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

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

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

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?

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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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.

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

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

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?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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?

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

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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?

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?

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

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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.

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

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

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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.

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

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.

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Investigate the different ways you could split up these rooms so that you have double the number.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.