Algebra, Local variables and the Module function

Question

Algebra is so much fun but I'm struggling to understand how MMA executes local variables within Module. Reading the Module documentation page I suspect it has something to do with "Before evaluating expr, Module substitutes new symbols for each of the local variables that appear anywhere in expr except as local variables in scoping constructs". But as I work through possible examples of how this works, the output behaviour continues to unpredictable. For example, what is causing the difference in output below? How would you explain this to a kindergartener? Or one's great grandmother?

In1= x=2;y=x+2;y+x

6

in2= Module[{a=2,b=a+2},a+b]

4+a

Answer 1

Look at the syntax coloring:

Note that the a in b = a + 2 (blue) is colored differently than the a in a = 2. (green). That is because they are different variables. The a in b = a + 2 references the variable Global`a (usually displayed as just a). You can also Trace the evaluation to see this:

Trace[Module[{a=2,b=a+2},a+b]]

{Module[{a=2,b=a+2},a+b],{a\$40801=2,2},{a+2,2+a},{b\$40801=2+a,2+a},{{a\$40801,2},{b\$40801,2+a},2+(2+a),2+2+a,4+a},4+a}