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

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

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

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

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

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

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

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

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?

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

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 ...

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

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

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

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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?

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

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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?

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

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Can you fit the tangram pieces into the outline of the house?

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

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

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?

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?

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

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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.

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

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?

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

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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

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

Can you fit the tangram pieces into the outline of Mah Ling?

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

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?

Can you fit the tangram pieces into the outline of Granma T?

Can you fit the tangram pieces into the outlines of the camel and giraffe?