Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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.

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

Exchange the positions of the two sets of counters in the least possible number of moves

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Can you fit the tangram pieces into the outline of Little Ming?

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

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

What is the greatest number of squares you can make by overlapping three squares?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Can you work out what is wrong with the cogs on a UK 2 pound coin?

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?

Can you make a 3x3 cube with these shapes made from small cubes?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Can you cut up a square in the way shown and make the pieces into a triangle?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of these convex shapes?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.