How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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 faces can you see when you arrange these three cubes in different ways?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
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 is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
Can you create more models that follow these rules?
How many models can you find which obey these rules?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?
An activity making various patterns with 2 x 1 rectangular tiles.
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Can you find ways of joining cubes together so that 28 faces are visible?
Have a go at this 3D extension to the Pebbles problem.
It starts quite simple but great opportunities for number discoveries and patterns!
These pictures show squares split into halves. Can you find other ways?
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
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.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Explore one of these five pictures.
In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.