How many centimetres of rope will I need to make another mat just
like the one I have here?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Measure problems at primary level that require careful consideration.
Measure problems for primary learners to work on with others.
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.
Measure problems for inquiring primary learners.
Measure problems at primary level that may require determination.
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.
A simple visual exploration into halving and doubling.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Use the information on these cards to draw the shape that is being described.
This article for teachers gives some food for thought when teaching
ideas about area.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What do these two triangles have in common? How are they related?
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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
How would you move the bands on the pegboard to alter these shapes?
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?
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?
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .
Derive a formula for finding the area of any kite.
An investigation that gives you the opportunity to make and justify
Determine the total shaded area of the 'kissing triangles'.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Can you work out the area of the inner square and give an
explanation of how you did it?
What fractions of the largest circle are the two shaded regions?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
What happens to the area and volume of 2D and 3D shapes when you
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?
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.
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?
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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.