Fall
2013 |
Mtg
# |
Agenda For Class Meeting
(What is Planned / What Happened) |
Homework Assignment/Tasks
(To be completed before the next class
meeting) |
|
AUG
19 |
|
|
|
[1] |
Activity
Getting a
Little Discrete
examples of Discrete Math topics, background information
|
Assignment
Getting
There ASAP
|
|
[2] |
Activity
Game of
Sprouts
how to play, rules
Notes Graph Theory Basics:
Vocabulary &
Counting
vertex, edge, degree, strategies for counting |
Question
How Many Moves?
Is it possible to predict the maximum number of moves that can be
played in a Sprouts game based on how many vertices at the start of the
game?
|
- introduce game by having a student come up and play
-
|
|
AUG
26 |
[3] |
Activity
Analyzing The Game of Sprouts
quantifiable differences,
collecting data, looking for patterns, describing patterns
|
Research
Leonard Euler
write 3 sentences about Euler's life and why he is considered one of
the greatest mathematicians
|
- preview the week
- ask if anyone got the answer to the question "How many Moves?"
- frame the activity as to get a glimpse of how mathematicians work in
analysis...emphasis on quantifiable differences...and sometimes we have
to look for more than just the obvious
- interesting number of available vertices as a quantifiable
difference...does this lead anywhere?
- find patterns in the columns -- pretty easy
- find relationships among the columns...some are easy, some not so
easy -- Algebra to the rescue
- introduce the adjective "complete" -- what do you think it means?
show some examples
- look at the pattern for complete graphs in respect to the
relationship between number of vertices & number of edges -- is there a
simple way to move ahead in the pattern without knowing the previous??
** might be a nice connection point to Pascal's Triangle
|
[4] |
Activity
The Euler
Characteristic
quantifiable
differences, pieces, regions, looking for patterns
Notes
Describing Structures in Graphs
adjacent, planar, complete
|
Crossword
Talking Graph Theory
facts about Euler, graph theory vocabulary
|
- begin working at the structure of graphs -- ask students
to draw graphs that are planar
- Review/Discuss Euler's life and achievements with the
assistance of these videos
An Evening with Leonard Euler:
http://www.youtube.com/watch?v=h-DV26x6n_Q
The Euler Identity (New Age-ish):
http://www.youtube.com/watch?v=zApx1UlkpNs
Euler Biography (CloudBiography):
http://www.youtube.com/watch?v=Ty6ejK1rAkg
- have students draw a graph that is planar with between 8
and 15 vertices and at least 20 edges but no more than 40
edges
- practice counting -- have students count vertices &
edges
- introduce the idea of regions or trapped space --
quantify these
- Euler "saw" a relationship here....what was it?
- motivate the Euler Characteristic by having
|
[5] |
Notes The
"Essence" of a Graph
equivalent representations,
isomorphisms,
planar
Activity
Water Puzzle
draw a graph that represents the game
|
Research
The Four-Color Problem
6 sentences (who, what, where, when, why,
how)
|
- start with a picture of K4 drawn
with and without edges crossing then a picture of K4 with
edges crossing -- now ask "Are the graphs different?" and
remind students to look for/identify quantitative
differences between the graphs
- sane number of edges, same number of vertices, same
adjacency relationships...hmmm, so the only difference is
the position of the vertices and how the edges were drawn --
so essentially the structure of the graphs are identical --
the DNA or the fingerprint
- introduce the term Isomorphic and relate to the term
Equivalent -- give examples of each
- move to the Isomorphic applet at
http://webspace.ship.edu/deensley/discretemath/flash/ch7/sec7_3/isomorphism.html
- while working through the applet, focus students on
