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
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 Overlapping Triangles (see below)
Promethean flipchart download
Smartboard notebook download