Volume of a cylinder inquiry

The prompt

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 inquiry

The teacher chooses to structure the inquiry depending on the students' prior knowledge of the topic and their experience of mathematical inquiry.

To initiate a structured inquiry the teacher changes the prompt:

The new prompt 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 points out that the volume might be greater at a point between six and seven and students try r = 6.1, 6.2, 6.3, and so on.

If students are sceptical that the maximum volume lies between six and seven, they can plot the graph of radius against volume or the teacher shows them the curve (see Lines of inquiry below).

Then students explore n = 9 or n = 12. In both cases the assertion in the prompt is true. The maximum volume for n = 9, for example, 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 (when n is a multiple of three) and those for which it is false.

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?

An alternative prompt

The contention in the alternative prompt is that the maximum volume of the cylinder occurs when the radius equals the height. This idea is easier for younger students to understand.

The prompt can be presented after students have explored in two dimensions. If you have rectangles of a given perimeter, what dimensions give the maximum area?

The greatest area of a rectangle with a perimeter of 12 cm, for example, is given by the 3-by-3 square. Indeed, the greatest area for any perimeter is given by a square.

This two-dimensional exploration leads directly into the three-dimensional case of the cylinder.

As the side lengths of a square are equal, the idea that the maximum volume of the cylinder occurs when the radius and height are also equal can seem plausible.

Classroom conjecture

During the question, notice, and wonder phase of the inquiry, Josh, a year 8 student, argued that the prompt is false by drawing an analogy with a cube:

"The maximum volume for cuboids with the same surface area happens when the cuboid is a cube. The cylinder is most like a cube when the diameter equals the height."

Josh's conjecture that the maximum volume occurs when 2r = h also seems plausible. However, as the inquiry developed, students were intrigued to find that the relationship is, in fact, r = 2h.

March 2025

Resources

Prompt sheet

PowerPoint

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