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Math Skills – Week 1

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Introduction – Me: Education ????????????? - UC Davis – 2004 - Highest Honors - BS Physics - Minor Mathematics - Senior thesis – Organic Superconductors Aggie - CSULB - 2005 to Now - 4.0 - MS Physics - Materials physics research on Graphene multi-layers

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Introduction – Me: Career SW Engineer @ the Boeing Co. High School Boeing 1999 787 Dreamliner

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Introduction - Me: Why FIDM? 1. How many have never been good at it? 2. How many dislike it? 3. How many think it is useless to you?

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Introductions / Administrative Stuff Welcome to Math Skills Introductions ○ You 1. Name? 2. Major? 3. Do you like math? Why or Why not? 4. Highest level math class taken? 5. What do you hope to get out of this class? Attendance @ 8:45 Syllabus Review Questions? Books

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Math Skills How to succeed in this class Collaboration is key! ○ Study Groups ○ Participation ○ Ask questions Lots of practice – Do homework! Study for quizzes Do class examples If you need extra help Come to office hour Free tutoring in Ideas Center

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Lecture Flow and Logistics Quiz at the beginning of each class on all material from previous week Presentation of notes for each section in these slides For each concept discussed… Do examples together Do class examples on separate sheet These are to be turned in at the end of class. Questions?

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Math Skills – Week 1

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Week 1 – Whole Numbers What are Whole Numbers? Section 1.1 Arithmetic with Whole Numbers Sections 1.2 – 1.5 Exponentials and Order of Operations Section 1.6 Prime Numbers and Factoring Section 1.7

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What are Whole Numbers? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …, 10,000,000 Standard Form 234,321,456 Each digit is in a given place value Hundred Millions Ten Millions Hundred- Thousands Ten Thousands Hundreds Tens Ones 234321456 PLACE VALUE CHART – pg. 4

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What are Whole Numbers? - Rounding Rounding whole numbers allows us to approximate the number to any place value Steps 1. Write out the number to be rounded in a place value chart (i.e 37) 2. Look at the number to the right of the place value you want to round to. (7) 3031 32 3334353637383940 Hundreds Tens Ones 37 1.If the number is > or = 5, increase the digit in the place value by 1, and replace all digits to the right of it by zeros 2.If the number is < 5, replace all of the digits to the right of the digit in the place value with zeros

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What are Whole Numbers? – Rounding Round 525,453 to the nearest ten-thousand. Steps 1. Write 525,453 in place value chart 2. What is 1 st digit to the right of the ten-thousands place value? 1. Is it > or = 5? Is it < 5? 3. Answer = 530,000 Class Exercise: Round to Nearest Hundreds and Tens Hundred- Thousand s Ten Thousand s Hundreds Tens Ones 525453

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What are Whole Numbers? – Rounding Examples: Round the following numbers to the given place value 1. 926 – Tens ○ 930 2. 1,439 – Hundreds ○ 1,400 3. 43,607 – Thousands ○ 44,000 4. 71,834,250 – Millions ○ 72,000,000

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Math Skills – Week 1

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Arithmetic with Whole Numbers Addition w/ and w/out carrying Subtraction w/ and w/out borrowing Multiplication Division

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Arithmetic with Whole Numbers - Addition Addition is the process of finding the total of two or more numbers Properties (pg. 9) Addition Property of Zero ○ Zero added to a number does not change the number 4 + 0 = 4, 0+ 7 = 7 Commutative ○ Order of adding numbers doesn’t matter 4 + 8 = 8 + 4, 12 + 2 is the same as 2 + 12

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Arithmetic with Whole Numbers – Addition Properties Contd. (pg. 9) Associative ○ Grouping the additions in any order does not change the result (3 + 2) + 4 = 3 + (4 + 2) (5) + 4 = 3 + (6) 9 = 9 Summation Phrases (for study pg. 10) 3 Added to 5 equals 8 10 More than 87 is 97 18 Increased by 1 is 19

