I cut this square into two different shapes. What can you say about the relationship between them?
What do these two triangles have in common? How are they related?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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.
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Explore the triangles that can be made with seven sticks of the same length.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
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 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?
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?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
These pictures show squares split into halves. Can you find other ways?
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 create more models that follow these rules?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
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?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
Why does the tower look a different size in each of these pictures?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
How many triangles can you make on the 3 by 3 pegboard?
Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
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 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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
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 one of these five pictures.