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?
Explore one of these five pictures.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
An investigation that gives you the opportunity to make and justify predictions.
These practical challenges are all about making a 'tray' and covering it with paper.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
How many tiles do we need to tile these patios?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
A follow-up activity to Tiles in the Garden.
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?
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 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?
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?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
How many centimetres of rope will I need to make another mat just like the one I have here?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
Use the information on these cards to draw the shape that is being described.
A simple visual exploration into halving and doubling.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
In how many ways can you halve a piece of A4 paper? How do you know they are halves?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Are these statements always true, sometimes true or never true?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?
We started drawing some quadrilaterals - can you complete them?
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 happens to the area and volume of 2D and 3D shapes when you enlarge them?
Measure problems for inquiring primary learners.