If I could pick one thing to be able to teach and magically have the kids understand, it would be integers. But unfortunately, I've got no magic here. So instead I have them record the rules and examples so they can refer back to it as much as necessary in the hopes that eventually it'll stick.
On the right side we record the rules in a foldable:
|Outside of the foldable. I let kids write the words and symbols and color however they want, they like it.|
|Inside of the foldable|
This is an example of a foldable that I complete over a number of days.
This is what it looks like with the left side pages. Notice that the left side is actually three pages taped one on top of the other so that they flip up.
On the first day we record the rules for multiplication and division. I will usually have them come up with the examples on their own giving them a little bit of direction (for same signs I tell them to pick one with two negatives and one with two positives). After recording the rules we tape down the bottom page on the left side. This is just a practice worksheet with multiplying and dividing.
On day two we do addition rules. Then we practice on the left side. The addition worksheet is the middle one.
On the last day we do subtracting. I use "keep, change, change" because that's what most of my kids are used to. I always tell them though that if they use "keep, flip, change" or "add the opposite" then they should write that instead. Then they tape the top worksheet in on the left side and practice.
Inside the subtraction flap, I have them use a highlighter to change the signs and label the "keep, change, change" so that if they were to look back in the future, they can see exactly what they did. Otherwise, I worry that if they just copy down what I'm doing they will just look back in the future and see an addition problem and have no idea what happened.
This is a set of pages where my left side is not spectacular or creative, it's just the practice that I felt they need. There are tons of other things you could put over here depending on what your kids needed.
Now the other part of this foldable is that after we make it, I refuse to just tell kids how to do these types of problems. If I come over and they ask something like, "Wait, so 2 + -10 is -8, right?" instead of just saying yes I will open their ISN and guide them through the problem always asking the same questions:
"Are you adding or subtracting?" adding ... I open the addition tab
"Are the signs the same or different?" different ... I point to the different sign rule
"So are you finding the sum or difference?" difference
"Is that what you did?" yes
"So is it the right answer?" yes
About this time they usually get annoyed that I didn't just tell them yes. After we go through this process a bunch times they won't bother asking me because I know I won't tell them and instead just look up the rules themselves. Exactly what I wanted :)
Also a note on differentiating just in general- for most of my kids the rules is sufficient. For something like this, I've found that most of them prefer to just have specific rules to refer to. There are always some though that I can tell are totally lost and with them I will have them use a number line to work through addition/subtraction problems.