What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

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?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Can you hit the target functions using a set of input functions and a little calculus and algebra?

Derive an equation which describes satellite dynamics.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Can you find the area of a parallelogram defined by two vectors?

A task which depends on members of the group noticing the needs of others and responding.

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Can you find the value of this function involving algebraic fractions for x=2000?

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Can you find a rule which connects consecutive triangular numbers?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Can you find a rule which relates triangular numbers to square numbers?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

How many winning lines can you make in a three-dimensional version of noughts and crosses?

By proving these particular identities, prove the existence of general cases.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

An algebra task which depends on members of the group noticing the needs of others and responding.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Show that all pentagonal numbers are one third of a triangular number.

Can you make sense of these three proofs of Pythagoras' Theorem?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?