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Arithmetic with Whole Numbers – Addition Addition of larger numbers Steps 1. Arrange the numbers by vertically aligning them by place value 2. Add digits in each column 3 2 1 + 6 4 7 2 ones tens hundreds thousands 6 7 9 3

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Arithmetic with Whole Numbers – Addition If sum of digits in each column is > 9 need to use carrying Steps 1. Arrange the numbers by vertically aligning them by place value 2. If sum > 9 write ones digit on the right below the equal bar, and carry the tens digit to the first place value on the left. 3. Continue this procedure From right to left 3 3 9 + 6 4 7 2 ones tens hundreds thousands

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Arithmetic with Whole Numbers – Addition 3 3 9 + 6 4 7 2 1 1 1 1 86 11

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Arithmetic with Whole Numbers – Addition – Contd. Examples: Solve the following addition problems Together (Ex. 3 pg. 11) 1. 4,561 + 56 2. 41,395 + 4,327 + 497,625 + 32,991 Class Examples 1. 658 added to 831 equals ? 2. 2,038 + 2,243 + 3,139. ○ Round your answer to the nearest thousand 3. Word problem: pg.15 #69

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Arithmetic with Whole Numbers – Subtraction Subtraction is the process of finding the difference of two or more numbers 8 – 5 = 3 MinuendSubtrahend Difference 8 9 5 5 - 2 4 3 2 ones tens hundreds thousands 6 5 2 3 Minuend Subtrahend

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Arithmetic with Whole Numbers – Subtraction When a digit in the Minuend is less than its corresponding digit in the Subtrahend, borrowing is necessary 1 2 3 4 - 4 8 5 ones tens hundreds thousands Subtrahend Minuend

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Arithmetic with Whole Numbers – Subtraction 1 2 3 4 - 4 8 5 947 2 1 1 1 01 Minuend Subtrahend Difference

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Arithmetic with Whole Numbers – Subtraction Subtraction Phrases (study pg. 18) 5 minus 5 equals 0 (5 - 5) 10 less than 87 is 77 (87 – 10) 18 less1 is 17 (18 - 1) Examples: Together ○ You had $415 on your student debit card. You use your card to buy $197 in books, $48 in art supplies, and $24 in concert tickets. What is the new balance on your debit card? ○ 146 Class Examples 1. 70,702 – 4239 = ? 66463 2. 46,005 minus 32,167 is ? Round your answer to the nearest thousand 13838, 14000 3. Word problem: pg.23 #110

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Arithmetic with Whole Numbers - Multiplication Multiplication is the process of finding the product of 2 or more numbers Multiplication notations: 7 x 8, 7. 8, 7(8), (7)(8), (7)8 8 x 5 = 40 Factor Product

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Arithmetic with Whole Numbers - Multiplication Multiplication properties 1. Multiplication Property of Zero ○ Product of a number and zero is 0. 534 x 0 =0 2. Multiplication Property of One ○ Product of 1 and any number is that number 2345 x 1 = 2345 3. Commutative ○ Order of multiplication is insignificant 2 x 3 = 3 x 2 since 6 = 6 4. Associative ○ Grouping of numbers in any order gives same result (Make sure to use Order of Operations..later) (4 x 2) x 3 = 4 x (2 x 3)

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Arithmetic with Whole Numbers - Multiplication Multiplication of large numbers Steps 1. Beginning with the ones digit in one of the factors, multiply it by each of the digits in the other factor. 2. Carry the tens digit in the subsequent product (if necessary) to the next digit in the other factor. 3. Repeat this process for each digit moving to the left, adding the carried tens digit each time Easier when viewed by example

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Arithmetic with Whole Numbers - Multiplication The product of 735 and 9 is ? To multiply larger numbers, we need to just repeat these steps for each digit in the lower product 7 3 5 x 9 5 4 1 3 6 9 x 5 = 45 9 x 3 = 27 + 4 = 31 9 x 7 = 63 + 3 = 66

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Arithmetic with Whole Numbers - Multiplication Examples 1. What is 439 x 206 ? ○ 90434 2. Find the product of 23 and 123 ○ 2829 Class Examples 1. 1050 x 4 = ? ○ 4200 2. 693 x 91 = ? ○ 3063 3. Word problem. Pg 31 #100 ○ 24 x 15 = 360 m 2

