In Algebra I, we drill into student’s heads the formula for the difference of two squares:

While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:

Students should be able to recognize the pattern (perhaps with a little prompting):

Students should hopefully recognize the perfect squares:

,

so that they can guess the answer to something like without pulling out their calculators.

Continuing the exploration, students can use a calculator to find

Students should be able to recognize the pattern:

,

and perhaps they can even see the next step:

.

From this point, it’s a straightforward jump to

,

leading students to guess that .

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