The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.
This article extends and investigates the ideas in the problem "Stretching Fractions".
Can you work out the parentage of the ancient hero Gilgamesh?
We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.
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.
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
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.
What fractions can you find between the square roots of 65 and 67?
A Sudoku with clues as ratios or fractions.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?
Relate these algebraic expressions to geometrical diagrams.
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled ropes...
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
I need a figure for the fish population in a lake. How does it help to catch and mark 40 fish?
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.
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. . . .