# Recurring decimals inquiry

# The prompt

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

Dan Walker, a secondary school teacher, devised the prompt for his year 8 class. He anticipated that the students would know some fraction to decimal equivalents and even how to carry out conversions, but doubted whether they had met recurring decimals.

He relates that he was "hoping for some interesting discussions and inquiries about the meaning of the dots, how to convert fractions to decimals, which fractions recur, the lovely patterns within the sevenths and probably a few misconceptions about fractions and decimals."

The prompt, Dan continues, helps give students the idea of terminating and recurring decimals without being directed by the teacher. Introducing the concept of reciprocals gives the inquiry more depth.

February 2019

See also the Fraction and decimal conversion inquiry.

# Classroom inquiry

Dan reports on how the inquiry developed:

The inquiry went well. I'm new to the school and starting to train this class in the ways of inquiry, but they got into it eventually.

The students' observations were fairly superficial at first because they are not used to this style of lesson. They did not come up with many questions, which might be a sign that the prompt was insufficiently interesting for the class.

As a result I slightly led proceedings by suggesting a few possible questions. I asked the students to carry on investigating for homework, writing down and testing any conjectures.

There were some nice findings which I shared with the class:

One student systematically converted 1/7, 2/7, 3/7, etc into decimals and discovered the nice patterns.

Another student created separate lists of fractions that terminate and fractions that recur, and made the conjecture that when fractions have denominators that are factors of 10 the decimal equivalent will terminate.

It was also useful to discuss a few things pupils missed. For example, one pupil had randomly converted 3/4 and 6/8 (by short division). Although these are equivalent decimals, the pupil did not spot the connection.

# Lines of inquiry

1. Explore and find patterns

Students explore the sevenths and note the repetend is made up of the same cycle of six digits starting with a different digit each time. The reciprocals give a mixture of terminating and recurring decimals depending on the denominator (see the equivalent decimals inquiry).

Students can go on to explore other 'fraction families' where the denominator is an odd or prime number.

2. Length of the decimal repetend

Another line of inquiry is to study the set of unit fractions with prime numbers as denominators. Students see a pattern emerge for p > 13. The length of the repetend is (p - 1) where p is the prime denominator. Does the pattern continue? If students explore further they start to build up sequence A002371 in the OEIS.