Level 3 (Formal Deduction): The nat

Level 3 (Formal Deduction): The nature of deduction is understood. . . .
Provide students opportunities
1. to identify what is given and what is to be proved in a problem
• For the following problem, identify what is known and what is to be proved or shown. Do NOT complete the proof. “The perpendicular bisector of the base of an isosceles triangle passes through the vertex of the triangle.”
2. to identify information implied by a figure or by given information
• Figure ABCD is a parallelogram. Discuss A B
what you know about this figure. Write a
problem in “If. . . then . . form based on
this figure. D c
3. to demonstrate an understanding of the meaning of undefined term, postulate, theorem, definition, etc.,
• Which of the following statements is a postulate, a theorem, a definition? Why?
a) Points that lie on the same line are called collinear. (D)
b) Two points determine a line. (P)
c) Every segment has exactly one midpoint. (T)
d) The midpoint of a segment is said to bisect the segment. (D)
4. to demonstrate an understanding of necessary and sufficient conditions
• Write a definition of a square that begins
a) A square is a quadrilateral...
b) A square is a parallelogram . . .
c) A square is a rectangle . . .
d) A square is a rhombus . . .
5. to prove rigorously the relationships developed informally at level 2
6. to prove unfamiliar relationships
7. to compare different proofs of a theorem—for example, the Pythagorean theorem
8. to use a variety of techniques of proof—for example, synthetic, transformations, coordinates, vectors
9. to identify general strategies of proof
• If a proof involves parallelism, try “saws,” “ladders,” or rotations of 180°.