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

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

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?

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

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

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

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

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

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?

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

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

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

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

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?

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

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?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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

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?

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?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these three-digit numbers to the nearest 100?

What happens when you round these numbers to the nearest whole number?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

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.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

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

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

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?

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.