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

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

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

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

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

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?

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?

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

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

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?

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?

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?

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

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?

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

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

In how many ways can you stack these rods, following the rules?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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.

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. . . .

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?

This article for primary teachers suggests ways in which to help children become better at working systematically.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

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?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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

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?

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.

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?