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This problem offers an excellent context for observing, conjecturing and thinking about proof. It also offers an opportunity to discuss the relationship between geometrical and algebraic methods for representing how numbers behave - an engaging introduction to number theory.
Along the way, students have the opportunity of practising routine algebraic procedures. The first problem involves factorising a quadratic expression to get a perfect square, and the second problem involves factorising a quartic expression.
Give students some time to discuss with their partner why the answers are always square numbers. Circulate and listen out for interesting insights.
"We've worked out what happens when you find the product of two numbers that differ by 2, and then add 1.
I'd like you to explain what happens when you:
"Start by testing some numerical examples before trying to generalise using the representations we've looked at. Can you prove your results?"
Is there a way to represent the product of the two numbers that will explain the patterns you noticed?