Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY THE GOLD BUG S2.1

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1 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY THE GOLD BUG S2.1 In The Gold Bug by Edgar Allan Poe, the character William Legrand stumbles across what appears to be a coded message written by the famous Captain Kidd. Legrand wonders if the message will lead him to the fabled buried treasure that has never been found. Decipher the message and discover the clues. "53 305))6*;4826)4.)4 ); 806*; ))85; 1 (;: *8 83(88)5* ; 46(;88*96*?;8)* (;485); 5* 2:* (;4956*2(5* 4)8 8*; );)6 8)4 ; 1( 9;48081;8:8 1; 48 85;4) *81( 9;48; \(88;4(?34;48)4 ;161; :188;?;" 265

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3 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY TOOL TIME S2.3 Your car refuses to start. You call a mechanic who shows up with just a hammer. Annoyed, you ask the mechanic, Why did you bring only a hammer and no other tools? Without hesitation the mechanic replies, A hammer is useful for fixing many problems. I know how to use the hammer, and I use it well. I don t know how to use the other tools. page 1 of 4 Would you let this mechanic work on your car? A good mechanic acquires proper tools and knows how to use the tools to fix your car. A professional baseball player works hard to develop the tools of throwing, batting, and fielding a baseball. Artists become skilled at using pencil, watercolor paint, oil-based paints, ink, or chalk to create images. The intern who hopes to become a skilled surgeon practices for hours and hours to perfect the ability to use surgical tools. Throughout Mathematics: Modeling Our World, you will use mathematical tools to solve real-life problems. In this activity, you review several important mathematical tools: words, tables, arrow diagrams, and graphs. You develop these tools while you create secret messages. You will practice using them in other lessons, with codes and with other real-life situations. WORDS One way to represent a coding process is to describe it in words: Replace each letter of the alphabet with a letter three positions to the right of the letter. 267

4 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.3 page 2 of 4 TOOL TIME TABLES A table matches each letter to a new letter. 1. a) Complete the table in Figure 1. The third entry reveals the pattern to follow. Figure 1. A coding table. Plaintext letter A B C D E F G H I J K L M Coded letter G Plaintext letter N O P Q R S T U V W X Y Z Coded letter b) How do you code the letter X? c) Use the table to encode the message Secret codes are fun. So far you have coded letters with other letters. The Zimmermann telegram is a historical example of a secret message coded with numbers (Figure 2). The Zimmermann telegram dates from 1916 and contains an offer to Mexico from German foreign minister Arthur Zimmermann. If Mexico would side with Germany against the United States, Germany would award Texas, New Mexico, and Arizona to Mexico if Germany won the war. The message was intercepted and cracked by the British, who turned it over to the United States. Suppose you match the 26 letters of the alphabet with the numbers 1 through 26 (Figure 3). Figure 2. The Zimmermann telegram. The number 1 is the position number for the letter A. The number 19 is the position number for the letter S. The table represents a cipher that matches a letter to a position number. There is no shift in the position. Figure 3. The position numbers for the letters of the alphabet. 268 Letter A B C D E F G H I J K L M Position number Letter N O P Q R S T U V W X Y Z Position number

5 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY TOOL TIME S a) Use the table in Figure 4 to represent a cipher that shifts each number +6. page 3 of 4 Plaintext letter A B C D E F G H I J K L M Coded value Figure 4. A shift of +6. Plaintext letter N O P Q R S T U V W X Y Z Coded value b) The word ace becomes Now use the shifted table to encode the message I like mathematics. ARROW DIAGRAMS Arrow diagrams may be used to represent the process of encoding a message. The arrow diagram in Figure 5 describes two steps: a change from a letter to a position number, followed by a shift of a) Use the coding process represented by the diagram in Figure 5 to encode the word arrow. b) Prepare an arrow diagram to represent a coding process that converts letters to numbers and shifts each position +8 units. Plaintext letter Assign position number Add 3 Coded value Figure 5. A coding process represented in an arrow diagram. 269

6 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.3 page 4 of 4 Coded value GRAPHS TOOL TIME A graph is a visual tool used to match a letter or position to a new coded letter or coded value A B C DE FG HI JKLMNOPQR STUVWXY Z Original position Figure 6. A coding graph. 4. a) The graph in Figure 6 represents the shift cipher of +2. Refer to the graph and code the word voting as numbers. b) Draw a shift +4 cipher on the graph in Figure 6. c) Use your graph to encode the message secret codes. 5. a) If you were asked to encode a message, which representation would you use: words, arrow diagram, table, or graph? Explain. b) If you were asked to communicate your shift-coding process to another person, which representation would you use: words, arrow diagram, table, or graph? Explain. 270

