This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

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?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Can you make the birds from the egg tangram?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

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

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?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

These practical challenges are all about making a 'tray' and covering it with paper.

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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

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?

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

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

What do these two triangles have in common? How are they related?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

An activity making various patterns with 2 x 1 rectangular tiles.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

Here is a version of the game 'Happy Families' for you to make and play.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Explore the triangles that can be made with seven sticks of the same length.

How many models can you find which obey these rules?

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.