Light Blue - Dark Blue
2 out of 3
Investigate the successive areas of light blue in these diagrams.
Problem
Look at each of these five squares.
What is the pattern?
How much of the second square is light blue? Can you write this as a fraction?
For each of the five squares, write the area of the square that is light blue as a fraction.
Can you work out what the next two diagrams would look like? What fraction of these squares will be light blue?
Getting Started
It might help to try and re-create the images using squared paper and coloured pens/pencils or squares of coloured paper.
Student Solutions
Francesca investigated this problem. She imagined that each time the big square was split up into little blocks that looked like the light blue ones. Then she counted how many light blue ones there were, and how many overall. This is what she got :
1, $\frac{2}{3}$, $\frac{4}{9}$, $\frac{8}{27}$, $\frac{16}{81}$, $\ldots$
$\frac{2^n}{3^n}$.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
Possible approach
Key questions
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