Why do this problem?
Unit fractions are the first fractions children meet, and
here we discover some very surprising and interesting
characteristics of these familiar numbers. Some of these
characteristics were known to the ancient Egyptians whilst
other conjectures are yet to be proved.
Whilst meeting both old and new mathematical ideas, students
can improve their fluency in addition and subtraction of
fractions and be challenged to generalise and explain their
findings.
Possible approach
You could choose to set the scene briefly by asking students
what they know about mathematics throughout history,
establishing the idea that some historical maths is distinctly
odd to our modern view point.
Explain that the ancient Egyptians didn't write fractions with
a numerator greater than 1 but expressed every fraction as the
sum of different unit fractions.
Work through the example of $\frac{2}{3}$ as the problem
suggests, asking students to lengthen each successive row by
substituting each unit fraction by a different pair, using
methods the students met in
Keep It Simple
Establish that we can keep lengthening the expression for any
$\frac{2}{n}$ fraction but what would have been of real value
to the Egyptians would have been a method for expressing these
fractions in the shorteset possible way, i.e. using just two
different unit fractions.
Confirm that this is possible for$\frac{2}{3}$ and then set the
challenge to choose their own $\frac{2}{n}$ fraction and
express it as the sum of just two unit fractions. Any that
can't be done can be written up on the board for the rest of
the class to attempt.
Stop the class, and ask them to step back from number crunching
and share any discoveries. Listen for any generalisations and
record them for discussion.
Students could follow this up by exploring fractions of the
form$\frac{3}{n}$, $\frac{4}{n}$ etc and be challenged to
express them in as short a way as possible.
Key questions
What do we already know that could help?
Possible extension
Some students might wish to undertake research about the
Rhind Mathematical Papyrus.
Possible support
Suggest to students that they work systematically by building
on the example of $\frac{2}{3}$ by going on to $\frac{2}{5},
\frac{2}{7}, \frac{2}{9}$... and look out for patterns.