Today in my remedial algebra class, I thought I would make an inequalities game. I had this great idea that I would put up on the board a whole bunch of inequalities, and each group would add or subtract or multiply or divide different things to these inequalities, and we would see if the result came out still true, or false.

(For example, it’s true that 2 < 4, and if you add 5 to both sides of this, you get a still true statement, 7 < 9, or if you multiply both sides by 2, it’s still true, 4 < 8, BUT, if you multiply both sides by a negative number, like -2, it’s not still true: -4 < -8 is NOT true. Which leads to rules about how you solve inequalities.)

So… the first problem was that only about a third of the class was there on time…. So I went over inequalities and how to graph them for a bit first, vamping…..

Once more students had arrived, I put them in 6 groups, and put 6 TRUE inequalities on the board, like this:

Group 1: 2 < 4 Group 2: 5 < 8 Group 3: -2 < 5 etc.

I was going to ask each group to do different things to their inequality — one group would add a number to both sides, another would multiply both sides by a number AND ONE GROUP would multiply both sides by a negative, and they would be the mystery group where it would turn out that this gives a false result!!

HA.

Never under estimate the degree to which following directions is difficult, especially in a remedial class. I put the problems up for each group and I could tell pretty quickly that most students were baffled.

SO… I had them all do more or less the same thing each round — first I had them all add or subtract the same number to both side of the inequality, then we discussed it, then I had each group multiply both sides of their inequality by 2, and we discussed it… and then I had each multiply by -2, and we discussed it. The last one — multiplying both sides by a negative — results in a statement that is NOT TRUE.

And then we went over the results: adding or subtracting by the same number — results in a true statement. Multiplying by a positive, results in a true statement. Multiplying by a negative number…NO! The statement turns out false… this lead us to how to solve inequalities.

Well, so the game failed… but the experiment worked out! They were asking questions, arguing with me, protesting, working problems, THINKING. It was pretty awesome, actually. Good class!

Susan, thanks for the comment! Yes, sometimes the exploration just doesn’t connect up to the “why we solve it this way” idea. This time, the connection seemed to work, I think because I laboriously wrote out on the board all the conclusions of our experiment: “add or subtract the same number to both sides of an inequality, still true! Multiply both sides of an inequality by the same *positive* number, still true! Divide both sides by a negative number, not true any more! To make it true, turn the inequality symbol around!” Then I worked with them to solve an inequality with all those “rules” still up on the board, and I kept referring to them — “Why can we do this? Because we tried it out, and it’s okay — that’s this one that we did.” Etc.

(By the way, I think the less than symbol reads as the start of html code, that’s why it made your numbers disappear! :-))

Weird the numbers didn’t show up… what I said in the parentheses was is two is less than four and we multiply both sides by negative three, we end up with neg. six is greater than neg. eight.

Thank you for sharing that info with us! I have tried that technique with my classes in the past and while they seemingly understand it there (if 2 -8), they do not make the connection to why that works when there is a variable. What have you done to connect it better? Thanks!