Clocks and Codes Patterns in Modular Arithmetic A Group Project for Math 104

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1 Clocks and Codes Patterns in Modular Arithmetic A Group Project for Math

2 Clocks and Codes Patterns in Modular Arithmetic. The project is due as noted in the syllabus and will be presented in class. Late work will not be accepted on this project for any reason, so please arrange to get your work here on time. You may, of course, turn the project in early.. Your grade will be based on the following calculations: a. Worksheets and application problems points b. Essay points c. Poster points. I do expect that this project will be done in a neat and scholarly manner. I do expect that the essay will either be typed or written in blue or black ink with conventions of spelling and grammar followed. You may complete the worksheets in pencil; however, all questions should be answered with complete sentences.. Your group task is to complete a. all the operation charts, work all nine application problems. b. What is for Lunch c. to create a Modular Arithmetic Poster d. to write a Modular Arithmetic Essay. e. The application problems and the worksheet What s for Lunch should be completed on your own paper. You should show your work for each question and explain your solution.. The poster will include: a. A title that tells the operation, the mod used and the transformation. b. Any of the three operations (addition, subtraction, or multiplication) with Mod or higher or division with any prime Mod 7 or higher and any transformation. c. A color or symbol key and your Mod operation table. d. Your poster will be graded on neatness and complexity. For example, division mod 7 would receive a higher grade than addition mod, a rotation would receive a higher grade than a slide, a circular poster would receive a higher grade than a square one. The back of this packet contains poster masters that you can enlarge and use, if you wish.. The essay will include: a. An explanation of what modular arithmetic is. b. An example of modular arithmetic in real life. c. The mod you used for your poster. d. The operation you used for your poster. e. An explanation of how you calculated your operation. g. An explanation of your transformation. i. If you use symbols instead of solid colors, explain how or why you chose the symbols to replace the number in your mod operation chart.

3 j. The answers to Thoughts on Modular Arithmetic (below) () Did you have to compute every problem in each of the operation charts? Explain. () What were the easiest problems to calculate? Why? () Did you find short cuts? If so what were they, and how did you verify that they always worked? () What is the commutative property? () Does modular addition have this property? Modular subtraction? Modular multiplication? Modular division? How could you verify your answers? (Remember to show something true you must show each and every case or give a proof. To show something false all you need is one example.) () What is the definition of inverse? Does modular arithmetic have additive inverses? Multiplicative inverses? If so what are they? 8. You may work on this project in groups of no more that. For groups of or less, each group member will receive % of their group grade. However, should you choose to form a larger group, your grades will be reduced accordingly. For example: each member of a group of would receive of their group grade, so a / grade of 8% would net an individual grade of %. 9. For each group, I expect one set of worksheets, one set of application problems, one essay, one poster, and a signed statement from each group member delineating how the work was accomplished.. I strongly encourage group work on this project. Have fun and be creative.. You may ask questions in class or during office hours about any part of this project.

4 Application Problems The following are all application problems of modular arithmetic. Your solutions must include all your work. Try not to make them too hard.. What time was it seven hours ago, if it is five o clock now?. Today is Thursday. What day of the week will it be in, days?. Today is Monday. What day of the week was it, days ago?. This month is October. What month will it be in months?. This month is March. What month was it months ago?. This month is July. What month will it be in, days? 7. Mrs. Kucheck s Egg Basket Mrs. Kucheck has a basket of eggs. If she takes the eggs out,,,, or at a time there is always one egg left in the basket. However, if she takes the eggs out 7 at a time, no eggs are left in the basket. The basket can hold eggs. How many eggs does Mrs. Kucheck have? 8. Dividing Time Marty said to Doc, So we re going to travel back in time. What year did you set the Delorean for? Ohhhh. I forgot, said Doc. What do you mean? exclaimed Marty. We could be lost in a time continuum! Well, what I remember is...if you divide the year by two, Doc said, you get a remainder of one. Great, Doc, that s just every odd-number year. You got to remember more! Great Scott! I just got a jolt of. gigabytes of brain power. Now I know if you divide the year by,,,, 7, or 9 you will also get a remainder of one. What about 8? If I divide by 8, do I get a remainder of one? asked Marty. No, Marty, replied Doc. All right, Doc, said Marty, that s it. Let s go. What year are Marty and Doc off to? 9. The Eccentric Clockmaker An eccentric clockmaker built three different clocks. The first clock was a five-minute clock and designed with an alarm set to sound each time the hand reached the number. The second clock was a six-minute clock designed to sound each time the hand reached the number. The third clock was a seven-minute clock designed to sound each time the hand reached the number. The clockmaker started the clocks simultaneously, and each clock begin to sound at its appropriate time. Was there a time when all three clocks sounded their alarms together? If so, tell when it occurred and explain why. If not, explain why not.

