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Alphabitia

Alphabitia. Read the introduction. Use the “artifacts”. Unit Long Flat. Presentation. Use a large white board to present your group’s numeration system. Include the information needed for someone to understand the system and be able to use it.

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Alphabitia

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  1. Alphabitia • Read the introduction. • Use the “artifacts”. Unit Long Flat

  2. Presentation • Use a large white board to present your group’s numeration system. • Include the information needed for someone to understand the system and be able to use it. • Select two members of the group to explain and answer questions.

  3. Show how you would write the number 4,628 in your system.

  4. Alphabitia Numeration System Proposals What characteristics do the systems have in common and how are they different? What are the pros and cons of your group’s system and the other groups’ systems?

  5. What makes an efficient numeration system?

  6. Alphabitia A numbering system is only powerful if it can be reliably continued. Ex: 7, 8, 9, … what comes next? Ex: 38, 39, … what comes next? Ex: 1488, 1489, … what comes next?

  7. The Numeration System we use today:The Hindu-Arabic System Zero is used to represent nothing and as a place holder. Base 10 Why? Any number can be represented using only 10 symbols. Easy to determine what number comes next or what number came before. Operations are relatively easy to carry out.

  8. In Base 10… Digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 We can put the digit 9 in the units place. Can we put the next number (ten) in the units place? Only one digit per place Placement of digits is important! 341 ≠ 143. Can you explain why not?

  9. Alphabitia • If we apply the aspects of our system to the artifacts of the Alphabitians, what would we come up with? • Symbols: A, B, C, D, 0

  10. Exploration 2.9 • Different Bases

  11. In another base… • We need a 0, and some other digits • So, in base 10, we had 0 plus 9 digits • What will the digits be in base 9? • What will the digits be in base 3? • Which base was involved in alphabitia?

  12. So, let’s count in base 6 • Digits allowed: 0, 1, 2, 3, 4, 5 • There is no such thing as 6 • When we read a number such as 2136, we don’t typically say “two hundred thirteen.” We say instead “two, one, three, base 6.”

  13. Count! In base 6 • 1, 2, 3, 4, 5, … • 10, 11, 12, 13, 14, 15, … • 20, 21, 22, 23, 24, 25, … • 30, 31, 32, 33, 34, 35, … • 40, 41, 42, 43, 44, 45, … • 100, … • 100, 101, 102, 103, 104, 105, … 110

  14. Digits 0,1,2,3,4,5,6,7,8,9 New place value after 9 in a given place Each place is 10 times as valuable as the one to the right 243 = 2 • (10 • 10) + 4 • 10 + 3 • 1 Digits 0, 1, 2, 3, 4, 5 New place value after 5 in a given place Each place is 6 times as valuable as the one to the right. 243base 6 = 2 • (6 • 6) + 4 • 6 + 3 • 1 or 99 in base 10 Compare base 6 to base 10

  15. 312 =3 • 100 + 1 • 10 + 2 • 1 312base 6 = 3 • 36 + 1 • 6 + 2 • 1 = 116 in base 10 Compare Base 6 to Base 10

  16. How to change from Base 10 to Base 6? • Suppose your number is 325 in base 10. • We need to know what our place values will look like. • _____ _____ _____ _____ 6•6•6 6•6 6 1 Now, 6•6•6 = 216. 216 = 1000 in base 6.

  17. Base 10 to Base 6 • ___1__ _____ _____ _____ 6•6•6 6•6 6 1 • Now, 325 - 216 = 109. Since 109 is less than 216, we move to the next smaller place value: 6 • 6 = 36. • 109 - 36 = 73. Since 73 is greater than 36, we stay with the same place value.

  18. Base 10 to Base 6 • __1___ ___3__ _____ _____ 6•6•6 6•6 6 1 • We had 109: 109 - 36 - 36 - 36 = 1. We subtracted 36 three times, so 3 goes in the 36ths place. • We have 1 left. 1 is less than 6, so there are no 6s. Just a 1 in the units place.

  19. Base 10 to Base 6 • __1___ ___3__ __0___ __1___ 6•6•6 6•6 6 1 • Check: 1 • 216 + 3 • 36 + 1 • 1 = 325 • So 325 = 13016

  20. Homework for Thursday 1/28 For Exploration 2.8, write up the following in an essay format: Describe the process your group went through to come up with a numeration system for Alphabitia. Explain your system. Describe your thinking about this project. Turn in your descriptions, along with the table on p. 41 and your answers to Part 3: #2,3,5

  21. Count! In base 16 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 1c, 1d, 1e, 1f, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 2c, 2d, 2e, 2f

  22. Homework for Tuesday 2/2 • Exploration 2.9: Part 1: for Base 6, 2, and 16, do #2; Part 3: #2, 3, Part 4: #1, 2, 4. For the base 16 section, change all the base 12 to base 16 (typo) • Read Textbook pp. 109-118 • Do Textbook Problems pp. 120-121: 15b,c, 16b,d, 17a,i, 18b,f, 19, 29

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