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 ...

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

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?

The challenge for you is to make a string of six (or more!) graded cubes.

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

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?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

A game in which players take it in turns to choose a number. Can you block your opponent?

A jigsaw where pieces only go together if the fractions are equivalent.

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

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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.

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

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.

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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 is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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?

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

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?

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

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

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

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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

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

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Build a scaffold out of drinking-straws to support a cup of water

Can Jo make a gym bag for her trainers from the piece of fabric she has?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

What shape would fit your pens and pencils best? How can you make it?

What shape and size of drinks mat is best for flipping and catching?

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

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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

How many models can you find which obey these rules?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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?

How can you make a curve from straight strips of paper?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.

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