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 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?

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

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

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

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?

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.

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?

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?

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?

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

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

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.

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

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 a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

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

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

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

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

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

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?

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?

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?

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

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

We have a box of cubes, triangular prisms, cones, cuboids, cylinders and tetrahedrons. Which of the buildings would fall down if we tried to make 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?

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

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

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

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

If you count from 1 to 20 and clap more loudly on the numbers in the two times table, as well as saying those numbers loudly, which numbers will be loud?

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

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

How many models can you find which obey these rules?

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

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.

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?

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

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

Can you make the birds from the egg tangram?

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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?