Inscribed shapes inquiry
Mathematical inquiry processes: Interpret; analyse structure; geometrical reasoning; conjecture and proof. Conceptual field of inquiry: Area; constructions; Pythagoras' theorem; fractions and percentages; similarity; surds.
This prompt is suitable for older students in secondary school and classes with high prior attainment in lower years. It may be presented to the class with the diagram below, particularly if the teacher wants to restrict speculation about the position of the shapes in the orientation phase of the inquiry.
Students' initial questions and observations usually focus on the meaning of 'fits better' and how a 'fit' can be measured. They have suggested using percentages or fractions to find the proportion of the shape 'filled in' by the inscribed shape or the region of the shape left 'unfilled'.
Conjecture and convince
By comparing the shaded regions as a proportion of the larger shape, students often claim that the square looks like it fills less of the circle than the circle fills of the square. This is true (and so it follows that the prompt is false), but students do not have a convincing answer to the question 'How do you know for sure?'
In a structured inquiry the teacher directs the class to find the areas of the shapes. Students use the formulae they already know or seek the teacher's help for those that they do not. The teacher might also co-construct the procedure to find a dimension using Pythagoras' theorem.
To calculate the percentage of the one shape covered by another, students set one dimension (the side of the square or radius of the circle) to, for example, one unit and work out the other dimensions.
On completing the calculation, they will often suggest doing the same for larger shapes, setting the dimension to two units or more. The fact that the percentage is the same in the new case gives the teacher an opportunity to discuss the concept of mathematical similarity.
Inviting the class to participate in directing the inquiry through the use of the regulatory cards has led to alternative approaches in the early phases. Students have suggested:
Using a different visual representation of the prompt in which the squares in the diagrams are the same size.
Attempting to construct accurate diagrams with a ruler and pair of compasses from which to measure the dimensions.
At the end of a lesson a year 10 class at Haverstock School (Camden, London, UK) made observations and posed questions about the prompt (see the picture). They cover five approaches:
Draw and analyse diagrams (black);
Consider the dimensions of the shapes (green);
Reason geometrically about which shape fits better (blue);
Decide how to work out whether the prompt is true (red); and
Make a conjecture about the prompt and prove (purple).
During a brief discussion orchestrated by the teacher, the students decided that 'fit better' means a higher percentage of the area of the larger shape is covered by the smaller one.
Lines of inquiry
Once the class has understood that all the particular cases of each diagram are mathematically similar, then a variable can be used to denote one dimension for the general case.
In the mathematical notes all dimensions are shown in terms of the radius (r) in a proof that a circle fits into a square better than a square fits into a circle.
2. Circle and polygons
Is it always the case that the circle fits into an n-sided polygon better than the inscribed polygon fits into the circle? (See the mathematical notes for the case of the circle and hexagon).
3. Combine other shapes
Does, for example, an equilateral triangle fit into a square better than a square fits into an equilateral triangle? The diagrams (see picture) can be used to initiate a discussion about which one shows an equilateral triangle inscribed in a square. Regardless that the first one does not, students often decide to find the percentage of the square filled by the triangle along with the others.
Mathematical notes for square, hexagon and circle.