Daniela Vasile (a Head of Mathematics in Hong Kong) sent the equation (right) to Inquiry Maths. She tweeted that, at first, "students say it's not true, then see it is! Are there other fractions that multiply similarly?" The equation is intriguing. We might start an inquiry by looking for more equations of this type and, when that search does not yield anything quickly, move on to using algebra in the form a/b x c/d = (10a + c)/(10b + d). It soon becomes clear that the algebra also fails to provide an obvious route to generating more examples - obvious, that is, in terms of GCSE-level maths. Kier Tipple (a maths teacher in Brighton, UK) used a formatted spreadsheet to ascertain that there is only one other non-trivial example: 1/2 x 5/4 = 15/24. Kier's report on his inquiry can be read here.
Sarah Beaumont (a Head of Mathematics in Kenya) has subsequently said that for a prompt to work it must have "less to it, but more in it." The intriguing equation certainly qualifies on the basis that it has less to it, but it does not have more in it. In failing to yield more than one other example of the same type, the equation could not provide a secondary school class with enough material to maintain interest and motivation for an open inquiry. As it does not allow for a successful exploratory phase, it would not work as a prompt in a 11-16 school.
Daniela replied to the comments above: "First of all, I agree with everything. My main concern is also that the statement does not develop into an inquiry that would keep the majority of students interested until the full solution is reached. However, I am not
concerned by the fact that there is a very limited number of similar statements. In real life we pose ourselves questions that at times do not have even a single solution. Solving a problem means finding all solutions or proving that there is no solution.
I trialled this inquiry with my Year 10 class in June 2013 and it was successful, but this success was mainly due to the profile of the high-attaining class. I thought, when playing with the equation, that an Excel-based way of finding all solutions could be developed by some students, especially as we are a 1-1 laptop school and the students are used to using Excel in their Maths classes. Throughout their schooling we give students some tools and it is interesting to see if they are able to pick the right tool when trying to solve a problem – this is why I was not worried by such an approach. In fact, some students started with Excel, but then proved algebraically that they had found all the solutions, whereas others went in the opposite direction.
I re-state that this inquiry is not of interest for the majority of the students, but it is appropriate for a small segment of them.
This document shows you the algebraic approach taken by some of my students. It does not provide a full solution, but just a couple of ways to begin. The inquiry took us one hour. Students were fully engaged, showing resilience, naturally embedding technology to enhance their approach to the task, practising algebraic manipulations, and self-organising into groups in order to avoid repetitive tasks."