Absolute Value Statements (and: Could You Generalize That?)

There was a question once on the PSSA (Pennsylvania state test) when I taught in Philadelphia that presented a range of acceptable values for the width of a nail head produced by a factory. There was an ideal (target) width that the nails were supposed to have, and then there was a certain amount of error that was allowed. The students were supposed to recognize that the interval of acceptable values could be expressed by an absolute value inequality, and then write that inequality.

It struck me that this was a great way to teach absolute value statements in the first place. With this, we could show that such statements actually have some utility in the world. Here are two (long) lessons that resulted. 

The Nail Factory (download): This is a four-part lesson that approaches the PSSA question's premise from different directions.

In the first part, the students measure nails (or at least virtual nails, on paper) and decide which ones satisfy a certain inequality and which don't.

In the second part, the students draw their own nails to show examples of what would be acceptable and what would be rejected. This is the kind of lesson that would require short discussions along the way; if you have a document camera, you could show student work at this point. The savvy student would draw the good nails all about the same size, and draw the rejected nails by exaggerating short and long lengths.

In the third part, the inequality changes, but is not revealed. Students measure examples of accepted and rejected nails, and then select from four possible choices for what the new inequality might be. You may want to ask if any of the students have an idea for making this process more efficient- and you may notice some students measuring the shortest and longest nails out of the "accepted" pile to match up with the possible statements. We're getting closer to the essential idea here- that the value we're subtracting from x in the statement is actually the center of the interval of acceptable lengths.

In the fourth part, the inequality changes again. The students measure the nails once again, but this time, they must write the statement themselves, and are not given choices. Hopefully by this point, there have been enough discussions surrounding this idea that this task will be within their reach. Of course, students may produce slightly different versions of the inequality. In fact, there's plenty of room for controversy at more than one point in this lesson; some of the nails are very close to the edge of being acceptable, and students may disagree over whether a certain nail should be accepted or not.

Shot Putters (download): This lesson (officially titled "Exploring Absolute Value") gets at the same essential idea that an absolute value statement has a solution set with a center and a radius, but it also asks us to use that knowledge to write a general expression. I've had great discussions with students about the mystery guest "Rufus," and how we could write an expression for his shot put "score" in terms of the a, b, and c that appear in his statement. I realize that this is a highly manufactured scenario. Perhaps inspiration will strike one day and I'll think of a more authentic way to teach this. In the mean time, I like this lesson anyway, because there just aren't enough opportunities for the students to use logic to build variable expressions- at least, not enough non-routine examples. Most of them are exactly the same and therefore pose no challenge (e.g. "Farrah makes t-shirts, she has a one-time expense of $___, and spends $___ making each one..." etc.).