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David M. Bressoud Macalester College, St. Paul, MN Project NExT-WI, October 6, 2006 •Do something that is new to you in every course. •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •What you grade is what counts for your students. •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •What you grade is what counts for your students. Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •What you grade is what counts for your students. Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •What you grade is what counts for your students. Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood •Do something that is new to you in every course. •Try to avoid doing everything new in any course. •What you grade is what counts for your students. Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood What you grade is what counts for your students. • Homework 20% • Reading Reactions 5% • 3 Projects 10% each • 2 mid-terms + final, 15% each If you hold students to high standards and give them ample opportunity to show what they’ve learned, then you can safely ignore cries about grade inflation. 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. “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 “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 Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, … random Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, … 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: ??? 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 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: ??? TEST YOUR GUESS Work with simpler problem: regions formed by lines on a plane: 0 lines: 1 region 6 1 line: 2 regions 1 2 lines: 4 regions 3 lines: ??? 2 5 7 3 4 START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD n Space cut by n Plane cut by n planes lines Line cut by n points 0 1 1 1 1 2 2 2 2 4 4 3 3 8 7 4 4 5 5 6 START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD n Space cut by n Plane cut by n planes lines Line cut by n points 0 1 1 1 1 2 2 2 2 4 4 3 3 8 7 4 11 5 4 5 Test your guess 6 START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD n Space cut by n Plane cut by n planes lines Line cut by n points 0 1 1 1 1 2 2 2 2 4 4 3 3 8 7 4 4 15 11 5 5 Test your guess 6 GUESS A FORMULA n 0 1 2 3 4 5 6 points on a line lines on a plane planes in space 1 2 3 4 5 6 7 1 2 4 7 11 16 22 1 2 4 8 15 26 42 n k GUESS A FORMULA n 0 1 2 3 4 5 6 points on a line lines on a plane planes in space 1 2 3 4 5 6 7 1 2 4 7 11 16 22 1 2 4 8 15 26 42 k n 0 1 2 3 4 5 6 0 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 2 0 0 1 3 6 3 0 0 0 1 4 10 10 4 0 0 0 0 1 5 15 20 15 5 0 0 0 0 0 1 6 6 0 0 0 0 0 0 1 GUESS A FORMULA n k–1-dimensional hyperplanes in k-dimensional space cut it into: n n n n 0 1 2 L k regions. GUESS A FORMULA n k–1-dimensional hyperplanes in k-dimensional space cut it into: n n n n 0 1 2 L k regions. Now prove it! GUESS A FORMULA n k–1-dimensional hyperplanes in k-dimensional space cut it into: n n n n 0 1 2 L k regions. Now prove it! Show that if R n, k # of regions with n hyperplanes in k-dimensional space, then R(n, k) R(n 1, k) R(n 1, k 1). What do you have to assume about k 1-hyperplanes in k-dimensional space? Stamp Problem: What is the largest postage amount that cannot be made using an unlimited supply of 5¢ stamps and 8¢ stamps? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 X 0 1 2 3 4 5 6 7 8 9 X X 10 11 12 13 14 X 15 16 17 18 19 20 21 22 23 24 X X 25 26 27 28 29 X 30 31 32 33 34 X 0 1 2 3 4 5 6 7 X8 9 X X X 10 11 12 13 14 X X 15 16 17 18 19 20 21 22 23 24 X X X X 25 26 27 28 29 X X 30 31 32 33 34 X 0 1 2 3 4 5 6 7 X8 9 X X X 10 11 12 13 14 X X X 15 16 17 18 19 20 21 22 23 24 X X X X X X X X 25 26 27 28 29 X X X X X 30 31 32 33 34 X 0 1 2 3 4 5 6 7 X8 9 X X X 10 11 12 13 14 X X X 15 16 17 18 19 20 21 22 23 24 X X X X X X X X 25 26 27 28 29 X X X X X 30 31 32 33 34 Stamp Problem: What is the largest postage amount that cannot be made using an unlimited supply of 5¢ stamps and 8¢ stamps? 4¢ and 9¢? 4¢ and 6¢? a¢ and b¢? How many perfect shuffles are needed to return a deck to its original order? In-shuffles versus out-shuffles In-shuffles in a deck of 2n cards is the order of 2 modulo 2n+1. Out-shuffles is the order of 2 modulo 2n-1. Tips on group work: •I assign who is in each group, and I mix up the membership of the groups. •No more than 4 to a group, then split into writing teams of 2 each. Have at least one project in which each person submits their own report. •Each team decides how to split up the grade.