# Equivalent decimals inquiry

# The prompt

Mathematical inquiry processes: Compare and contrast; test cases; find rules. Conceptual field of inquiry: Conversion of fractions to decimals; terminating and recurring decimals.

Amanda Kirby, a teacher of mathematics at St Clement Danes School, Hertfordshire (UK), designed the prompt after thinking about how she had previously encouraged students to discover recurring decimals. She investigated the decimal equivalents of ninths using a calculator before moving on to fractions with a denominator of 99.

Amanda created the prompt for her high attaining year 9 class. At the start of the inquiry, students made observations and asked questions:

A prime number has only two factors, 1 and itself.

19/58 recurs.

Two is even and prime so both parts of the statement cannot be true.

Terminates means ‘stops’.

1/2 = 0.5 and 1/3 = 0.333 recurring.

Could we plot a graph or put the results in a table?

What is the fraction for the decimal 0.363636 recurring?

The class went on to make and test conjectures about recurring decimals:

If the denominator is a multiple of 3, the decimal will recur.

Prime denominators give recurring decimals, except 2 and 5.

A fraction has a terminating equivalent decimal if its denominator is a factor or multiple of 10.

September 2018

See also the Fraction and decimal conversion inquiry.

# Explore and generalise

The steps (illustrated) were devised by a year 8 class to determine if the equivalent decimal of a fraction terminates or recurs.

The class had worked out the decimal equivalents of the unit fractions up to a thirtieth. After examining the prime factors of the denominators, the students made generalisations that they shared and refined during a feedback session orchestrated by the teacher.