ILLUSIVE CODES
AUTHOR: Jamie Sloat
SUBJECT: Algebra
LESSON LENGTH: 3-4 50 minute class periods
SOURCE: Mathematics: Modeling our World. Annotated Teacher’s Edition; Course 1. © 1998
KEY CONCEPT(S): Equivalent expressions; Solving Equations
LESSON OVERVIEW:
The purpose of this lesson is to modify the two-step process and examine a multi-step coding process to see if it makes messages harder to crack. Also, to use knowledge of equivalent expressions to analyze number tricks and multi-step coding processes.
OBJECTIVES: The student will be able to:
· Recognize that a multi-step coding process can be modeled and decoded using a two-step process.
· Use symbolic means to show the equivalence of two expressions.
· Apply their knowledge of solving equations to decoding messages
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
Suppose you apply the tricks of the magician to secret codes. Some magicians use number tricks. One goes like this:
· Pick a number(share it with the class but don’t let the teacher know what it is)
· Multiply by 3
· Subtract 1
· Multiply by 2
· Add 3
What did you get? The magician can quickly tell you the original number, but how does he do it? Is it really a trick? Does the magician really reverse all those steps in his head?
The magician’s number trick is similar to a multistep coding process. If you increase the number of steps in the coding process, will it be more difficult to crack? Will the increase in the number of steps eliminate the clues that made the two-step process less effective?
You tried the shift cipher last week, and it passed two of the three criteria for an effective code. You modified the shift cipher, and the resulting two-step process also passed two of the three criteria. Modify the model again. Increase the number of steps, and determine if a multistep coding process is effective. Is it possible that the illusive tricks of the magician may lead to elusive codes for the code cracker?
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STUDENTS:
Students will work in groups of 2 or 3 on the activity entitled “Number Tricks”. 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 show that multistep linear processes reduce to at most two-step processes.
TEACHER:
The teacher will do rounds throughout the classroom and check the progress of each group, providing guidance and correcting misunderstandings when they occur.
Questions:
Is a multistep coding process difficult to crack?
What makes it more difficult?
Is there a way to simplify the problem?
Why do we want a coding process that is harder to decode?
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Have group 1 present their solution to the first problem in the activity, group 2 present their solution to the second problem, etc up to group 4 presenting their solution of problem four.. Have the class discuss the following question:
Is a multistep coding process effective? Why or why not?
EXPLORATION: Day 2&3
STUDENTS:
Have students work in their same groups as the previous day on the Activity entitled “Pick a Number”. In this activity students use symbolic means to show the equivalence of two expressions. They prepare a short report on various methods of showing equivalence. Groups who finish this exercise early may continue on to work on the supplemental activity entitled “Coding Surprises”.
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:
What are useful methods for showing equivalence?
Make sure that students understand that symbolic methods using the distributive property, manipulatives, tables, and graphs are all good methods, but that arrow diagrams are not useful for demonstrating equivalence.
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Begin with any groups that didn’t present previously and select groups to present different ways to show that a multistep process reduces to just two steps. Have different groups share their solution to number 3 on “Pick a Number” and number 4 on the supplemental activity “Coding surprises”.
Have the class discuss the methods for showing equivalence and when certain methods are more useful than others.
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Continuous evaluation should be happening while talking with the groups during the activity and during the class discussions of solutions.
On Day 1 pass out the individual work entitled “Number Trickery” as homework due the following day. In this assignment students examine multistep processes using arrow diagrams, graphs, tables, and equations.
On Day 2 pass out the individual assignment activity entitled “The Magic of Algebra” to be turned in on the day after the last day of the completion of this entire lesson. In this assignment students work with multistep processes and find their equivalent two-step processes.
On the last day of the lesson, give students the short assessment quiz entitled “Multi-Stepping”.
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In groups of 3, have the students develop their own number tricks and ways to reverse them. Challenge them to develop a multi-step process with 10 steps and to then find the two-step process that models it for decoding.
SAFETY: Be respectful of calculators and of your classmates.