Try continuing these patterns made from triangles. Can you create your own repeating pattern?

These pictures show squares split into halves. Can you find other ways?

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 do these two triangles have in common? How are they related?

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?

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 find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

This problem is intended to get children to look really hard at something they will see many times in the next few months.

Explore the triangles that can be made with seven sticks of the same length.

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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?

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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.

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles 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.

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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?

How many faces can you see when you arrange these three cubes in different ways?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Sort the houses in my street into different groups. Can you do it in any other ways?

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?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?