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By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

Use the applet to explore the area of a parallelogram and how it relates to vectors.

Can you work through these direct proofs, using our interactive proof sorters?

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Can you find a way to prove the trig identities using a diagram?

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

A preview of some of the beam deflection mechanics you will look at in the first year of an engineering degree

What have Fibonacci numbers got to do with Pythagorean triples?

Can you find pairs of differently sized windows that cost the same?

A space craft is ten thousand kilometres from the centre of the Earth moving away at 10 km per second. At what distance will it have half that speed?

Can you interpret this algorithm to determine the day on which you were born?

Get to grips with the variables behind the motion of a simple pendulum

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

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

Derive Euler's buckling formula from first principles.

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

How good are you at finding the formula for a number pattern ?

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

Relate these algebraic expressions to geometrical diagrams.

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

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

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

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

What is an AC voltage? How much power does an AC power source supply?

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.

What happens when you straighten out concentric circles?

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.

An advanced mathematical exploration supporting our series of articles on population dynamics for advanced students.

Explore the energy of this incredibly energetic particle which struck Earth on October 15th 1991