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Overlapping shapes inquiry

The overlapping shapes prompt was designed by Colm Sweet (a maths teacher in Horsham, UK). The inquiry that develops from the prompt has the potential to encompass different concepts in the maths curriculum, from creating and naming shapes to calculating areas and ratio. Depending on the complexity of the inquiry, the prompt is suitable for all year groups in secondary school. Students have asked the following questions in the orientation stage:
  • What is the name of the shape created by the overlapping triangles?
  • How many shapes can you create in the overlaps?
  • Can you only create one kind of shape with two circles? 
  • Is it possible to make a triangle with the overlap of the two triangles?
  • Are the triangles equilateral?
  • Does it matter if you use a different type of triangle (or a rectangle)?
  • Does it matter if the two shapes are different sizes?
  • Is there a rule for the number of different shapes you can make with the overlaps?
  • Can you overlap the squares (circles / triangles) so the area of the three parts are equal?
  • Could we try overlapping pentagons?
  • What shapes could be made with three overlapping shapes of the same type?
Students have set out to answer their own questions about the shapes they can make with the overlap. (The teacher is advised to 
Overlapping circles prompt
The overlapping shapes prompt started out as a standard problem. The problem, based on the diagram above, related to the areas of the three regions. The potential of the diagram to form a stimulus to inquiry was identified by Colm Sweet. The overlapping circles prompt is the result of a series of iterations during which the prompt became progressively more open (see creating a prompt page for details).
prepare the shapes in paper and tracing paper beforehand). The cards chosen in the regulatory phase of the inquiry have often been about 'working with another student' and 'sharing our results' in order to find all the possible shapes made by the overlaps. Older students have attempted to solve the area questions, especially the one about creating three squares with equal areas. They start with a particular numerical example. Then they might, under the teacher's supervision, move on to the general case. If the area of a square is A, then the length of one of its sides is A. When the three areas are equal, the area of the overlap is A/2 and the length of its side is A/2. Thus, the ratio of the length of a side of the square to the length of a side of the overlap is A:A/2 or, in simpler terms, 1:1/2.

Colm Sweet is second in charge of the mathematics department at Tanbridge House School, Horsham (UK). You can follow him on twitter @MrCSweet

Prompt sheet
Prompt sheet Overlapping triangles inquiry (see below)

Alternative prompt
Overlapping Triangles
This prompt uses only one part of the overlapping shapes prompt. It might be more suitable for less experienced inquirers who would find the three sets of diagrams difficult to analyse in one go. It could form a starting point before moving onto the main prompt. For a structured approach to this prompt, see the box on the right.
Structured inquiry 
he prompt (right) was devised for a structured inquiry by Adam Otulakowski. He, along with colleagues, used the prompt as an assessment-style activity for year 10 classes. Initial discussions focussed on the assumptions underlying the diagram, and on specific concepts such as area, Pythagoras' Theorem, trigonometry and proof. As part of the structure, Adam indicated what grade the students could expect to achieve by following certain lines of inquiry. After the first lesson students received personalised feedback (see the example below) before setting out, in the second lesson, to respond to the feedback either on their own or in a group of students looking at a similar problem. Adam evaluated his experience of using inquiry: "I found this method of teaching highly effective, differentiating by topic and also by level without any ceiling as I prompted students to delve further and further into the maths."

Adam Otulakowski was head of mathematics at Tanbridge House School, Horsham (UK) until August 2014. In 2014-15 he teaches at Camberwell High School in Melbourne, Australia.

Students' initial questions and comments  
Notes and calculations
Feedback sheet
Circle prompts   
The prompt (right) has led to students attempting to construct the diagram, but mostly it leads to questions about the area of the circles. As R = 2r (where R is the radius of the large circle, and r is the radius of the smaller ones), the area of the large circle is twice the area of the two smaller ones. Students have gone on to inquire into diagrams of this type with similar polygons. What fraction of the large shape is covered by the two smaller ones? Is the relationship the same for all polygons?

This second prompt was first spotted as a standard mathematical problem. The diameter of the large circle was given as 20cm, and the question required students to find "the radius of the largest circle that fits in the middle." As an inquiry for older secondary school students, the prompt has given rise to classes finding their own dimensions and working out the areas of the large and smaller circles. They have also generalised the result for any length of the diameter. If R is the radius of the large circle and r the radius of a smaller circle, then R = 2√2r and the answer to the original problem can be found by using the expression 2r(√2-1).