So I made a handout about adding and subtracting integers that’s a visual nightmare so I’m not even sharing it. In the process, I’ve got a hypothesis about why every semester students seem to get adding integers reasonably well and then explode when subtraction gets dumped on them.
I realized that there are four possibilities when you’re adding integers if students understand the commutative property: putting two positives together — old school –, putting two negatives together (not a positive; the biggest hurdle here…), putting a big positive and small negative together, and putting a big negative and a small positive together. I’m thinking for the big picture people that it might just be peachy to show them that.
With subtraction, there are five possibilities — and five’s too big for lots of people… unless you chunk it up. It’s also coming right smack ON TOP of the adding negatives. It’s kinda like how much harder it is to juggle “just one more thing.”
So I’m wond’ring if starting out with “big minus little” and “little minus big” and getting that down pat … and *then* “negative minus positive” as being the same as negative plus negative — more negative, thank you…
and *finally,* the (-)(-) situations… where either it’s a positive number in front — so you just add — or negative, the nastiest… but *hopefully* if this has been done systematically, by now the student will have adding different signed integers at least hovering near the elusive mastery/automaticity.
So… assuming I’ll successfully breach what was my nemesis developing the app the first time, and be able to construct quizzes and store that information, etc… that’s my plan.
Resource Room Dot Net Blog 
howardat58
September 25, 2015
One of the dangers of saying “let us have negative numbers as well, they will come in useful” (some parody here!) is that the real nature and purpose of having “signed” numbers , positive and negative, is to have numbers for describing position and change. this means that the arithmetic operations of “add” and “subtract” need to be interpreted in those terms
So firstly we can see the integers as labels for whole number spaced points on a number line.
Secondly we can see them as measures of movement along the line. +3 is 3 steps in the increasing positive direction and -5 is 5 steps in the decreasing positive, or increasing negative direction. This is more cumbersome to write than to explain directly!
Now for arithmetic:
adding
A + B now means, as a question, where do I end up if I start at A and take B steps
A – B now means , again as a question, how far is it from B to A
This takes care of all the otherwise apparently arbitrary and meaningless rules, and with the number line in front of you(that is, the student) becomes completely logical and meaningful
xiousgeonz
September 25, 2015
The issue is, though, that the direction gets really confusing if all the combinations of positive and negative differences and motions are learned too closely together. Then it becomes completely illogical and loses the meaning.
howardat58
September 25, 2015
My thinking was that if they do enough “sums” using the description of movements then they would see the sense in some rules, and then not need to (rote) learn them. You are quite right that sometimes too much stuff is shoved down the students’ throats in too short a time, with the ideas getting pushed aside. This is true of a lot of fraction teaching.
xiousgeonz
September 25, 2015
Yes!! Essentially the same problem, though I think it’s a little more severe with fractions.