Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
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 involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Sort the houses in my street into different groups. Can you do it in any other 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.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
What do these two triangles have in common? How are they related?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
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?
How many triangles can you make on the 3 by 3 pegboard?
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
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 is the largest cuboid you can wrap in an A3 sheet of paper?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
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?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
These pictures show squares split into halves. Can you find other ways?
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.
Explore the triangles that can be made with seven sticks of the same length.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
How many faces can you see when you arrange these three cubes in different ways?
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?
How many models can you find which obey these rules?
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?
Have a go at this 3D extension to the Pebbles problem.
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.
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?
Can you find ways of joining cubes together so that 28 faces are visible?
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?