Mathematical inquiry processes: Set constraints; test cases and find counter-examples; generalise and prove. Conceptual field of inquiry: Graphs of functions; integration.
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.
Questioning, noticing and wondering
Students in a year 13 Further Maths A-level 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
The teacher guides students to consider the simplest case first. When f(x) = x1 the prompt is true for any point on the line because shapes A and B would be congruent triangles.
For f(x) = x2, the position of the vertical line depends on the coordinates of the point chosen - (1,1), (2,4), (3,9) and so on. If the equation of the vertical line is x = a, then the value of a is always greater than a half when n is greater than one. After exploring the quadratic function, students could generalise for that case before generating more examples with a different value of n and their own coordinates.
The area of A is the integral of the function between the limits a and 0. The area of B can be calculated by subtracting the value of the integral between x and a, which gives the area under the curve, from the area of the rectangle (see the illustration).
Through calculating the value of a in different cases, students will develop an understanding of the general procedure. From there, they might be able to attempt a proof of the general case.
The mathematical notes contain particular cases and the general case.