I just reread what I had written last year about inequalities – I should have used some of those ideas this year! Anyhow, I am always trying to figure out another way to have the students understand WHY they need to change the direction of the inequality symbol when they are multiplying or dividing by a negative, so this is what I tried this year in my Math 7CP classes…
We’ve talked a lot about how “-” means subtract when seen like this: 5 – 3, five minus three, or means negative when looking at this: -8, negative eight, and how it also means the opposite as in this: the opposite of 12 is -12. So this year I concentrated on having the kids look at what they had as the “solution” line vs. the “problem” with just the terms in the answer, before they wrote down the inequality symbol. For example, if the problem was “-3x > 9”, and the answer line had “x -3” (missing the > or < on purpose), are the signs the opposite of how they started out? So the kids would look at the -3x compared to the x and see that the first was negative, and the second was positive, and the same with the 9 and the -3. Then I asked the class, “If the signs are the opposite, what do you think happens with the inequality symbol?” And a lot said, “It should be the opposite too!” We did a few problems together, (2 with negative variable terms, 1 with a positive term) then played our version of Grudge to get more practice, and have the kids talk to each other.
This is our version of Grudge, as modified by our first-year-fabulous-we are so happy we hired her-teacher, Maddy:
The students are shown the problem, and work quietly on their white boards for 1 – 2 minutes, depending on the type of problem. Then when the timer goes off they talk in their groups for about a minute (or longer if necessary), and this is when they are comparing their work and results and helping each other (critiquing the reasoning of others) so that everyone has the same result (and hopefully understands what to do). The teacher is walking around during all of the above time, and if a group is working particularly well together the group can get an “X” added to the board for their team. When this timer goes off it’s quiet again and one person from each group holds up their board, and the teacher tells them if they are correct (or not) and one student can go up and erase an X from another team on the board. This usually is pretty quick because there are only about 8 boards to look at. The class is supposed to be relatively quiet during this too, as it got too crazy with the teammates telling the person at the board who to X out! (So they need to plan this in advance.) To keep it moving, the teacher starts counting down from 10 about 5 seconds after the last team gets to go up, and everyone needs to be back in their seats by the end of the count. When the “round” is finished, the teacher calls on a student to explain how they solved the problem, and either the teacher or the students is solving it on the paper for the class to see. Then it’s time for the next problem!
We don’t play Grudge every week, but it’s a fun way to do some practice and have the kids working together. Note to self – it’s better to have team numbers than a student’s name! I had rearranged my tables and hadn’t moved the numbers around (I have them hanging from the ceiling tiles over the table groups) so I just put up a name of a person from each team. Apparently one of the names was for a person everyone wanted to eliminate – I don’t think it would have been the same if I’d written another name from the team. Anyhow, the X’s were gone on the first round, and the next round the name was erased (and I was looking at the class, not the board, so didn’t see it happen). Then when it was brought up that the name was gone, NO ONE would tell me who did it. (Honestly I’m a little surprised, usually someone will come up and tell me afterwards. I need to keep my eye on this a little better.) I’ll make sure I’ve rearranged the table numbers before we play it again!
I just graded the tests we gave for Inequalities, and the last question was “Explain how solving an inequality is different from solving an equation when the variable term is negative.” 18/32 of the students wrote something about “flipping the sign”, 2 left it blank, and the other 12 wrote about how equations have = and inequalities have > or < but didn’t specifically address “the negative” so got 1/2 credit. I would have preferred 100% of the class writing about changing the direction of the inequality symbol, but I think they’re off to a good start! So try having your class look at the problem and the solution and see if the terms are “opposites”. Because if they are, then the symbol needs to be the opposite as well!