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Gnomon Dimensions
Age 14 to 16
Challenge Level 
Why do this problem?
This problem continues to investigate the sequence of shapes introduced in Building Gnomons Learners will need to use their skills of representation to communicate their ideas and justify their findings. This problem builds on learners' prior knowledge of how the Fibonacci sequence grows.Possible approach
Ask learners to work in small groups to investigate areas and
dimensions of gnomons.
After a short time, draw all the groups together to share
ideas about how they might organise their approaches and record
findings. Which groups are working systematically, which have used
effieicent recording methods?
It is most desirable for learners to develop their own
representations to justify any patterns they find. However, if they
are struggling the Hint contains one way of recording edge lengths
in terms of Fibonacci numbers that might be a useful stimulus for
discussion.
Is it possible to predict the dimensions of the gnomons in the
sequence?
Encourage the use of diagrams and notation to explain how the
pattern will continue and why.
Sharing findings and justifications might be achieved by the
use of posters which groups present to the rest of the group. Use
the opportunity for other learners to feedback on the clarity of
what is presented.
Key questions
How does the approach in the hint work?
How does the way you put pairs of gnomons together result in
new Fibonacci numbers?
Possible extension
The article Whirling
Fibonacci Squares explores some of the ideas of the Fibonacci
numbers in more detail.
Possible support
Try Building Gnomons first. Sheep Talk could be used as an introduction to Fibonacci numbers.You may also like
Searching for Mean(ing)
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
Building Gnomons
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.