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

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

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?

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?

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

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?

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?

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

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?

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

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

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

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

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

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?

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?

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?

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

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

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

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

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

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

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

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

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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?

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

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?

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?

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?

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....?

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

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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?

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