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 at primary level that require careful consideration.

Measure problems for primary learners to work on with others.

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

Measure problems at primary level that may require resilience.

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

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.

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

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.

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?

Measure problems for inquiring primary learners.

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

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

In how many ways can you halve a piece of A4 paper? How do you know they are halves?

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

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?

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 simple visual exploration into halving and doubling.

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?

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.

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

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.

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

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

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?

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

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

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

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

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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

A task which depends on members of the group noticing the needs of others and responding.

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?

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?

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?

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?

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?

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

A follow-up activity to Tiles in the Garden.

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

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .