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

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

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

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

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

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

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

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?

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

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

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

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?

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

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?

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

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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?

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

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?

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.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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?

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?

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?

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

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?

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

How many possible necklaces can you find? And how do you know you've found them all?

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?

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?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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?

Four small numbers give the clue to the contents of the four surrounding cells.

Find out about Magic Squares in this article written for students. Why are they magic?!

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

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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....?