I am looking for simple examples of situations where a good notation/diagram was fundamental for solving an elementary problem in mathematics (I am looking for examples accessible to a basic school student).
"General interest" mathematics journals like Mathematics Magazine or The Mathematical Intelligencer sometimes have "proofs without words", and the Wikipedia article is also helpful. These are diagrams that more or less immediately prove a mathematical statement. For instance, there are a number of "rearrangement proofs" of the Pythagorean theorem. More examples are Cantone (2020) or Chakraborty (2018) or Mukherjee (2021), all of which are unfortunately paywalled. Roger Nelsen also published collections of such proofs in book form.
I think tables are a very powerful method for making sense of logic problems. Before, I knew the method, found these GRE/LSAT* problems much harder. I think the benefit is in (1) overcoming the working memory bottleneck and (2) making sure all cases are covered.
For example: https://www.youtube.com/watch?v=loyetRWK3VU
P.s. Tables are not the only form of diagram for logic problems, but are pretty common. Sometimes just linking networks are the diagrams. If you check out an LSAT prep book, they will show several types of diagrams.
*It looks like the GRE Analytical section has been changed into a writing sample. Used to consist of LSAT style questions. (Not sure if the writing sample benefits from tables/not.)
Function Notation f(x)
$\forall x,y: [(x,y)\in f\implies x\in A \land y\in B]\\ \land ~ \forall x\in A: \exists y:[(x,y)\in f ~ \land ~ \forall z: [(x,z)\in f \implies z=y]]$
We can write:
$\forall x\in A: f(x)\in B$
Both fully and formally describe an arbitrary function mapping elements of a domain $A$ and to elements of a codomain $B$.