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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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
An investigation that gives you the opportunity to make and justify
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
How would you move the bands on the pegboard to alter these shapes?
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.
A simple visual exploration into halving and doubling.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Explore one of these five pictures.
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?
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 practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Use the information on these cards to draw the shape that is being described.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
Can you put these shapes in order of size? Start with the smallest.
These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
A follow-up activity to Tiles in the Garden.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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
Can you draw a square in which the perimeter is numerically equal
to the area?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
What do these two triangles have in common? How are they related?
Measure problems for primary learners to work on with others.
Measure problems at primary level that require careful consideration.
Measure problems at primary level that may require determination.
Are these statements always true, sometimes true or never true?
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
Measure problems for inquiring primary learners.