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?

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?

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

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

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

How many trapeziums, of various sizes, are hidden in this picture?

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.

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?

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?

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

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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.

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

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

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

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

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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

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.

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 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?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Can you fill in the empty boxes in the grid with the right shape and colour?

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

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.

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

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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.

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?

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

How many trains can you make which are the same length as Matt's, using rods that are identical?

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?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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

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

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

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?

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?

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?

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