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There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

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Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Fair Shares?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem starts with a simple situation which can be analysed quickly using mental methods, but which provides a starting point for tackling a more challenging problem. The challenge can be tackled at many different levels, using trial and improvement (perhaps using spreadsheets), looking for number patterns, or with a more formal algebraic approach.

Possible approach

Introduce the first problem. Solve it together with the class. Can anyone explain why everything works out so neatly?

Now share the second problem. Students may wish to use trial and improvement to solve it (using spreadsheets might make things quicker). Alternatively, they may use insights from the first problem to suggest starting points. For those who are confident using algebra, the problem can be expressed and solved using equations.

Share approaches and answers to the second problem and encourage the students to use insights from this discussion to help them solve the third problem.

After solving the third problem, students who are less confident with algebra could be encouraged to look at the patterns in the results so far and to suggest generalisations based on what they have seen. Then they could test their conjectures with a variety of examples of their choice.

Key questions

How can we express each part of this problem algebraically?

Possible Extension

Students who are more confident can be challenged to write their generalisation algebraically, and manipulate the algebraic expressions to produce a rigorous proof that the results will always hold.

Possible Support

Peaches Today, Peaches Tomorrow.... might offer a suitable introduction to the fraction skills required in this problem.