Challenge Level

This problem offers an excellent opportunity for students to practise converting fractions into decimals, while also investigating a wider question that connects their knowledge of prime factors and place value.

Start by writing a list on the board of the following sequence of fractions:

$$\frac1{40}=$$ $$\frac2{40}=$$ $$\frac3{40}=$$... up to $$\frac{20}{40}=$$

"Do you know how to write any of these fractions as decimals?"

Give students a little time to figure out which fractions they recognise, perhaps using equivalent fractions as an intermediate step. Then fill in the decimal equivalents on the board, inviting students to share the thinking they did to work out the decimal forms. For example:

*"I know that $\frac{10}{40}$ is $\frac14$ which can also be written as 25% or $\frac{25}{100}$ so it's $0.25$."*

*"If $\frac4{40}$ is $0.1$, then $\frac2{40}$ must be $0.05$ because it's half as big."*

Now introduce the main problem. Write up or display these eight fractions:

$$\frac23 \qquad \frac45 \qquad \frac{17}{50} \qquad \frac3{16} $$ $$\frac7{12} \qquad \frac58 \qquad \frac{11}{14} \qquad \frac8{15}$$

"Which ones do you think can be written as a terminating decimal, and which ones do you think have to be written as a recurring decimal?" (If students have not yet met the idea of terminating and recurring, clarify the meanings.)

Give students a bit of time to make their predictions, and then invite them to work out the decimal equivalents and see if they are right. They might do this by using equivalent fractions, a written division calculation, or using a calculator. Once everyone has worked out the decimal equivalents, take time to discuss whether students' predictions were correct and whether there were any surprises.

"In a while, I am going to give you some fractions. Your challenge is to devise a method for working out straight away whether a fraction is equivalent to a terminating or recurring decimal."

Give students some time to try some examples of their own to test any conjectures that they make. You could collect examples on the board in two columns: terminating and recurring.

In the last few minutes of the lesson draw together the insights and methods that have emerged, and test students' methods with some carefully chosen examples.

Think about the denominators of fractions that you know will terminate. What do they have in common?

Why is the prime factorisation of the denominator important?

When you collect examples on the board in two columns (terminating and recurring) consider writing the fractions in their lowest terms and then writing the denominators as a product of their prime factors.

Students could go on to explore recurring decimals in Tiny Nines and Repetitiously.

For a challenging extension, some students may wish to consider the idea of terminating and recurring representations in other number bases.