Light blue - dark blue
Investigate the successive areas of light blue in these diagrams.
Age
7 to 11
Challenge level
Problem
Look at each of these five squares.
Image
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.
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, $\frac{2}{3}$, $\frac{4}{9}$, $\frac{8}{27}$, $\frac{16}{81}$, $\ldots$
She noticed that
the number on top got multiplied by 2 each time, and the number on
the bottom got multiplied by 3 each time.
Some of our more advanced readers
might know that we could write this as
$\frac{2^n}{3^n}$.
$\frac{2^n}{3^n}$.
Francesca also
noticed that the amount of light blue got smaller and smaller each
time. She thinks that if we could do this forever, in the end the
whole square would be dark blue.
Teachers' Resources
Light Blue - Dark Blue
Look at each of these five squares.
Image
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.
Why do this problem?
This problem gives children the opportunity to explore fractions in a practical context. They will be identifying and explaining patterns, and justifying their ideas.
Possible approach
Show the images to the group, perhaps one at a time, inviting them to look carefully at each. Then, hide the images and ask learners to talk to each other about what they saw. After some time, bring them together to discuss as a whole group and at this stage you might encourage them to think about the fractions of each colour if it doesn't come up naturally. It will be necessary to reveal
the images again at certain moments, in order to check ideas and facilitate the discussion.
Then ask children to work in pairs to recreate the images and extend the sequence. Squared paper and squares of coloured paper might be useful, but encourage each pair to make their own decisions about the materials they will use. This will be an ideal time for you to wander around the room listening to the conversations. Listen out for those learners who are challenging their partner's
assumptions and justifying their answers.
Key questions
What fractions can you see?
Is there a pattern to the fractions of area in each image?
Can you convince us that your fraction is correct?
Possible extension
Learners could also make up other patterns of their own for partners to extend.
Possible support
Some children might benefit from having copies of the images available on a sheet to which they could refer when necessary.