### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

### Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

### Special Sums and Products

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

# Egyptian Fractions

### 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

This problem follows on from Keep it Simple

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 support

Keep It Simple provides a good introduction to this activity.

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.

### Possible extension

Some students might wish to undertake research about the Rhind Mathematical Papyrus.

Students could be directed to The Greedy Algorithm