One of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.
Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.
The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.
What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?
What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.
What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.
Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.
You may want to view another similar lesson on quadrilaterals.