I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for novices or younger students to understand. From what I understand you can alternatively argue with case by case analysis but there's a lot of cases and it doesn't look that satisfying.
I have came up with some geometric proofs of my own and would happily share them since I have found nothing like this online, but I get the feeling that some already exist and are as refined and streamlined as they will ever be, so if you happen to know any please do share :)
P.S. Just to make myself clear I am not talking about just natural numbers here, there's plenty of that stuff online for sure.
Edit: Thanks for your comments, I attached a picture and will explain my arguments for commutativity and associativity. I am aiming to be as elementary as possible and give a good reason to believe the theorems rather than a rigorous proof.
I am modelling integers as arrows pointing to equally spaced points on a line, addition defined in the obvious way.
The top of the image below demonstrates commutativity. Call the blue arrow x and the orange arrow y. Drawing x and y next to each other makes a combined arrow of length |x| + |y|, moving the arrows so that they start from the ends of x and y preserves the length, thus they meet at the same point. This shows that going along blue then orange gets you to the same place as orange then blue, in other words x+y = y+x. Similar argument should hold for any two integers with different signs.
Next is associativity, equivalent to show (x+y)+z = (z+y)+x by commutativity. The first part is (x+y)+z, then (z+y)+x can be represented by travelling in reverse with the left direction increasing as shown next (read upwards). The arrow representing (z+y)+x is the reflection of (x+y)+z, so it has the same length, thus in this interpretation it is the same integer.