January%2020,%202009

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January 20, 2009

• Hope you enjoyed the long
• weekend!

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January 20, 2009

• Sec 1.3 Patterns (Exploration 1.1)
• Sec 1.4 Representations (Exploration 1.4)
• Sec 1.7 Connections
• Sec 2.3 Numeration

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1.3 - Patterns

• Ex Find the next term in the sequence.
  Describe the pattern you found (use a sentence).
• a) 1, 2, 4, 8, 16,
• b) 25, 5, 1, 0.2, 0.04,
• c) 1, 3, 6, 10, 15,
• 1, 0, -1, 0, 1, 0, -1,

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1.3 (contd)

• A little math history
• Carl Friedrich Gauss (1777-1855)

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Exploration 1.1 (handshakes)

• Goal determine how many handshakes will occur
  if each student in this class shakes hands with
  every student.
• Work on this alone for a few minutes. Can you
  apply something weve discussed today?

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Exploration 1.1 (handshakes)

• Discuss your approach to the problem with others
  at your table (the answer is not so important
  its the solution that you should discuss). What
  were some of the ideas you had not thought of?

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Exploration 1.1 (handshakes)

• For homework, answer the following two questions
• How many handshakes will there be in our class
  (24 people)?
• How many handshakes will there be in a group of n
  people?
• Explain what you did to solve, and why you did it.

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1.4 - Representations

• Diagrams
• Graphs
• Tables
• Sketches
• Equations
• Words
• Etc.

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Exploration 1.4 (Darts)

• Purpose think about solution
• strategies as opposed to trying random things.
• If you have 4 darts and the dart board shown
• above, which of the scores below are possible,
• assuming that every dart hits the board?
• (Explain your reasoning)
• 6 10 13 15 20 28

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Exploration 1.4 (contd)

• What did you notice?
• Part 4a asks you to list all scoring outcomes,
  e.g. 7, 7, 7, 7 or 1, 3, 5, 7 etc. With your
  table, discuss how you could do this
  systematically.

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Exploration 1.4 (contd)

• What if you only had 3 darts? What scores would
  then be possible? Answering this and explaining
  why will be part of your homework for next time.

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2.3 - Numeration

• Have you ever thought about what it was like for
  people to invent counting?
• Abstract notion of number
• Representing discrete amounts and measures

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2.3 (contd)

• As societies became more complex, so did number
  systems.
• Tally systems
• Roman (widely used for a long time still has
  some usage)
• Egyptian (earliest known system)
• Babylonian (very advanced for its time)
• Mayan (probably first to develop concept of zero
  as we know it)
• Hindu-Arabic (system we use)

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2.3 (contd) Roman system
Value Symbol Value Symbol
1 I 100 C
5 V 500 D
10 X 1000 M
50 L
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2.3 (contd) Egyptian system
1 (staff) 10,000 (pointing finger)
10 (heel bone) 100,000 (fish or tadpole)
100 (scroll) 1,000,000 (astonished person)
1000 (lotus flower)
             
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2.3 (contd) Babylonian system

• Positional system
• The number 60
• Symbol for zero
• (developed later)

Value Symbol
1 ?
10 ?
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2.3 (contd) Mayan system

• The number 20
• Used the concept of zero, but only for place
  holders
• Used three symbols 1 5 0
• Wrote their numbers vertically is 5 5
  10,
• is 4 5 5 14

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2.3 (Mayan system contd)

• Mayans left a vertical gap to represent place
  value
• is one 20, and 0 ones 20.
• is two 20s 6 46

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2.3 (contd)

• What similarities/differences do you see in these
  systems?
• Advantages/disadvantages of each system?

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Homework

• Read text section 1.7
• Write up the handshakes exploration.
• Write up the darts exploration.
• Do text assignment posted on web pg
• Due Thursday, 1/22 at the beginning of class.