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

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

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. . . .

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.

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.

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?. . . .

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

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

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

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?

Make the twizzle twist on its spot and so work out the hidden link.

A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry