Absolute Zero Part II: Applying Piecewise Function Approach for Algebra 2

[Update: Read Eric Jablow's profound comments on this post and some general discussion of graphing calculators...]

As promised, here's another installment of a piecewise function development of the absolute value function, suitable for advanced Algebra 1 students but more appropriate for Algebra 2. You may not agree with the target audience or the approach, but I have used it with mixed effectiveness. Of course you can redesign it to meet the needs of your students but the key ingredient is the use of function tables. Do you see the Rule of Four being utilized? I apologixe in advance for the klutzy formatting of the tables and the inequality symbols. I will eventually clean this up.

1. Consider the functions, f(x) = |x|, g(x) = x, and h(x) = -x.

(a) Complete the following function table.

x ............... Y1=f(x)=|x|...............Y2=g(x)=x...............Y3=h(x)=-x
-3.............. 3 ................................ -3 .......................... 3
-2............... ___ ............................ ___ ..................... ___
-1............... ___ ............................ ___ .................... ___
0............... ___ ........................... ___ .................... ___
1................. ___ ........................... ___ ................... ___
2................ ___ ........................... ___ ................... ___
3................ ___ ........................... ___ ................... ___

(b) Sketch the graphs of f(x), g(x) and h(x) on the same set of axes in THREE different colors on the domain [-3,3].

(c) Answer the following based on the table and graphs:
f(x) = g(x) when x is _________
f(x) = h(x) when x is _________
Now, rewrite this symbolically as:
|x| = x when x is __________ and
|x| = -x when x is _________.

[Note: This could easily have been handled on a graphing calculator, which is why the functions are labeled Y1 and Y2. This is one of the best uses of this technology. However, I'm a believer in doing it by hand the first time around - your choice! Also, note the heuristic of repeating the function on each line rather than the standard braces used for piecewise definition. Later on the student can abbreviate the format. ]

2. Consider the function f(x) = |x| - x
(a) Complete the table:

x..............Y1=f(x)=|x|-x
-3
-2
-1
0
1
2
3

(b) Sketch the graph of f(x) on the domain [-3,3].
(c) From the table and/or the graph we conclude that
f(x) = _____ for x < 0;
f(x) = _____ for x ≥ 0

3. [More difficult] Consider the function f(x) = |x-2| + |x-4| + |x-6|
(a) Make a table of values for f using the ten integer values from x = -2 to x = 7 inclusive.
(b) Sketch the graph of f.
(c) Define f piecewise, similar to 2(c).
(d) Determine the coordinates of the minimum point of f. Justify.

4. [The Generalization] Consider the function f(x) = |x-a| + |x-b| + |x-c|,
where a < b < c
(a) Define f piecewise as in 3(c).
(b) Determine the coordinates of the minimum point of f. Justify.

Absolute Zero Part I: Do Absolute Values Leave Some Out in the Cold?

[If you absolutely can't wait for a challenge problem, go to the bottom, but it might be worth reading through this first...]

Absolute value equations and inequalities, in particular, are notoriously difficult for most students. If you are a math educator, is this a topic you relish?

So what method of solution usually works best for the student? What method of presentation is most effective for the instructor? If the equations are straightforward such as |x+3| = 7, most students seem comfortable with expressing the equation as a disjunction: x+3 = 7 or
x+3 = -7. Some instructors, in preparing students for the technical definition (using cases), require the student to express this as x+3 = 7 or -x-3 = 7. Rarely have I observed instructors introduce the full-blown piecewise definition using cases early on in algebra:
x+3 = 7 if x>=-3
OR
-x-3= 7 if x<-3. This is generally believed to be too sophisticated for an introductory treatment. Motivating the technical definition of |x| usually comes later on in Algebra 2. However, I have always been a bit uncomfortable teaching the traditional algorithm for absolute value inequality problems such as |x+3| "<" 7 which leads to the conjunction x+3 ">" -7 AND x+3 "<" 7 or, in combined form, -7 "<" x+3 "<" 7. Math instructors devise creative mnemonics to help students recall the procedure. This all begins when prealgebra students are exposed to the verbal description of the piecewise definition of the absolute value function:
The absolute value of a positive number is that number and the absolute value of a negative number is its opposite. The absolute value of zero is zero.

This is immediately followed by a number of numerical examples, guided practice with some more complicated variations involving mixed operations and an assignment. The student sees this as another example of something they're supposed to learn in math without much meaning attached. Most catch on to the idea by repetition and errors are generally caused by weaknesses with signed numbers or order of operations.

However, some educators prefer the 'distance' interpretation of absolute value to make this notion more meaningful. Technically one has to distinguish between the real number, x, and the graph of x on a number line, but this distinction is often sacrificed for clarity:

The absolute value of x is its distance from zero.

Thus, both 8 and -8 have the same absolute value because they are the same distance from zero on a number line. Since distance cannot be negative, we have a powerful visual model that students can use to make sense of this idea, although it seems limited when solving more complicated equations or inequalities later on.

Stay tuned for more on this topic (including a 2-dimensional graphing approach using functions), but, since some readers are disappointed if there is no challenge problem, here's one for you. It's not that difficult, a version of this has recently been tested on the SAT and it can be approached in a variety of ways (formal algorithm, guess-test, etc.), but I will challenge you to solve it using the distance model! Don't hesitate to take strong exception to my comments above!

For how many positive integer values of x is
|x-2007| > x?