Open inquiry

As a more open inquiry, this prompt could lead in different directions. Students' initial questions or observations usually focus on the meaning of 'fits better' and how a 'fit' can be measured. Students have suggested using percentages or fractions to find the proportion of the shape 'covered' by the inscribed shape or the region of the shape left 'uncovered'. 

By considering the regulatory cards, the class might decide to construct the inscribed shape accurately with a ruler and a pair of compasses. For others, the inquiry has continued with an inductive phase based on sketches of one or more diagrams. The inquiry has moved quickly to a deductive proof when students understand the mathematical similarity of the diagrams, especially if they are accustomed to expressing areas in general terms.

The inquiry can be extended by changing the shapes in the prompt to include different pairs (equilateral triangle and circle, hexagon and circle, and so on). This can lead to a discussion of why some pairs are not possible (that is, why one shape cannot be inscribed in another) such as an equilateral triangle inside a square.

Areas of the curriculum that have been covered in this inquiry:

• Calculating areas of polygons (including hexagons, etc.)

• Areas of circles

• Constructions

• Similarity

• Pythagoras' Theorem

• Percentages and fractions

• Surds

• Proof.