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Building Gnomons

Age 14 to 16 Challenge Level:

Why do this problem?

This problem is about investigating a numerical sequence using pictorial representations. Learners investigate the reflections and rotations needed to combine two consecutive pictures in the sequence to make the next, thus gaining a deeper understanding of the structure of the Fibonacci numbers. Learners may describe patterns in the sequence using their own symbolic representations.

Possible approach

Write up the sequence $1, 1, 2, 3, 5$ on the board. Ask for suggestions as to what might come next, with reasons why. Reveal that the sequence we are investigating is the Fibonacci sequence made by adding together the previous two terms of the sequence (starting with 0,1).

Show the picture of the gnomons and allow the class some time to experiment until they can find ways of transforming two consecutive gnomons into the next one. It is important to have some sense of the orientation of the gnomon - the diagrams in the problem all have the missing part of the rectangle in the top left, so this convention could be adopted. The idea is to come up with a set of instructions which describes how to manipulate two consecutive gnomons to make the next gnomon in the sequence. Learners may adopt shorthand ways of writing any reflections and rotations they have used, but it is important that they are accurate in their recording in order to spot (and later justify) patterns. One way of checking the accuracy of their recording is for everyone to swap their instructions with someone else to see if they make sense when followed to the letter.

The second part of the problem starts to look at the lengths of the sides of the gnomons and investigate the Fibonacci sequence. The key is to come up with a systematic way of recording the edges in terms of the Fibonacci numbers; there is a diagram in the Hint which may help.

Key questions

Can you describe how you have joined your gnomons to make the next one in the sequence in a way that other people can understand?
Is there more than one way of joining gnomons?
Can you generate all the gnomons using the same process each time?

Possible extension

Gnomon Dimensions looks at more patterns and generalisations that can be made.

Possible support

Cutting out and physically manipulating the gnomons or making gnomons from cubes can help learners to describe patterns that emerge.