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.

Explore the relationships between different paper sizes.

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

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

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

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

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

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

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.

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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

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

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

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

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

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?

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

Which countries have the most naturally athletic populations?

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

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

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?

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...

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?

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

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?

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?

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...

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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?

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

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?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

Which set of numbers that add to 10 have the largest product?

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

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

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?

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