7 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY DECODER SKILLS S2.5 1.Suppose you use a coding process described by the arrow diagram in Figure 1. a) Encode the letter P. b) Decode the number 18. c) Add the decoding process to the arrow diagram above. Plaintext letter Assign position number Add 6 page 1 of 2 Coded value Figure 1. Arrow diagram for a coding process. 2. a) Complete the table in Figure 2. Plaintext letter A B C D E F G H I J K L M Coded value Plaintext letter N O P Q R S T U V W X Y Z Figure 2. A coding table. Coded value b) Encode the letter L. c) Decode the number 28. d) Describe how you decoded the number 28. e) Identify the domain and the range for this coding process. 271

8 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.5 page 2 of 2 3.Suppose you used a coding process described by the word equation Coded value = Original position + 12 (or c = p + 12). a) Encode the letter Q. DECODER SKILLS b) Decode the number 16. c) What equation are you solving when you decode 16? Coded value A B C DE FG HI JKLMNOPQR STUVWXY Z Original position 4. Suppose you used a coding process described by the partial graph in Figure 3. a) Encode the letter H. b) Decode the number 27. c) Describe how you decoded the number 27. Figure 3. Part of a coding process. d) The point (9, 25) is a point on the graph. Describe the meaning of the point in the coding context. 272

9 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY TESTING 1, 2, 3 S2.9 The purpose of this activity is to determine if a combination of a shift and a stretch is an effective coding process. page 1 of 4 1. A two-step coding process must involve more than just two additions. A shift of 3 followed by a shift of 2 is the same as a single shift of 5. A twostep process combines multiplication and addition, a stretch and a shift. Plaintext letter Assign position Original position Multiply by 3 Suppose you create a coding process consisting of a stretch by a factor of 3 followed by a shift of +1 (Figure 1). The equation is c = 3p + 1 or, on a graphing calculator, Y1 = 3X + 1. Stretch value a) Use the arrow diagram or equation and complete the table in Figure 2. Add 1 Coded value Interpret position Coded letter Figure 1. A stretch of 3 followed by a shift of +1. Plaintext letter A B C D E F G H I J K L M Original position Coded value 4 Coded letter D Plaintext letter N O P Q R S T U V W X Y Z Original position Coded value Figure 2. A coding table. Coded letter b) The smallest position number is 1. The largest is 26. The domain is the numbers from 1 to 26. What are the smallest and largest coded values? What is the range for this coding process? 273

10 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.9 Coded value page 2 of 4 32 E F 31 D 30 C 29 B 28 A 27 Z 26 Y 25 X 24 W 23 V 22 U 21 T 20 S 19 R 18 Q 17 P 16 O 15 N 14 M 13 L 12 K 11 J 10 I 9 H 8 G 7 F 6 E 5 D 4 C 3 B 2 A A B C DE FG HI JKLMNOPQR STUVWXY Z Original position Figure 3. A coding-process graph. TESTING 1, 2, 3 c) Use the values from the table to build a partial graph for the coding process (Figure 3). d) How does the graph for this two-step stretch-and-shift cipher differ from the graph of a shift cipher? e) The point (1, 4) is on the graph. It means the position number 1 (for the letter A) is matched with the coded value 4. How much does the coded value change when you move from one position number to the next on the graph? f) Use the table or the graph to encode the message Cover your tracks. g) Is a stretch-and-shift coding process easy to encode? Defend your answer. 2.The arrow diagram in Figure 4 shows a two-step coding process and decoding process. The decoding process reverses the steps. Assign position number CODING Multiply by 5 Add 3 Figure 4. A coding/decoding arrow diagram. Plaintext letter Original position Divide by 5 Stretch value Subtract 3 Coded value 274 DECODING

11 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY TESTING 1, 2, 3 S2.9 Code by multiplying by 5 and then adding 3. Decode by subtracting 3 and then dividing by 5. a) Make an arrow diagram of the coding/decoding process in Item 1. page 3 of 4 b) The equation for this coding process is c = 3p + 1. Suppose the coded value is 46. Decode by solving the equation 46 = 3p + 1. Find the plaintext letter. c) Use the graph, table, arrow diagram, or equation to decode this message: d) Is a stretch-and-shift coding process easy to decode? Defend your answer. 275

12 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.9 page 4 of 4 TESTING 1, 2, 3 3.The following message is encoded with a two-step coding process: a) Statistics provide clues for the code cracker. Use a tally chart and do a frequency study with the coded values. Remember that the letters E, T, N, R, I, O, A, and S occur most often in the English language. b) If the message is coded with a function, the table and graph reveal clues for the code cracker. Build a graph or table from the results of your tally chart. Look for linear patterns in the graph or spacing patterns in the table. c) Crack the code. Decipher the message. d) Does a two-step coding process pass the tests for an effective code? 276

