Last week I took my class on a field trip. I had everyone in class think of the last digit of their phone number. I told half of them to put a minus sign in front of that number. Then we went out into the hall. If this term wasn’t so short and if the weather was better, I would have taken them outside. I had the students line up in numerical order lowest number to my left. I told them I was forming several clubs and that they should step forward if they qualified. I went through the following equalities and inequalities. They soon got the hang of it and stepped forward correctly when their number satisfied the relationship. We saw the tails of the line step forward and a middle interval step forward and no one step forward and everyone step forward. At the end I told them that they now knew the concepts for the next three sections of the text – inequalities and absolute values.
Note that the varied colors are not an affectation. Students like and respond to color. I didn’t use the last two inequality statements but I wish I had. The next day the class spent ten minutes trying to get the difference. I had drawn the following diagrams on the board but they, or at least a vocal minority, weren’t getting it. Finally I showed them for the intersection that if as my hand passed down the diagram it crossed both arrows that section was part of the answer and for union if as my hand passed down and crossed one or the other or both arrows that section was part of the answer. This worked. When they took their quiz the next day many of them got these type of problems correct.
As an afterthought I asked them to express as one set and . Again a vocal minority had trouble. I should have drawn this diagram and saved myself three minutes of class frustration.
Now the “pass the hand down” technique works.
As I told my students, the distinction between “and” and “or” is important to know since they will find purposeful and non-purposeful misuses of the words in politics and advertising.