Taking Risks

One mental block to solving problems is the fear of taking a risk: making an attempt at a solution before you know if it will in fact solve the problem.  I try to encourage risk-taking with younger students and easier problems.  For example, if you are asked to solve for $x$ given this equation:  $3x-2=2x$, the next logical step is to subtract $2x$ from or add $2$ to both sides.

$3x-2=2x$

$A_{2}: 3x=2x+2$

$S_{2x}: x=2$

Some students inexperienced in algebra may begin by dividing both sides by $3$.  I won’t stop them from trying this approach.  As long as it is legal to divide both sides by $3$ we should give it a try and see if it brings us closer to a solution.

$3x-2=2x$

$D_{3}: x-\frac{2}{3}=\frac{2}{3}x$

The we proceed to solve for $x$:

$S_{\frac{2}{3}x}: \frac{1}{3}x-\frac{2}{3}=0$

$A_{\frac{2}{3}}: \frac{1}{3}x=\frac{2}{3}\rightarrow x=2$

We see that we can reach a solution, but it introduces some ugly fractions.  Only after pursuing this path do I then suggest the more elegant fraction-free solution.