1. Alphabitia Read the introduction. Use the “artifacts”. UnitLong Flat

2. Your job: See if you can create a numeration system, like the ones we have just seen. Make notes, and be ready to explain it to others who are not part of this tribe. It should be logical, and be able to be continued past Z. Keep the materials for Monday.

3. Alphabitia Numeration System Proposals What did you come up with in your group? What are the pros and cons of your group’s system and the other groups’ systems?

4. Write your group’s numeration system in the table.

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 10Why? 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. Exploration 2.9 Different Bases

10. 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?

11. 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 213 6, we don’t typically say “two hundred thirteen.” We say instead “two, one, three, base 6.”

12. 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

13. Compare base 6 to base 10 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. 243 base 6 = 2 (6 6) + 4 6 + 3 1 or 99 in base 10

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

15. 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. _____ _____ 666 66 6 1 Now, 666 = 216. 216 = 1000 in base 6.

16. Base 10 to Base 6 ___1__ _____ _____ _____ 666 66 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.

17. Base 10 to Base 6 __1___ ___3__ _____ _____ 666 66 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.

18. Base 10 to Base 6 __1___ ___3__ __0___ __1___ 666 66 6 1 Check: 1 216 + 3 36 + 1 1 = 325 So 325 = 1301 6

19. Homework for Thursday 1/29 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

20. 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

21. Homework for Tuesday 2/3 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