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

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

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

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

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

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

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

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?

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.

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

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

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?

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

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

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?

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

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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?

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?

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

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 ...

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?

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?

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

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

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

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

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?

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

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

Reasoning about the number of matches needed to build squares that share their sides.

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.

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

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

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

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?

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

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 dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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.

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?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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

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.

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?