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Adding fractions inquiry

   
This prompt was devised by Mark Greenaway (an advanced skills teacher in Suffolk, UK) to encourage students to analyse the sum of two unit fractions in which the denominators are in the form n and n+1. I am a recent convert to this prompt. My initial reaction was to have misgivings about the prompt's potential to sow misconceptions in students' minds or to focus their thinking on the operations rather than underlying concepts. However, in a recent lesson study with two departmental colleagues (Helen Hindle and Hugh Salter), I have changed my mind completely. Indeed, the value of the prompt lies precisely in the way it exposes students' misconceptions and procedural thinking that already exist. The inquiries that developed from the prompt featured hugely valuable discussions in which entrenched notions were challenged and an understanding of the concept of a fraction was reconstructed by students and the teacher.
  
I
n
the lesson study cycle, which involved year 7 classes, I went first. I decided to use a number line as a tool with which to approach the concepts of a fraction and then of adding fractions. Before showing the class the prompt, we started by locating fractions on a number line. This led immediately to our first misconception about representing 1/6 (see box below). 
   
The students' questions and comments about the prompt provided a strong foundation for inquiry. In particular, the
speculation around the solution to
 1/+ 1/5 motivated the students to request instruction in how to add fractions. Should we continue the sequence 5/67/12 to 9/18 or should we use the 'rule' derived from the denominators of the unit fractions (add them for the numerator; multiply for the denominator)?
   
Misconceptions identified during the lesson study
  • 1/6 should be placed half way along a line of 12 units because the 'number' six stands for the length along the line. The student who said this had no problem marking a quarter. Thus, while students might have a sense of 1/4, they might not have developed a conceptual understanding.
  • Having started with number lines of length 12 units, students refused to use a line of six units to show the first calculation in the prompt. This revealed an inability to conceive of a fraction as part of any whole. Once a third was represented by an arrow four units along a line of length 12, students would not accept it could also be shown as two units along a shorter line of length six.
  • To add two fractions, students claimed, you add the numerators and then the denominators separately. Thus, 1/2 + 1/3 = 2/5. This shows a misconception of fractions as two unrelated 'numbers'.
  • To add two fractions, you add the denominators to get the numerator in the answer and multiply them to get the denominator. (As students realise during their inquiry, this works for unit fractions, but not when the numerator is greater than one.)
  • When showing the solution to 1/2 + 1/on a number line, students start both fractions at zero, rather than place one fraction after the other (see illustration). This idea that the fraction of a line can only be shown from zero was surprisingly common.

One 'lower ability' class that was part of the lesson study posed meaningful questions and made insightful observations (above). The responses show the potential of the prompt to promote questioning and noticing in all classes.
  
As the inquiries developed, students were taught to link the number of intervals on the number line with the product of the denominators. The students then showed the sum of any two fractions on a number line by using equivalent fractions. So, typically, a student went on to show 1/+ 1/5 on a number line of length 20, explain why it is equivalent to 5/20 + 4/20, and give the solution 9/20.
Andrew Blair  June 2014

Resources
Prompt sheet
Promethean flipchart   download
Smartboard notebook 
 download

   
Research on representing fractions
This paper presented at a 2009 conference of the British Society for Research into Learning Mathematics discusses lower secondary students' attempts to represent fractions using diagrams. The authors' conclusion stresses the importance of students drawing their own diagrams: "Using one's own diagrams effectively is much more demanding, and more indicative of a sound understanding, than using ready-made diagrams."
   
An alternative prompt  
Terry Patterson, a maths teacher in London, contacted Inquiry Maths about a prompt she had devised (below). Terry's first experience of an inquiry lesson came when she used the prompt with her year 8 class. She commented on the emotional impact an inquiry can have: "The students' questions are moving and revealing. They loved running the lesson.... I was quite choked up after my first lesson yesterday - an eye-opener." The class had low prior attainment in maths and the prompt gave Terry an insight into the students' level of understanding: "Every question they posed revealed the group's bafflement." The questions included:
  • Why does 1 - 1 = 1?
  • Why does 2 - 3 = 6?
  • Is it to do with times tables?
The last question could follow from identifying supposed links between the numerators and denominators - that is, 1 x 1 = 1 and 2 x 3 = 6 respectively. The questions reveal the kinds of misconceptions that are common when students are faced with fractions prompts.

You can follow Terry Patterson on twitter @Terry_Patt. 
The inquiry with a mixed attainment class   
After year 7 (grade 6) students asked the questions and comments above, the teacher offered them six regulatory cards (below). A third of the students were
confident enough in adding fractions to 'make up more examples' or attempt to answer the problem about how the right-hand side continues. A few students requested an explanation; the teacher invited them to the front for a short tutorial. The others selected either 'work with another student' or 'ask the teacher for something to do'. The teacher had three levels of tasks linked to a number line: marking fractions on a number line, finding the fraction of a number to develop an understanding of equivalent fractions, and adding fractions. At each stage, students were required to make up their own examples to assess their level of confidence.

   
The students who set out to answer the problem concluded, with the teachers help, that the general form of the fraction on the right-hand side (with n being, for them, the denominator of the first fraction) is:
At the start of the second lesson, the same students explained on a number line why the equations in the prompt are correct. They then decided to change the prompt and made the numerators two and then three, proceeding to adapt the generalisation. Other students made up more difficult equations involving the addition and subtraction of fractions with co-prime denominators, which they took great delight in presenting to their peers.
 
Questions and observations
Above is a picture of the questions and observations from year 7 students during a lesson study. Students wrote on whiteboards mounted on the walls around the classroom. Mark Greenaway posted the questions and observations below from one of his classes on his website, which is highly recommended for its comprehensive coverage of ideas about teaching maths. You can follow Mark on twitter @suffolkmaths.