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

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?

Can you make the birds from the egg tangram?

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

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

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

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

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

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

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?

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

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

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?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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?

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

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

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

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.

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

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?

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?

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

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

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?

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?

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

Here are some ideas to try in the classroom for using counters to investigate number patterns.

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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?

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

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

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

How many models can you find which obey these rules?

You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

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

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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.

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