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?

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

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

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

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

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

Exchange the positions of the two sets of counters in the least possible number of moves

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

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?

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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.

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

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

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

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

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

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.

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.

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

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.

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

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!

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

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

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

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?

Square It game for an adult and child. Can you come up with a way of always winning this game?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

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 Square It?

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?

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?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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?

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

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

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

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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 mark 4 points on a flat surface so that there are only two different distances between them?

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.