The number line prompt is suitable for most secondary school classes, although it has been developed mainly with those in years 7 and 8 moving from arithmetic to algebraic reasoning. It invites students to generalise about the difference between two products. In the initial phase of inquiry, students (under the teacher's guidance if necessary) explain the procedure shown in the prompt. They often ask if the difference is always two, but cannot believe the result will be the same with larger numbers .
All-attainment classes have readily set about exploring more examples before taking great pleasure in proving that, with four consecutive numbers, the difference between the products will always be two. Individual students or pairs have gone on to create their own inquiry by changing the prompt. The following changes have led to algebraic proof:
- Combining different pairs of numbers on the number line;
- Extending the number line to six consecutive numbers;
- Using different intervals between the numbers; and
- Using different sequences, such as quadratic, cubic, and Fibonacci.
This prompt has been developed in sessions with Richard Goodman (Principal Lecturer, University of Brighton) and cohorts of teachers training to teach maths on the Mathematics Development Programme (2009-2012) and the Developing Mathematical Practice course (2013-14). In 2014, Liam Richman (Oakwood School, Horley, UK) produced this paper on the prompt to fulfil part of the course requirements.