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

How many centimetres of rope will I need to make another mat just like the one I have here?

Measure problems for inquiring primary learners.

Measure problems for primary learners to work on with others.

Measure problems at primary level that may require determination.

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

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

Measure problems at primary level that require careful consideration.

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.

Use the information on these cards to draw the shape that is being described.

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

How would you move the bands on the pegboard to alter these shapes?

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 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?

This article for teachers gives some food for thought when teaching ideas about area.

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

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

A simple visual exploration into halving and doubling.

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

Look at the mathematics that is all around us - this circular window is a wonderful example.

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

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

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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

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

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

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

Can you work out the area of the inner square and give an explanation of how you did it?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

These practical challenges are all about making a 'tray' and covering it with paper.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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

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

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

A follow-up activity to Tiles in the Garden.

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