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Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this process starting with T2 to form a sequence of nested triangles and circles. What happens to the triangles? You may like to investigate this interactively on the computer or by drawing with ruler and compasses. If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by f(x) = (90 - x/2) where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens. Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)

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Unusual Long Division - Square Roots Before Calculators

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem takes more able students into the realm of 'non-calculator' methods that lie beyond the arithmetic they became familiar with when they were much younger. It usefully draws attention to the need for validation in any algorithm whether carried out electronically or 'by hand'.

Possible approach

  • Spend a little time looking at the validity of the standard method for 'Long Division'. Discuss the historical need for efficient algorithms before electronic calculators, when computation was manual, and point out that calculators and computers aren't 'magic' and there still has to be a valid algorithm.
  • Find some square roots of two-digit numbers to 2dp by trial and improvement.
  • Spend time understanding what this new method involves (maybe use the audio link on the Problem page ,while keeping the working still on view), practise and then organise a time trial.
  • Alternate between this method's algorithm and trial & improvement, for the square roots of 30, 50, 60, 70, 80, and 90, all to two decimal places. Record the calculation time for each one and compare methods.

Key questions

  • How do we find the square root of 40 on a calculator that only does simple '4 rules' arithmetic?

  • What exactly is the method here?

Possible extension

Explain that a mathematician will always want to justify or validate a procedure and leave that challenge with the group.

Possible support

For less able students raising awareness that methods of calculation need justifying can lead to a stronger and more satisfying grasp of arithmetic procedures like 'long multiplication' (traditional and alternative) and 'long division'.