Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Can you split each of the shapes below in half so that the two parts are exactly the same?
Can you describe a piece of paper clearly enough for your partner to know which piece it is?
Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?
We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?
Have a go at this 3D extension to the Pebbles problem.
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?
Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.
Move four sticks so there are exactly four triangles.
Make a flower design using the same shape made out of different sizes of paper.
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
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?
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?
Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you fit the tangram pieces into the outline of the butterfly?
Can you fit the tangram pieces into the outline of the candle?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Can you fit the tangram pieces into the outline of the telephone?
Can you fit the tangram pieces into the outline of the sports car?
Can you fit the tangram pieces into the outlines of the telescope and microscope?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
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 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?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these two grids?
Can you fit the tangram pieces into the outline of Little Ming?
Can you fit the tangram pieces into the outline of Mah Ling?
Can you fit the tangram pieces into the outline of this teacup?
Make a cube out of straws and have a go at this practical challenge.
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?
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?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Can you fit the tangram pieces into the outlines of the convex shapes?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
This article looks at levels of geometric thinking and the types of activities required to develop this thinking.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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?
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 this area?
This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.
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?