# The prompt Mathematical inquiry processes: Set conditions; generate examples; test particular cases; represent; reason. Conceptual field of inquiry: Volume of geometrical solids; formulae; graph results.

This inquiry is suitable for older secondary school students who are familiar with the inquiry process. They need resilience to unpick the prompt - to explain the formula (or request a teacher's explanation) and to identify the constraint. Once the prompt is clear, students seem quick to speculate that it must be false because there are "so many types of numbers". However, when n is defined as a whole number, the prompt is true if n is a multiple of three. Students have reached this conclusion through the realisation that the maximum volume occurs when r = ⅔n.

In the first phase of this inquiry, students' questions often far outweigh statements or speculations:

• What does the prompt mean?

• Why is the volume of a cylinder πr2h? Is it always the same?

• Can you explain the formula?

• What are r, h and n?

• Is n a whole number?

• Does the answer depend on the value of n?

• Why do r and h have to equal n?

After the orientation phase, classes have decided to test the prompt by developing a collaborative inquiry with groups or pairs of students exploring for different values of n (restricted to whole numbers at first). The results for the values of r and h that give the maximum volume can be collected together in a table from which students induce the relationship between r, h and n by spotting patterns. Alternatively, students have graphed the volume given by each radius for a particular value of n, paying particular attention to the stationary point when it lies between whole number values of the radius. The class has gone on to compare the graphs, discussing why each one looks similar no matter what the value of n. During the inquiry, students often become so involved in calculating the volume of cylinders that they forget the aim of the inquiry set at the start - that is, to evaluate if the prompt is true. It can be necessary, therefore, for the teacher to guide the class towards reflecting on exactly what has happened and what it means in regard to the aim. Furthermore, students should be encouraged to consider why the maximum volume occurs when the ratio of radius to height is 2:1.

# Inquiry pathways

### Proof that the maximum volume occurs at r = ⅔n

While there are a number of pathways to pursue at this point, each one could result in another round of exploratory calculations leading to the induction of a new result. The inquiry teacher might judge this to be too repetitive and introduce calculus to prove the general result (illustration). By working out a derivative, students can deduce the new results, rather than continue to substitute numbers into a formula.

### Further pathways

(1) Change the constraint in the prompt - for example, 2r + h = n.

(2) Search for the values of r and h that give the minimum volume.

(3) Find the maximum volume of other solids given a constraint - for example, the volume of a pyramid with the sum of the base side length and height equal to some n. # Alternative prompts

The teacher can make the prompt more accessible by restricting the values that r, h or n could take. When n = 10, for example, students have fewer variables to think about. The first alternative prompt (above) is more open. By 'giving' students n = 10, it suggests n could have taken other values. The second alternative prompt (below) suggests the sum of r and h is fixed. 