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?
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.
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.
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.
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
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
I cut this square into two different shapes. What can you say about
the relationship between them?
This article for teachers gives some food for thought when teaching
ideas about area.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A simple visual exploration into halving and doubling.
How would you move the bands on the pegboard to alter these shapes?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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 is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What do these two triangles have in common? How are they related?
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?
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?
Use the information on these cards to draw the shape that is being described.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
Here are many ideas for you to investigate - all linked with the
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
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
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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?
An investigation that gives you the opportunity to make and justify
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?