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?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
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?
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?
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?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
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?
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?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate the different ways you could split up these rooms so that you have double the number.
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?
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?
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?
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?
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?
Can you create more models that follow these rules?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
In how many ways can you stack these rods, following the rules?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Why does the tower look a different size in each of these pictures?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Explore one of these five pictures.
How many models can you find which obey these rules?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
Have a go at this 3D extension to the Pebbles problem.
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.