13 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY CIRCLES AND SQUARES S2.10 page 1 of 3 represents p, the position number for a letter. O represents the number 1. O represents p + 1 OOO represents 2p Write the coding process that is represented. a) OOOO b) OOO OOO 2. Draw squares and circles to represent the expression. a) p + 4 b) 4p c) 3p + 1 Suppose you add another step to the process: Represent p + 3 as OOO. Multiply by 2: 2(p + 3), OOO OOO. Write another way: 2p (p + 3) is equivalent to 2p + 6. In other words, a coding process that shifts 3 and then stretches 2 is equivalent to a coding process that stretches 2 and then shifts

14 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.10 page 2 of 3 3. Write an equivalent expression. a) 3(p + 1) CIRCLES AND SQUARES b) 2(p + 2) c) 4(2p + 3) 4. Change an expression without parentheses to an equivalent expression with parentheses. For example, OOOOOOOOO, or 3p + 9, can be written as three identical groups, OOO OOO OOO or as 3(p + 3). 5. Change each of the following to an equivalent expression with parentheses. a) 3p + 6 b) 2p + 8 c) 6p Circles and squares can be used to determine if two coding processes are equivalent. a) Write an equation to represent the coding process: multiply by 2, then add 5. b) Represent the process with circles and squares. c) Write the equation to represent the coding process: add 5, then multiply by 2. d) Represent the process with circles and squares. e) Are the coding processes in (a) and (c) equivalent? 278

15 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY CIRCLES AND SQUARES S2.10 Mathematicians use a symbolic method for changing an expression like 3(p + 1) to the equivalent expression 3(p) + 3(1). It is called the distributive property and is generalized in the formula a(b + c) = ab + ac. The a is multiplied by b and c separately. The examples below illustrate the use of the distributive property. 5(p + 4) = 5(p) + 5(4) = 5p (2p + 1) = 9(2p) + 9(1) = 18p + 9 6(5 + 3p) = 6(5) + 6(3p) = p (2p + 4)3 = 3(2p + 4) = 3(2p) + 3(4) = 6p + 12 Each can be verified with circles and squares. Here is a representation of 5(p + 4) = 5p page 3 of 3 p + 4 OOOO 5(p + 4) OOOO OOOO OOOO OOOO OOOO 5p + 20 OOOOOOOOOOOOOOOOOOOO 7. Use the distributive property to change each of the expressions with parentheses to an equivalent expression without parentheses. a) 2(3p + 8) b) 4(p + 6) c) 5(2 + 3p) d) (8 + 2p)6 The distributive property is used throughout Mathematics: Modeling Our World. In other lessons and units, you will determine whether the distributive law applies to subtraction, to negative numbers, and to multiplication with several variables. 279

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17 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY SOLVING EQUATIONS S2.11 page 1 of 2 1. a) Write an equation for the coding process shown in Figure 1. Then describe the decoding process. Assign position number Multiply by 2 Add 8 b) Write an equation for the coding process in Figure 2. Then diagram or describe the decoding process. Plaintext letter Coded value Figure 1. Arrow diagram of a coding process. 2. a) c = 4p + 1 describes a coding process. What is the coded value for D? b) Decode 33. Plaintext letter Assign position number Add 8 c) c = 2(p + 3) describes a coding process. What is the coded value for K? Multiply by 2 Coded value Figure 2. Arrow diagram of a coding process. d) Decode 34 if it was coded with the process in part (c). 3. a) c = 3p 4. Solve to find p when c = 17. b) y = 2x 5. Solve to find x when y = Solve for x: a) 63 = 9(x + 4) b) 25 = 3x 8 5. Use the distributive property or circles and squares to create equivalent expressions without parentheses. a) y = 2(3x + 1) b)z = 6(w + 4) c) y = 9(3x + 1) 5 281

18 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.11 page 2 of 2 6. Solve these equations, using an arrow diagram and the process of decoding, or using a symbolic method. One equation has been solved for you using a symbolic approach. Solve the equation 28 = 5x 2. In the context of codes, you are decoding 28 for the coding process that multiplies a variable by 5 and then subtracts 2. You decode or solve using inverse operations in reverse order. Solve by adding 2, then dividing by = 5x 2 Add = 5x Replace with = 5x Divide by 5. 30/5 = x Replace 30/5 with 6. 6 = x The solution to the equation is x = 6. You can check your solution by replacing x with 6 in the original equation and verifying that the solution produces a true statement. 28 = 5(6) 2 28 = = 28 The solution x = 6 produces a true statement. Solve for x. a) 17 = 3x 4 SOLVING EQUATIONS b) 4x + 13 = 33 c) 5(2x + 1) = 45 (Hint: Solve by dividing by 5, subtracting 1, dividing by 2.) d) 5(2x + 1) = 45 (Hint: Use the distributive property and change 5(2x + 1) to an equivalent expression.) e) 0.8x + 24 = 60 f) 2x + 12 = 4 282

