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.

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

Numbers vs Numerals

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.

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

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

Egyptian Numeration 300 + 10 = 310 (Notice no ones) 20,000 + 300 + 4 = 20304

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

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

Mayan Numerals Used the concept of zero, but only for place holders Used three symbols: • --- 1 5 0 Wrote their numbers vertically: ••• is 3 + 5 = 8, --- is 5 + 5 = 10

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

Roman Numerals No zero. Symbols: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1000 If the Roman Numerals are in order from greatest to least, then add: VII = 5 + 1 + 1 = 7; XVI = 10 + 5 + 1 = 16

Roman Numerals I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1000. 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; XCII = 100 - 10 + 1 + 1 = 92

Exploration 2.8 Exploration 2.8--We will not do this entire thing--it is too long! But we will do parts. Today, we will begin with manipulatives.

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

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. Be ready to present your group’s numeration system for the Alphabitians on Tuesday.

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?