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?

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

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

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?

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?

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

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

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

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

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

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

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

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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?

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?

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

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.

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

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?

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.

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

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.

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?

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?

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

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

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

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

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

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Can you find all the different ways of lining up these Cuisenaire rods?

Number problems at primary level that require careful consideration.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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

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?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.