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?

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

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

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

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

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

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

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

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

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?

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

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?

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?

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?

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

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

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?

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?

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?

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

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?

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

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

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

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?

This train line has two tracks which cross at different points. Can you find all the routes that end at Cheston?

My coat has three buttons. How many ways can you find to do up all the buttons?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Can you fill in the empty boxes in the grid with the right shape and colour?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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?

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

How many models can you find which obey these rules?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Find all the numbers that can be made by adding the dots on two dice.

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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