Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Can you find a way to identify times tables after they have been shifted up?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

An environment which simulates working with Cuisenaire rods.

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

There are lots of ideas to explore in these sequences of ordered fractions.

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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?

What's the greatest number of sides a polygon on a dotty grid could have?

Make some loops out of regular hexagons. What rules can you discover?

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

A introduction to how patterns can be deceiving, and what is and is not a proof.

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

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

Find the smallest value for which a particular sequence is greater than a googol.

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Use Farey sequences to obtain rational approximations to irrational numbers.

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

Investigate the different sounds you can make by putting the owls and donkeys on the wheel.

A story for students about adding powers of integers - with a festive twist.

Investigate these hexagons drawn from different sized equilateral triangles.

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?