Cryptography and Using Matrices By Sanjar Hekmati CURRICULUM
Cryptography and Using Matrices By Sanjar Hekmati
CURRICULUM STANDARDS AND BENCHMARKS • Standard: Students will understand algebraic concepts and applications • 9 -12 Benchmark. A. 1: Represent and analyze mathematical situations and structures using algebraic symbols. • 9 -12. A. 1. 1 Use the special symbols of mathematics correctly and precisely. • 9 -12 Benchmark A. 3: Use mathematical models to represent and understand quantitative relationships. • Standard 9: Understand how mathematical ideal build on one another • Standard 10: Use representations to solve problems • 9 -12 Benchmark D. 1: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
OBJECTIVES • • • To identify matrices and their elements To organize data into matrices To solve certain matrix equations To multiply two matrices To use inverse matrices in solving matrix equations • To use real-world connection
MATERIALS AND RESOURCES • A graphing calculator a. TI-83, TI-84 series • Game set 1, 2, and 3 • Video on Cryptography • Video on matrix multiplication • Internet access (or any graphing calculator -homework)
PREREQUISITES • • Algebra 1 Knowledge of using a graphing calculator Be able to scramble letters to make words Be able to use logic, strategy, work well with a group
TASK Essential questions: a. What is Cryptography? b. Writing activity- Construct a “Frayer Model” on Cryptography c. What is matrix multiplication d. Solve varies matrix multiplications • The students (groups of 3 to 4) will start by solving simple system of equations to obtain a information on which game set (1, 2, 3) to try first • The students will use the encoding matrix to decode another matrix • The students will use the new matrix to obtain numbers that correspond to certain letters from a table (hints will be given) • The students will scramble(decode) the letters to make out certain words and phrases
The Matrix
PROCEDURES A = 26, 0 = INDICATES A BREAK N = 13 B = 25 0 = 12 C = 24 P = 11 D = 23 Q = 10 E = 22 R = 9 F = 21 S = 8 G = 20 T = 7 H = 19 U = 6 I = 18 V = 5 J = 17 W = 4 K = 16 X = 3 L = 15 Y = 2 M = 14 Z = 1
Example: •
Sample from game set 2 (game image) # 4. Hint – Part of a parabola 6. 5 16. 5 20. 5 23. 5 15 1 Matrix C = 1. 75 2. 25 1. 25 4. 75 -3. 5 2. 5 4. 5 -2 -3. 5 -7 0 8 21 14 7 8 0 S F M T S o = 2 0 9 26 0 2 = Y o R A o Y= AXIS OF SYMMETRY 3 12 18 14 22 0 X O I M E o
ASSESSMENT • The game is designed for students to learn through problem based learning, the teachers role is to proctor the groups as their solving the puzzles • In order for fairness, the teacher will not help the groups with errors and give extra help • Individual assessment will take place at home or outside of the classroom where students are required to visit the teacher website to play the game and turned in a hard copy of the puzzles they finished.
Free online graphing calculator
Everyday cryptography
Implications: • Proficiency with the use of graphing calculators raises test scores • Students are allowed to use graphing calculators on SBA • When graphing calculators are effectively used in the mathematics classroom, they are a powerful tool to assist teachers in providing their students with an environment to help them construct their mathematical knowledge and understanding • A graphing calculator is one of the most portable and affordable(only on ebay) technology in mathematics education • A graphing calculator quickens the mechanical procedure in solving mathematical problems and creates a highly interactive learning environment, which makes learning a seemingly difficult subject, easy • The game of cryptography is something new to students and interesting enough to keep them engaged while learning math.
SWBAT, CONTENT AREA AND CONNECTION • • PLAY THE GAME USING MATRIX MULTIPLICATION PLAY THE GAME USING MATRIX INVERSES PLAY THE GAME USING LOGIC, PLAY THE GAME USING HINTS TO CONNECT TO MATH CONCEPTS • PLAY THE GAME CONNECTING SOLUTIONS TO REAL-WORLD SITUATIONS • MAKE THE CONNECTION BETWEEN THE GAME AND THE MATH CONCEPT
REFERENCES • Secondary school teachers’ conceptions and their teaching practices using graphing calculators, Jane A. Lee and Douglas E. Mc. Dougall, International Journal of Mathematical Education in Science and Technology, 2010. • Effects of the application of graphing calculator on students’ probability achievement, Choo-Kim Tran, Computers and Education, May of 2012 • Instructional efficiency of the integration of graphing calculators in teaching and learning mathematics, Wan Zah Wan Ali, International Journal of Instruction, July 2009. • Fostering positive attitude in probability learning using graphing calculator, Lau Siong-Hoe and Madhubula Bava, November 2011
ONLINE RESOURCES Game on line • http: //hekmati. bhs. beleneagles. org Free graphing calculators online • http: //my. hrw. com/math 06_07/nsmedia/tools/Graph_Calculator/graph Calc. html • http: //www. calculator-grapher. com/calculators/matrix-calculator. html • http: //www. numberempire. com/matrixcalculator. php • Help on matrix multiplication • http: //www. youtube. com/watch? v=_f. Fj 4 Nb. Lc. TU • http: //www. prenhall. com/divisions/esm/app/calc_v 2/ • http: //www. prenhall. com/divisions/esm/app/graphing/ti 83/Home_Scre en/Menu_Keys/Matrix/matrix. html
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