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?

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

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

A follow-up activity to Tiles in the Garden.

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

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?

An investigation that gives you the opportunity to make and justify predictions.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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?

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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.

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

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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. . . .

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Investigate what happens when you add house numbers along a street in different ways.

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

This article for teachers suggests ideas for activities built around 10 and 2010.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?