SHIFT CIPHERS
Author: Tom Abraham
Source: Mathematics: Modeling Our World, Course 1
Date: Day 3
Class: 9th Grade Algebra
Duration: 1 hour
Goals: Students will be able to:
· Encode messages using a simple shift cipher.
· Represent a shift cipher as a function and a graph.
· Understand that math can simplify processes and make encoding messages more efficient.
TEKS: b.1
(C) The student describes functional relationships for given problem situations
and writes equations or inequalities to answer questions arising from the
situations.
b.1 (D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
b.1 (E) The student interprets and makes inferences from functional relationships.
b.2
(C) The student interprets situations in terms of given graphs or creates
situations that fit given graphs.
b.3 (B) Given situations, the student looks for patterns and
represents generalizations algebraically.
c.1 (A) The student determines whether or not given situations can
be represented by linear functions.
c.1 (C)
The student translates among and uses algebraic, tabular, graphical or verbal
descriptions of linear functions.
c.3 (A) The
student analyzes situations involving linear functions and formulates linear
equations or inequalities to solve problems.
c.3 (C)
For given contexts, the student interprets and determines the reasonableness of
solutions to linear equations and inequalities.
c.4 (A) The
student analyzes situations and formulates systems of linear equations to solve
problems.
ENGAGEMENT
|
|
STUDENTS
SAY |
GUIDING
QUESTIONS |
|
Write a message on the
board using a shift +3 cipher - "PDWK LV VR FRRO," which translates
to "MATH IS SO COOL" |
This
cipher should be a bit harder to solve, but many students should still be
able to do it. |
“How
does a shift cipher work?” “How
does this cipher work?” |
EXPLORATION
|
|
STUDENTS
SAY |
GUIDING
QUESTIONS |
|
“In your groups, use the
graph paper and come up with your own shift ciphers. Draw the relationship between your letters
and their coded forms. For the sake of
simplicity, keep your shifts to 10 or less.” “Draw two graphs, one with
numbers and one with letters.” |
Students should come up
with linear graphs that relate a letter to its coded symbol. Students should be able to
come up with relationships similar to y = x + c, where ‘y’ is the coded
symbol, ‘x’ the original letter, and ‘c’ the shift value. Students may be confused at
first by negatives, zeros, and numbers greater than 26. It may take some time, but the students
should realize that a value of 0 is equivalent to 26, a value of -2 is
equivalent to 24, and a value of 28 is equivalent to 2. A shift +34 cipher just
shifts a letter 8 places to the right (34 – 26 = 8). |
“How can a shift cipher be
represented as a pictorial relationship?” “If this were a graph, what
would you put on the x-axis and the y-axis?” “What kind of relationships
can you see between your pictures and your ciphers?” In a shift cipher, what
happens when your coded numerical value ends up being less 0, -2, and/or 28? What would a shift +34 cipher look like? |
EXPLANATION
|
|
STUDENTS
SAY |
GUIDING
QUESTIONS |
|
“You
have just seen three ways that you can encode a message: a table, a graph,
and a function.” “The
table shows the original letter and its coded form. The graph conveys the same information but
as a picture. The function is the
simplest and most efficient method of describing the cipher.” “A
function is a mathematical way of showing the relationship between two sets
of numbers.” (Have students put their functions and
graphs of the numerical relationships on the board for the class.) “Look at the different ciphers that have
been placed on the board. How are the
graphs and the functions related? Does each graph and its corresponding function convey the
same message?” “Graphs,
functions, and tables are different ways of conveying the same information.” |
Students
should be able to respond that they can represent codes using tables,
pictures, and equations. They
should be able to state that tables contain the original letters and/or their
number values, coded values, and the coded letter forms. The
picture or graph should show the same relationships as above, but as the
graph of a line. A
function simply states the mathematical relationship between a letter’s
numerical representation and its coded numerical form. Students
should understand that the function shows the relationship between the
numerical values of the original letters and their coded forms. Students
should be able to say that the original value and the coded value are ‘c’
values apart, and that holds true for all the letters in their ciphers. |
“Which
are the three ways that we have learned to code messages?” “What
does the table contain?” “What
would a graph look like?” “Is
a function an efficient way to code messages?
Why or why not?” “What
are the two sets of values that your functions relate?” “What
relationships were you able to come up with?” |
EXTENSION
|
|
STUDENTS
SAY |
GUIDING
QUESTIONS |
|
“The
shift cipher you have just learned is in the form y = x + c. Let’s take out our graphing calculators and
see what your graphs would look like if your function were to look like y =
ax + c, where ‘a’ is any old number.” |
Students
should know that a constant is a number who’s value does not change and a
variable is a symbol that can stand in for any number or for an unknown
number. Students
should realize that the graphs change steepness and/or get shifted
vertically. |
“What
is it called when a letter or symbol can stand in for a number and where its
value won’t change?” “What
is it called when a letter or symbol stands in for an unknown number or
quantity?” “What
do your graphs look like? Change the
‘a’ and the ‘c’ values around. What
happens to the graph?” |
EVALUATION
Ask students the following questions: