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.