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?

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 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?

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

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?

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

How many models can you find which obey these rules?

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?

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?

These pictures show squares split into halves. Can you find other ways?

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

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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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.

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?

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

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

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?

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

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

Can you make the birds from the egg tangram?

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?

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

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

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

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?

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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

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

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.

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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?

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

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

Can you split each of the shapes below in half so that the two parts are exactly the same?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?