Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Can you adjust the curve so the bead drops with near constant vertical velocity?

Looking at the graph - when was the person moving fastest? Slowest?

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

If you plot these graphs they may look the same, but are they?

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Explore the relationship between resistance and temperature

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Build series for the sine and cosine functions by adding one term at a time, alternately making the approximation too big then too small but getting ever closer.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .