How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
Investigate these hexagons drawn from different sized equilateral triangles.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
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?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
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.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
What do these two triangles have in common? How are they related?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
A follow-up activity to Tiles in the Garden.
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?
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.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Explore one of these five pictures.
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.
Why does the tower look a different size in each of these pictures?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
Can you create more models that follow these rules?
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?
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?
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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!
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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 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?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?