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Arithmetic with Whole Numbers - Division Division is the process of dividing a number into a desired number of sets (example divide 24 objects into 4 evenly distributed groups) Note: the quotient x divisor = dividend 4 24 Dividend Divisor 6 Quotient

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Arithmetic with Whole Numbers - Division Properties One in division ○ Any number divided by itself is 1: 7/7 = 1 ○ Any whole number divided by 1 is itself : 8/1 = 8 Zero in division ○ Zero divided by any whole number is zero 0/354 = 0 ○ Division by zero is not allowed: 354/0 Can’t do this! Division phrases: the quotient of 30 and 10 is ? (30 ÷ 10) 6 divided by 2 is ? (6 ÷ 2)

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Arithmetic with Whole Numbers - Division Long division example (no remainder) 2808 divided by 8 = ? Check the answer 351; check 351 x 8 = 2808 ○ Class Example Find the quotient of 4077 and 9. Check your answer by multiplying the dividend by the quotient. Check your answer 453 check 453 x 9 = 4077

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Arithmetic with Whole Numbers - Division What happens if divisor does not divide evenly into the dividend? We have “leftovers” or, a remainder Long Division Example with a remainder ○ Find the quotient of 2522 and 4. Check the answer. ○ 630 r2; check 630 x 4 + 2 = 2522 ○ Class Example: Find the quotient of 5225 and 6. Check your answer. 870 r 5; check 870 x 6 + 5 = 2522

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Arithmetic with Whole Numbers - Division Long division example (larger numbers) 1598 ÷ 34 ○ 47 4578 ÷ 42 ○ 109 Class Example ○ 7077 divided by 34 ? Check your answer. 208 r 5; check (208 x 34) + 5 = 7077

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Math Skills – Class 1

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Exponential Notation Exponentials are just multiplication We say 2 raised to the power of 3 or “two cubed” The exponent (or power) indicates how many times to multiply the factor by itself 2 3 Exponent (a.k.a power) Factor = 2 x 2 x 2 = 8

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Exponential Notation Terminology 6 1 – “Six to the first power” or “six” 6 2 = 6 x 6 – “Six to the second power” or “six squared” 6 3 = 6 x 6 x 6 – “Six to the _____ power” or “Six _______“ 6 4 = 6 x 6 x 6 x 6 – “Six to the fourth power” third cubed

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Exponential Notation Examples Write the following in exponential notation 1. 3 x 3 x 3 x 5 x 5 3 3 x 5 2 2. 10 x 10 x 2 x 2 10 2 x 2 2 Simplify the following 1. 2 3 x 5 2 2 x 2 x 2 x 5 x 5 2. 3 2 x 5 3 1. 3 x 3 x 5 x 5 x 5

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Order of Operations We have just discussed most the basic arithmetic operations: Addition: + Subtraction: - Multiplication: (), [ ], { }, x Division: ÷, / Exponentiation: 2 2 3 + 66 - 2 x = ?

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Order of Operations What are some of the issues we face in solving this problem? 1. Where do we start? 2. Where to next? Turn out with 4 possible answers! 36, 27, 37, and 52 Which one is right? Can you imagine: ○ 12 ÷ 32 + 3 2 + 2 (10 x 2 - 1) x 9 3 + 12 x (101 - 9 + 31) 3 + 6 x 6 – 2 = ?

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Order of Operations To answer this question, we need a set of rules called Order of Operations Please – Parenthesis (Brackets or braces) When multiple sets, work with the innermost pair first then move outward. ○ Example: (2 - 40 (3 x 2 -1)) do (3 x 2 - 1) first Excuse – Exponentials My – Multiplication Dear – Division Aunt – Addition Sally – Subtraction The Left to Right Rule ○ Do multiplication and division in order from left to right. ○ Do addition and subtraction in order from left to right

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Order of Operations Examples: Simplify the following Simple examples: 1. 15 + 5 x 10 65 2. Class Examples: 1. 5 + 8 ÷ 2 -9-9 2. 34 + 2 x 10 -54 PEMDASPEMDAS

