The first few days of my discrete math class are rolling out as expected. Besides covering operational issues, the opening of the class pushes students to break their perception of what a math class looks like and how it operates.
I have been pleased with how responsive the class has been. When pushed to think of problems in new and different ways or to look for patterns and connections to physical representations, the class has really pushed themselves.
After working through some Pascal triangle investigations, we started working with figurate numbers. I opened with square numbers, as usual, to work on thinking of familiar entities differently. When we transitioned to triangular numbers, the class easily dove into working with the representations and patterns. The exciting part was that students saw how to calculate triangular numbers but wanted to have a more efficient way to perform the calculation. I'll be interested to see what processes and formulas they come up with.
For those interested, here is the table of contents from my discrete math ebook. You can get a sense of the material that is covered and how the course progresses through the topics.
Table of Contents
- Counting and Discrete Probability …………………………………………………….… page 3
- Initial Investigations ……………………………………..………………………… page 3
- Figurate Numbers ……………………………….………..………………………… page 4
- Finite Differences ……………………………….………..………………………… page 7
- Polygonal Numbers ………..……………………………..………………………… page 9
- Combinations and Permutations ……………………...……………………… page 10
- The Pigeonhole Principle ……………………………………..…………………. page 12
- Advanced Counting ……………………………………..………………………..… page 13
- Discrete Probability ……………………………………..………..……………..… page 15
- Conditional Probability …………………………..……..………………………… page 17
- Probability Practice and Mastery Quiz ………………………………………. page 20
- Graph Theory ……………………………………..…………………………………….……… page 21
- Introduction to Graphs ………………………………..…………………..……… page 21
- Graph Theorems ………...………………………………..………………………… page 27
- Mail Route Practice ……………..……………..………..………………………… page 31
- Hamilton Paths and Circuits …………..……………..………………………… page 32
- Planar Graphs …………….………………………………..………………………… page 35
- Complete and Complementary Graphs …………………………………….. page 39
- Graph Coloring ………………………………………..…..………………………… page 43
- Elementary Number Theory ………..…………………………..………………………… page 50
- Prime Numbers …………………………..………………..………………………… page 50
- Prime Number Distribution ………...………………..………………………… page 53
- Prime Number Sequences ……………...……………..………………………… page 57
- Relative Primes ………..…………………………………..………………………… page 59
- Prime Sums …………………………………………..……..………………………… page 62
- Prime Factorization ……….……………………………..………………………… page 63
- Perfect Numbers ………………………...………………..………………………… page 66
- Mersenne Primes and Prime Formulas ……...…..………………………… page 69
- Euclidean Algorithm ………………………………..…..………………………… page 71
- Cryptography ……………………………………..…………………………………..………… page 74
- Caesar Ciphers ……………………...……………………..………………………… page 74
- Cipher Functions …………..……………………………..………………………… page 76
- Affine Ciphers …………………....………………………..………………………… page 80
- Chinese Remainder Theorem ……….………………..………………………… page 81
- Congruences ……………………………….………………..………………………… page 83
- Solving Congruences ……………………………………..………………………… page 88
- Congruences and Divisibility ………………..………..………………………… page 90
- Modular Arithmetic …………….………………………..………………………… page 93
- Primality Tests ……………………………………...……..………………………… page 95
- Cryptography Revisited ………………………………...………………………… page 97
- Diffie-Hellman Exchange ……….……………………..………………………… page 98
- Cracking the Diffie-Hellman Exchange ……………...…………………...… page 101
- RSA Protocol ……………………………...………………..………………………… page 103
- RSA Practice ………………………….……………………..………………………… page 105
- Set Theory and Boolean Algebra ……...………………………..………………………… page 108
- Understanding Sets ……………………..………………..………………………… page 108
- Set Operations ……………………………….……………..………………………… page 111
- Venn Diagrams ……………………………...……………..………………………… page 115
- DeMorgan’s Laws and Cartesian Products …………….…………………… page 116
- Boolean Algebra …………………………..………………..…………………..…… page 118
- Boolean Functions ……………….………………………..………………..……… page 120
- Boolean Identities ……………..…………………………..………………………… page 122
- Truth Tables ……………………………………..…………..……………..………… page 124
- Logic Gates ……………………………………………….…..……………………..… page 128
- Logic and Proof …………………………………………………….....…………………..…… page 132
- Statements and Negation ………………..……………..………………………… page 132
- The Language of Logic …………………………….……..………………..……… page 134
- Showing Truth …………………………….………………..………………………… page 136
- Methods of Proof ……………………...…………………..………………………… page 139
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