteristics disquieting.
Page 8: One of these candidates seems to be a function on part of its domain, but is clearly not a function once students encounter the problematic part of the domain. Students may wonder whether we can just wall off that part of the domain, and say that you're only allowed to drag the independent variable past the wall.
Page 9: All of the candidate dependent variables move discontinuously, and students may want to disqualify variables that behave this badly. This might be a good time to point out one reason we need clear definitions is to provide guidance in difficult cases like this one.
Page 10: Point `h'` never moves, so how can it be a dependent variable? Yet it does seem to fit the definition based on the fact that there are never multiple values of `h'` for the same value as `h`. This is a good question for students to continue to wonder about; right now it's best to avoid giving them the simple expert answer that `h -> h'` is a constant function.

These hard cases give students a chance to clarify and simplify their definition. Try to end the discussion with at least two and preferably more candidate definitions displayed, and ask students to continue thinking about these definitions. Over the next few class sessions, return to the candidate defintions on occasion, with an aim of arriving at a concensus definition within a week or so.