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'The Greedy Algorithm' printed from http://nrich.maths.org/
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Why do this problem?
This problem follows on from Keep It Simple and Egyptian FractionsThese three problems together offer students an opportunity to engage with some mathematical ideas in depth and not just with the rather mechanical process of adding and subtracting fractions.
This problem in particular requires students to compare fractions and may deepen their understanding of their relative sizes.
Possible approach
Students should already have worked with fractions of the form
$\frac{1}{n}$, $\frac{2}{n}$ and possibly $\frac{3}{n}$ and
$\frac{4}{n}$ in Keep
It Simple and Egyptian
Fractions .
To give students a 'feel' for the difficulty of expressing
fractions as the Egyptians did, ask students to work in pairs for a
short time finding an Egyptian sum for any fraction of their
choice. Note, all the unit fractions in the summation must be
unique. Any that they can't do can be written on the board for
others to attempt.
Draw the group together and share strategies. Compare their
efficiency. Do they always give the same result?
If you start with $\frac{4}{5}$, for example, and apply
students' different strategies, you may end up with different
outcomes:
$\frac{4}{5} = \frac{1}{2} + \frac{1}{4} + \frac{1}{20}$
or
$\frac{4}{5} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10}$
or
$\frac{4}{5} = \frac{1}{3} + \frac{1}{5} + \frac{1}{6} +
\frac{1}{15} + \frac{1}{30}$
....
If no-one has suggested it, introduce the Greedy Algorithm and
check to see if anyone has used it already.
Ask students to choose fractions of their own and
apply the Greedy Algorithm. Does it always work?
Can they find a convincing explanation?
Key Questions
Is our fraction larger or smaller than $\frac{1}{2}$? How do
we know?
Is our fraction larger or smaller than $\frac{1}{3}$? How do
we know?
Is our fraction larger or smaller than $\frac{1}{4}$? How do
we know?
....Possible extension
Students could try to prove why Fibonnacci's Greedy Algorithm
always terminates (the numerators always decrease and must
therefore reach one).
Does the Greedy Algorithm always result in the sum with the
fewest possible terms?
Can anyone find a counter example?The Eye of Horus: often it was good enough to use only the fractions
that represent $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$.... to get a fraction that is close enough to any specific fraction. Suggest that students pick some fractions and convert them to this form of Egyptian fraction.
How close does this method get to the target fraction?
Students might like to research how these particular fractions were written down in pictorial form.
Possible support
Students who need support in comparing the size of fractions
might have to do some preliminary work on equivalent
fractions.
Alternatively, they could convert them to decimal equivalents
using a calculator.