Specifying Exercise: Chop a Shape Into Simpler Ones

Parent Strategy: Solve a Simpler Problem

Geometry Strategy: Chop a Shape Into Simpler Ones

Slightly More Specific Geometry Strategy – Keep An Eye Out For Simpler Shapes, Especially Right Triangles

Area Version #1: If you’re trying to find the area of a complex figure, it’s often helpful to chop it into shapes whose areas you can more easily find.

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Area Version #2: If you’re trying to find (or remember) the formula for the area of a type of shape, try cutting and rearranging that shape’s pieces into a more familiar one. In fact, there are two cuts that often help: cutting off a right triangle and making a midline cut.

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This has a natural extension to limits, as we pass to the infinite.

Trigonometry Version: When you’re trying to find missing sides or angles of a polygon, it’s often helpful to chop that shape into triangles and to chop that triangle into right triangles.

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Quadrilateral Proof Version: If you’re trying to prove (or remember) something about the sides or angles of a quadrilateral, it’s often helpful to draw one or both diagonals and divide the quadrilateral into two (or four) triangles.

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Volume Version: Much like with area. Version 1 says, try to chop up the volume of a complex solid into simpler solids you can tackle individually. Version 2 says, if you’re trying to derive or remember the formula for the volume of a solid, try to chop and rearrange its pieces into a box. There is a natural extension of volume to limits, as we pass to the infinite. (Cavalieri’s Principle)

Auxiliary Lines Version: If you’re stuck on a geometry problem, try adding lines that create right triangles, equilateral triangles or isosceles triangles, because you can often use their properties to illuminate other aspects of the problem.

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Questions:

  • How does proof by dissection fit into all of this?
  • Is this a helpful analysis?
  • How would I use this during the year in the course of my planning? (I think this could help me focus on better mathematical goals, give better hints and identify better things to talk about in whole-group discussions. that’s my prediction, anyway.)
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2 thoughts on “Specifying Exercise: Chop a Shape Into Simpler Ones

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