shapes

Writing variables as shapes simplifies the concepts by reducing clutter. We can define a variable by filling its shape with an expression. For example, in solving linear systems by substitution, it's easier for a child to see a substitution as shapes rather than letters. Typically, an alphabet souped version requires more careful reading to distinguish letters from each other, whereas a quick glance can distinguish a square from other letters. For instance,

y = 2x ,       3y = x+3  --->   3 * 2x   = x+3   is harder to read than

+----+     +----+                 +----+
| 2x |    3|    | = x+3  --->   3 | 2x | = x+3   
+----+ ,   +----+                 +----+

substitution method

We approach solving linear systems by the Method of Substitution. The teacher must be sure that the student has mastered the mechanics of Spread, the Forward-Reverse Verb Equivalence discussed in the Peel section, and equivalence of verb centers (-1 = -1/1 = 1/-1, x = 0+x = x+0, -x = -1*x, ...). Verb Centers are group-theoretic identities, mentioned in the Adjectives section.

After these 2 pitfalls, the hardest part of linear systems is organizing the work so that the child does not lose sight of the substitution strategy while focusing on the details of algebraic manipulation. To that goal, we propose organizing the Method of Substitution as in the following diagram:

Note the alternating sequence of vertical (nudify) and horizontal (substitution/swap) steps: nudify-swap-nudify-swap. Using shapes helps the child to visualize the swap.

This Step diagram is just a way of organizing work. In a 3 variable system, the Steps picture will look like

           (1)   (2)   (3)
            +     
            |     
            |                          nudify x in (1) to get x(y,z)
            |      
(1) x(y,z)  +-->--+ ...,=,y,z...       swap x in (2) to get an equation in y,z
                  |
                  V                    nudify y to get y(z)
                  |
(2) y(z)          +-->--+  ...=,z...  swap x(y,z) & y(z) into (3) to get equation in z
                        |
                        V             nudify z to get constant
                        |
 (3) ...y,=...    +--<--+  z=const.   then replace z=const into (2)
                  |        
                  V        tidy
                  |
(2) ..x,=.. +--<--+        y=const.   then replace y=const into (1)
            |
            V              tidy
            |
(1) x=const +