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Math Skills – Week 1. Introduction – Me: Education  ????????????? - UC Davis – 2004 - Highest Honors - BS Physics - Minor Mathematics - Senior thesis.

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Presentation on theme: "Math Skills – Week 1. Introduction – Me: Education  ????????????? - UC Davis – 2004 - Highest Honors - BS Physics - Minor Mathematics - Senior thesis."— Presentation transcript:

1 Math Skills – Week 1

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

3 Introduction – Me: Career  SW the Boeing Co.  High School  Boeing Dreamliner

4 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?

5 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?  8:45  Syllabus Review Questions?  Books

6 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

7 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?

8 Math Skills – Week 1

9 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

10 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 PLACE VALUE CHART – pg. 4

11 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) 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

12 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

13 What are Whole Numbers? – Rounding  Examples: Round the following numbers to the given place value – Tens ○ ,439 – Hundreds ○ 1, ,607 – Thousands ○ 44, ,834,250 – Millions ○ 72,000,000

14 Math Skills – Week 1

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

16 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+ 7 = 7 Commutative ○ Order of adding numbers doesn’t matter = 8 + 4, is the same as

17 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 Increased by 1 is 19

18 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 ones tens hundreds thousands

19 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 ones tens hundreds thousands

20 Arithmetic with Whole Numbers – Addition

21 Arithmetic with Whole Numbers – Addition – Contd.  Examples: Solve the following addition problems  Together (Ex. 3 pg. 11) 1. 4, , , , ,991  Class Examples added to 831 equals ? 2. 2, , ,139. ○ Round your answer to the nearest thousand 3. Word problem: pg.15 #69

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

23 Arithmetic with Whole Numbers – Subtraction  When a digit in the Minuend is less than its corresponding digit in the Subtrahend, borrowing is necessary ones tens hundreds thousands Subtrahend Minuend

24 Arithmetic with Whole Numbers – Subtraction Minuend Subtrahend Difference

25 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 = ? ,005 minus 32,167 is ? Round your answer to the nearest thousand 13838, Word problem: pg.23 #110

26 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

27 Arithmetic with Whole Numbers - Multiplication  Multiplication properties 1. Multiplication Property of Zero ○ Product of a number and zero is x 0 =0 2. Multiplication Property of One ○ Product of 1 and any number is that number 2345 x 1 = 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)

28 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

29 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 x x 5 = 45 9 x 3 = = 31 9 x 7 = = 66

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

31 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

32 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)

33 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

34 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 = 2522 ○ Class Example: Find the quotient of 5225 and 6. Check your answer. 870 r 5; check 870 x = 2522

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

36 Math Skills – Class 1

37 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

38 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

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

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

41 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 ÷ (10 x 2 - 1) x x ( ) x 6 – 2 = ?

42 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: ( (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

43 Order of Operations  Examples: Simplify the following Simple examples: x Class Examples: ÷ x PEMDASPEMDAS

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

45 Math Skills – Week 1

46 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

47 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 / 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!!!

48 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, 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 ends with 0 thus 5 is a factor of 520 (520 ÷ 5 = 104)

49 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

50 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

51 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

52 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

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

54 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 x 3 x 5 x x 53

55 Prime Factorization  Class Examples: Find the prime factorization of the following numbers: 1. 9 ○ 3 x ○ ○ 2 x 3 x ○ 3 x ○ 3 x 3 x 5 x 5 First few prime numbers: 2, 3, 5, 7, 11, 13, 17


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