How to Teach Absolute Value
As an introduction to absolute values, my eighth-grade daughter was taught that, for example, |-4| equals 4. That sounds like a good start. She learned that the absolute value means to take away the minus sign. That sounds like a good start.
But there are no concepts here. That is a bad omen.
I asked her the meaning of |x - 4|. She took away the minus sign and said it equalled x + 4. Guess what? She didn't have the concept! (To be fair to my daughter, her textbook seems to have a completely misleading definition of the concept, at least as I understand it.)
How to Teach
One of the basic techniques for teaching a concept is to ask, what problem is this concept an answer for? Then start with the problem. There isn't any huge concept here, but there is a useful concept. In this case, we could have equally started with the question, "Why did mathematicians develop this convention?
Problem. On a ruler, there is a mark at 9 and a mark at a. What is the distance between the two marks?
Of course, for the usual skill of a student at this level, it is useful to start with the marks between at defined numbers, like 6 and 3. Then I put one mark at 0 and one at a. Given that the value of a is unknown, the best that can be said is that the distance is a. Or I put a mark at 12 (on an imaginary 12 inch ruler) and a mark at a. Here the answer is 12-a.
Then you have a problem like one mark is at 9 and the other mark is at a. What is the difference between the two? If the student answers a-9, it is easy to show by specific examples of a that this probably works for values less than 9. (e.g., it works when a = 6.)
Then it is easy to show that this formula does not work for a = 10, because it produces a distance of -1, when the student knows the correct answer is 1.
The student should be able to work to this answer:
a - 9 when a > 9
9 - a when 9 > a
You can decide if you want to discuss the fact that this leaves the answer undefined when a = 9. There are several ways to solve this minor difficulty.
Now you have done all the work, time to add the factoid the student cannot figure out on his/her own. To make the answer a little easier to write down, mathematicians have a symbol for absolute value, which is for this problem |a - 9|. ("Absolute" is probably a misnomer.)