looking at structures -- i.e the number of vertices, the
degree of the vertices, adjacencies, and triangles -- these
features will help in determining if two graphs are
isomorphic
- return to the K4 example and discuss the concept of
planar in light of isomorphic graphs -- make sure to define
planar as a graph that cannot be drawn without edges
crossing
- determining whether a graph in non-planar is difficult,
but there are two classic examples of non-planar graphs,
namely K5 or greater or K3,3
- make the connection with the idea of looking for these
structures within graphs as a way to decide if the graph is
planar or not by using the applet at
http://webspace.ship.edu/deensley/discretemath/flash/ch7/sec7_3/planargraphs.html
- time permitting, discuss how these graphs get used? what
is the practical reason for studying graphs? by introducing
the Water Puzzle and how a graph can "map" the game
- *** might be a good way to transition to game theory
|
[6] |
Audio Clip
Solving or Proving? The 4-Color Problem
Activity
Coloring Maps
Four Color
Problem, algorithms
|
Activity
Coloring Maps
Four Color
Problem, algorithms
|
- ask for the who, what, where,
when, how, why for their 4-color problem homework and write
on the board -- verify & correct where necessary
- introduce the audio clip and
Keith Devlin
- as the clip plays, check
off/correct the w, w, w, w, h, & w written on the
board
- dntart with a picture of K4 drawn
with and without edges crossing then a picture of K4 with
edges crossing -- now ask "Are the graphs different?" and
remind students to look for/identify quantitative
differences between the graphs
- sane number of edges, same number of vertices, same
adjacency relationships...hmmm, so the only difference is
the position of the vertices and how the edges were drawn --
so essentially the structure of the graphs are identical --
i.e. the DNA, the fingerprint
|
|
SEP
02 |
|
LABOR DAY HOLIDAY |
[7] |
Activity
Tracing a Graph
Seven Bridges of Königsberg & Eulerian paths,
necessary & sufficient conditions
|
Research
The Traveling Salesman Problem
6 sentences (who, what, where, when, why,
how)
|
- spend a few minutes reviewing: count
the number of vertices, count the number of edges (count
degrees), adjacent vertices, non-adjacent vertices, planar
- sane number of edges, same number of vertices, same
adjacency relationships...hmmm, so the only difference is
the position of the vertices and how the edges were drawn --
so essentially the structure of the graphs are identical --
i.e. the DNA, the fingerprint
|
[8] |
Notes/Video
Movement in Graphs:
Paths & Circuits
path, circuit, necessary &
sufficient conditions
|
|
[9] |
Notes
Traveling Salesman Problem & Algorithms
Nearest Neighbor, Cheapest Link,
strategies, establishing simple repeatable
steps/clear process
|
Activity
TSP in Florida
find Hamiltonian circuits using the Nearest Neighbor algorithm
first then the Cheapest Link algorithm
|
|
|
SEP
09 |
[10] |
Review
TSP in Florida
Discuss
TSP Applications
routing, construction
|
Read
Splitting Terrorist Cells
|
- start with a "quiz" on Canvas that refers to the
homework assignment from the weekend
- project the distance matrix and demonstrate how to apply
the Cheapest Link algorithm
-
|
[11] |
Discuss
Connectivity in Graphs
how to measure, bridges, cut vertices, pieces,
applications
|
Crossword
Graph Theory Vocabulary Review
|
- review vocab
- re-introduce new vocab & practice
- use campus connection as example for discussion -- how
to connect
- find Hamiltonian or Euler path? -- find the hamiltonian
- How well connected? no bridges and no cut verticies --
yay
- how many edges to remove before causing problems? 2 --
how bad a problem?
- after removing 1 edge, go from zero bridges to YIKES!
