A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
This problem starts by asking students to find which numbers can be expressed as the difference of two square numbers, and then suggests some possible avenues for exploration. This can then be used as a springboard to generalisations and the use of algebra for justifications and proof. Along the way, students have the opportunity to make use of the
important identity $a^2 - b^2 = (a + b)(a - b)$.
An alternative to this problem which some students may find more accessible is Hollow Squares.
The plenary can involve students presenting their findings to the rest of the class. Expect students to be clear and rigorous in their justifications. Encourage students to challenge any proofs that lack clarity and rigour, and suggest ways of improving them.
One way to round off the work on this problem could be:
"In a while, I'm going to give you a number and ask you to quickly find one or more ways to write it as the difference of two squares, or to convince me that it can't be done."