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.

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

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?

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

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?

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

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?

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?

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

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

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

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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?

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

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

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

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

If you had 36 cubes, what different cuboids could you make?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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.

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?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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

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

How many models can you find which obey these rules?

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?

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

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

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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?

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.

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

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

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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?

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

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

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

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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.