5 Addition Operation Charts Mod + Mod Mod 7 Mod 7 + Mod + Mod

6 Mod + Addition Operation Charts Mod Mod 9 Mod Mod Mod 8 7

7 Subtraction Operation Charts Mod - Mod Mod 7 Mod - Mod 7 - Mod

8 Mod - Mod 9 Subtraction Operation Charts 7 Mod Mod 8-7 Mod 9-7 8

9 Mod 8 Multiplication Operation Charts Mod X Mod 7 7

10 Mod X Mod Mod Mod 7 X 8

11 X Multiplication Operation Charts Mod Mod Mod 9 Mod 8 X 7 7 Mod 9 X Mod 8 9

12 Division Operation Charts Note: While the charts are modified to show not all divisions are defined, all the numbers must be on the clock to calculate the division correctly. Mod Mod Mod 7 Mod 7 Mod Mod

13 Division Operation Charts Note: While the charts are modified to show not all divisions are defined, all the numbers must be on the clock to calculate the division correctly. Mod Mod Mod 9 Mod Mod Mod 8

14 . Complete the mod addition chart below. +. Assign a different color to each of the numbers,,,, and. Addition Patterns Coloring Chart. Complete the mod addition chart below. +. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod addition chart into the grid below and substitute the color you assigned for each number.. Copy the interior of the mod addition chart into the grid below and substitute the color you assigned for each number.

15 . Complete the mod 7 addition chart below. Addition Patterns Coloring Chart +. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod 7 addition chart into the grid below and substitute the color you assigned for each number.

16 Subtraction Patterns Coloring Chart. Complete the mod subtraction chart below. -. Complete the mod subtraction chart below. -. Assign a different color to each of the numbers,,,, and.. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod subtraction chart into the grid below and substitute the color you assigned for each number.. Copy the interior of the mod subtraction chart into the grid below and substitute the color you assigned for each number.

17 Subtraction Patterns Coloring Chart. Complete the mod 7 subtraction chart below. -. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod 7 subtraction chart into the grid below and substitute the color you assigned for each number.

18 Multiplication Patterns Coloring Chart. Complete the mod multiplication chart below. X. Complete the mod multiplication chart below. X. Assign a different color to each of the numbers,,,, and.. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod multiplication chart into the grid below and substitute the color you assigned for each number.. Copy the interior of the mod multiplication chart into the grid below and substitute the color you assigned for each number.

19 Multiplication Patterns Coloring Chart. Complete the mod 7 multiplication chart below. X. Assign a different color to each of the numbers,,,,, and.. Copy the interior of the mod 7 multiplication chart into the grid below and substitute the color you assigned for each number. 7

20 Division Coloring Patterns. Complete the mod division chart below.. Complete the mod division chart below.. Assign a different color to each of the numbers,,, and.. Assign a different color to each of the numbers and.. Copy the interior of the mod division chart into the grid below and substitute the color you assigned for each number.. Copy the interior of the mod division chart into the grid below and substitute the color you assigned for each number. 7

21 Division Coloring Patterns. Complete the mod 7 division chart below.. Assign a different color to each of the numbers,,,, and.. Copy the interior of the mod 7 division chart into the grid below and substitute the color you assigned for each number. 8

22 Transformations Slide. Complete the mod division chart below.. Assign a different color to each of the numbers,,, and.. Copy the interior of the mod division chart into the grid below and substitute the color you assigned for each number. I. Using the table above slide the color pattern to quadrants II, III and IV. II I 9

23 Transformations III IV Reflection. Complete the mod addition chart below. +. Assign a different color to each of the numbers,,,, and. Copy the interior of the mod addition chart into the skewed table below. I. Using the table above reflect the color pattern to quadrants II, III and IV.

24 Transformations II II I IV Rotation. Complete the mod subtraction chart below. -. Assign a different color to each of the numbers,,,, and. Copy the interior of the mod subtraction chart into the circular table below.

25 Transformations I. Using the table above rotate the color pattern about the origin to quadrants II, III and IV. II I III IV

Problem of the Month. Once Upon a Time

Problem of the Month. Once Upon a Time Problem of the Month Once Upon a Time Level A: When it is four o clock, how many minutes must pass before the big hand (minute hand) gets to where the little hand (hour hand) was at four o clock? How did