Introduction E9F5FC

Questions FFFFC0

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Ideas

• Distinguishing relevant and irrelevant data.
• Distinguishing known and unknown information.
• Analogy.
• Structural similarity.
• Pattern.
• Consider the meaning of the answer arrived at by algebra. Is it meaningful? Is it unique? Is each answer acceptable?
• Estimate the answer in advance.
• Study the effect of restrictions on the data upon the final result.
• Specialize a problem.
• Generalize a problem.

Examples

Implication

• PS-pg.9. Poet's way vs. Peasant's way. Peasant's way is direct, calculating going forwards. Poet's way is elegant, perhaps calculating going backwards.
  • If the sum of two numbers is 2, and the product of these same two numbers is 3, find the sum of the reciprocals of these numbers. The peasant's way is to solve linear equations, which requires the use of imaginary numbers. The poet's way is to work backwards, starting with {$\frac{1}{x} + \frac{1}{y}$}, and rephrasing that as {$\frac{y+x}{xy}$} and then realizing this is {$\frac{2}{3}$}.
• PS-1-1
  • Variable. Give names to every angle.
  • Lattice of conditions. Triangles group angles together.
  • ...? Note that the sum of angles is the same for each triangle.
  • Challenge 1 and 2
    • Extend the domain. By moving the point E to C, we are setting the related bisecting angle to 0. By moving the point further down the line, outside of the triangle, we are likewise replacing the bisecting angles with their negatives.

读物

• PS Alfred Posamentier, Charles Salkind. Challenging Problems in Geometry.