MemberJuly 7, 2020 at 4:03 pm
OK. You have done the impossible, which is to model “negative groups.” I appreciate that! I usually punt and use the commutative property on that kind of problem (-2 x 3 = 3 x -2). I really like zero pairs; they make sense to me. But I find that they do not help students–that is, they help when they have the cubes in front of them, but then there is no middle step that helps them notate what they are doing when they add that zero pair. I don’t think the number line adds anything because you’re just placing the resulting cubes on the number line. So the jump to abstract is too abrupt. I would like to invent a middle step where students could keep track of that step, such as
-2 – 5
-2 – 5 + 5 – 5
but I don’t think that helps anything. So even though we do a lot with zero pairs, I find myself relying on memorized rules, such as “subtraction is adding the opposite.” I find it makes a lot of sense to kids to change -2 – 5 to -2 + -5. But then all the work with the cubes flies out the window. The cubes support the answer we get using the rule, but they don’t give us a way to get that answer without using cubes.
Another approach I would like to start using is distinguishing between “take-away” subtraction and “difference” subtraction. If you can take away easily, such as 4 – 3, then use that. But if you cannot take away, such as 5 – (-3) (you cannot take -3 away from 5 because there are no negative cubes), then think of it as “difference.” The difference between 5 and -3 is 8 on a number line. So simple! No guessing, no trying to remember a rule like “two negatives make a positive,” which is going to mess them up when they add two negatives. BUT. What about -3 – 5? How do we acknowledge that the difference between -3 and 5 is -8, not 8? On a number line, it looks like exactly the same problem. So that makes me want to go back to re-writing subtraction as adding the opposite, or using take-away and zero pairs.
If you come up with a way to think about difference and distinguish between 5 – (-3) and -3 – 5, I would love to hear!