This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
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 find ways of joining cubes together so that 28 faces are
Which of these dice are right-handed and which are left-handed?
Here are shadows of some 3D shapes. What shapes could have made
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
Can you cut a regular hexagon into two pieces to make a
parallelogram? Try cutting it into three pieces to make a rhombus!
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.
Can you fit the tangram pieces into the outlines of the watering can and man in a boat?
Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
A hundred square has been printed on both sides of a piece of
paper. What is on the back of 100? 58? 23? 19?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Can you fit the tangram pieces into the outlines of the workmen?
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
Can you fit the tangram pieces into the outlines of the candle and sundial?
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.
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?
Can you arrange the shapes in a chain so that each one shares a
face (or faces) that are the same shape as the one that follows it?
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 fit the tangram pieces into the outlines of the chairs?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Exploring and predicting folding, cutting and punching holes and
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?
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?
Can you fit the tangram pieces into the outlines of these clocks?
How many balls of modelling clay and how many straws does it take
to make these skeleton shapes?
Where can you put the mirror across the square so that you can
still "see" the whole square? How many different positions are
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!
You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.
Reasoning about the number of matches needed to build squares that
share their sides.
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Make a cube out of straws and have a go at this practical
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?
Which of the following cubes can be made from these nets?
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?
Each of the nets of nine solid shapes has been cut into two pieces.
Can you see which pieces go together?
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 teachers describes a project which explores
thepower of storytelling to convey concepts and ideas to children.
Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some
of the small cubes in the ways described, how many will have holes
drilled through them?
This problem invites you to build 3D shapes using two different
triangles. Can you make the shapes from the pictures?
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 work out what is wrong with the cogs on a UK 2 pound coin?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
Can you fit the tangram pieces into the outline of Little Ming?