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

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

This article for teachers suggests ideas for activities built around 10 and 2010.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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?

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?

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.

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

A follow-up activity to Tiles in the Garden.

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

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

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?

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

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?

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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

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?

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

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?

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

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

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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?

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

A challenging activity focusing on finding all possible ways of stacking rods.

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?

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

Which way of flipping over and/or turning this grid will give you the highest total? You'll need to imagine where the numbers will go in this tricky task!

In how many ways can you stack these rods, following the rules?

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.