Thursday, April 26, 2007

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:


(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.


Eric Jablow said...

The last and most general question, part 4, might amuse your students most if you were to ask the question, “How can you solve this doing the least amount of work?”

Without figuring out the form of the equation on any of the intervals, a student can say:

For very negative x, the function is a linear one. When x passes a, the form of the equation changes, but the new form is also linear. Ditto for b and c.

Now, the extrema of a linear function on an interval always occur at its endpoints. Also, as x goes to either +∞ or −∞, the function increases without bound. So, the only values of x one needs to worry about are a, b, and c.

f(a) = b + c − 2a.
f(b) = c − a.
f(c) = 2c − b − a.

Notice that f(c) = f(b) + (c − b), and f(a) = f(b) + (b − a). So, the answer is x=b. There's no need for actually computing the form of f. Applied laziness—that's the spirit!

There has to be something more than this, however, and we can find a good lesson for your students by changing the focus.

Instead of finding the extrema of a function defined by absolute values on the entire plane, let's find the extrema of a linear function on a region defined by linear inequalities. In other words, consider linear programming.

Try maximizing f(x, y) = 7x + 6y given x ≥ 0, y ≥ 0, x + y ≤ 4, 2x + y ≤ 5.

Without using graphing calculators (evil machines!), draw the region. Then, persuade the students that the solution can't be in the interior of the region. Just go up, and f will increase. Then, depending on which boundry line segment you get to, show that traveling one way or the other on it will increase the value of f. Experiment, and show you get to (1, 3) in the end.

Now, discuss what this would mean if you had many variables and many constraints. Explain where this happens in real life—deciding what products to manufacture, devising rationing systems in wartime, etc. Mention George Danzig and his simplex algorithm, discussing how little time it usually takes, but that it could be exponential in the number of variables and equations. Modern advance have created impractical polynomial-time algorithms for this, but simplex is still used the most. Finally, discuss P=NP.

You don't need discuss anything your students can't understand here; even though the techniques may be beyond them, the problems are not.

Dave Marain said...

very nice observation that the extrema will occur at endpoints or 'critical' points (the vertices of the angles formed by the pieces) and the connections to linear programming and simplexes are wonderful...
The fact that the min will always occur at 'b' should be of interest as well. Although one does not need to work out the individual 'cases' for this purpose, I was attempting to provide practice for a 'case' analysis, which I believe needs to be part of their training. Even more interesting is to provide a linear piecewise graph first and have the student develop the function both with and without absolute values! After all, students see line graphs every day without ever considering there are functions representing them - mathematical models!

I get the sense that your opinion of the graphing calculator is comparable to some people's view of the 'evil empire' of some baseball team! My take is that, as long as the instructor requires students to do simple graphs and solutions by hand first, it is a useful tool to allow students to probe further, gain better visualizations and explore 'what if' questions that I was not able to consider when I was learning all of this. Your comment about finding the 'easy' way unfortunately is viewed differently by students in the 21st century! What we see every day in our classes is that students' mental math skills and recall of important knowledge has declined rapidly over the past few decades. It can't all be blamed on the calculator, but it is part of the picture. Kind of like a paraphrase of a well-known comment we hear: "Calculators don't kill kids' minds, ..."

Eric Jablow said...

There's a difference between what a student needs to learn and what a working mathematician or engineer needs to do. I agree: go through all the cases in at least one example, but then talk about how an engineer or scientist or statistician would learn to do it in practice. I think that no one who needs to use graphing calculators should be permitted to use one.

I mentioned LP because it was a good way to connect your topic with a practical problem done every day in the outside world. So often, I hear, "Why do I need to learn this? I'll never use it!" That may be true for finding minima of sums of absolute values. But some of the techniques are cruical in industry, the military, public health, and other practical fields. Even if you couldn't discuss the matrix-based method of the simplex-algorithm, you could point out and prove that the solution is at a corner. You could then point out how many corners exist in general, and then talk about complexity. That's an important issue, and it doesn't require rigor to usefully think about.

Dave Marain said...

That's why there should be even more networking between science, industry and the math/science classrooms. There is some group of engineers that reaches out to students here in NJ and I suspect there are groups in every state. We need much more of this to bring the real world into the classroom.
I have always tried to make math meaningful for my students but, not being an applied mathematician, means I have to do a great deal of research to find these applications. I always need help!

I think you know the primary goal of this blog, other than ego-fulfillment, is to provide models of actual lessons teachers can use/revise in the classroom. Lessons that have been classroom-tested...
Even if most educators do not like them, at least, it might plant some seeds for developing their own lessons that go beyond the routines of classroom mathematics. I love investigations that lead to understanding of important concepts. They are not easy to do, but I have been trying to write these for 37 years and continue to try...
Thanks again for your valuable insights and practical views.

jonathan said...

I like this set of exercises. There is application far beyond absolute value, to any situation where separate cases are to be considered under separate conditions, including both outside of math, and outside of industry... You've isolated a valuable critical thinking skill.

Dave Marain said...

thanks jonathan!
piecewise functions and analysis by cases is a crucial topic for precalculus, calculus, and beyond. If you check NCTM's standards and even the benchmarks for Algebra 2 in the American Diploma Project, this does not get much attention. This is why the 'vertical teams' approach makes sense. The Calculus teacher works with the precalculus teacher to develop a curriculum that makes sense. This approach would then filter down to Algebra 2, Algebra 1, prealgebra, etc. with geometry included as well. Some districts are doing this but more often districts rely on curricula developed by some external group that catches someone's attention. We need more consistency...

Eric Jablow said...

Here's a HTML hint, Dave. &le; gives ≤, and &ge; gives ≥. &ne; gives ≠, and &equiv; gives ≡. Here is a good general reference.