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RULES Classroom Rules 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 Homework Rules 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) 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) 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) 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? 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? DEAL WITH NEGATIVE NUMBERS How many integers are between -50 ~ 50 inclusive? How many integers are between -50 ~ -10 inclusive? 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 How many even numbers are between 2 ~ 200 inclusive? 2, 4, 6, …., 198, 200 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? 3(2x2-1), 5(2x3-1), …., 121(2x61-1), 123(2x62-1) 3(2x1+1), 5(2x2+1), …, 121(2x60+1), 123(2x61+1) 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 Slightly Trickier … 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 ?