An examination of fundamental concepts of numbers, including divisibility, congruencies, the distribution of Primes, Pythagorean triples, the Euclidean Algorithm, the Fundamental Theorem of Arithmetic, Diophantine equations, and Goldbach’s conjecture. Emphasizes active student involvement in posing and testing conjectures, formulating counter examples, logical arguments and proofs.
- Describe a rationale and a set of strategies for the activity of problem-generation. Students will compare and contrast the framing of questions; e.g., those that attempt to get at the internal vs.; those that analyze the external character of the problem; those that require an exact vs. an approximate answer; those that make use of a pseudo-historical vs. metaphorical context.
- Discuss the processes of inquiry; including how to get started, the processes of specializing and generalizing, the elements of justification.
- Discuss the roles of affect in mathematical problem-solving.
- Discuss divisibility properties of integers, the greatest common divisor, the least common multiple, and the Fundamental Theorem of Arithmetic.
- Discuss prime numbers, gaps between primes, the twin prime conjecture, Goldbach’s conjecture.
- Describe properties of congruencies (a is congruent to b modulo n) residue systems and examples (odometers, clocks).
- Investigate Pythagorean triples and Diophantine equations.
- Discuss the distinctions between rational vs. irrational number, geometrically and algebraically.
- Demonstrate knowledge of the properties of the real number system and of its subsets
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