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Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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Pole Vaulting

Consider the mechanics of pole vaulting

Five Circuits, Seven Spins

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This is a good problem for discussion and developing clear visualisation and mathematical communication. It relates the angle of rotation of a circle to a distance and is therefore of use in exploring radians and the formula $s=r\theta$.

Possible approach

Students' abilities to visualise the meaning of this problem might vary considerably. As such, this problem can appear to be difficult until a clear approach to the solution is found. The behaviour of the disc at the corners is likely to cause the most difficulty in imagining the rotation. As a result, students might need to be given a variety of visual devices to allow them to get started. For example:
  • Imagine looking down onto the tray and watching the disc rotate about its centre.
  • Imagine breaking the journey into a series of straight line trips.
  • Imagine that the disc is pinned down in the centre and the tray is a track moved around the disc.
  • Imagine that the edge of the disc is coated in ink. Which parts of the tray would be coloured following a lap of the track?
  • Roll a coin around a book and use the head on the coin as a reference. Does the head rotate as it moves through a corner (i.e. when moving from a horizontal to a vertical part of the the journey).
This is the sort of problem which becomes much clearer once a solution has been found. Once students have solved the problem they should try to rewrite their answer and method as clearly as possible, in a way which is both simple but complete.

It is possible to tackle this problem using degrees and the formula for the circumference of a circle, but it is much simpler to solve using radians and the formula $s=r\theta$.

Key questions

How far does the centre of a disc move in one revolution when the plate is in contact with a straight edge?

What mathematics allows us to relate this distance to an angle?

What units should we measure the angle of rotation in? Why?

If the disc has rotated $7$ full times, how far must it have rolled?

As the disc makes a single lap of the tray, what parts of the tray will have made contact with the disc? How far is this?

Possible support

Consider the distance a bicycle travels when the wheels rotate once.

Read the article A Rolling Disc - Periodic Motion.

Possible extension

Try the problem Contact


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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