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

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Presentation on theme: "Future Middle School Teachers Janet Beissinger and Bonnie Saunders University of Illinois at Chicago"— Presentation transcript:

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4 Future Middle School Teachers Janet Beissinger and Bonnie Saunders University of Illinois at Chicago beissing@uic.edu saunders@uic.edubeissing@uic.edusaunders@uic.edu Cryptography and Mathematics for and inservice The Cryptoclub Project is funded by the National Science Foundation Grant # 0840313. Prior funding was from NSF Grant # 0099220.

5 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

6 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

7 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

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

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

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

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

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

13 Additive Cipher: key 7 – switch to numbers November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 13 crypto 21724151914 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

14 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 21724151914 92431222621 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

15 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 21724151914 924522021 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

16 Additive Cipher: key 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 16 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

17 Additive Cipher: key 7 – To decrypt, subtract 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 17 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

18 Additive Cipher: key 7 – To decrypt, subtract 7 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 18 211-61 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

19 Additive Cipher: key 7 – To decrypt, subtract 7 – Reduce mod 26 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 19 211201 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

20 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 211201 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

21 Additive Cipher: key 7 – To decrypt, add 26 – 7 = 19 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 21 2111 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

22 Additive Cipher: key 7 – To decrypt, add 26 – 7 = 19 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 22 211201 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

23 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 211201 91818 abcdefghijklmnopqrstuvwxyz 012345678910111213141516171819202122232425

24 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

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

26 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

27 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

28 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 28 Number of 12-oz weights 12144149154159164169174179184189194199204209214219 11132137142147152157162167172177182187192197202207 10120125130135140145150155160165170175180185190195 9108113118123128133138143148153158163168173178183 896101106111116121126131136141146151156161166171 784899499104109114119124129134139144149154159 6727782879297102107112117122127132137142147 56065707580859095100105110115120125130135 44853586368737883889398103108113118123 336414651566166717681869196101106111 224293439444954596469747984899499 112172227323742475257626772778287 0051015202530354045505560657075 0123456789101112131415 Number of 5-oz weights Combination Chart

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

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

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

32 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 32 Number of 12-oz weights 983889398103108113118123128133138143148153158 8717681869196101106111116121126131136141146 7596469747984899499104109114119124129134 64752576267727782879297102107112117122 535404550556065707580859095100105110 423283338434853586368737883889398 311162126313641465156616671768186 24914192429343944495459646974 1-13-8-3271217222732374247525762 0-25-20-15-10-505101520253035404550 -37-32-27-22-17-12-7-238131823283338 -2-49-44-39-34-29-24-19-14-9-41611162126 -3-61-56-51-46-41-36-31-26-21-16-11-64914 -5-4-3-2012345678910 Number of 5-oz weights Combination Chart

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

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

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

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

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

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

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

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

41 November 29, 2012 National Council of Teachers of Mathematics 2012 Regional Conference 41 21 is the multiplicative inverse of 5: multiplying by 21 “undoes” multiplying by 5. To decrypt the times 5 cipher, multiply by 21

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

43 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

44 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

45 Resources Bonnie Saunders saunders@uic.edusaunders@uic.edu Materials for preservice and/or inservice course: Problem Set, Project descriptions, syllabus, etc Number Theory for Teachers workbook Janet Beissinger beissing@uic.edubeissing@uic.edu For more information about CryptoClub Project and Summer Leader Workshop opportunities. www.crcpress.com 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 # 0840313. Prior funding was from NSF Grant # 0099220.

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