19 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY CODING FROM ALPHABET TO ALPHABET 2 Coding the alphabet into other letters is always possible if only addition or subtraction are used in the coding process. In this activity, you look at what happens when multiplication is used. The process consists of several actions (Figure 1). S2.12 page 1 of 4 Action 1 Action 2 Action 3 Action 4 Original letter Position Coded value Position Coded letter Figure 1. A coding process. Action 1: replace the plaintext letter by its position number. Action 2: use the coding process to find the code number. In a shift cipher, for example, add the shift number to the position number. Action 3: subtract (or add) 26 as many times as needed to arrive at a number between 1 and 26. This is a position number. Action 4: replace the position number with the code letter. For a coding process that involves multiplication, you need four similar actions. Here are three examples for the coding process c = 5p. D T H N W K For the letter D, action 3 is not needed. For the letter H, you have to subtract 26 one time. For the letter W, you have to subtract 26 two times to arrive at a number between 1 and

20 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.12 Coded value page 2 of 4 Z 26 Y 25 X 24 W 23 V 22 U 21 T 20 S 19 R 18 Q 17 P 16 O 15 N 14 M 13 L 12 K 11 J 10 I 9 H 8 G 7 F 6 E 5 D 4 C 3 B 2 A A B C DE FG HI JKLMNOPQR STUVWXY Z Original position Figure 2. c = 2p. CODING FROM ALPHABET TO ALPHABET 2 Not every multiplication can be used to code the alphabet into code letters. Observe the graphs of two different processes that use multiplication. Figure 2 illustrates multiplication by 2, or c = 2p; Figure 3 illustrates multiplication by 3, or c = 3p. The graphs are incomplete, but enough points have been graphed to demonstrate the pattern. In both coding systems, every letter of the alphabet has a code letter, so every letter can be coded in both systems. But there is a big difference in the two systems. The first one cannot be used for coding, but the second one is a good coding process. 1. a)use the process c = 2p to find the coded letters for F and S. Coded value Z 26 Y 25 X 24 W 23 V 22 U 21 T 20 S 19 R 18 Q 17 P 16 O 15 N 14 M 13 L 12 K 11 J 10 I 9 H 8 G 7 F 6 E 5 D 4 C 3 B 2 A A B C DE FG HI JKLMNOPQR STUVWXY Z Original position Figure 3. c = 3p. b)use the process c = 3p to find the coded letters for F and S. 2. Decode Q for each process. a)c = 2p b)c = 3p 3. What problems arise from using c = 2p as a coding process? 284

21 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY CODING FROM ALPHABET TO ALPHABET 2 S2.12 Multiplication does not always result in a good system for coding letters as letters. In the remainder of this activity, you will determine which multiplications result in good coding processes. Because several multiplications are possible with an alphabet of 26 letters and each possible multiplication requires a large graph, simplify by looking first at the problem for a smaller alphabet. Discover why some multiplications are useful and others are not. Transfer your discoveries to an alphabet of 26 letters. Study a small alphabet consisting of only five letters: A, B, C, D, and E. Their position numbers are 1, 2, 3, 4, and 5. There are four different coding systems for an alphabet of five letters if only addition is used. They are c = p + 1, c = p + 2, c = p + 3, and c = p + 4. Determine how many different coding systems are possible in an alphabet of five letters when only multiplication is used. 4.In Figure 4, the graph for multiplication by 2 has been completed for you. Complete the graphs for c = 3p, c = 4p, and c = 5p. page 3 of 4 c = 2p c = 3p c = 4p c = 5p E 5 D 4 C 3 B 2 A A B C DE E 5 D 4 C 3 B 2 A A B C DE E 5 D 4 C 3 B 2 A A B C DE E 5 D 4 C 3 B 2 A A B C DE Figure 4. Graphs of several coding processes. 5. Which of the four multiplications do not represent valid coding processes? 6. a) Prepare graphs for the coding processes c = 6p, c = 7p, c = 8p, and c = 9p. b) Compare with the graphs of c = 2p, c = 3p, and c = 4p. Describe similarities and differences you observe. 285

