Volume of a cylinder inquiry
Mathematical inquiry processes: Set conditions; generate examples; test particular cases; represent; reason. Conceptual field of inquiry: Volume of geometrical solids; formulae; graphs and turning points.
The inquiry is suitable for older secondary school students. In the orientation phase, students often require support to unpick the meaning of the prompt.
The teacher should ensure that the class identifies the four components of the prompt:
(1) The formula for the volume of a cylinder. If the formula is new to the class, the teacher might build on students' knowledge of the formula for the area of a circle by explaining how the height builds a third dimension on a circular cross section.
(2) The constraint placed on r and h. The idea of a constraint might also be new to the class. In a structured inquiry (see below), the teacher starts by specifying a particular constraint, such as the sum of the radius and height is 10.
(3) The meaning of n. The idea that students can assign any number to the constraint is important in empowering them to follow their own lines of inquiry. However, when exploring the prompt, some numbers are more sensible than others. Initially, the teacher should restrict the inquiry to whole numbers.
(4) The aim of the inquiry. The inquiry aims to test an assertion about the values of r and h that give rise to the maximum volume of a cylinder for any value of n. The maximum occurs, the prompt asserts, when the radius and height are whole numbers.
The assertion turns out to be true if n (defined as a whole number) is a multiple of three. That is because, in general, the maximum volume occurs when the radius is two-thirds of n (or, put another way, the ratio between radius and height is 2:1). So, for example, if n = 9, then the maximum volume is achieved by setting r = 6 and h = 3.
Unlike in a discovery model of learning, it is not a requirement of successful inquiry that every student reaches this result. With or without the generalisation, they will have become more fluent in applying the formula for the volume of a cylinder.
Starting the inquiry
The inquiry might start with students speculating on the type of cylinder that gives the greatest volume - tall and thin, short and wide or somewhere in between.
On seeing the prompt, students' questions often outweigh any statements or conjectures:
What does the prompt mean? What does 'integer' mean?
Why is the volume of a cylinder πr2h? Is it always the same?
What are r, h and n?
Is n a number? What type of 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 used the regulatory cards to decide to test the prompt by developing a collaborative inquiry with groups or pairs of students exploring different values of n. The results for the values of r and h that give the maximum volume can be collected together in a table from which students might attempt to spot patterns.
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 decide if the assertion in the prompt is true. The teacher might, therefore, stop the exploration at intervals and encourage the class to reflect on what the results mean in relation to the aim.
Structured and guided inquiries
Depending on the students' prior knowledge and experience with mathematical inquiry, the teacher might choose to structure or guide the inquiry.
For a structured inquiry the teacher removes the idea of a variable (n) that could stand for different values and restricts exploration to one case - that is, n = 10.
By recording the results in a table, students conclude that the maximum volume occurs when r = 7.
However, the case contradicts the prompt because the maximum volume lies between r = 6 and r = 7. The teacher could point out that the volume might be greater between six and seven - a possibility that students readily accept.
When trying r = 6.1, 6.2, 6.3 and so on, some students will need to be reminded of the constraint (r + h = 10).
If students are sceptical that the maximum volume falls between six and seven, the teacher could require them to plot the graph of radius against volume or show them the curve (see below).
The teacher then directs students to try n = 9. In this case the assertion in the prompt is true - the maximum volume occurs when r and h are integers (r = 6 and h = 3).
The inquiry now becomes about classifying the cases for which the prompt is true and those for which it is false.
For a guided inquiry, the prompt retains the idea of a variable that can stand for different values, but the teacher directs the class to start the inquiry with n = 10.
This is more open than the prompt for the structured inquiry because, in 'giving' students the first value of n, they are aware that it could take different values.
This realisation is more likely to lead to students choosing their own values and directing the inquiry more independently.
Lines of inquiry
1. Represent the results graphically
On graphing radius against volume (on the x-axis and y-axis respectively) for a particular value of n, students can see that the maximum volume occurs when r is two-thirds of n.
A graph can also suggest that the maximum volume lies between whole number values of the radius. For example, in the case of n = 10, students might conclude that the maximum volume occurs at r = 7 cm (V = 461.81 cm2). However, when they draw the graph as a curve, they appreciate that the maximum volume lies between r = 6 and r = 7.
By expressing h in terms of n and r (h = n - r), students are able to generate curves for different values of n using Desmos (see illustration).
Classes have gone on to compare curves, discussing why each one looks similar. The teacher focuses their attention on the shape of the curves: Why do they rise more slowly than they descend? How can we explain the curves going below the x-axis and to the left of the y-axis?
2. Generalise and prove
After the class has generated a number of results, students often make a generalisation about the relationship between r, h and n.
Students can test their generalisation with more cases, but, after a while, the process becomes repetitive.
Depending on the prior knowledge of the class, the teacher might decide to introduce calculus to prove the generalisation.
3. Change the prompt
(a) Change the constraint in the prompt
If the constraint were 2r + h = n, what is the new relationship between r and n? How is the relationship different to the one for the constraint r + 2h = n? Is there a pattern if we consider 3r + h = n and r + 3h = n? What about kr + h = n and r + kh = n (where k is a positive integer)?
(b) Change the solid from a cylinder to a cone or a pyramid
Given the similarity between the formulae for the volumes of cylinder and cone, is the relationship between r and n the same?
What happens if we explore the maximum volume of a pyramid using l + h = n where l is the side length of the base and h is the height of the pyramid?