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Any fraction can be converted to a decimal and any decimal can be converted to a fraction.

This 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 promote students' questioning, the question in O'Connor's article is written as a statement. This allows the teacher to draw on and simultaneously develop students' curiosity around of the issue of whether all, part or none of the statement is true. It also allows the teacher to assess the students' existing knowledge through the types of questions they pose or comments they make about the prompt. 
   
The fir
st part of the prompt can be presented on its own. In one case when a teacher took this approach with a year 7 class, a student suggested in the orientation phase of the lesson that the class inquire into the reverse statement. The teacher decided to restrict the first part of the inquiry to conversion from fractions to decimals, but opened a new inquiry pathway at a later date when the class went on to consider converting decimals (including recurring decimals) to fractions.
      
Areas of the curriculum that have been covered in this inquiry are:
  • Simplifying fractions
  • Short division
  • Converting fractions to decimals
  • Categorising fractions that give terminating and recurring decimals
  • Factors and multiples
  • Explaining why fractions give terminating and recurring decimals (referring to the prime factors of the denominators)
  • Converting decimals, including recurring decimals, to fractions
  • Irrational numbers
Framing question
Although only fraction to decimal conversion is discussed in 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)

Notes
   
Resources
Promethean flipchart    download

Generating questions
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.

When Amelia used the inquiry prompt, she was a PYP teacher at the Vientiane International School (Lao PDR). She is now at the Luanda International School (Angola). Amelia has run workshops on using Inquiry Maths prompts in the primary classroom. You can follow her on twitter @_AmeliaOBrien.
  
Questions and comments from a year 8 class   
Caitriona Martin's year 8 class asked these questions 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 using the prompt - this time with a low ability year 9 class - 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! :D"
   
Update
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."

Caitriona
, a maths coordinator at St. Andrew's School, Leatherhead (UK), has been using inquiries since she became a qualified teacher in 2011. You can follow her on twitter @MrsMartinMaths.