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Counting

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.

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Counting

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  1. 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 Numerals

  5. Early numerals • 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.

  6. Numeration Systems • Tally systems: |||| • 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

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

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

  9. Roman Numerals Symbols: I V X L C D M 1 5 10 50 100 500 1000 If the Roman Numerals are in order from greatest to least, then add:VII = 5 + 1 + 1 = 7 XVI = 10 + 5 + 1 = 16

  10. 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

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

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

  13.    = 1 • ____ + 11 • 60 + 21 = 1 • 60 • 60 + 11 • 60 + 21 = 3600 + 660 + 21 = 4281

  14. 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.

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

  16. 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.

  17. 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

  18. Mayan Numeration Uses base 20 New place value… left a vertical gap. • is one 20, and 0 ones = 20. •• • is two ____ + 5 + 1 = _____

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