I cut this square into two different shapes. What can you say about the relationship between them?
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 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.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What do these two triangles have in common? How are they related?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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 find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
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?
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
How many triangles can you make on the 3 by 3 pegboard?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Why does the tower look a different size in each of these pictures?
Explore one of these five pictures.
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
Can you create more models that follow these rules?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
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?
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.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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?
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?
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What is the largest cuboid you can wrap in an A3 sheet of paper?