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?

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.

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

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?

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?

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

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

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?

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

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

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

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

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

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

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

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

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

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

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

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

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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

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

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

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

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?

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.

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

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

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?

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

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

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?

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?

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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....?

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

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 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?

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.

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.

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.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?