Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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 centimetres of rope will I need to make another mat just like the one I have here?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
How many tiles do we need to tile these patios?
Here are many ideas for you to investigate - all linked with the number 2000.
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?
This article for teachers gives some food for thought when teaching ideas about area.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Measure problems for inquiring primary learners.
In how many ways can you halve a piece of A4 paper? How do you know they are halves?
Measure problems at primary level that require careful consideration.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Measure problems at primary level that may require resilience.
Look at the mathematics that is all around us - this circular window is a wonderful example.
Measure problems for primary learners to work on with others.
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?
Use the information on these cards to draw the shape that is being described.
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
A simple visual exploration into halving and doubling.
An investigation that gives you the opportunity to make and justify predictions.
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
What do these two triangles have in common? How are they related?
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.
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?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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 this 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?
How would you move the bands on the pegboard to alter these shapes?
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.
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 rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?