2.1 Inequalities

This is pretty easy, isn’t it? You can just start by having them work through “Inequalities” with no preamble at all.

After they’ve worked on it for a while, you might want to interrupt the class to talk about it for a while. #1 is obviously an attempt to get at the old “you have to reverse the inequality when you multiply or divide by a negative number” thing. But stress as loudly as you can, the idea that they shouldn’t just take anyone’s word for it . The question we’re trying to get at is, why do you have to switch the inequality, then and only then? I usually illustrate this point by drawing a number line and showing how, on the left of the zero (in “negative land”) the numbers are going backward (an observation that every second grader notices, but that they have forgotten by high school). So you can visually show how $2<3$ becomes $\mathrm{–2}>-3$ when you move it to the other side.

Also, tell them how important it is to distinguish carefully between ANDs and ORs—this will become a major issue later. I have two pet peeves on this topic.

One pet peeve is people who memorize a facile rule (such as “less-than problems become AND and greater than problems become OR”) without having the slightest idea what they are doing. I go out of my way to create problems that frustrate such rules (which isn’t hard to do). They have to see and understand what these conjunctions mean .

The other pet peeve is $x>\pm 4$ . This is, for all intents and purposes, meaningless. I want them to realize that on #9, and then I warn them that I will always take off points if they answer any question this way. (This comes up in the context of $\left|x\right|>4$ and is a good example of how wrong you go if you answer mechanically instead of thinking.) $x$ can equal $\pm 4$ , if you know what that means (shorthand for $x=4$ or $x=-4$ ), but it cannot be greater than , or less than , ±anything.

Homework: