The inspiration for the prompt comes from James Tanton. His problem considers one case, whereas the prompt invites students to explore the general case. Nevertheless, in the exploration phase of the inquiry, the teacher might guide the class to consider specific cases.

The statement in the prompt is true in only one case - that is, in the simplest case when n = 1. However, in the classroom students have assumed it is true for all cases and set out to prove it so from the start of the inquiry. In fact, the equation of the vertical line changes depending on the function and coordinates.

Question, notice and wonder

Students in a year 13 A-level Further Maths class started the inquiry in different ways. Some dived into inquiry, using the equation of the vertical line in the prompt to determine the limits of integration. Other students held back to think about their approach. 

After a few minutes, Zahra observed, "The areas in the diagram look about the same, but the vertical line is more like three-quarters of the x-value." She asked if the equation of the vertical line was correct. The question led onto speculation: surely, reasoned Percy, the position of the line would vary depending on the value of n. 

The teacher suggested it might be better to move from the particular to the general and guided the inquiry in that direction (see below). 

Andrew Blair, February 2023