This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
Can you create more models that follow these rules?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
Sort the houses in my street into different groups. Can you do it in any other ways?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Explore the triangles that can be made with seven sticks of the same length.
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.
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
These pictures show squares split into halves. Can you find other ways?
Can you find ways of joining cubes together so that 28 faces are visible?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
An activity making various patterns with 2 x 1 rectangular tiles.
How many models can you find which obey these rules?
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?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
What do these two triangles have in common? How are they related?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A challenging activity focusing on finding all possible ways of stacking rods.
In how many ways can you stack these rods, following the rules?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
Have a go at this 3D extension to the Pebbles problem.
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?
Investigate the different ways you could split up these rooms so that you have double the number.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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 challenge extends the Plants investigation so now four or more children are involved.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Investigate the number of faces you can see when you arrange three cubes in different ways.
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
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.