Make a flower design using the same shape made out of different sizes of paper.

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?

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 visualise what shape this piece of paper will make when it is folded?

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

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?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

Join pentagons together edge to edge. Will they form a ring?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Make a cube out of straws and have a go at this practical challenge.

Exploring and predicting folding, cutting and punching holes and making spirals.

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 ...

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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

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?

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

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?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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

What shape is made when you fold using this crease pattern? Can you make a ring design?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

This article for teachers describes a project which explores the power of storytelling to convey concepts and ideas to children.

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Can you fit the tangram pieces into the outlines of the convex shapes?

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.

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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?

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?

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.

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

Which of these dice are right-handed and which are left-handed?

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.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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!

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!