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Scale factors inquiry

The surface area and volume of cube B are twice the surface area and volume of cube A.
James Thorpe
(a UK mathematics teacher) devised this prompt about the scale factors involved in enlarging mathematically similar shapes. It has been skilfully designed to require students to separate the mathematical features of the diagram from its 'coincidental' features. The surface area of cube B is four times as large as the surface area of cube A; and the volume of cube B is 8 times as large as the volume of cube A. Students might link these scale factors of enlargement to the side lengths of the two cubes (4 and 8 respectively).
Thus, students might attempt to conjecture 
generalisations involving the side lengths of the cubes, such as the scale factor of enlargement for the surface area equals the side length of the smallest cube and the scale factor of enlargement for the volume equals the side length of the largest cube. Further inquiry, however, leads to a different conclusion. The scale factors of enlargement for 3-dimensional shapes whose lengths are enlarged by a scale factor k are:
 Surface area Volume 
Scale factors of enlargement  k2 k3
The prompt might be used after students have already learnt how to calculate the surface area and volume of 3-dimensional shapes. However, the prompt could be used to introduce students to the concepts of surface area and volume. Such an inquiry might involve students in selecting regulatory cards that lead to a phase of instruction, as well as to phases of class discussion in order to develop a conceptual understanding and of practice to become fluent in procedural calculation. The prompt might also involve students in drawing cuboids on isometric paper and in designing nets as a pictorial representation of the formula for the surface area.
Structural reasoning
After this inquiry was posted on the website, one person on twitter (@ProfSmudgecommented: “Odd to take a science (empirical) rather than a mathematics (structural/analytical) approach to maths!” The suggestion is that the student is involved exclusively in empirical induction - that is, drawing more cubes, collecting and tabulating results, developing a hypothesis, and testing the hypothesis with another case. This is the the method of science.
Andrew Blair (@inquirymaths) replied that inquiry is about linking everyday empirical induction to mathematical forms of deduction. As secondary school students often prefer empirical and narrative arguments (see the Longitudinal Proof Project by Hoyles and Küchemann), the aim of the inquiry teacher should be to develop structural arguments out of inductive observations. As Polya says, deductive reasoning “completes” inductive reasoning. If the teacher attempts to impose mathematical deduction over empiricism, there is no guarantee that the inadequacies of induction are challenged. The two forms of thinking could co-exist without ever being linked.
The person on twitter continued: "You can test your ideas empirically, and you can explore if you need to develop a feel for the situation. Often it is about seeming to be busy but not having to think (till later, perhaps)." On another occasion, the interlocutor explained that teachers "tend to put too much faith in patterns and do not go further." This is a danger that inquiry maths teachers should avoid.
While there is often an extended period of orientation at the start of an inquiry when a student might choose to practise a mathematical procedure, the ultimate aim is to develop a conceptual understanding of the prompt. For the scale factors inquiry, this would mean understanding how the scale factors for surface area and volume relate to the structural elements of the diagram in the prompt. Thus, eight of the small cubes fit inside the large one and four of the faces on the small cube make up a face on the large one.

For more on inductive and deductive reasoning in inquiry, see Andrew Blair's article for Mathematics Teaching.


Classroom inquiry
These are the questions and observations of James Thorpe's year 8 (grade 7) class: 
The students have calculated the surface area and volume of both cubes. James structured the inquiry by giving students diagrams of cubes with side lengths from one to 6. He then collected their results into a table before the class identified the links between the scale factors of enlargement for length, surface area, and volume. 
The questions and observations suggest further pathways of inquiry. They provide the teacher with the opportunity to guide students towards new aims (see table).
 Question or observation Potential inquiry pathwayAim
The formula:
volume = width x height x depth
Extend the use of the formula to cuboids.Determine if the links between the scale factors for length, surface area, and volume are the same for cuboids. 
Do we need π?Introduce similar cylinders (and cones) by enlarging the height and radius.Determine if the links between the scale factors for length, surface area and volume are the same for cylinders.
Does Pythagoras' Theorem help?Consider what happens to any diagonal across a face and any diagonal through the cube under enlargement.Determine if other lengths behave in the same way as the enlarged edges.
How can we prove/disprove?Use a cube with side length x and consider various scale factors. Then consider side length x and scale factor k.Prove the result is consistent for cubes, cuboids, and other 3-d shapes.

James Thorpe was a maths teacher at John Taylor High School, Staffordshire (UK) at the time of the inquiry. He taught parts of the maths curriculum through inquiry and devised his own prompts.