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.
I cut this square into two different shapes. What can you say about
the relationship between them?
How many tiles do we need to tile these patios?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Here are many ideas for you to investigate - all linked with the
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
These practical challenges are all about making a 'tray' and covering it with paper.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
These pieces of wallpaper need to be ordered from smallest to
largest. Can you find a way to do it?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
What do these two triangles have in common? How are they related?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
An investigation that gives you the opportunity to make and justify
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 draw a square in which the perimeter is numerically equal
to the area?
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?
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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 thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Can you put these shapes in order of size? Start with the smallest.
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.
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Use the information on these cards to draw the shape that is being described.
A simple visual exploration into halving and doubling.
This article for teachers gives some food for thought when teaching
ideas about area.
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
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
Explore one of these five pictures.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.