1. Warm Up Activity Four 4s Use four 4s and the arithmetic operations (+, -, x, /) plus grouping symbols to create each of the counting numbers from 0 to 10. Use exactly four 4s for each number.

2. Counting Did humans always do math as we do it today? Did they always use the symbols we use?

3. Numbers vs Numerals

4. Early numbers Numbers did not start out as 1, 2, 3, … The first numeration systems were often 1, 2, many. More complex number systems arose out of need.

5. Numeration Systems Tally systems: |||| at a time. Egyptian--probably earliest known Babylonian--only two symbols Roman--most widespread, all over Europe from the Roman Empire Mayan--only three symbols Hindu-Arabic--what we use today

6. Egyptian Numerals Used at about the same time as Babylonian Numbers--many similarities More symbols: 110 100 1000 10,000 More symbols for 100,000 and 1,000,000

7. Egyptian Numeration 300 + 10 = 310 (Notice no ones) 20,000 + 30 + 4 = 20,034

8. Roman Numerals Symbols: IV X L C D M 1510501005001000 If the Roman Numerals are in order from greatest to least, then add: VII = 5 + 1 + 1 = 7 XVI = 10 + 5 + 1 = 16

9. Roman Numerals If the Roman numerals are NOT in order from greatest to least, then subtract where the order is wrong. IV = 5 - 1 = 4 IX = 10 – 1 = 9 XCII = 100 - 10 + 1 + 1 = 92

10. Babylonian Numerals Initially, no zero. Later developed: Two symbols only:  = 1; = 10. Additive when written from greatest to least:    = 10+10+1+1+1 Use 60 as a base--there is a break after 60.     means 60+60+60+10+1

11. Babylonian Numeration    = 1 ____ + 11 60 + 21 = 1 60 60 + 11 60 + 21 = 3600 + 660 + 21 = 4281

12. Exploration 2.8 This exploration takes you out of your comfort zone and encourages you to think about what is involved in developing a numeration system.

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

14. Your job: Create a numeration system for your tribe using the artifacts left by the member who died. Think about what features are desirable in a numeration system. 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. Write your group’s numeration system in the table. When you have a system you are happy with, put the details on a white board and be ready to present your group’s numeration system for the Alphabitians.

15. Mayan Numerals Used the concept of zero, but only for place holders Used three symbols:--- 150 Wrote their numbers vertically: is 3 + 5 = 8, --- is 5 + 5 = 10

16. Mayan Numeration New place value… left a vertical gap. is one 20, and 0 ones = 20. is two ___ + 6