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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

Can you draw a square in which the perimeter is numerically equal to the area?

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

This activity investigates how you might make squares and pentominoes from Polydron.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

An activity making various patterns with 2 x 1 rectangular tiles.

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Find out what a "fault-free" rectangle is and try to make some of your own.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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. . . .

These practical challenges are all about making a 'tray' and covering it with paper.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

How many possible necklaces can you find? And how do you know you've found them all?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

How many different triangles can you make on a circular pegboard that has nine pegs?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?