REVERSING
SHIFT CIPHERS
Author: Tom Abraham
Source: Mathematics: Modeling Our World, Course 1
Date: Day 8
Class: 9th Grade Algebra
Duration: 1 hour
Goals: Students will be able to:
· Decode messages using a simple shift cipher.
· Given a coded message, be able to represent the cipher as a function and a graph.
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.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
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STUDENTS
SAY |
GUIDING
QUESTIONS |
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Begin by speaking Pig
Latin to the students. Ask them about their
day, how classes are going, how they like school, did they have fun at the
pep rally, the football game, etc.
Most students should already be familiar with Pig Latin. Ask them to respond similarly. “How
were you able to understand what I was saying? What is Pig Latin? How do you translate back and forth from
English to Pig Latin?” “The
reason you were able to understand me is because you knew the process
involved in going form English to Pig Latin.
Describe that process in as much detail as you can.” “This
process is similar to being able to decode a message. Coders and decoders go back and forth when
coding and encoding a message. They
need to be able to understand each other, and so, both people know how the
message was coded in order to be able to decode it.” |
Students
should explain that Pig Latin is when the first letter of a word is moved to
the end of the word and the sound “ay” is added to it. If the first letter of the word is a vowel,
the vowel stays at the front of the word, but the “ay” sound is still added
to the end of the word. (ex: “longhorn” becomes “onghornlay”
and “eight” becomes “eightay”) |
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EXPLORATION
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STUDENTS
SAY |
GUIDING
QUESTIONS |
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Someone come up to the
board and draw in graphical form, what a shift +2 cipher
looks like. What does it look like as
a function? “Split up into groups of
two. I want each person to come up
with a unique shift cipher and a coded message to decode. Make sure that your cipher is expressed as
a function, a graph, and a table. Once
each person has come up with their own ciphers, write a coded message using
that cipher.” “Exchange ciphers and
messages and decode your partner’s message.
You will need either their function, graph, or table in order to do
it.” |
Students should explain
that a shift cipher is one a letter or its numerical equivalent is replaced
by another letter or number a few places down from it. A coded value of 0 or -2 is
when the shift value subtracts passed ‘A’ or the value ‘1.’ Any value less than 1 or greater than 26
loops back around. Hence, a value of 0
or -2 is actually a value of 26 and 24.
A shift +30 cipher will
actually just shift a letter 4 places to its right (30 - 26 = 4). Students should say that a
function expresses a relationship between two sets of values. A variable is a symbol that
can take the place of unknown values or numbers whose values can change. A constant is a number
whose value does not change. |
What is a shift cipher? What happens if your coded
numerical value ends up being 0 or -2?
What about a value of 28? What would happen if you
had a shift +30 cipher? What is a function? What is a variable? What is a constant? |
EXPLANATION
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STUDENTS
SAY |
GUIDING
QUESTIONS |
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Have students put example
ciphers and coded messages on the board for the class to decipher. “What
is the process that you used to go from the coded form to the original form?” “Reversing
the encoding process is called decoding.
In math, when ever you reverse a process (or a function) you perform
an inverse action.” |
Students
should say that they took either coded numerical value, added or subtracted
the cipher value, and then found the original letter from that value. By
subtracting, adding, dividing, and multiplying. |
How
would you reverse addition, subtraction, multiplication, and division,
respectively? |
EXTENSION
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STUDENTS
SAY |
GUIDING QUESTIONS |
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“Come
up with a function that reverses the function y = x + 12.” |
Students
should write something in the form of y = x – 12 or x = y – 12. |
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EVALUATION
Ask students the following questions: