All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

What's the chance of a pair of lists of numbers having sample correlation exactly equal to zero?

Numbers arranged in a square but some exceptional spatial awareness probably needed.

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

How much peel does an apple have?

Explore the properties of combinations of trig functions in this open investigation.

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Work with numbers big and small to estimate and calculate various quantities in physical contexts.

Use trigonometry to determine whether solar eclipses on earth can be perfect.

An introduction to a useful tool to check the validity of an equation.

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

A follow-up activity to Tiles in the Garden.

How many different sets of numbers with at least four members can you find in the numbers in this box?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

Get some practice using big and small numbers in chemistry.

Investigate constructible images which contain rational areas.

Work out the numerical values for these physical quantities.

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?

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

Where we follow twizzles to places that no number has been before.

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

A challenging activity focusing on finding all possible ways of stacking rods.

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Formulate and investigate a simple mathematical model for the design of a table mat.

It starts quite simple but great opportunities for number discoveries and patterns!

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

A description of some experiments in which you can make discoveries about triangles.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Where should runners start the 200m race so that they have all run the same distance by the finish?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.