Recurring decimals inquiry

The prompt

Mathematical inquiry processes: Compare and contrast; test other cases; find rules. 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. The concept of reciprocals potentially gives the inquiry more depth.

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."

An alternative prompt

Amanda Kirby, a teacher of mathematics at St Clement Danes School, Hertfordshire (UK), designed the prompt after thinking about how she encourages students to discover recurring decimals. She investigates 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. A selection is shown below:

  • 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.