Start with a preliminary investigation into which fractions make recurring decimals.
Allow students to use electronic calculators and also (essential) to experiment with a standard non-calculator method for the division, to acquire a feel for the elements involved in 'decimalising' a fraction.
Once the group can see which fractions will lead to recurring decimals, and can really understand why, present the Tiny Nines problem.
Activity which concentrates on changing decimals to fractions and fractions to decimals to important work. Ask students to make up a set of eight fractions which don't seem particularly related. Ask then to guess the order of the fractions, smallest to largest.
Direct them to convert each fraction to its decimal form and compare, and then challenge them to find a way to get this ordering right every time, whatever the fractions.