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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Measure problems at primary level that require careful consideration.
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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
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.
How would you move the bands on the pegboard to alter these shapes?
What do these two triangles have in common? How are they related?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
An investigation that gives you the opportunity to make and justify
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
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
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?
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?
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
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Explore one of these five pictures.
How many tiles do we need to tile these patios?
A follow-up activity to Tiles in the Garden.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you draw a square in which the perimeter is numerically equal
to the area?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.