Joshua Newton

Harder Codes

Objectives:

Students will be able to:

            1. Learn how to modify a shift cipher by using a stretch.

            2. Discuss the distributive property and why the order of operations is important.

TEKS:

Algebra I

(c) Linear functions: knowledge and skills and performance descriptions.

(1) The student understands that linear functions can be represented in different ways and translates among their various representations. Following are performance descriptions.

(A) The student determines whether or not given situations can be represented by linear functions.

(B) The student determines the domain and range values for which linear functions make sense for given situations.

(C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions.

(2) The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. Following are performance descriptions.

 (C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b.

 (F) The student interprets and predicts the effects of changing slope and y-intercept in applied situations.

Algebra II

(c) Algebra and geometry: knowledge and skills and performance descriptions.

(1) The student connects algebraic and geometric representations of functions. Following are performance descriptions.

(B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions.

(C) The student recognizes inverse relationships between various functions.

Equipment List:

                  Graphing Calculators

Five-E Organization

Teacher Does                                               Student Does

Engage: Review what a shift cipher is and why it is an easy cipher to break. Ask students if they can think of ways to modify it that would make it more difficult to crack. Remind students that the operations performed need to be simple and that it must be easy to decode. Introduce students to the idea of a stretch, or multiplication and explain how it can be used to modify the shift cipher. Give students a simple message to encode and decode and ask them how easy a stretch is to use.

Students will discuss various ways that the shift cipher could be modified.

Explore: Review what the algebraic equation for a shift cipher is and ask students what a stretch would look like in an equation. Is order of operations important when encoding and decoding a message when using a stretch and shift? Have students use a graphing calculator to determine what the effect of a stretch is on the graph of an equation. What does a stretch of 3 do to the equation? What changes? What effect does changing the shift have on the graph? From looking at a graph of a cipher, how can you tell if a stretch was used? Have students do the Individual Work 8 activity in unit two of Mathematics: Modeling Our World.

It would take the form of a coefficient. Yes, if you do not use the correct order you may get the wrong values. A stretch of 3 changes the slope of the graph from 1 to 3. It changes the y-intercept of the graph, It will have a slope different than one.

Explain: Why does order of operations matter when you are encoding or decoding a message? Do you use the same equation for encoding and decoding? Have students do the Individual Work 9 activity in unit two of Mathematics: Modeling Our World.

If you do not use the right order, the message will not be coded correctly. You will have different equations for encoding and decoding because to decode a message you need to do the reverse what you did to encode it.

Source:

Mathematics: Modeling Our World