Matrix Code Mathematics

Length of lesson: 2 Days

Description of the class:

                     Name of course: Algebra

Source of the lesson:

            Mathematics: Modeling Our World

            Unit Two: Secret Codes

            Lesson Seven

TEKS

(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.

I. Overview

            This lesson gives a way to use matrices to encode messages. It begins with the students recalling what a matrix is and what operations you can perform on them. Then they will explore using matrices to encode messages and practice decoding them. Then the students will explain how they used the matrices for encryption. Finally there will be a class discussion about what the benefits and shortcomings of using matrices for encryption are.

II. Performance or learner outcomes

            Students will be able to perform simple operations on matrices, such as adding two matrices, subtracting two matrices, and multiplying a matrix by a constant.

            Students will be able to convert a message into a matrix, apply matrix operations to the matrix to get an encoded matrix. They will also be able to apply matrix operations to decode the matrix and then convert it into the original message.

III. Resources, materials and supplies needed

            Graphics calculators

IV. Supplementary materials, handouts. (Also address any safety issues

      Concerning equipment used)

            Handouts A2.18 and S2.15 from Mathematics: Modeling Our World.

Five-E Organization

Teacher Does                     Probing Questions                      Student Does       

Engage: The teacher makes sure each student has a graphics calculator. Then asks the students questions to determine what they know about matrices. The teacher should determine if the students are familiar enough with matrices to go ahead with the lesson on using codes or if more time should be spent developing their matrix skills. The teacher should then introduce the topic of encoding and see if the students can come up with a way to convert a message into a matrix.

What is a matrix? What are the elements? What is the size? How do you define a matrix in the calculator? How do you add two matrices? What must be true to add the matrices? How do you subtract two matrices? What must be true? How do you use the calculator to add and subtract two matrices? How do you multiply a matrix by a constant? How do you convert a message into a matrix? How can a cipher be done using matrix operations?

The students should give answers to the questions developing both their knowledge of matrices and also practice on using matrices on calculators. The students should also come up with a way to convert messages into matrices and then encode them. The students then work on the A2.18 activity on keyword coding.

Explore: The students then will be given some practice in making and using matrix codes by being split into groups of two and working on the S2.15 handout. Then each pair will be asked to come up with a secret message of their own and encode it using a matrix code.

How do you build the matrix from a keyword? What operations are being used to encode the matrix? What do you need to know to decode it? Why does the decoding work?

The students work on the handout in pairs and develop their skills with matrix coding.

Explain: Students will go up to the board and describe how they did the problems on the handouts and the class will discuss any misunderstandings. Then each pair will describe how they decoded the message from the other group.

How did you solve the problem? What why did you take [insert step here]? How did you know you should to do it? What if the problem was [a modification]? What would you have had to do?

The students should give their explanations of the problems in the handout. The other students should listen attentively and discuss any problems they encountered while solving the problems.

Extend / Elaborate: The class will discuss what was easy and what they liked about using matrix codes. They will discuss whether they think matrix codes are more secure and what makes them so. Then they will discuss ways of breaking matrix codes.

What was easy about using matrix codes? Why did using matrices make this type of code easier? Are matrices useful? Do the definitions of the matrix operations make sense? What was difficult about using matrix codes? How does the security of matrix codes compare other ciphers? Are they easier or more difficult to break? Why are they more secure? How can someone break a matrix code?

The students should discuss the benefits and drawbacks of using matrix codes. This part of the lesson should solidify in the students’ minds why using matrix codes is better than a simple cipher and also allow them to more fully understand the abstract concept of a matrix. Students should spend a fair amount of time posing different methods of breaking matrix ciphers. It may be valuable to see if their methods work with examples or have other students critique their idea.

Evaluate:

Each student will hand in their A2.18 and S2.15 worksheets. Each pair will also hand in the message they encoded and the message they decoded. This in addition to the answers given to questions in the other phases will be used to assess the students.