Decoding
AUTHOR: Jamie Sloat
SUBJECT: Algebra
LESSON LENGTH: 4-5 50 minute class periods
SOURCE: Mathematics: Modeling our World. Annotated Teacher’s Edition; Course 1. © 1998
KEY CONCEPT(S): Inverse of a Function, Coding Methods: stretch cipher, piecewise cipher, Families of Functions, Solving Equations, Modular Arithmetic
LESSON OVERVIEW:
The purpose of this lesson is to decide whether a shift cipher is easy to decode and to explore the mathematical concept of inverse by doing reverse of coding (decoding). This lesson will use multiple representations including the graphing calculator, to solve simple equations while decoding secret messages encoded with a shift cipher and will introduce the mathematics of modular arithmetic.
OBJECTIVES: The student will be able to:
· Use multiple representations to solve simple equations
· Perform basic modular arithmetic
· Solve simple linear equations for an unknown
· Solve for the inverse of a linear equation
· Recognize the relationship between a function and it’s inverse
· Use a graphing calculator to represent and solve equations
· Determine whether or not a shift cipher is easy to decode
TEKS:
111.32 Algebra I
(A) Basic understandings:
(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities.
(3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships.
(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations.
(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.
(B) Foundations for functions: knowledge and skills and performance descriptions.
(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.
(E) The student interprets and makes inferences from functional relationships.
(2) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.
(3) The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. Following are performance descriptions.
(A) The student uses symbols to represent unknowns and variables.
(B) Given situations, the student looks for patterns and represents generalizations algebraically.
MATERIALS LIST:
ENGAGEMENT
TEACHER:
What is the reverse of each of the following processes?
What operation reverses each of the following mathematical operations?
Sometimes you need to find half of a number or 25% of a number. What is the reverse of finding half of a number? What is the reverse of finding 25% of a number?
STUDENTS:
Should be participating and answering the above questions with appropriate responses (i.e. unwrapping a package, draining the pools water, etc.)
Students might find the first questions elementary, but might get stumped for a second when asked what the reverse of finding half of a number or 25% of a number is. Make sure to give them time to think about it before moving onto the exploration.
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STUDENTS:
Students will work in groups of 2 or 3 on the activity entitled “Codes in Reverse”. Each group will be assigned a group number in order to easily keep track of groups during the explanation process. The purpose of this activity is to determine if messages coded with a shift cipher are easy to decode. Students determine whether the representation affects the difficulty of the decoding process.
Students should be working productively in their groups. Since this is the introduction activity to the lesson, students should not have too many difficulties with the activity.
TEACHER:
The teacher will do rounds throughout the classroom and check the progress of each group. Possible questions for the groups might include:
Do tables and graphs provide every solution pair for an equation or situation?
How can you find every solution?
Which way is easier: tables or graphs? Why?
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Have group 1 present their solution to decoding “Message 1”, group 2 present their solution to “Message 2”, etc up to group 4 presenting their solution of “Message 4”. Have the class discuss the following questions:
EXPLORATION: Day 2
STUDENTS:
Have students work in their same groups as the previous day on the Activity entitled “Symbolic Decoding”. The purpose of this activity is to introduce equation solving. Groups who finish this exercise early may continue on to work on the supplemental activity entitled “Decoder Skills”.
TEACHER:
While students are working on this activity in their groups, the teacher should walk around and evaluate each group’s progress, addressing misconceptions when needed. Ask the students:
How is solving an equation like decoding with a table?
How is solving an equation like decoding with a graph?
How is solving an equation like decoding with an arrow diagram?
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Begin with any groups that didn’t present previously and have each group present their solutions to numbers 3, 4, and 5 from the Activity.
Ask the class to discuss what would happen if constants were changed or added to the equations.
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STUDENTS:
Rearrange the class groups so that students will have a chance to work with different classmates. Label the groups 1, 2, 3, etc so as to keep track of them during the explanation process. Have the groups work on the activity entitled “Coding from Alphabet to Alphabet”. The purpose of this activity is to formalize the process of coding from alphabet to alphabet. Students are introduced to modular arithmetic through addition and subtraction modulo 26.
Students might have trouble making the transition from “normal” arithmetic to modular arithmetic. If needed, give the students more examples (i.e. what is 3mod8, 2mod8, 5mod8, etc.)
TEACHER:
As the students are working, the teacher should check the progress of each group making sure they are on task and correcting any misconceptions and providing guidance along the way.
Make sure students understand that you can work modulo n, where n is any number.
If students have trouble with modular arithmetic it might be beneficial to talk about it as “clock” arithmetic.
Ask each group to pick 2-3 problems from the activity that they would like to go over as a class and to hand it in before leaving.
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From the problems the students picked to go over as a class, start with Group 1 and have each group go up and work on a problem. If the group didn’t get a solution to the problem have the class discuss the difficulties that occurred and how you might get around those difficulties.
It is expected that students might have trouble transitioning from positive values mod n to negative values. Give extra attention to these examples.
Bring attention to the term “congruent” modulo 26. In what other ways do mathematicians use the word “congruent”? What do you think “congruent” means (equal in measure)?
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Continuous evaluation should be happening while talking with the groups during the activity and during the class discussions of solutions.
Pass out the worksheet entitled “Reverse That Process” as homework for Day 1. This individual work reviews the idea that decoding is the reverse of coding, while allowing students to practice using the various tools developed thus far.
The worksheet entitled “Step by Step” will be homework for Day 2. This individual work reviews encoding and decoding.
For the last two days of the lesson, students will work on the assessment problem entitled “Shifting Rings” individually to be handed in the day after the last day of the lesson.
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Give each group 5 minutes to develop their own cipher and write an encrypted message to another group in the class. Exchange messages among the groups and have a competition to see who can decode their message the fastest.
What methods make decoding the hardest? Do we want harder or easier decoding methods? What kind of situation might you be in when you are in a real race for time to decode a message?
You can then have the students research the use of cryptography in World War II.
SAFETY: Be respectful of calculators and of your classmates.