Seven oranges at the table

A puzzle given to a ninth-grader: share 7 oranges among 12 people so everyone gets the same, without cutting any orange into twelfths. One answer is to cut four oranges into thirds and three into quarters. Everyone gets a third and a quarter, and 1/3 + 1/4 = 7/12.

It is also the best answer: no way of cutting at all leaves a bigger smallest piece than a quarter. Mathematicians call this "the muffin problem", and nobody has a formula for it in general. Scribes in ancient Egypt wrote shares the same way, as sums of different unit fractions.

At dinner, share something awkward together and ask: can we make the smallest piece bigger? Siegler and colleagues (2012) found that how well ten-year-olds understood fractions and division went with their algebra five or six years later, in the US and the UK. That is a correlation, not a promise.