Overlapping shapes inquiry
Mathematical inquiry processes: Explore; generate and classify more examples. Conceptual field of inquiry: Area; describe shapes; fractions and percentages.
The overlapping shapes prompt was designed by Colm Sweet (a head of mathematics in West Sussex, UK). The inquiry that develops from the prompt has the potential to encompass different concepts in the mathematics curriculum, from creating and naming shapes to calculating areas and ratio. For inexperienced inquirers, the teacher might choose only one pair of overlapping shapes as the prompt.
Students' questions and observations
Students have asked the following questions about the prompt 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 prepare the shapes on paper or provide tracing paper). 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. For inexperienced inquirers, the teacher might choose only one pair of overlapping shapes as the prompt.
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.
At the time he designed the prompt, Colm was second in charge of the mathematics department at Tanbridge House School, Horsham (UK).
The prompt 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 was head of mathematics at Tanbridge House School, Horsham (UK) until August 2014. He then moved to Camberwell High School in Melbourne, Australia.