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?

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

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

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

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

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

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

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

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

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?

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

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

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?

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

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?

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

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?

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

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?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Can you find all the different triangles on these peg boards, and find their angles?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

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?

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.

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

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.

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?

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

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

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?

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.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

How many models can you find which obey these rules?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?