Can all unit fractions be written as the sum of two unit fractions?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

It would be nice to have a strategy for disentangling any tangled ropes...

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. . . .

Here is a chance to play a fractions version of the classic Countdown Game.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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.

An environment which simulates working with Cuisenaire rods.

Can you match pairs of fractions, decimals and percentages, and beat your previous scores?

Can you find the pairs that represent the same amount of money?

How much of the square is coloured blue? How will the pattern continue?

Can you work out how many lengths I swim each day?

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?

Is there a quick way to work out whether a fraction terminates or recurs when you write it as a decimal?

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Take a look at the video and try to find a sequence of moves that will untangle the ropes.

Written for teachers, this article describes four basic approaches children use in understanding fractions as equal parts of a whole.

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

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...

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

There are lots of ideas to explore in these sequences of ordered fractions.

In a certain community two thirds of the adult men are married to three quarters of the adult women. How many adults would there be in the smallest community of this type?

When I type a sequence of letters my calculator gives the product of all the numbers in the corresponding memories. What numbers should I store so that when I type 'ONE' it returns 1, and when I type. . . .

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

What do you notice about these families of recurring decimals?

Who first used fractions? Were they always written in the same way? How did fractions reach us here? These are the sorts of questions which this article will answer for you.

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

I need a figure for the fish population in a lake. How does it help to catch and mark 40 fish?

What fractions can you find between the square roots of 65 and 67?

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

Pick two rods of different colours. Given an unlimited supply of rods of each of the two colours, how can we work out what fraction the shorter rod is of the longer one?

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you find its length?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?