We finished our 2 days of inequalities today. I used the questions from the last blog https://www.dropbox.com/s/20m9558h0aoatzd/Inequalities%202014%202015.docx?dl=0 but we didn’t get through all of the examples yesterday – but I actually think that was fine. I have 3 pre-algebra honors classes this year and in 2 classes we ended with #9, but in my last class we only finished through #8. When we solved #8 we talked about money coming out of your pocket is “negative” for you, so each ticket would cost -$7. Then about half of the class thought the total, $50 would be positive, so we talked about that too. I had asked them what the answer would be after a student had read the problem out loud, so “finding the answer” wasn’t what we were concentrating on. I drew a number line showing -50 through 20 so we could look at it and decide if the inequality would be -7x > 50 or -7x < 50. They understood by looking at the number line that it would be greater than -50, but logically they thought it should be less than -50. We talked about it being less than or equal to $50 to spend, but this is one of those instances where you have to really think about how to write the problem! The kids wrote the inequality then we did it together – and talked about what happens to the answer if we just bring down the > symbol: it was x > 7 kids, and we all knew it should be x < 7 kids. In one class one student said, “But don’t you have to change the direction of the symbol if you multiply or divide by a negative number?” but this didn’t happen in the other classes. We talked about how we were changing the relationship of the problem a little, then the class was over and they had some inequality hw with only positive variables (from FACEing Math Algebra).
Today we went over any questions from the hw, then I had the kids stand up at their tables and in their groups of 4 line up, from left to right, from shortest to tallest. Then I “changed the rule” and said tallest to shortest, and they switched places. Then I “changed the rule” and had them line up from youngest to oldest, left to right. Then the whole class lined up in height order around the classroom. Finally I asked them what would be the easiest way to have them line up the other way in height order, and all 3 classes said “Just walk half way around the class.” Cool! I told them to remember they had to change their positions when I changed the rules on them, and now we’d do the math. We did a few equation examples, multiplying or dividing by positive and negative numbers and saw that everything was still true. Then we did a couple of inequality examples multiplying and dividing with positive numbers…then I had them multiply both sides of the inequality by -2, and just write the new “numbers”. They looked at it and saw if they just brought down the symbol the inequality wouldn’t be true. We talked about how with inequalities there is a relationship, and we had just changed the problem from positives to negatives, we’d “changed the rules” so therefore to keep the relationship true we had to change the direction of the inequality symbol. We did another problem dividing by -3, and I think it was clicking with a lot of them. We then did a few problems on the FACEing math page with negative variables, and showed how we were changing the problem – the negative had been on the left (-3x > 12), then changed to the right (x < -4), so the relationship with the symbol had to change too, to keep it true.
We’ll see tomorrow if the lesson was as good as I hope. I do think lining up in their groups and then rearranging was helpful (or at least fun!).