In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

Delight your friends with this cunning trick! Can you explain how it works?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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.

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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?

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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 ...

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

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

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of Little Fung at the table?

A game to make and play based on the number line.

Use the tangram pieces to make our pictures, or to design some of your own!

Can you fit the tangram pieces into the outline of this telephone?

Can you fit the tangram pieces into the outline of Mai Ling?

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?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Here's a simple way to make a Tangram without any measuring or ruling lines.

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outlines of the workmen?

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

Can you fit the tangram pieces into the outline of this plaque design?

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outlines of the candle and sundial?

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

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

Can you fit the tangram pieces into the outline of this junk?

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

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of the rocket?

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

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

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?

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

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?