Why do this problem?

The idea that a decimal can 'go on forever' is subtle and interesting and confusing.

Possible approach

"I'm thinking of two numbers, I add 6 to the smaller one to get the larger one, OR I could multiply the smaller one by 4 to get to the larger one. What are my numbers?"

Continue with more examples to wake everyone up and establish that a difference and a ratio define two unknowns (but not necessarily in those words).

Include examples that lead to simple fractions, eg d=5 r=3, and insist on fraction, not decimal answers.

Put 0.222222... and 2.22222.... on the board, and ask how to get from the smaller to the larger in two ways. Work through the processes for finding these "unknowns", alert to all opportunities for students to talk about the meaning of these recurring decimals.

Ask students to choose and work on their own pairs of related recurring decimals, from those in the problem, or later, to make up their own. It's easy for students to verify their final fraction on a calculator.

Key questions

What does 0.22222222 .... actually mean? How many decimal places are there?
Would multiplying by 10 give a decimal that ended? Why not?
Can I do 2.2222222... divided by 0.222222222... on a calculator?

Possible extension

What is the fraction equivalent of 0.999999....? 0.4999999....?
How would you find the fraction for 0.225225225... or 0.222522252225....
Do you think that all recurring decimals will correspond to a fraction?
Can you work out which fractions will correspond to a recurring decimal and which fractions will not?

Possible support

Experimentation with a calculator for small numbers can help students to get into the problem.

Students could be asked to catalogue decimal equivalents of many common fractions, classifying the decimals as terminating, recurring and "no obvious repeats". This data set can be used to check work later, or to suggest recurring decimals to convert back. Encourage students to classify and describe families of decimals with clear recurring patterns (eg ninths and elevenths).

Encourage students to spot patterns and then to make a conjecture about the result when dividing two recurring fractions.

$\frac{2.2}{0.22}, \frac{2.22}{0.222}, \frac{2.222}{0.2222}, \frac{2.2222}{0.22222},$ etc.

Can they extend this to the second part of the question?