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Why do this problem?
In this problem
students have the opportunity to create quadratic equations and solve them by factorisation or by using the quadratic formula. The follow-up questions offer the chance for some interesting generalisations and justifications.
This task is ideal for students who have already been introduced to solving quadratic equations.
Set the initial challenge to the whole class to work on individually or in pairs:
"On my last birthday, my friend said to me:
'In 15 years' time, your age will be the square of your age 15 years ago!'
Can you work out how old I am?"
As they are working, move around the class to identify the different methods students are using to solve it (eg trial and improvement, algebraically using factorisation, algebraically using the quadratic formula).
Once students have had a chance to make some progress with the problem, bring the class together and share approaches. If nobody has tried an algebraic approach, model it with the class on the board.
"I wonder whether this sort of thing could happen for other ages."
Suggest that they choose a statement of the form
"In n years' time, my age will be the square of my age n years ago"
for values of n between 2 and 30.
Those who are confident might be encouraged to choose larger values of n.
Ask students to report, as they go along, which values of n lead to a special age, and which don't - these could be collected on the board for all to see.
Once results have been collected
, a pattern should emerge as to what is special about the ages that work.
"Can anyone suggest a large value of n that you think might work?"
(For example, if I am 210, my age in 190 years' time is the square of my age 190 years ago!)
To prove that the pattern will continue to hold
, it may be necessary to introduce an algebraic representation for triangular numbers. Expressions for two consecutive triangle numbers can be used as the basis for the proof. Picturing Triangle Numbers
provides a nice way to introduce the expression
Alternatively, the pictorial representation of triangle numbers can be used to construct a visual proof.
If I am $x$ years old now how old was I 15 years ago?
If I am $x$ years old now how old will I be in 15 years' time?
Can you use these expressions to form an equation to solve?
The proof that has been suggested shows that triangle numbers are always special ages; to prove that it is only
the triangle numbers that have this property is a suitable challenging extension.
A simpler route into the task could be to start by considering someone whose age in 3 years' time is the square of their age 3 years ago.
An alternative strategy to solve the problem without needing to solve quadratic equations involves thinking about the difference between numbers and their squares.