Fraction and decimal conversion inquiry
Mathematical inquiry processes: Test particular cases; identify different types of cases; conjecture and generalise. Conceptual field of inquiry: Fraction-decimal conversion; product of prime factors; terminating and recurring decimals.
The prompt was inspired by Mary O'Connor's article published in the research journal Educational Studies in Mathematics in 2001. In the article, O'Connor analyses the discourse between a teacher and her grade 5 class that is initiated by the framing question: ‘Can all fractions be turned into decimals?’ What results is a 'position-driven discussion' in which the teacher skilfully manages the pupils' examples, counter-examples, conjectures and conceptions.
In line with the aim of the Inquiry Maths model to develop students' curiosity, the question in O'Connor's article is written as a statement. This allows the teacher to draw on and simultaneously develop students' own questioning about the prompt: Is all, part or none of the statement true? It also allows the teacher to assess the students' existing knowledge through the types of questions they pose or observations they make about the prompt.
The teacher can structure the inquiry by breaking the prompt into two parts, focusing initially on the first part: Any fraction can be converted to a decimal. Students test the assertion to see if they can find any counter-examples. By dividing a fraction’s numerator by its denominator (either by using short division or with a calculator), it can become quickly evident that there are two types of resultant decimals, those that terminate and those that recur.
Lines of inquiry
Listing decimals that terminate and recur
Students create two lists of unit fractions that convert to decimals that terminate and those that recur. They then search for rules based on the prime factors of the denominators, explaining why a unit fraction is in one list or the other.
Finding all decimal equivalents
Students find all the decimal equivalents of fractions from 1/2 to 19/20 by formatting a spreadsheet.
Inquiring into fraction 'families'
Students inquire into different ‘families’ with the same denominator by increasing the numerator one each time. The decimal equivalents of the sevenths, which is possibly the most interesting 'family', recur with a cycle of the same six digits.
Once students have established the truth of the first part of the statement in the prompt, they often accept the truth of the second by reasoning that, "if you can turn a fraction into a decimal, then you can turn it back again." It is difficult for students to conceive of decimals existing independently of a fraction equivalent (see irrational numbers in the mathematical notes). Terminating decimals can be converted to fractions with denominators that are powers of 10. Students might try denominators of 9, 99, 999 and so on for recurring decimals, but the approach will work in only a few cases.
Amelia O'Brien used the prompt with her grade 5 class. The pupils generated a wealth of questions that include the procedural (how do you ...?), the connectionist (how are ... connected to ...?), the creative (can you mix ...?), and the conceptual (is ... a special case?). In the conceptual category, the question about whether a 'never-ending' decimal can be converted into a fraction has the potential to take the inquiry into complex mathematics even for secondary school students. The question that introduces negative numbers raises another important issue. As pupils often meet fractions for the first time in the context of concrete manipulatives or 'pieces of pie', they can find it difficult to conceive of a negative fraction.
The pupils in Amelia's class have come up with a comprehensive set of questions that encompass different types of mathematical thinking. Each one contributes to the potential learning during the inquiry. Below are some of the prompt sheets used by pupils to note down their initial thoughts about the statement. Amelia puts them on display to generate further discussion and promote new pathways in the inquiry.
Developing a culture of inquiry
Caitriona Martin's year 8 class asked the questions (picture) during a 'FIG' Friday. In September 2013, Caitriona introduced Functional, Inquiry and Group work into Friday maths lessons for Key Stage 3 classes (children aged between 11 and 14). In another inquiry with a low attaining year 9 class using the prompt, Caitriona reports high levels of motivation: "My year 9s all left the lesson being able to convert from a fraction to a decimal and vice versa because they needed to have that skill in order to answer their own questions. What’s better, they were the ones who asked me to teach them how to convert from a fraction to a decimal without using a calculator – I said I would teach it to them if they really wanted me to!"
After a term of FIG Fridays, Caitriona conducted a departmental review. She reports, "I asked how the staff distributed the lessons between functional, inquiry and group work. Inquiry came up as the FIG style that teachers used the most, which is super!
Also, one of my colleagues ran a session sharing a lesson that had gone well and he said that the pupils responded so well to that lesson because they were used to the inquiry lessons – hence they were trained to ask good questions and make connections and ‘go with the flow’, rather than that being an out of the ordinary thing to do."
Questioning and wondering
These are the questions and observations of a year 7 class at Carre's Grammar School in Sleaford (Lincolnshire, UK). Rachel Mahoney, the teacher, started a new term with the prompt, reporting that the inquiry was a "very worthwhile activity."
Although only fraction to decimal conversion is discussed in Mary O'Connor's article, the full framing question she presents is ‘Can any decimal be turned into a fraction and can any fraction be turned into a decimal?’ O'Connor writes:
The specific formulation of the mathematical question plays a role in the teacher’s actions. Because the question is asking whether all fractions can be turned into decimals, and vice versa, the students are required to utilize various computational methods in evaluating the ‘transformability’ of classes of cases. Many students will know that some fractions can be converted: benchmark fractions like one half or one fourth have a decimal equivalent that these students will have encountered many times in previous lessons.
So for these numbers at least they will believe that the quantities named by the two expressions are equivalent, and they will not have to call on a computational algorithm to verify the possibility of a transformation. These benchmark equivalences will already have the status of ‘math facts.’ But knowledge of these facts will not provide a complete answer to the question. The students must be able to evaluate the question for any fraction (or any decimal, depending on which part of the question is being answered). (From Educational Studies in Mathematics 46, 143–185)