Volume and surface area inquiry
Mathematical inquiry processes: Explore; test particular cases; reason. Conceptual field of inquiry: Surface area; volume; geometrical solids.
Although the prompt is false, it often takes students some time to find a counter-example. Classes have decided to explore 'small' cuboids by drawing 2-d representations of the cuboids (on isometric paper) and their nets. The conclusion follows that the statement is true. However, the volume and surface area of a cube with a side length of 6 units are numerically equal. Larger cuboids show the statement is false.
This is another inquiry that teaches students to think about the structure of the mathematical object under inquiry. As one student has commented, "We need to visualise a case in which the space inside the cuboid is getting bigger while the surface area remains the smallest possible." This occurs when the cuboid is made 'more compact' – that is, when the values of the three dimensions are becoming closer to each other.
An important point to make in this inquiry (occasionally made by a student in the orientation phase of the inquiry) is that we are comparing two different units of measure - an observation that can lead to a productive discussion about dimensions.
Lines of inquiry
(1) Constraints on exploration
In the initial phase of the inquiry, the teacher might suggest placing limiting conditions on the cuboids that students draw. The class might agree that, for example, length (l) + width (w) + height (h) = 10, changing the number as the inquiry progresses. While the constraint makes calculations manageable for students, it denies them, at least in the short term, the opportunity to show the prompt is false.
(2) Substituting into formulae
It is unlikely that students suggest an algebraic approach spontaneously, but they could adopt one when shown after an initial concrete stage. Making the the formulae for the volume and surface area of a cuboid equal leads to a numerical exploration: Is it possible to find values of l, w and h, such that lwh = 2(lw + lh + wh).
(3) Inquiring into other solids
In the classroom, students have suggested extending the inquiry by considering other solids. Cylinders might be easier than triangular prisms to use because the latter require knowledge of Pythagoras' Theorem to calculate the surface area accurately. Older secondary school students have extended their inquiries into pyramids and even spheres. They are much more likely to employ an algebraic approach.
A new line of inquiry
The picture shows the questions and observations from a year 10 class at Haverstock School (Camden, London, UK). The inquiry started with a period of exploration during which students attempted to find a counter-example to the contention in the prompt.
However, the inquiry was to follow a new pathway after a pair of students noticed that, for a cuboid with dimensions l = 5, w = 5 and h = 10, the numerical value of the volume equals that of the surface area. The discovery opened the way to students seeking other examples in which l = w. For example, when l = w = 8
64h = 2(64 + 8h + 8h) and h = 4.
The class then moved onto general statements of the relationship between n and h when l = w = n and further cases when l = n and w equals a multiple of n. Students extended the inquiry to incorporate other solids, taking a consistently algebraic approach to deduce the relationship between the dimensions when the volume equals the surface area.
Exploration and reasoning
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), posted the pictures below on twitter. They show the students' initial questions and exploration of the prompt. Matthew reports that, "Students were looking at the concepts of volume and surface area for the first time this year and were able to discover so much. They used drawings, magnet tiles, and charts to show their thinking. Additionally, the students were not only able to prove the statement false, but they were able to develop a rule for the situation by just playing around with the arithmetic, which is the intention of the prompt (see more pictures below).Great conclusions were drawn and led us to ideas around notations for cubed and squared units. I was impressed with their work."
Initial questions and observations
Combining practice with reasoning
Rachel Mahoney blogs about using the prompt with her year 7 class here. She remarks how "the value that the students get from Inquiry Maths lessons is fantastic and I love that it makes them think for themselves and encourages them to ask questions." She continues that the inquiry was "a great way to expose my students to thinking skills and starting to make links and conjectures" while at the same time they were practising how to calculate the volume and surface area of cuboids. At the end of the blog, you can find the resources that Rachel prepared for the lesson.
The prompt in the secondary classroom
James Thorpe tried out the prompt with his year 10 class (set 4). The students' responses and questions are shown above. James reports that some students were, at first, a little sceptical about the inquiry process, but they demonstrated the resilience to make the inquiry a success. James was keen to praise the class: "The students were fantastic and proud to disprove the prompt. The really nice part was when one student posed this question: 'What if the surface area and volume are the same?' I was proud of the kids!"
James also tried the prompt with his bottom set in year 11. Again, the students managed to disprove the prompt and went on to consider what type of cuboids would be counter-examples. James commented on the type of learning that occurred during the inquiry: "As a tool to learn, review and practice area and volume skills, the inquiry was excellent and it also took the students' thinking skills further."
At the time of the inquiry, James was a teacher of mathematics at John Taylor High School, Staffordshire (UK).
Students' questions and observations
These are the initial questions and observations from year 8 mixed attainment classes. The inquiry pathways that could develop from these questions and comments include:
Discussion or instruction on the the meanings of volume and surface area and how to calculate them;
Distinguishing between area and volume and their units of measurement;
Developing and using a formula for the surface area and volume of a cuboid;
Drawing 2-dimensional representations of cuboids on isometric paper and their nets on squared paper;
Checking the contention in the prompt is true for cuboids and cubes;
Finding counter-examples and explaining the 'type' of cuboid that shows the contention to be false;
Modifying the contention in the prompt; and
Extending the contention to cylinders, pyramids and spheres.
These pictures show two displays of posters created in response to the prompt. The first one is from year 7 students at Melksham Oak Community School (Wiltshire, UK). Tim Lamb, the Head of Maths at the school, says that "both my pupils and I are very much enjoying Inquiry Maths." As well as using prompts from the website, the department has developed an inquiry toolkit that encourages students to reflect on their inquiries. The toolkit lists the following questions:
How did I do the inquiry?
How did I plan it?
What sort of questions did I ask?
What strategy worked?
What steps did I take?
Creating alternative prompts
Using exam questions to devise prompts
John Petry, a teacher of mathemtics at Parliament Hill school in Camden (London), designed this prompt for his year 10 class. It is based on a question from the GCSE exam that students in the UK sit at the age of 16. John set some questions for students to think about, but the diagram could work as an inquiry prompt on its own. Alternatively, a statement might be added to focus students’ reasoning on particular properties of the solids:
The volume of the cuboid will always be greater than the volume of the cylinder.
As x increases, the surface area and volumes of the solids remain in proportion.
Changes to the prompt could involve the ratio of the cylinder’s radius and height and the ratio of the cuboid's length, width and height.
From problem to prompt
On reading about this prompt, Mike Ollerton suggested this problem: Find integer dimension solutions for a cuboid with a surface area of 100 square units? The problem could be extended, he continues, into proving there are n solutions. To turn this into an inquiry, with students having the opportunity to ask the questions at the outset, the problem could be turned into one of the following statements: It is impossible to have integer dimensions if a cuboid has a surface area of 100 square units.
An alternative prompt
Pat Doe, head of a science department in Brighton (UK), suggested the diagram above, which he uses to discuss the rate at which solids dissolve, could be used as a prompt in maths classrooms. The statement acts to focus students' thinking on the mathematical features of the diagram - including area, surface area, and volume. Their questions and comments invariably relate to what is increasing and how it is increasing. The inquiry has led into students extending the diagrams to the next cases and trying to give expression to the number sequences that arise. Another approach is to inquire into rectangles and cuboids by stipulating a ratio for, respectively, length and width, such as 2:1, and for length, width and height, such as 2:1:1. (The rectangles could be split into squares and cuboids into cubes to avoid repeating the sequences generated from the original diagram.)