Sum of unit fractions inquiry
Mathematical inquiry processes: Search for examples that satisfy the condition; conjecture, generalise and prove. Conceptual field of inquiry: Addition of fractions; unit fractions.
The prompt, which can be presented in words or as an equation, has often revealed deep misconceptions that students hold about algebra and fractions. For example, they will assert:
1/4 + 1/4 does not work because p and q have to be different.
It does not work because if you add the numerators you get two, not one.
The first statement can be addressed through a discussion guided by the teacher, but the second reveals a misconception about fractions. An explanation of why 1/3 + 1/6 = 1/2 can begin to address the issue.
During 15 minutes of exploration after the initial orientation phase of the inquiry, a year 8 mixed attainment class at Varndean School (Brighton, UK) devised a list of equations, which the teacher arranged on the board in this way:
The teacher called on individuals to explain each example, thereby reinforcing the procedure for adding fractions. In the discussion that ensued, students showed how the examples were related to the 'originals', as they became known, in the left-hand column. Students who had been quiet up to that moment were invited to fill in the blanks and extend the table vertically or horizontally. The inquiry then took two separate directions: students could continue to extend the table, justifying their new examples, or they could focus on generalising from the 'originals' (see next section).
Extending the inquiry
(1) Research Egyptian fractions.
(2) Is it possible to find three unit fractions that sum to a unit fraction? What about four unit fractions? Five? And so on.
Pattern spotting, reasoning and proof
Once students have found a third, a sixth and a half (or the fractions have been introduced by the teacher), they can begin to spot patterns. Students might conjecture that pr = q and, when that condition is not sufficient to satisfy the prompt in all cases, they add that r + 1 = p. More examples follow (see the table). The teacher then encourages the class to express p, q and r in terms of one variable (n) in preparation for a proof.
A proof that the relationships pr = q and r + 1 = p are sufficient to fulfil the conditions in the prompt follows:
Rachel Mahoney, a teacher of mathematics at Carre's Grammar School in Sleaford (Lincolnshire, UK), used the prompt with her year 7 class. She posted this blog about the inquiry. Rachel explains how she structures the inquiry:
"Once again I am pleased that I used an inquiry with my students. I always think that they gain a lot from these tasks in terms of both skills and developing their own thoughts and independence. I do use the guided sheets (right) to help structure the inquiry as at present, this is only the second inquiry my year 7s have completed. As they undertake more inquiries it would be good to slowly take the scaffolding away and allow the students more freedom to inquire such as changing the prompt."
Questions and observations
These are the questions and observations from a year 7 class at the start of the inquiry.