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

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?

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?

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?

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

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

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?

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 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?

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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

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

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

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?

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

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?

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?

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 challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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

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

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

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

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?

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?

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.

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

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?

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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

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?

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?

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

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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?

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

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 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?

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