OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.
The objectives of the problem below include:
• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)
THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.
Answer: (Pi-2)/2
Note: Please verify!
REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]
• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?
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2 comments:
A related problem, no trig required:
http://fivetriangles.blogspot.com/2013/02/45-overlapping-circles.html
Very nice variation and brings in rotation in a natural manner. As you can tell I'm more interested in teacher reaction to the reflections following the problem. It's important for teachers to have a resource of problems like yours or mine but I see these as a springboard for generating dialog and developing deeper conceptual understanding. Mathematical habits of mind so to speak.
Do you think maybe one student in a classroom would be curious about the general rotation problem or the general intersection problem? I believe it is our obligation to engender that kind of curiosity and NOT just with the gifted students.
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