- How long will the entire construction job take? the sum
of all the times is 31, but does the structure of the graph
indicate shorter?
|
[12] |
Review
Graph Theory Fundamentals
Cheapest Link & Nearest Neighbor algorithms, planar, paths, circuits, complete,
adjacent, isomorphic
|
Sample Test Questions
Graph Theory Fundamentals
[Solutions]
|
- distribute copies of last year's test and provide about
10 minutes for students to work on answering the questions
without any assistance
-
- students allowed to use all materials (notes, hw,
classwork, iPads, computers)
- remind students about the hw assignemtn
|
[13] |
Test
Graph Theory Fundamentals
Collect Homework & Classwork
|
Assignment
Sudokus
- A Glimmer of Algorithms
|
- allow the entire period for the test
- students allowed to use all materials (notes, hw,
classwork, iPads, computers)
- remind students about the hw assignemtn
|
|
SEP
16 |
[14] |
Discussion
Navigating Sudoku Using Algorithms
practice the sudoku algorithm discussed in class
Video/Activity
Getting There Efficiently
Dijkstra's Algorithm
|
Assignment
Finding the Shortest Path in Graphs [Numb3rs]
complete
questions #1-4
|
- start by making sure that everyone understands the goal
of a sudoku and the parameters that must be satisfied --
also note that sudokus are not mathematical...the numbers
could be replaced with letters or pictures...so it's more of
a logic puzzle
- solicit one or two students to explain how they start a
sudoku -- have them tell what to do first using a sudoku
projected on the board
- invite students to follow the 8 step algorithm to work
through the sudoku -- which quadrant to start with?
- spend time to work through at least 2 quadrants --
emphasize that repeating the same procedure over and over --
eventually it should lead to a solution or pretty darn close
to a solution
- show the start of NUMB3RS episode "Money for Nothing"
until Charlie has talked about Dijkstra's Algorithm and the
possible paths out of LA show up on the computer screen
- project the hw assignment and work through #1 together
|
[15] |
Notes
Examining Dijkstra's Byproduct
subgraphs,
trees, spanning trees
Activity
Modeling: Getting Things Done
flow, directed
edges & digraphs, modeling
|
Assignment
Turner Construction
find the least
amount of time it will take to complete the construction project
|
- review homework -- complete #3 on the board
- "what are the chances that you'll have to find a
hijacked semi in your professional life?"...so let's talk
about a related idea that might be more "realistic" -
construction industry anyone? remember the construction of
SLC and the aux gym?
- Like many things in life, there is usually a sequence to
how things happen -- sometimes some things can happen in
parallel, other times it must be sequential -- sometimes a
combo of both
- Turner Construction -- draw a graph that represents the
construction process
- introduce directed edges and digraphs
- how to quantify the weeks -- start with week 0 or week
1? -- how to represent in the graph
- How long will the entire construction job take? the sum
of all the times is 31, but does the structure of the graph
indicate shorter?
|
[16] |
Review
Turner Construction
critical paths, flow analysis
Notes
Navigating Flow in Graphs
source, sink, quantifying capacity, max flow
|
Assignment
Finding the Flow
map the max
flow for a digraph
|
- continue/finish discussion of Turner Constructions -- emphasize critical paths, the analysis the graph allows
for evaluating slowdowns, etc.
- introduce flows in networks by using the conveyor belt
analogy with this video:
- discuss what other real life phenomenon are examples of
flow -- include electricity and discuss why it cannot be
stored
- show the start of NUMB3RS episode "Blackout" -- need to
edit the beginning but focus on Charlie's examination of the
network to identify the next substation to target if the
goal is a cascading failure of the network
- use the NUMB3RS handout to work through an example of
determining maximum flow - be sure to define source & sink
- introduce hw assignment
|
[17] |
Notes
Navigating Flow in Graphs [Numb3rs]
quantifiable differences, starting backwards, exclusionary approach
|
Assignment
How Much Can You Flow?
