Transcript Day 2

January 20, 2009
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Hope you enjoyed the long
weekend!
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January 20, 2009
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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
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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, …
d) 1, 0, -1, 0, 1, 0, -1, …
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1.3 (cont’d)
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A little math history…
Carl Friedrich Gauss (1777-1855)
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Exploration 1.1 (handshakes)
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Goal: determine how many handshakes will
occur if each student in this class shakes
hands with every student.
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• Work on this alone for a few minutes. Can
you apply something we’ve discussed
today?
Exploration 1.1 (handshakes)
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• Discuss your approach to the problem with
others at your table (the answer is not so
important – it’s 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)
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For homework, answer the following two
questions:
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1.
How many handshakes will there be in our class (24
people)?
2.
How many handshakes will there be in a group of n
people?
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Explain what you did to solve, and why you did it.
1.4 - Representations
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Diagrams
Graphs
Tables
Sketches
Equations
Words
Etc.
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Exploration 1.4 (Darts)
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Purpose: think about solution
strategies as opposed to trying random things.
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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)
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20
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Exploration 1.4 (cont’d)
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• 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 (cont’d)
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• 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
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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 (cont’d)
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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 (cont’d) Roman system
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Value
Symbol
Value
Symbol
2
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I
100
5
V
500
1
10
X
1000
M
50
L
C
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D
2.3 (cont’d) Egyptian system
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10
=
(staff)
=
(heel
bone)
100 =
(scroll)
1000 =
(lotus
flower)
10,000 =
100,000 =
1,000,000=
(pointing
finger)
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(fish or
tadpole)
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(astonished
person)
2.3 (cont’d) Babylonian system
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•Positional system
•The number 60
•Symbol for zero:
(developed later)
Value
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10
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Symbol
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2.3 (cont’d) Mayan system
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• The number 20
• Used the concept of zero, but only for place
holders
• Used three symbols:
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1
5
0
• Wrote their numbers vertically:
is 5 + 5 = 10,
is 4 + 5 + 5 = 14
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2.3 (Mayan system cont’d)
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Mayans left a vertical gap to represent place
value:
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is one 20, and 0 ones = 20.
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is two 20’s + 6 = 46
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2.3 (cont’d)
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What similarities/differences do you see in
these systems?
Advantages/disadvantages of each system?
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Homework
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• 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.
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