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
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
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. . . .
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.
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 outlines of the candle and sundial?
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?
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
This article looks at levels of geometric thinking and the types of
activities required to develop this thinking.
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outline of this sports car?
What happens when you turn these cogs? Investigate the differences
between turning two cogs of different sizes and two cogs which are
Here's a simple way to make a Tangram without any measuring or
Which of these dice are right-handed and which are left-handed?
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.
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,. . . .
A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
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?
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?
Make a cube out of straws and have a go at this practical
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!
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
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?
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.
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.
Billy's class had a robot called Fred who could draw with chalk
held underneath him. What shapes did the pupils make Fred draw?
Can you cut a regular hexagon into two pieces to make a
parallelogram? Try cutting it into three pieces to make a rhombus!
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.
Each of the nets of nine solid shapes has been cut into two pieces.
Can you see which pieces go together?
Here are shadows of some 3D shapes. What shapes could have made
Can you fit the tangram pieces into the outline of these convex shapes?
I've made some cubes and some cubes with holes in. This challenge
invites you to explore the difference in the number of small cubes
I've used. Can you see any patterns?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
This second article in the series refers to research about levels
of development of spatial thinking and the possible influence of
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Paint a stripe on a cardboard roll. Can you predict what will
happen when it is rolled across a sheet of paper?