develop an
algorithm to determine the maximum flow in a digraph
|
- work through the hw modeling an algorithmic approach to
determining the max flow -- emphasize finding quantitative
differences between the verices and use those to determine
flow through individual vertices -- working backwards from
the sink
- introduce/explain hw assignment / start in class
|
|
SEP
23 |
|
FACULTY IN-SERVICE
DAY |
[18] |
Video
Applications of Graphs & Algorithms
work flow &
queues
|
Assignment
Connecting the Campus
|
- preview the week
- review hw from the weekend - remind of sink, source,
establishing bounds, quantifying vertices with flow in &
flow out, reverse analysis
- watch For All Practical Purposes Episode #3 starting
with the Bell Laboratories piece on connectivity -- stop
video along the way to make connections
- show/discuss NUMB3RS handout Critical Maths from "End of
Watch" -- highlight space race need to identify critical
paths, cooking a meal example, & share experience of Pizza
Hut 5 minute guarantee with an 8 minute oven
- explain hw assignment / start in class
|
[19] |
Activity
Increasing Electricity Flow
|
Sample Test Questions
More Graph Theory & Algorithms
[Solutions]
|
- indicate that one of the two activities will be selected
as a 5-point possible for the hw
- introduce/explain Increasing Electricity Flow
activity -- students will have 25 minutes in class to work
together to provide a solution
- regroup as a class and discuss
- distribute old quiz and answer the multiple-choice &
true/false questions
- remind students to prepare for tomorrow's test
|
[20] |
Test
More Graph Theory & Algorithms
Collect Homework & Classwork
|
Assignment
Prove that the number xxx is a prime number.
|
- allow the entire period for the test
- students allowed to use all materials (notes, hw,
classwork, ipads, computers)
- remind students about the hw assignemtn
|
|
SEP
30 |
[21] |
Notes
Getting Into Algorithms
multiplication & division, efficiency, prerequisites, Rubik
Cubes, divisibility
|
Research
The Sieve of Eratosthenes
|
- preview the week
- start with soliciting ways to multiply 142 and 58 -- be
sure to highlight at least 4 options (traditional, repeated
addition, partitioning into smaller/easier multiplications
and addition, lattice multiplication, and peasant's
multiplication)
- engage in discussion with questions about prerequisite
knowledge (times tables), how these algorithms are similar
(lattice & traditional), etc.
- give special attention to Peasant's because of the
simplicity and relative quickness of it's method -- tie into
Rubiks Cube
- transition into algorithms for determining if a number
is prime by introducing divisibility rules
- tease out the divisibility algorithm for multiples of 3
-- connect to the partitioning algorithm for multiplication
|
[22] |
Notes
Setting the Stage for Number Theory
partitioning & factoring, uniqueness of prime factorization, sifting
for primes
|
Research
Goldbach's Conjecture
Who? What?
Where? When? Why? How?
|
- have "infinity" playing as students settle into to class
- start by formally defining prime and divisible --
provide examples & stress that we are only talking about
positive integers
- transition to discussing the Sieve of Eratosthenes -- an
algorithmic way to find prime numbers
- will we eventually run out of prime numbers? gut feeling
no because there are infinite positive natural numbers, but
how do we know for sure the primes don't fizzle out?
- introduce the video Lecture 7: Numbers of Prime
Importance (from Zero to Infinity: A History of Numbers)
- stop video to emphasize why 1 is not considered a prime
number
- stop video to provide concrete numerical example of
showing the existence of prime numbers beyond a finite list
- end class by discussing the idea of finding patterns in
primes -- look at the Sieve table to see that primes must
end in a 1, 3, 7, or 9, what about spacing -- then define
Twin Primes
-
https://www.youtube.com/watch?v=iFuR97YcSLM
|
[23] |
Video Getting
Primed for Number Theory
distribution of primes, π(n)
function, establishing bounds, natural log, comparing quantities
|
Research
Mersenne Primes
Who? What?
