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Can you explain why a sequence of operations always gives you perfect squares?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.


The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Difference of Two Squares

Age 14 to 16
Challenge Level

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
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 $3$?"