Difference of Two Squares

This problem is available as a printable worksheet: Difference of Two Squares 

Why do this problem?

This problem is an excellent context for observing, conjecturing and thinking about proof. It offers an opportunity for purposeful practice of algebraic manipulation of quadratic expressions.

Possible approach

"Choose any multiple of 3, take the numbers on either side of your chosen number, square them, and find the difference.
For example, if I chose 33, I would work out
$34^2=1156$
$32^2=1024,$
so the difference is $1156-1024=132$."

Collect some of the students' responses on the board.
"Is there anything special about all our answers?"

They are all multiples of 2.
They are all multiples of 3.
They are all multiples of 4.
They are all multiples of 12.

"With your partner, see if you can explain what you've noticed."

Give students some time to think about explanations, and circulate to listen to what they come up with. If no-one thinks of using algebra, pose the questions:

"Is there a way I can use algebra to represent a multiple of 3?"
"How can I use my algebraic representation to prove that the answer will always be a multiple of 12?"

Once students have engaged with the algebra, bring the class together once more and invite a couple of students out to the board to show their proofs.

Then set the follow-up challenge:
"What if we started with a multiple of 5 instead of a multiple of 3?" and invite them to explore and prove their cojectures in a similar way.

"Is there a similar relationship for other times tables?"

Possible support

Pair Products is a similar problem, but with a little more structure and support.

Possible extension

"What if we started with a multiple of $k$ instead of a multiple of $3$?"