Where? When? Why? How?
|
- start by discussing Goldbach's Conjecture -- who, what,
where, when, why, and how -- be sure to include discussion
of $1,000,000 prize
- work through several examples -- 30=13+7+5+3+2
- emphasize the idea of partitioning -- breaking a
quantity into smaller quantities -- a way to find out about
the "nature" of the number, similar to in chemistry and
distillation
- introduce and show the video Lecture 9: The Prime Number
Theorem and Riemann
- stop to highlight/explain the function
π(n) -- emphasize
that it's a function like functions in algebra but the
difference is there is no quick and easy plug and chug way
of getting the values
- mathematicians have been trying to figure out a formula
for this function with the hopes that it will give us
insight/the ability to predict where prime numbers are, i.e.
how they are distributed
- breakthrough was that n divided by the natural log of n
served as an upper bound for
π(n) -- but how close does this upper bound get
to the actual value of
π(n)
- end class with discussion of taking into account size of
numbers when discussing how close
|
[24] |
Video
Getting to "Know" Numbers
divisibility, visualizing patterns, proof,
|
Research
Perfect Numbers & Pentagonal Numbers
|
- finish discussion about comparing numbers via ratios
- ask about Mersenne numbers -- write the pattern on the
board and see how well it works
- go to the GIMPS website online and highlight how the
program works and how big these numbers are/how many digits
- introduce video The Primes from Mathematics Illustrated
-- stop after the introduction of figurative numbers
- end class by discussing figurative numbers
|
|
OCT
07 |
[25] |
Notes Figurative Numbers:
Patterns Galore
triangular, square, pentagonal, Gauss & partitions, Collatz
conjecture
|
Research
Fibonacci
6 sentences (who, what, where, when, why,
how)
|
- start class by reviewing the prior week and giving some
examples of questions types that could be asked on the test
- review vocabulary -- prime, composite, odd, even
- re-introduce figurative numbers -- numbers that are
generated by arranging objects into regular polygons
- triangular numbers: 3, 6, 10, 15, 21,... -- draw shape
on board and then increase the number of edges on each side
-- look for pattern to how many new vertices get added at
each juncture
- square numbers: 4, 9, 16, 25, 36,... -- draw shape on
board and then increase the number of edges on each side --
look for pattern to how many new vertices get added at each
juncture
- pentagonal numbers: 5, 12, 22, 35, 51,... -- draw shape
on board and then increase the number of edges on each side
-- look for pattern to how many new vertices get added at
each juncture
- show how the patterns can be reversed back to include 1
and 0 for each type of number
- now look for patterns that exist among the sets of
numbers -- i.e. square numbers and triangular numbers,
pentagonal numbers and triangular numbers, etc.
- Gauss -- Prince of Mathematics -- theorem that every
positive integer can be written as the sum of at most
3 triangular numbers -- some sense that triangular numbers
are in a way like prime numbers except for the uniqueness
|
[26] |
Notes Sequences:
Gauss & Fibonacci
recursive definitions, partial sums
Activity
Generating Sequences in Excel
|
Crossword
Number Theory Review
|
- historical notes on Gauss -- the famous adding up
integers punishment story
- introduce Fibonacci numbers -- highlight the idea of a
recursive pattern/definition
- look at partial sums of Fibonacci numbers -- any
patterns?
- transition to laptop cart etiquette -- how to remove and
not remove / how to replace and not replace
- login using BCP credentials -- if you want to save work,
several options: on your BCP network drive, personal USB
drive, email to yourself
- Microsoft Excel -- explain that a spreadsheet is a bunch
of individual calculators that can talk to each other, etc.
- establish columns with headings Triangular, Square,
Pentagonal, Fibonacci -- introduce Excel commands/language
while generating these number sequences
|
[27] |
Notes Number
Symbols & Zero
counting methods, number systems, alternative bases
Activity
Generating Sequences in Excel
|
Sample Test Questions
Number Theory Basics & Algorithms [Solutions]
|
- start with slide showing triangular numbers and
Fibonacci numbers using Roman Numerals -- discuss the nature
of Roman Numerals / positional? how many symbols necessary?
