Combined operations with algebraic fractions
When addition, subtraction, multiplication, division and raising to a power appear in one expression, the usual order is kept: powers first, then multiplication and division, then addition and subtraction; the contents of brackets are done first. We work step by step: reduce the fractions inside brackets to a single fraction, then multiply or divide, and finally reduce the result. At each step, factoring denominators opens the way to cancellation. For example (1/x + 1/y) : ((x + y)/(xy)) = ((y + x)/(xy)) · (xy/(x + y)) = 1. The final result is often a simple expression or even a number, which is a sign of correct work.
“Expression chain”: a long expression is cut into parts. Each pupil simplifies their part and passes it on; the last pupil writes the final answer. The team wins if the chain does not break.