Keep it simple
Problem
Keep it Simple printable sheet
Unit fractions (fractions which have numerators of 1) can be written as the sum of two different unit fractions.
For example
$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$
Charlie thought he'd spotted a rule and made up some more examples.
$\frac{1}{2} = \frac{1}{10} + \frac{1}{20}$
$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$
$\frac{1}{3} = \frac{1}{7} + \frac{1}{21}$
$\frac{1}{4} = \frac{1}{5} + \frac{1}{20}$
Can you describe Charlie's rule?
The denominator of the last fraction is the product of the denominators of the first two fractions.
Are all his examples correct?
What do you notice about the sums that are correct?
Find some other correct examples..
How would you explain to Charlie how to generate lots of correct examples?
Alison started playing around with $\frac{1}{6}$ and was surprised to find that there wasn't just one way of doing this.
She found:
$\frac{1}{6} = \frac{1}{7} + \frac{1}{42}$
$\frac{1}{6} = \frac{1}{8} + \frac{1}{24}$
$\frac{1}{6} = \frac{1}{9} + \frac{1}{18}$
$\frac{1}{6} = \frac{1}{10} + \frac{1}{15}$
$\frac{1}{6} = \frac{1}{12} + \frac{1}{12}$ (BUT she realised this one didn't count because they were not different.)
Charlie tried to do the same with $\frac{1}{8}$. Can you finish Charlie's calculations to see which ones work?
$\frac{1}{8} = \frac{1}{9} + ?$
$\frac{1}{8} = \frac{1}{10} + ?$
$\frac{1}{8} = \frac{1}{11} + ?$
..........
Can all unit fractions be made in more than one way like this?
Choose different unit fractions of your own to test out your theories.
Getting Started
Try working systematically through all the possibilities.
$\frac{1}{8} = \frac{1}{9} + ?$
$\frac{1}{8} = \frac{1}{10} + ?$
$\frac{1}{8} = \frac{1}{11} + ?$
$\frac{1}{8} = \frac{1}{12} + ?$
$\frac{1}{8} = \frac{1}{13} + ?$
$\frac{1}{8} = \frac{1}{14} + ?$
$\frac{1}{8} = \frac{1}{15} + ?$
$\frac{1}{8} = \frac{1}{16} + ?$ (but this won't count)
Why is $\frac{1}{9}$ the first one you can use?
Why don't you need to go further than $\frac{1}{16}$?
Student Solutions
Catherine and Poppy from Stoke by Nayland Middle School made a good start on this problem, and Kijung from Wind Point Elementary School found that:
Not all of Charlie's examples were right.
To be correct, one of the unit fractions must have a denominator which is 1 more than the denominator of the original unit fraction, and the other unit fraction must have a denominator which is the product of the other two denominators:
$$ \frac{1}{n} = \frac{1}{n+1}+\frac{1}{n(n+1)}$$
Here are some other examples that work:
$ \frac{1}{5} = \frac{1}{6}+\frac{1}{30}$
$ \frac{1}{6} = \frac{1}{7}+\frac{1}{42}$
$ \frac{1}{105} = \frac{1}{106}+\frac{1}{11130}$
$\frac{1}{8}$ can also be expressed as the sum of two unit fractions in several ways:
$\frac{1}{8} = \frac{1}{9} +\frac{1}{72}$
$\frac{1}{8} = \frac{1}{10} +\frac{1}{40}$
$\frac{1}{8} = \frac{1}{11} + \frac{1}{n}$ is not possible
$\frac{1}{8} = \frac{1}{12} +\frac{1}{24}$
Felix from Condover Primary acutely observed that unit fractions with denominators which are prime numbers can only be written in one way as the sum of two distinct unit fractions.
Rose, from Claremont Primary School in Tunbridge Wells, Kent worked out a general formula:
$ \frac{1}{z} = \frac{1}{y}+\frac{1}{x}$ (where $z$, $y$ and $x$ are positive integers and $y < x$)
Using $\frac{1}{10}$ as an example:
$ \frac{1}{10} = \frac{1}{11}+\frac{1}{110}$
$ \frac{1}{10} = \frac{1}{12}+\frac{1}{60}$
$ \frac{1}{10} = \frac{1}{14}+\frac{1}{35}$
$ \frac{1}{10} = \frac{1}{15}+\frac{1}{30}$
I listed the values of $yz$ that provide solutions:
$1$, $2$, $4$ and $5$
These are also the factors of $z^ 2$ (i.e. $100$) that are smaller than its square root: $1\times100$
$2\times50$
$4\times25$
$5\times20$
$10\times10$
This pattern also occurred for $\frac{1}{12}$:
$ \frac{1}{12} = \frac{1}{13}+\frac{1}{156}$
$ \frac{1}{12} = \frac{1}{14}+\frac{1}{84}$
$ \frac{1}{12} = \frac{1}{15}+\frac{1}{60}$
$ \frac{1}{12} = \frac{1}{16}+\frac{1}{48}$
$ \frac{1}{12} = \frac{1}{18}+\frac{1}{36}$
$ \frac{1}{12} = \frac{1}{20}+\frac{1}{30}$
$ \frac{1}{12} = \frac{1}{21}+\frac{1}{28}$
Here $y  z = 1, 2, 3, 4, 6, 8, 9$
and the factors of $z ^ 2$ (i.e.$144$) are:
$1\times144$
$2\times72$
$3\times48$
$4\times36$
$6\times24$
$8\times18$
$9\times16$
$12\times12$
$\frac{1}{10}$ can be written as the sum of two different unit fractions in $4$ ways.
In this case $z ^ 2$ has $9$ factors and $yz = 4$
$\frac{91}{2}=4$
$\frac{1}{12}$ can be written as the sum of two different unit fractions in $7$ ways.
In this case $z ^ 2$ has $15$ factors and $yz = 7$
$\frac{151}{2}=7$
Conclusion:
If $n$ is the number of factors of $z ^ 2$,
$\frac{1}{z}$ can be written as the sum of two different unit fractions in $\frac{n 1}{2}$ ways.
Rose's conclusion draws on her two examples, but when we generalise in mathematics, we need to be sure that what we have noticed will be true in all other cases.
Can anyone provide a convincing explanation for why Rose's conclusion is, or is not, correct?
Teachers' Resources
Why do this problem?
This is the first problem in a set of three linked activities. Egyptian Fractions and The Greedy Algorithm follow on.
It's often difficult to find interesting contexts to consolidate addition and subtraction of fractions. This problem offers that, whilst also requiring students to develop and analyse different strategies and explain their findings.
Possible approach
Pose the initial part of the problem as it is set and ask the students to suggest what Charlie's rule might be. Allow some time for them to work out which sums are correct and ask them to modify Charlie's rule so that it always generates correct solutions. Working in pairs, invite students to generate some more examples that confirm their new rule. Collect some of these on the board for a
general discusion. (With some classes this could lead to an algebraic explanation/proof.)
Alison's question offers an opportunity to involve the whole class in a collaborative activity. Talk through what Alison might have been thinking as she generated different pairs which worked. This might be an opportunity to talk to the class about the value of working systematically. How can Alison be sure that she has found all the possible pairs?
In pairs, ask students 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?
What is the strategy for finding all the combinations of two unit fractions that add up to a third unit fraction?
Possible support
Some students 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
$\frac{1}{6} + \frac{1}{7} = \frac{13}{42}$  Image

$\frac{1}{6} + \frac{1}{8} = \frac{7}{24}$  Image

...  
$\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$  Image

Possible extension
Ask students to produce an algebraic or visual proof of Charlie's revised rule.
You may wish to move students on to Egyptian Fractions.