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.
In 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 1/4 + 1/5 motivated the students to request instruction in how to add fractions. Should we continue the sequence 5/6, 7/12 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)?
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 1/4 + 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