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

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?

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

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?

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?

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

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

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

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

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

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

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.

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?

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

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

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

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

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

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?

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?

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.

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

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

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

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?

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

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?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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....?

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?

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

Use these head, body and leg pieces to make Robot Monsters which are different heights.

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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?

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

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?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

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?

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?

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

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?