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

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

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

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

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

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?

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

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.

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

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?

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?

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.

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

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?

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

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

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?

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?

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

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

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?

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.

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

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

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'?

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?

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

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

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?

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?

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.

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

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?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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?

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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....?

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?

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?

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?

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?

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

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