grouping? compare with Arabic numerals
- discuss counting -- one-to-one correspondence / the idea
of keeping track by grouping (use a shepard and sheet
analogy with small pebbles) /
- using a deck of cards, count the cards emphasizing
grouping first in the traditional 10s / then recount the
cards using base 6 grouping
- go to the GIMPS website online and highlight how the
program works and how big these numbers are/how many digits
- emphasize the idea of partitioning -- breaking a
quantity into smaller quantities -- a way to find out about
the "nature" of the number
- return to
- label a column Collatz -- introduce the =if( function to
check
- end class by distributing number grid -- complete by
identifying triangular, Fibonacci, etc.
|
[28] |
Test Number Theory Basics & Spreadsheets
Collect Homework & Classwork
|
Assignment
Avoiding Friday the 13th
|
- allow the entire period for the test
- students allowed to use all materials (notes, hw,
classwork, ipads, computers)
- remind students about the hw assignment
|
|
OCT
14 |
[29] |
Activity
Using Modular Arithmetic:
Friday the 13th
modeling
repeating cycles, finding congruences
|
Assignment
Predicting the Future
|
- direct students to the pdf for this activity
- depending on time and level of student, use the chart to
develop the idea of repeating cycles
|
|
FROSH SERVICE DAY / PSAT / SENIOR WORKSHOPS |
|
OCTOBER BREAK - Classes Do Not Meet |
|
|
OCT
21 |
[30] |
Notes
Into the Mod: Check Digit Applications
algorithms, congruence, ISBN-10 & ISBN-13 numbers
|
Research
UPC Number Check Digits
|
- solicit ISBN numbers and ask to
read all but the last digit / do the magic on the board
- solicit UPC numbers and ask to read all
but the last digit -- UPC have 5 and 5 inside the barcode
and 1 on each end / do the magic on the board
|
[31] |
Notes/Activity
Sneaky Scrambling Algorithms
counting methods, scrambling via modular
arithmetic & geometry
|
Read
Cryptography: Secret Writing pp 11-28
|
- end with rectangular transposition challenge -- offer to
answer questions about the method / depending on
amount of time available, offer extra credit points for the
first correct solution
|
[32] |
Notes Only Two
Options: Transposition & Substitution
scrambling vs. replacing, algorithms, reversibility
|
Read
Cryptography: Secret Writing pp 29-56
|
- start with second rectangular transposition challenge --
very similar enciphering method as the previous example
- explain expectations for the reading homework -- you are
responsible and there will be questions on the test that are
drawn from the reading, but it's your responsibility to read
- highlight the idea of "stock phrases" on p14 -- give
example of at BCP the phrase "Go Bells" is often used at the
conclusion of talks/etc. so a message between BCP students
might end with "Go Bells" -- this gives enemies something to
look for/a way into the system / connect this to the Enigma
Machine and the allies using "Hail Hitler" as the in for
breaking German messages
- highlight the only two actions that can be performed,
namely scrambling or replacing / pp17-18
- show the NSA recruitment advertisement and highlight
that they are not looking for experts in Calculus :)
- return to the scramble MHRNRRKBEOUTYSEOC and reveal that
the message is NUMBERTHEORYROCKS -- the scramble is not
random, but generated by using both addition and
multiplication in mod 17
- begin a process of trying to figure out what was added
and multiplied to generate the scramble -- why focus on
unique letters vs. non-unique?
|
[33] |
Notes/Activity
Going Backwards: Unscrambling
modular arithmetic & inverse operations
|
Assignment
Solving Equations in Modular Arithmetic
|
- assign every student two numbers
-- they will be responsible for the letters in those
positions in the plaintext message
- display a message on the board
with corresponding position values -- ask students as a
group to encipher the message by scrambling the positions
using an affine scramble / once students have determined
where their letters will be moved, have them come up to the
board and write their letter...no students should be
fighting for the same box!