22 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.12 page 4 of 4 CODING FROM ALPHABET TO ALPHABET 2 Before you draw conclusions about the number of different processes possible in an alphabet of 26 letters, study another small system. 7.Consider an alphabet of six letters: A, B, C, D, E, and F with position numbers 1, 2, 3, 4, 5, and 6. The graphs for c = 2p, c = 3p, c = 4p, and c = 5p are drawn in Figure 5. Figure 5. Graphs of several coding processes. F E D C B A c = 2p A B C DE F F E D C B A c = 3p A B C DE F F E D C B A c = 4p A B C DE F F E D C B A c = 5p A B C DE F Which of the four multiplications do not represent valid coding processes? 8. How many different coding processes that use only multiplication are there in a complete alphabet of 26 letters? You may want to investigate alphabets of 7, 8, or 9 letters before you draw conclusions. 286

23 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY CODING SURPRISES S2.14 The problems in this activity are typical of problems that arise with coding processes and number tricks. There is no single right way to solve any of them. Use a calculator or pencil and paper. (All coding questions refer to processes that code letters as numbers.) page 1 of 2 1. Some magicians design number tricks by building the trick with several sets of parentheses and simplifying it at the end. Here is the symbolic form of such a number trick: (2(x 2) + 5) 4. a) Find a two-step symbolic form that is equivalent to this one. Tell why you think they are equivalent. b) Suppose someone gets 27 for an ending number. Show how you can solve an equation to get the starting number. 2.The coding processes c = 2p + 5 and c = 3p 8 look different. Do they code any letters the same way? If so, which ones? 3.Two coders change their coding process monthly to make it harder to read their messages. To make it even more difficult, they prefer to have no letters coded the same way they were coded the previous month. This month their coding process is c = 2p 1. Find several coding processes they could use next month. Explain why you think they work. 4.Can you invent two shift ciphers that code exactly two letters the same? Three letters? Is there a general rule? 287

24 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.14 page 2 of 2 CODING SURPRISES 5.Many magicians do a card trick that predicts a certain card is a of clubs. Devise a number trick that ends in regardless of the starting number. To be convincing, your trick should have at least four steps and all the constants involved should be whole numbers. 6.A code cracker has determined that the letter A is coded as 1 and the letter B as 4. Find more than one coding process that does this. 7.Here is a number trick. a) Pick a number between 1 and 10. b) Subtract the number from 16. c) Add 1 to your original number. d) Multiply the two numbers you got in steps (b) and (c). Suppose someone gets 70. Find the starting number. 8.One problem with coding processes is that the coded values can be quite large. Find the symbolic form of a coding process that does not give a coded value larger than 26, but that does give a coded value different from the position number. 9. Use your knowledge of code cracking to decode this message:

25 Mathematics: Modeling Our World Unit 2: SECRET CODES SUPPLEMENTAL ACTIVITY MATRIX ZOO S2.15 Suppose you want to code the message Meet me on Friday so that different letters shift different amounts. One way to do this is to use a matrix with different numbers (Figure 1). page 1 of Figure 1. Coding with a matrix. Not all the numbers have to be different, just some of them. For example, the matrix shown in Figure 2 also works Figure 2. Coding with a matrix. 1.Use Figure 2. a) What is the coded value of the first E in meet? b) What is the coded value of the second E in meet? c) Why is a message coded with this process hard to break? d) Take a closer look at the second matrix. The numbers in the first column are all 1s and the first letter of the alphabet is A. What is the 14th letter of the alphabet? What six-letter word is spelled by the letters whose position numbers are in the columns? e) What does the person receiving this message need to know to decode it? 289

26 SUPPLEMENTAL ACTIVITY Unit 2: SECRET CODES Mathematics: Modeling Our World S2.15 page 2 of 2 MATRIX ZOO 2.To code a message with this process, you need to select a word. Suppose you want to use tiger and that your message is Meet me in Paris. Because tiger has five letters, you should put the message in a matrix with five columns. a) Write the message in a matrix with five columns. Add as many blank spaces as you need to finish the matrix. Do not code blank spaces between words. b) Write the matrix from part (a) with numbers instead of letters. c) Write a matrix that represents tiger. Because T is the 20th letter of the alphabet, your matrix should have 20s in the first column. The matrix should have the same number of rows as the matrix from part (b). d) Add the matrices from parts (b) and (c). (You can do the addition yourself or use a calculator.) Write the matrix. Then remove the numbers from the matrix and write them in a straight line. e) Describe how a message coded with this process is decoded. 3.See if you are right. Decide whether a keyword-matrix coding process is easy to decode. The message below is coded with the word giraffe. Decode it