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Order of Operations “Difficult” examples (More operations) 1. 3 (2 + 1) – 2 2 + 4 ÷ 2 7 ○ Class Examples: 1. 64 ÷ (8 – 4) 2 x 9 – 5 2 -11 2. 5 x (8 – 4) 2 ÷ 4 – 2 -18 PEMDASPEMDAS

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Math Skills – Week 1

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Factoring Factors of a given number are the numbers that divide it evenly (no remainder) Ex ○ Factors of 20 are 20, 10, 5, 4, 2, 1 ○ Factors 15 are 15, 5, 3, 1 To find the factors of a number… Divide the number by 1, 2, 3, 4, 5,… Continue until the factors start to repeat The numbers that divide it evenly are its factors

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Factoring Example: Find all the factors of 42 ○ 1, 2, 3, 6, 7, 14, 21, 42 are all factors of 42. Class Example: ○ Find all the factors of 30 30 / 1 = 30, 1 and 30 are factors 30 / 2 = 15, 2 and 15 are factors 30 / 3 = 10, 3 and 10 are factors 30 / 4 = Doesn’t divide evenly 30 / 5 = 6, 5 and 6 are factors 30 / 6 = 5, 6 and 5 are factors -Repeat so stop!!!

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Factoring Easy for smaller numbers, what about finding factors of 1,078? A little help ○ 2 is a factor of a number if the last digit is 0, 2, 4, 6, 8 436 ends with 6 thus 2 is a factor of 436 (436 ÷ 2 = 218) ○ 3 is a factor of a number if the sum of the digits of that number is divisible by 3 evenly 489 – sum of digits is 21 and 21 ÷ 3 = 7. Thus 3 is a factor of 489 (489 ÷ 3) = 163 ○ 5 is a factor of a number if its last digit is 0 or 5 520 ends with 0 thus 5 is a factor of 520 (520 ÷ 5 = 104)

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Factoring Together: Find all the factors of 72 ○ 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Class Examples: Find all the factors of 40 ○ 1, 2, 4, 5, 8, 10, 20, 40

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Prime Factorization – Prime and Composite Numbers A Prime Number is a whole number whose only factors are 1 and itself 7 is prime because its only factors are 1 and 7. Prime numbers from 1 to 50 ○ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 A Composite Number is any whole number that is not prime (i.e. it has more factors than just 1 and itself) Note: 1 is not considered prime or composite

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Prime Factorization Prime Factorization of a number is the representation of a number as a product of its prime factors Example: The prime factorization of 60 is 60 = 2 x 2 x 3 x 5 Prime factorizations will be very useful tools in helping us solve many problems in upcoming chapters

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Prime Factorization To find the prime factorization of a number: 1. Divide the number by 2 If 2 divides the number evenly, it is the first prime factor 2. Divide the resulting quotient by 2 again If it divides evenly, it is the second prime factor. If it does not divide evenly, try dividing the result by the next prime number. If that number divides evenly, it is the second prime factor. 3. Continue this process until the resulting quotient is 1. Represent this in a T-Diagram

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Prime Factorization Example: Find the prime factorization of 60 Represent in a “T-Diagram” 60 230 215 35 5 1 The prime factorization of 60 = 2 x 2 x 3 x 5 First few prime numbers: 2, 3, 5, 7, 11, 13, 17

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Prime Factorization For larger numbers, prime factorization could be difficult because we don’t know when to stop We stop when the square of the trivial divisor (number on the left side of the T-Diagram) is greater than the number we are factoring Examples: Together ○ Find the prime factorization of 1. 315 -3 x 3 x 5 x 7 2. 106 -2 x 53

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Prime Factorization Class Examples: Find the prime factorization of the following numbers: 1. 9 ○ 3 x 3 2. 31 ○ 31 3. 78 ○ 2 x 3 x 13 4. 177 ○ 3 x 59 5. 225 ○ 3 x 3 x 5 x 5 First few prime numbers: 2, 3, 5, 7, 11, 13, 17

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