The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

There are nasty versions of this dice game but we'll start with the nice ones...

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

What's the largest volume of box you can make from a square of paper?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

You'll need to know your number properties to win a game of Statement Snap...

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

Where should you start, if you want to finish back where you started?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Use your skill and judgement to match the sets of random data.

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

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Can you find any two-digit numbers that satisfy all of these statements?

Can you work out which spinners were used to generate the frequency charts?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

Can you work out the probability of winning the Mathsland National Lottery? Try our simulator to test out your ideas.

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

Which countries have the most naturally athletic populations?

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

Play around with sets of five numbers and see what you can discover about different types of average...

How well can you estimate 10 seconds? Investigate with our timing tool.

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

If you move the tiles around, can you make squares with different coloured edges?

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?