AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE TEACHER NETWORK Title: EGYPTIAN FRACTIONS (Grades 7 to 10) Egyptian fractions are unit fractions, that is fractions with a numerator of 1. We are going to investigate methods of
reducing unit fractions to the sums of different smaller unit fractions as we see in this example:
1 1 1
= +
2 3 6
1 1 1
= +
3 4 12
1 1 1
…
= +
4 5 20
Try some for yourself. Does this always work? Could you explain to someone else how to use this method?
Perhaps we can find other ways to reduce unit fractions to the sums of different smaller unit fractions.
For example:
1 1 1
= +
6 7 42
1 1 1
= +
6 8 24
1 1 1
= +
6 9 18
1 1 1
= +
6 10 15
1 1 1
1
…
= + +
6 11 14 231
Can all unit fractions be made up in more than one way like this? Why? Try some examples of your own to test out
your ideas.
Solution
It is always possible to write a unit fraction as the sum of two unit fractions one with a denominator that is 1 more
than the denominator of the original unit fraction, and the other with a denominator that is the product of the other two
denominators:
1
1
1
=
+
n n +1 n(n +1)
Here are some other examples that work:
1/5=1/6+1/30
1/6=1/7+1/42
1/105=1/106+1/11130
But not all solutions
1/8 can also be expressed as the sum of two unit fractions in several ways:
1/8=1/9+1/72
1/8=1/10+1/40
1/8=1/11+1/n is not possible
1/8=1/12+1/24
and
1/12=1/13+1/156
1/12=1/14+1/84
1/12=1/15+1/60
1/12=1/16+1/48
1/12=1/18+1/36
1/12=1/20+1/30
1/12=1/21+1/28
Notice that denominators that are prime numbers can only be written in one way as the sum of two distinct unit
fractions.
Notes for teacher
Why do this activity?
It's often difficult to find interesting contexts to consolidate addition and subtraction of fractions. This activity offers
that, whilst also requiring learners to develop and analyse different strategies and explain their findings.
Possible approach
To start the lesson - Write the following on the board and ask learners to say which are correct and which are wrong.
1 1 1
= +
2 10 20
1 1 1
= +
3 4 12
1 1 1
= +
3 7 21
1 1
1
= +
8 10 110
1 1 1
= +
2 3 6
1 1 1
…
= +
4 5 20
Allow some time for learners to work out which sums are correct and ask them to try to
find a pattern in the correct results and a rule that generates correct solutions.
Invite learners to work in pairs to find more examples that confirm their new rule. Collect
some of these on the board for a general discussion. (With some classes this could lead to
an algebraic explanation/proof that
- call this RULE 1)
To continue the lesson – Write the following on the board.
1 1 1
= +
6 7 42
1 1 1
= +
6 8 24
1 1 1
= +
6 9 18
1 1 1
= +
6 10 15
1 1 1
1
= + +
6 11 14 231
Say that the learners should now try to find as many ways as possible to write each
fraction as a sum of other unit fractions in different ways. (RULE 2)
This might be an opportunity to talk to the class about the value of working
systematically. How can they be sure that they have found all the possible pairs?
Ask students to work in pairs and to choose their own unit fraction and find all the
correct pairs.
Collect all results on the board and encourage students to share their strategies for
finding all possible combinations.
…
Key questions
Can a unit fraction always be written as the sum of two different unit fractions?
Which unit fractions can only be written in one way?
Can you tell me how to find all the combinations of two unit fractions that add up to the unit fraction you chose?
Possible extension
Ask learners to give an algebraic or visual proof of RULE 1.
Can they predict how many different pairs of unit fractions will add up to any given unit fraction?
You could ask learners to do
The Greedy Algorithm https://aiminghigh.aimssec.ac.za/grades-7-to-10-the-greedy-algorithm/
Possible support
Some learners may find it easier to contribute to the class discussion by working systematically to generate lots of unit
sum calculations and highlighting any that result in a unit fraction as an answer.
For example
1/6+1/7=13/42
1/6+1/8=7/24
...
1/6+1/12=1/4