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Clock Hands

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

Watch the Clock

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

Take the Right Angle

How many times in twelve hours do the hands of a clock form a right angle? Use the interactivity to check your answers.

Overlapping Squares

Age 7 to 11
Challenge Level

two images of overlapping squares

Describe what is happening in these pictures.
Do the pictures appear to be coming towards you, or going away?

How many squares are there in each picture?
If the side of the smallest square is 1 unit, how many units is the side of the next square?
And the next?
How many units is the side of the largest square?
What is the area of the smallest square?
What is the area of the next square? And the next?
What is the sequence of the areas of the squares?


The smallest square is one unit of area. How many more units are in the second square?
How many more units of area are in the third square than in the second square?
What is the rule for building the squares?


Look at the top picture. Start from the largest square and think of it as shrinking as it turns.
Through what angle has it turned when it has changed to the smallest square?
Is it the same in both pictures?
What is the angle of rotation as one square changes to the next?
Is it the same in both pictures?
Make a similar picture. You could change the rules if you would like to.

This image, along with the questions and notes, features on the Exploring Squares CD-ROM, developed by members of the NRICH Team and published by Virtual Image Publishing Ltd. For more details, please see our publications page .


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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