This 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. In the classroom, the prompt has involved a number of different areas of the maths curriculum:
- Adding and subtracting fractions;
- Equivalent fractions;
- Spotting patterns in lists of equations that satisfy the constraints within the prompt;
- Inducing expressions for p, q and r from the patterns;
- Using negative indices to deduce relationships between p, q and r;
- Showing the relationships between p, q and r in equations;
- Developing algorithms to generate examples;
- Using algebraic fractions to prove the relationships between p, q and r must always hold;
- Summing unit fractions to find other rational numbers.
The lesson notes (below) describe how two classes approached the inquiry and also suggest different pathways for exploration. The mathematical notes are provided by Kier Tipple (Assistant Headteacher, Brighton Aldridge Community Academy, UK) who carried out his own inquiry into the prompt.