I think this needs to be the big practical project that I try to tackle. If I think that this habit/strategy/hint/whatever is too big to help (unless it’s somehow been specified for you) then it’s time to show exactly how complex and under-specified it is.
In doing this I’ll, for now, limit myself to a high school geometry class. And I’ll limit myself to studying the CME Geometry text, which I’m pretty familiar with. I’ll look towards my own problem-solving in geometry to support the search. After all is said and done, we’ll start looking to extend this throughout other grades to show how this idea develops.
The question is, what are some of the more specific versions of “Chop a Shape Into Simpler Shapes” that help students?
Some quick thoughts:
- Trig – Dropping an altitude in trig problems to create two right triangles is a crucial move that needs to be made explicit.
- Proof – It’s often helpful to add the diagonals to a quadrilateral and chop it up into a series of triangles. This often helps by letting us find some similar or congruent triangles. To prove the Isosceles Triangle Theorem it helps to drop an altitude.
- Area – This is an important subcategory. It has a few subcategories of its own. Make Right Triangles: Making right triangles in an equilateral triangle or regular polygon to find its area; making right triangles by dropping altitudes in triangles or trapezoids or parallelograms in order to cut and rearrange their pieces into rectangles/triangles/parallelograms and find their areas; cutting circles into triangle slivers. Chopping a complex shape into pieces whose area formulas you know is another. This is helpful for proof by dissection.
- Volume – Cavalieri’s principle, and natural extensions of lots of the Area stuff
- Advanced Geometry – A great deal of auxiliary lines problems are chopping problems. It’s often a good idea to chop a shape into an isosceles triangle or a right triangle if you can. (This was a crucial move in GoGeometry #8).