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RULES
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Classroom Rules
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No gum/food/drink
No cell phone/computer
No distracting other people
First offense – verbal warning
Second offense – 10 minute time out
Third offense – eat dinner with adults
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Homework Rules
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Weekly homework
HOW TO COUNT
WB Math Group
DON’T WE KNOW HOW TO COUNT
ALREADY?
Not Really…
•Ancient people could only count to three
•But they know how to compare numbers that
are larger than three
•How to count huge numbers? Can’t use your
fingers!
•How to count tricky numbers?
•Even majority of the adults don’t know how to
count certain things
HOW MANY PEOPLE ARE THERE?
Warren
 Basking Ridge
 New Jersey
~ 15,000
~ 21,000
~ 9,000,000 (6)
US
 World
~ 300,000,000 (8)
~ 7,000,000,000 (9)
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SOME BIG NUMBERS
Number of Web Pages Google
Searches ~ 50,000,000,000 (10)
 Number of stars in the Milky Way
Galaxy ~ 400 billion 400,000,000,000
(11)
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Number of atoms in a drop of water
~ 2,000 billion billions (21)
 Number of atoms in the universe
1,000… (80)
 Number of Permutations of a 7x7
cube 1,000,…(160)
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Now Let’s Dive In!
HOW TO COUNT NATURAL NUMBERS
Natural numbers
are positive
integers
How many integers are between 1 ~ 100 ?
 How many integers are between 1 ~ 100 inclusive?
 How many integers are between 1 ~ 100 exclusive?
 How many integers are between 1 ~ n inclusive?
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HOW TO COUNT NATURAL NUMBERS
How many integers are between 234 ~ 1011
inclusive?
 How many integers are between 234 ~ 1011
exclusive?
 How many integers are between 0 ~ 50 inclusive?
 How many integers are between r ~ r + n
inclusive?
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DEAL WITH NEGATIVE NUMBERS
How many integers are between -50 ~ 50
inclusive?
 How many integers are between -50 ~ -10
inclusive?
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ARITHMETIC SEQUENCE
An arithmetic (or linear) sequence is a sequence of numbers in
which each new term is calculated by adding a constant value
to the previous term
•2, 6, 10, 14, 18, 22, …
•-5, -3, -1, 1, 3, 5 …
•-1, 10, 21, 32, …
•1, 2, 4, 8, 16, …
ARITHMETIC SEQUENCE
a little algebra
ARITHMETIC SEQUENCE
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How many even numbers are between 2 ~ 200
inclusive?
2, 4, 6, …., 198, 200
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How many integers are in the sequence
3,6,9,…,300?
How many even numbers are between 50 ~ 199
inclusive?
ARITHMETIC SEQUENCE
How many odd numbers are between 1 ~ 101,
inclusive?
1, 3, 5, 7, 9, 11, …., 99, 101
1(2x1-1), 3(2x2-1), …, 99(2x50-1), 101(2x51-1)
 How many odd numbers are between 3 ~ 123
inclusive?
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3(2x2-1), 5(2x3-1), …., 121(2x61-1), 123(2x62-1)
3(2x1+1), 5(2x2+1), …, 121(2x60+1), 123(2x61+1)
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How many numbers are in the following
sequence: 5, 9, 13, 17, …, 401 ?
GEOMETRIC SEQUENCE
A geometric sequence is a sequence
of numbers in which each new term
(except for the first term) is
calculated by multiplying the
previous term by a constant value.
•1, 2, 4, 8, 16, 32, 64, 128
•1, 10, 100, 1000, 10000,
•1, 3, 9, 27, ??, ??
•2, 3, 5, 9, 17, 33, 65, 129
GEOMETRIC SEQUENCE
GEOMETRIC SEQUENCE
1, 2, 4, 8, … 1024
 3, 6, 12, 24, 48, …, 300
 64, 32, 16, 8, … ½, ¼
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Slightly Trickier …
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4, 5, 7, 11, 19, …, 2051
SOME EXERCISES
How many integers are there from 25 to
79 inclusive?
 What is the 53rd integer in the sequence
86, 87, 88, …?
 The largest of 123 consecutive integers is 307.
What is the smallest?
 The smallest of r consecutive integers is n, what
is the largest integer?
 The largest of r consecutive integers is k. what is
the smallest?
 How many integers are there in the sequence n,
n+1, n+2, …, n+r ?
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