- model one or two letters for them
- verify the scramble using the Excel
spreadsheet
- repeat the process but this time
give the students a message in its enciphered form & provide
the enciphering keys (additive and multiplicative) -- ask
students to decipher the message
- emphasize that going backwards is more difficult that
going forwards
|
|
OCT
28 |
|
OPEN HOUSE
HOLIDAY |
[34] |
Activity
Modular Arithmetic
Messages
congruence, avoiding negatives, finding remainders
|
Crossword
History
of Secret Writing
|
- begin by working through the homework assignment --
assume that only a handful of students even attempted the
assignment
- use the assignment to instruct key points for
working in modular arithmetic: avoid negatives, avoid
division, values should not exceed remainder values, etc.
- use remaining class time for students to work on
deciphering the message while practicing solving modular
equations -- offer extra points to the first person or team
to correctly decipher the message
|
[35] |
Activity
Going Backwards: Inverses
multiplicative inverses
|
Sample Test Questions
Modular Arithmetic & Introduction to Secret Writing [Selected Solutions] |
- begin by discussing the previous day's challenge --
assign each group an equation to solve and then
brainsorm how to use those answers to reveal the message
- focus on equation #9 and highlight the fact that it took
a long time (many additions of the mod) before arriving at a
value that was a multiple of the coefficient of x -- but IS
there an easier and/or more efficient way of finding the
solution?
- --working through the homework assignment --
assume that only a handful of students even attempted the
assignment
- use the assignment to instruct key points for
working in modular arithmetic: avoid negatives, avoid
division, values should not exceed remainder values, etc.
- use remaining class time for students to work on
deciphering the message while practicing solving modular
equations -- offer extra points to the first person or team
to correctly decipher the message
|
[36] |
Test
Modular Arithmetic & Introduction to Secret Writing
Collect Homework & Classwork
|
Read
Cryptography: Secret Writing pp 57-90
|
- allow the entire period for the test
- students allowed to use all materials (notes, hw,
classwork, ipads, computers)
- remind students about the hw assignment
|
|
NOV
04 |
[37] |
Notes
Mathematics of Substitution: Forwards
methods for scrambling the alphabet, one-to-one correspondence,
cipher charts vs. code systems
|
Read
Cryptological Mathematics pp 27-34
|
- start class by asking students to decipher the message
"Did you do your reading this weekend" that was enciphered
using a mono-alphabetic substitution where the keyword
'MEAT' was used to scramble the alphabet
- after about 5 minutes of time present Ceasar ciphers
(alphabet shifting) & give an example on the board / discuss
how this is not very secure-complicated
- compare/contrast Ceasar vs. Keyword vs. Modular methods
for scrambling the alphabet -- then present the idea of a
polyalphabetical substitution to reduce the "Wheel of
Fortune" effect
- end class with Cement Truck joke/punchline -- give time
to find patterns
|
[38] |
Activity
Automating Substitution Ciphers
text & chart referencing functions, number to text conversions,
uppercase vs. lowercase
|
Assignment
Monoalphabetic Ciphers
|
- anually create a substitution chart by typing one
letter per cell
- encipher 'bellarmine' manually making the distinction
between upper & lower case as presented in the text
reading
- introduce the idea of having the spreadsheet
"automatically" do the substitution -- present the vlookup
and hlookup functions
class by asking students to decipher the message
"Did you do your reading this weekend" that was enciphered
using a mono-alphabetic substitution where the keyword
'MEAT' was used to scramble the alphabet
- after about 5 minutes of time present Ceasar ciphers
(alphabet shifting) & give an example on the board / discuss
how this is not very secure-complicated
- compare/contrast Ceasar vs. Keyword vs. Modular methods
for scrambling the alphabet -- then present the idea of a
polyalphabetical substitution to reduce the "Wheel of
Fortune" effect
- end class with Cement Truck joke/punchline -- give time
to find patterns
|
[39] |
Activity
Automating Substitution Ciphers: Generating Alphabets
number to text conversions, replicating patterns using mod
|
|
|
[40] |
Activity
Automating Substitution Ciphers: Backwards
text & chart referencing functions
|
Video
Code-Breakers: Bletchley Park's Lost Heroes
|
|
|
NOV
11 |
[41] |
Quiz
Code-Breakers: Bletchley Park's Lost Heroes
Activity
Affine Ciphers
additive vs. multiplicative, keys, modular inverses
|
Read Cryptological Mathematics
pp 76-81
|
|
[42] |
Activity
Polyalphabetical Ciphers
keys, modular patterns, Enigma Machine
|
Assignment
Polyalphabetic Ciphers
|
|
[43] |
Notes
How Secure Is Your Cipher System?
