What can you see? What do you notice? What questions can you ask?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
What is the greatest number of squares you can make by overlapping
Can you make five differently sized squares from the tangram
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Square It game for an adult and child. Can you come up with a way of always winning this game?
Complete the squares - but be warned some are trickier than they
These points all mark the vertices (corners) of ten hidden squares.
Can you find the 10 hidden squares?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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
Cut a square of paper into three pieces as shown. Now,can you use
the 3 pieces to make a large triangle, a parallelogram and the
This activity challenges you to make collections of shapes. Can you
give your collection a name?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
A group activity using visualisation of squares and triangles.
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 practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
These pictures show squares split into halves. Can you find other ways?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Can you each work out what shape you have part of on your card? What will the rest of it look like?
What do you think is the same about these two Logic Blocks? What
others do you think go with them in the set?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.
Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
This interactivity allows you to sort logic blocks by dragging their images.
This activity investigates how you might make squares and pentominoes from Polydron.
Look at how the pattern is built up - in that way you will know how
to break the final pattern down into more manageable pieces.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
Can you use LOGO to create this star pattern made from squares.
Only basic LOGO knowledge needed.
Creating designs with squares - using the REPEAT command in LOGO.
This requires some careful thought on angles
Arrange the shapes in a line so that you change either colour or
shape in the next piece along. Can you find several ways to start
with a blue triangle and end with a red circle?
A Short introduction to using Logo. This is the first in a twelve part series.