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.

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

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

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?

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

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

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?

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?

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?

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

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

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?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

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

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

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

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

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?

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?

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

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Find all the numbers that can be made by adding the dots on two dice.

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Try this matching game which will help you recognise different ways of saying the same time interval.

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.

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. . . .

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

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

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.

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

How many possible necklaces can you find? And how do 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.

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

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?

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?

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?

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

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.

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....?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

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?