Sunday, September 17, 2006

Next Lesson: Absolute Value Equations

For Tuesday's lesson, we will be looking at the meaning of absolute value, and how to solve absolute value equations. I decided to go with the traditional idea of treating |x| as the distance between x and 0, and using that as the base for solving equations such as 2|x - 5| - 8 = 4. The method we will use is:
1) isolate the absolute value
2) write the meaning in words (just for now, until it sinks in)
3) draw a number line
4) use the number line to write 2 equations and solve them
5) plot the final solutions on a new number line

This is different than what we are doing in the regular (not honors) algebra 2 classes. There, the method is pretty much the same, but we are teaching them the fundamental concept that |x - a| is the distance between x and a. They are being taught to rewrite any absolute value equation in that form first. I think it is helping them solve the basic equations and inequalities, and it is definitely helping them with the "tolerance" word problems. That's ok for the regular class, because that satisfies the standards for absolute value, and they won't be doing anything else with it for the rest of the year.

My impression, though, is that teaching it this way somehow would make it harder if you go on to study absolute value functions. For example, I want my honors students, if given a graphical representation of a function (and not necessarily the equation), to be able to graph the absolute value of that function. I think that this requires you to think of the absolute value as something that always returns a positive value (i.e. the magnitude of the distance from 0), and not necessarily the distance between two non-zero numbers. I really believe that this type of representational fluency is necessary for them to be able to understand Calculus. Anyway, if anyone has any reactions or advice, please feel free to leave a comment.

The handout and sample problems I will use for this lesson can be found on ILoveMath.


Anonymous said...

From the point of view of more advanced math, I think both concepts are important. There are different situations in which one or the other seems more "fundamental." When dealing with convergence of series, the key fact is that -|x| <= x <= |x|. On the other hand, the notion of distance d(a,b) = |a-b| is the starting point for metric topology.

I agree with the way you're teaching it. As you say, if you start with |x-a|, it's not clear what is the conceptual meaning of |x| alone. (Also, things like |x+3| cause confusion.) If you start with |x| alone, it may be easier to figure out |x-a|.

I believe that absolute value in general doesn't really make sense, in a "why do we care about this" kind of way, until you get to complex numbers. In the complex plane, if you start with the definition of |x|, then the interpretation of |x-a| as the distance from x to a comes from the idea of head-to-tail vector addition. It's the same on the real line. Anyway, I think any presentation of absolute value in one dimension only will seem unmotivated and kind of pointless. Since talking about complex numbers would be a disaster in terms of too much complication, I don't see any way to fix this.

Dan Greene said...

Thanks for the response. One of the motivations that caused us to think of teaching |x - a| as the distance from x to a are the "tolerance" types of word problems that are seen on the standardized tests. For example: "There is a kind of fish that can survive between 45 and 65 degrees. Write an absolute value inequality that models this situation. What is the ideal temperature for this fish?". Now, admittedly, this does not seem like a very relevant problem, but it's what they are being asked to do. Teaching absolute value this way allows us to think of "a" as the "ideal" (found by taking the average of the extremes - i.e. 45 and 65 in the example), and the distance between "x" and "a" as the "tolerance", "d" . Then, the model will always be |x - a| <= d.

I felt that this was too limited of an application to build the concept around for the honors class, but this is the only application in our text.

In honors, we will see complex numbers later in the year, and use absolute value in that context.

Are there any ideas out there for other applications of absolute value at an Algebra 2 level?