brute force &
key options, strategies for breaking
|
Read
Cryptological Mathematics
pp 103-108
|
|
[44] |
Notes
Increasing Complexity & Subtle Twists
Hill
Ciphers/matrix algebra, steganography
|
Research
RSA Encryption
6 sentences (who, what, where, when, why,
how)
|
|
|
NOV
18 |
[45] |
Notes -
Understanding Public Key Encryption: RSA
modular arithmetic, primes, finding inverses, public and private
keys |
Sample Test Questions
Cipher Systems
& Spreadsheets [Selected
Solutions]
|
- preview the week -- remind students about being prepared
for the test
- begin by discussing the problem of key exchanges -- how
do you make sure that both parties have access to the same
keys?
- move into the ides of a public key -- a way for people
to send you a secure message where how to encipher is public
but the how to decipher is private
- use an example of Joe wants to send the letter J to you
-- you instruct hum to use mod 527 and key 7 -- takes the
letter J, converts to number, then uses multiplication in
the form of exponents where the number 10 will be
multiplied 10 times in a row -- this generates a large
number which is then reigned in by converting to its mod 527
equivalence, so that Joe send you the number 175 which
represents his original letter J / invite students to join
in the process
- how do you go backwards? solicit from students their
enciphered letter values -- let them know the plan is to
raise the number they sent to the 343 power! -- yikes a
ridiculously huge number...how can you compute that? luckily
you just want to know what the remainder is after dividing
by 527
- using an Excel spreadsheet, show how the modulu
congruence can be used to compute the remainder without
having to get that big number -- stress the algorithmic
approach of this
- verify that the reverse process is working correctly by
checking the values from the solicited student
numbers/original letters
- transitiion to the Youtube video
-
- end class with Cement Truck joke/punchline -- give time
to find patterns
|
[46] |
Test
Cipher Systems & Spreadsheets
Collect Classwork & Homework
|
Assignment
Design Your Own Cipher System (example)
-
must have
substitution and transposition
-
not more
than 6 steps in complexity
-
originality/creativity (5pts max)
-
communication/presentation (5pts max)
|
|
[47] |
Cipher System Project -
Introduction & Information
scope, expectations, guidelines, specifications |
Cipher System Project |
[48] |
Cipher System Project -
Organizational Workday
organize homepage, synthesize ideas for substitution &
transposition, establish timeline/key dates, assign responsibilities |
Cipher System Project |
|
|
|
DEC
09 |
[54] |
Notes Overview of Game Theory
quantifying strategies, zero-sum, minimax theorem, Nash Equilibrium |
|
[55] |
Activity Two-Player Games: Strategies
collecting data, looking for patterns |
Activity
Sprouts Strategy
Is there a winning strategy for the game of Sprouts?
|
[56] |
Activity Two-Player Games: Winning Strategies
rules that drive strategies, combinatorial analysis, Grundy's Game |
|
[57] |
Activity The Prisoners' Dilemma
rules that drive strategies, combinatorial analysis |
Sample
Final Exam |
|
DEC
16 |
[58] |
Semester
Reflection |
Study/Prepare
for
Final Exam |
|
SEMESTER EXAMS - SOCIAL SCIENCE &
MATHEMATICS |
|
SEMESTER EXAMS - ENGLISH & RELIGIOUS STUDIES |
|
SEMESTER EXAMS - LANGUAGE & SCIENCE |
|