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 at primary level that may require determination.
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?
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.
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.
Measure problems for primary learners to work on with others.
Measure problems at primary level that require careful consideration.
Measure problems for inquiring primary learners.
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?
Look at the mathematics that is all around us - this circular window is a wonderful example.
How would you move the bands on the pegboard to alter these shapes?
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 time?
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?
Use the information on these cards to draw the shape that is being described.
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?
What do these two triangles have in common? How are they related?
A simple visual exploration into halving and doubling.
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?
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.
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 predictions.
A task which depends on members of the group noticing the needs of others and responding.
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 into?
Determine the total shaded area of the 'kissing triangles'.
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
A follow-up activity to Tiles in the Garden.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
Explore one of these five pictures.
How many tiles do we need to tile these patios?