I cut this square into two different shapes. What can you say about the relationship between them?

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?

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

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

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

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.

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.

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?

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?

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?

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?

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?

An investigation that gives you the opportunity to make and justify predictions.

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

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

Here are many ideas for you to investigate - all linked with the number 2000.

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

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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?

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?

Why does the tower look a different size in each of these pictures?

A follow-up activity to Tiles in the Garden.

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

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?

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?

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 the number of faces you can see when you arrange three cubes in different ways.

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

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

What is the largest cuboid you can wrap in an A3 sheet of paper?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Investigate what happens when you add house numbers along a street in different ways.

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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?

A description of some experiments in which you can make discoveries about triangles.

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.