This prompt originated in a problem that Andy Strickland (a maths teacher in Worthing, UK) posed himself: Is it possible to find a pair of fractions to satisfy the statement? Since being turned into a prompt, the statement has provided the starting point for inquiry with all types of secondary school classes. Initial comments and questions include:
- What do 'sum' and 'product' mean?
- How do you add and multiply fractions?
- Does it always work?
- It will never work (sometimes accompanied by an example).
- Can you show or prove it is true?
Students are intrigued by this statement because, at first sight, it is not possible. However, that is only because students tend to define fractions in a limited way, concentrating on proper fractions in their exploration. Indeed, proper fractions cannot satisfy the condition in the prompt. Teachers can illustrate an explanation of why not by using a number line or diagram (see box).
With a move onto improper fractions (possibly under the teacher's guidance), the inquiry can develop along different pathways. (At this point, the teacher might choose to rule out the trivial solution of two fractions that both simplify to two.) In classrooms, the inquiry has passed through different phases, such as planning, exploration, generalising and evaluating (see phases of inquiry). Alternatively, it has zig-zagged between inductive and deductive reasoning (see forms of reasoning). Students as young as those in year 8 have been involved in adding and multiplying algebraic fractions. Older students have shown that fractions in the following form satisfy the condition in the prompt:
Another inquiry pathway develops by representing the solution set graphically (see the box).
The inquiry has been extended in classrooms by changing the prompt in the following ways:
Reflections on using the inquiry for the first time by Alison Browning (a secondary school maths teacher who is on twitter @browning_alison)