Future Middle School Teachers Janet Beissinger and Bonnie Saunders University of Illinois at Chicago

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Presentation transcript:

Future Middle School Teachers Janet Beissinger and Bonnie Saunders University of Illinois at Chicago Cryptography and Mathematics for and inservice The Cryptoclub Project is funded by the National Science Foundation Grant # Prior funding was from NSF Grant #

Common courses for Middle School Math Endorsement Geometry Calculus Number Theory November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 5 College level math Connections to middle school Peer-teaching experiences College level math Connections to middle school Peer-teaching experiences

Common courses for Middle School Math Endorsement Geometry Calculus Number Theory and Cryptography lcm, gcd, factoring, prime numbers combination problems, modular arithmetic reasoning and proof negative numbers -- number sense – solving linear equations -- functions November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 6

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 7 problem solving middle school teaching number theory emphasizing explanation moving from simple- minded, tedious solutions to efficient and elegant ones algebraic and abstract thinking

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 8 middle school teaching emphasizing explanation

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 9

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 10

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 11

Additive Cipher: key 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 12 crypto

Additive Cipher: key 7 – switch to numbers November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 13 crypto abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – switch to numbers – To encrypt, add 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 14 crypto abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – switch to numbers – To encrypt, add 7 – Reduce mod 26 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 15 crypto abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, subtract 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, subtract 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, subtract 7 – Reduce mod 26 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, subtract 7 – Reduce mod 26 – Switch to letters November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 20 club abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, add 26 – 7 = 19 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, add 26 – 7 = 19 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference abcdefghijklmnopqrstuvwxyz

Additive Cipher: key 7 – To decrypt, add 26 – 7 = 19 – Switch to letters November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 23 club abcdefghijklmnopqrstuvwxyz

Additive Ciphers – To encrypt add the key k – To decrypt add the additive inverse mod 26 of k 26 – k or – k November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 24

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 25 number theory algebraic and abstract thinking

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 26 problem solving moving from simple- minded, tedious solutions to efficient and elegant ones

Combination Problem – What weights can be measured with a balance and 5-oz and 12-oz weights? November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 27

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 28 Number of 12-oz weights Number of 5-oz weights Combination Chart

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 29 Negative Combination

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 30

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 31

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 32 Number of 12-oz weights Number of 5-oz weights Combination Chart

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 33 number theory algebraic and abstract thinking

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 34

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 35

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 36

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 37

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 38

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 39

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 40

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference is the multiplicative inverse of 5: multiplying by 21 “undoes” multiplying by 5. To decrypt the times 5 cipher, multiply by 21

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 42

Multiplicative Cipher key k – Encrypt by multiplying by k mod 26 – Decrypt by multiplying by the multiplicative inverse of k mod 26 – Find the inverse of k by solving: k  m – 26  n = 1 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 43

November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 44 problem solving middle school teaching number theory emphasizing explanation moving from simple- minded, tedious solutions to efficient and elegant ones algebraic and abstract thinking

Resources Bonnie Saunders Materials for preservice and/or inservice course: Problem Set, Project descriptions, syllabus, etc Number Theory for Teachers workbook Janet Beissinger For more information about CryptoClub Project and Summer Leader Workshop opportunities. The Cryptoclub: Using Mathematics to Make and Break Secret Codes by Janet Beissinger and Vera Pless, CRC Press cryptoclub.org Under construction but has lots of activities for students and others The CryptoClub Project is funded by the National Science Foundation Grant # Prior funding was from NSF Grant #