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

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.

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

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?

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?

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

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?

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

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

This task challenges you to create symmetrical U shapes out of rods and find their areas.

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

If you had 36 cubes, what different cuboids could you make?

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

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

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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?

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.

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?

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

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

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?

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

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

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.

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

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

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?

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?

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?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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

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?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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....?

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?

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

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

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

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?