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Teaching Combinatorics in a Discrete Math Class

Teaching Combinatorics in a Discrete Math Class. David M. Bressoud Macalester College, St. Paul, Minnesota MathFest, Albuquerque, NM August 5, 2005. MATH 136 DISCRETE MATHEMATICS

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Teaching Combinatorics in a Discrete Math Class

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  1. Teaching Combinatorics in a Discrete Math Class David M. Bressoud Macalester College, St. Paul, Minnesota MathFest, Albuquerque, NM August 5, 2005

  2. MATH 136 DISCRETE MATHEMATICS An introduction to the basic techniques and methods used in combinatorial problem-solving. Includes basic counting principles, induction, logic, recurrence relations, and graph theory. Every semester. Required for a major or minor in Mathematics and in Computer Science. I teach it as a project-driven course in combinatorics & number theory. Taught to 74 students, 3 sections, in 2004–05. More than 1 in 6 Macalester students take this course.

  3. “Let us teach guessing” MAA video, George Pólya, 1965 • Points: • Difference between wild and educated guesses • Importance of testing guesses • Role of simpler problems • Illustration of how instructive it can be to discover that you have made an incorrect guess

  4. “Let us teach guessing” MAA video, George Pólya, 1965 • Points: • Difference between wild and educated guesses • Importance of testing guesses • Role of simpler problems • Illustration of how instructive it can be to discover that you have made an incorrect guess • Preparation: • Some familiarity with proof by induction • Review of binomial coefficients

  5. Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, …

  6. random Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, …

  7. random Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, … Simpler problem: 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions 4 planes: ???

  8. random Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, … Simpler problem: 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions 4 planes: ??? Educated guess for 4 planes: 16 regions

  9. TEST YOUR GUESS Work with simpler problem: regions formed by lines on a plane: 0 lines: 1 region 1 line: 2 regions 2 lines: 4 regions 3 lines: ???

  10. TEST YOUR GUESS Work with simpler problem: regions formed by lines on a plane: 0 lines: 1 region 1 line: 2 regions 2 lines: 4 regions 3 lines: ??? 6 5 1 7 2 4 3

  11. START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD

  12. START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD Test your guess

  13. START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD Test your guess

  14. GUESS A FORMULA

  15. GUESS A FORMULA

  16. GUESS A FORMULA nk–1-dimensional hyperplanes in k-dimensional space cut it into:

  17. GUESS A FORMULA nk–1-dimensional hyperplanes in k-dimensional space cut it into: Now prove it!

  18. GUESS A FORMULA nk–1-dimensional hyperplanes in k-dimensional space cut it into: Now prove it!

  19. This PowerPoint presentation and the Project Description are available at www.macalester.edu/~bressoud/talks

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