Lesson Plan – Stretching a Shift Cipher

Name: Connie Aphonephanh

Title of the Lesson: Stretching and Shifting

Date: Day 17

Length of Lesson: 50 minutes

Description of the class: High school level students

Name of the Course: Algebra I

Grade Level: 9-10

Honors or Regular: Both

TEKS addressed:

(b) Foundations for functions: knowledge and skills and
 performance descriptions.
 
(4) 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.

 (A) The student finds specific function values,
  simplifies polynomial expressions, transforms
 and solves equations, and factors as necessary
 in problem situations.
 
 (B) The student uses the commutative, associative,
 and distributive properties to simplify
 algebraic expressions.

I. Overview

Using their knowledge about shift ciphers from previous lessons, students will modify the shift cipher by adding an additional step – a stretch. The students will compare previous coding process and examine whether adding an additional step meets the criteria for an effective coding process.

II. Performance or learner outcomes

Students will be able to:

(1) use their knowledge of shift cipher to determine which coding process

is more effective.

III. Resources, materials and supplies needed

IV. Supplementary materials, handouts

Five – E Organization

Teacher Does Probing Questions Student Does

Engage: (10 minutes) The teacher will ask 12 students to play the roles of the spreadsheet cells and have them form a 4 X 3 array. Label the left column of students “A,” the second column “B,” and the third column “C.” Ask students A1, the first student in column A, to select a number between 1 and 10. Give student A2 a card with a rule written on it. (e.g. “Add 16 to the number in the cell above you.” Then A2 passes the rule card to student A3, who applies the rule to the results announced by A2. Then A3 passes the rule card to student A4, who applies the rule to the results announced by A3. The teacher will ask the class to try and guess both the rule and the original secret number. Next, the teacher will repeat the same process with students in column B and C, but the rules will change. The students in column C will combine the results of column A and column B (e.g. “Multiply the numbers in columns A and B corresponding to your row number.”)

“Who would like to volunteer for a game we will play? I only need 12 volunteers.” “Student A1 (first student in column A), can you pick a number between 1 and 10? Don’t tell anyone that number yet.” “Student A1, will you whisper your number to Student A2?” “Student A2, what is the answer after you applied the rule? Just give me the answer and not how you got it.” “Student A3, what is the answer after you applied the rule? Remember just give me the answer and not how you got it.” “Student A4, what is the answer after you applied the rule? Remember just give me the answer and not how you got it.” “Can someone guess what the rule and the secret number is?”

Students will volunteer. Student A1 will pick a number between 1 and 10. Student A1 will whisper his/her number to Student A2. Student A2 will apply the rule to the number and announce the result to the class. Student A3 will apply the rule to the number and announce the result to the class. Student A4 will apply the rule to the number and announce the result to the class. A student will volunteer a rule and an answer. Students in column A will verify whether it is right or not.

Teacher Does Probing Questions Student Does

Explore: (20 minutes) The teacher will introduce to the students a modified version of the shift cipher that involves a combination code through examples. This whole process is known as a two-step process. The students already know how to shift left and right from previous lessons. The teacher will review ciphering with the students again. The teacher will ask the students what they think of the word “stretch.” The teacher will extend the definition of stretch to numbers and math. If the students respond with “add 4,” then the teacher will ask the students to recall that adding 4 move right 4. The teacher will ask the students for other suggestions. The teacher will have the students apply a stretch of 2 to the phrase, “Texas Longhorn.” Now, the teacher will have the students shift the phrase by +3. The teacher will have the students represent the last two mathematical operations as an arrow diagram. The teacher will go around and check if students are able to draw an arrow diagram. Now, the teacher will have the students try decoding a coded message to see if two-step process is difficult to crack.

“Code the phrase,’ Texas Longhorn’? What did you get?” “When you think of stretching, what comes to your mind?” “If stretch means to make bigger/larger, then what does stretch do to a number?” “How would you stretch a number, x, by 4?” “Remember that if you add 4, what happens to a shift cipher?” “Then, how would you make a number bigger besides adding 4?” “Apply a stretch of 2 to the phrase “Texas Longhorn,” what code do you get?” “Apply a shift +3 to the phrase “Texas Longhorn,” what code do you get?” “Represent the stretch of 2 and the shift of 3 as one arrow diagram.” Let’s work backwards and try decoding this code, 11 21 6 101 76 106 What is the code? What is the two-step process, what are the stretch and the shift? What is the equation that represents the code?

The students should answer with the following: 20 5 24 1 19 12 15 14 7 8 12 18 14 19 The students will respond with “to make something bigger or to expand something.” Again, the students should respond “to make bigger/larger.” The students will have a natural desire to answer “add 4.” The students should answer “move right 4.” The students should answer “multiply by 4.” The students should answer 40 10 48 2 38 24 30 28 14 16 24 36 28 38 The students should answer 43 13 51 5 36 27 33 31 17 19 27 39 31 41 The students should have this as an arrow diagram: Original Position à Multiply by 2àAdd 3 à Coded Value The students should answer “Beat OU.” Stretch:5 Shift: 1 c = 5p + 1

Teacher Does Probing Questions Student Does

Explain: (10 minutes) The teacher will review the two-step process with the students. If the students are not able to answer, the teacher will explain to the students a modified version of the shift cipher that involves a combination code. This whole process is known as a two-step process. A combination code modifies the shift cipher by combining two mathematical operations. These two operations combine a stretch and a shift in either order. The teacher will review the two-step process with the students. The teacher will discuss with the students if the two-step process is an effective coding process.

What do you think a two-step process? What does it mean to stretch? What does it mean to shift? Is it harder to code with the two-step process? Is it harder to decode with the two-step process? Is the two-step process better than the shift cipher, in terms of coding and decoding? If so, why?

The students should answer that a two-step process is a combination code that combines to mathematical operations: a stretch and a shift. The students should answer that to stretch is to make the number bigger by multiplying. The students should answer that to shift is to move the number either left or right. Answers will vary. The students should answer that it is not as simple as moving from the left or right. The process is a little more complex. The students should answer, “Yes, it is harder to decode because you have to figure out a pattern in order to decode the code.” The students should answer that “shift cipher is easier to code and decode, but the two-step process takes more math to code and decode is harder.” “Because it’s harder, the two-step process is a more effective way of coding.”

Teacher Does Probing Questions Student Does

Extend/Elaborate: (10 minutes) The teacher will extend the topic of coding by giving a brief introduction to the importance of order of operations. The teacher will have the students perform an example to see a glimpse of two different methods. The teacher will extend/elaborate on the topic of order of operations in depth in the next lesson.

“Does it matter if you shift first and the stretch or do you have to stretch first and then shift?” Consider two methods: Method 1: Stretches 2, then shifts +1. Method 2: Shifts +1 first, and then stretches 2. Does changing the order produce a different coding process? Use the phrase “Texas Longhorns”

The students should want to answer that “it doesn’t matter” “What are the differences between the codes?” Does it matter if you shift first and stretch second?