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A square fits into a circle better than a circle fits into a square.

This prompt is suitable for students in years 10 and 11 and classes with high prior attainment in lower years. The inquiry could follow a structured (see box, right) or more open pathway.
  
Structured 

The teacher directs the class to draw a number of diagrams of squares inscribed in circles and circles inscribed in squares. The students then calculate the percentage of the each shape covered by the inscribed shape. After identifying a general result, the teacher co-constructs a proof with the class. 
   
Open
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'. Through 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. 

Notes
   
Resources
Prompt sheet
Promethean flipchart    download
Smartboard notebook   download

Structured inquiry
A year 10 class at Haverstock School (Camden, London, UK) made the observations and posed the questions in the picture. The students decided that 'fit better' means a higher percentage of the area is covered. They went on to conduct a teacher-structured inquiry (see resources below).

Resources
Structured inquiry
Inquiry sheet