1. Math Skills – Week 1
INtroduction

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

3. Introduction – Me: Career
787
Dreamliner
SW the Boeing Co.
High School  Boeing 1999

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

5. Introductions / Administrative Stuff
Welcome to Math Skills
Introductions
You
Name?
Major?
Do you like math? Why or Why not?
Highest level math class taken?
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
CH.1 - Whole Numbers

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
PLACE VALUE CHART – pg. 4
Hundred Millions
Millions
Ten
Hundred-Thousands
Thousands
Hundreds
Tens
Ones 2 3 4 1 5 6
Whole numbers are those shown. A number that has no fraction.
When a whole number is written with the digits…said to be written in standard form with each digit in a given place value. Shown is a place vale chart. Describe exactly what a place value is.

11. What are Whole Numbers? - Rounding
Rounding whole numbers allows us to approximate the number to any place value
Steps
Write out the number to be rounded in a place value chart (i.e 37)
Look at the number to the right of the place value you want to round to. (7)
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
If the number is < 5, replace all of the digits to the right of the digit in the place value with zeros
Hundreds
Tens
Ones 3 7 30 31 32 33 34 35 36 37 38 39 40

12. What are Whole Numbers? – Rounding
Round 525,453 to the nearest ten-thousand.
Steps
Write 525,453 in place value chart
What is 1st digit to the right of the ten-thousands place value?
Is it > or = 5? Is it < 5?
Answer = 530,000
Class Exercise: Round to Nearest Hundreds and Tens
Hundred-Thousands
Thousands
Ten
Hundreds
Tens
Ones 5 2 4 3
Answers:
Nearest hundreds: 525,500
Nearest Tens: 525,450

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

14. Arithmetic with whole Numbers
Math Skills – Week 1
Arithmetic with whole Numbers

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 = 4, 0+ 7 = 7
Commutative
Order of adding numbers doesn’t matter
4 + 8 = 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 97
18 Increased by 1 is 19

18. Arithmetic with Whole Numbers – Addition
Addition of larger numbers
Steps
Arrange the numbers by vertically aligning them by place value
Add digits in each column
thousands
hundreds
tens
ones

19. Arithmetic with Whole Numbers – Addition
If sum of digits in each column is > 9 need to use carrying
Steps
Arrange the numbers by vertically aligning them by place value
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.
Continue this procedure
From right to left
tens
ones
thousands
hundreds

20. Arithmetic with Whole Numbers – Addition1 1 11 11 6 8 1 1

21. Arithmetic with Whole Numbers – Addition – Contd.
Examples: Solve the following addition problems
Together (Ex. 3 pg. 11) 4,
41, , , ,991
Class Examples
658 added to 831 equals ?
2, , ,139.
Round your answer to the nearest thousand
Word problem: pg.15 #69
Together
4617
Example 3 pg. 11 = 576,338
Class
Exercise 9 pg. 13 = 1489
Exercise 27 pg. 13 = 7420, 7000
Add 110, = 118,295

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

24. Arithmetic with Whole Numbers – Subtraction1 1 1 2 1
Minuend
Subtrahend 7 4 9
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
70,702 – 4239 = ?
66463
46,005 minus 32,167 is ?
Round your answer to the nearest thousand
13838, 14000
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
8 x 5 = 40
Factor
Factor
Product

27. Arithmetic with Whole Numbers - Multiplication
Multiplication properties
Multiplication Property of Zero
Product of a number and zero is 0.
534 x 0 =0
Multiplication Property of One
Product of 1 and any number is that number
2345 x 1 = 2345
Commutative
Order of multiplication is insignificant
2 x 3 = 3 x 2 since 6 = 6
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
Beginning with the ones digit in one of the factors, multiply it by each of the digits in the other factor.
Carry the tens digit in the subsequent product (if necessary) to the next digit in the other factor.
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 3 4 x
9 x 5 = 45
9 x 3 = = 31
9 x 7 = = 66
Everyone expected to know single digit multiplication
6 6 1 5

30. Arithmetic with Whole Numbers - Multiplication
Examples
What is 439 x 206 ?
90434
Find the product of 23 and 123
2829
Class Examples
1050 x 4 = ?
4200
693 x 91 = ?
3063
Word problem. Pg 31 #100
24 x 15 = 360 m2

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 6
Quotient
4 24
Divisor
Dividend

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 47
4578 ÷ 42
109
Class Example
7077 divided by 34 ? Check your answer.
208 r 5; check (208 x 34) + 5 = 7077

36. Exponentials and Order of Operations
Math Skills – Class 1
Exponentials and Order of Operations

37. 2 = 2 x 2 x 2 = 8 3 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 3
Exponent (a.k.a power) 2
= 2 x 2 x 2 = 8
Factor

38. Exponential Notation Terminology
61 – “Six to the first power” or “six”
62 = 6 x 6 – “Six to the second power” or “six
squared”
63 = 6 x 6 x 6 – “Six to the _____ power” or
“Six _______“
64 = 6 x 6 x 6 x 6 – “Six to the fourth power”
third
cubed

39. Exponential Notation Examples
Write the following in exponential notation
3 x 3 x 3 x 5 x 5
33 x 52
10 x 10 x 2 x 2
102 x 22
Simplify the following
23 x 52
2 x 2 x 2 x 5 x 5
32 x 53
3 x 3 x 5 x 5 x 5

40. 3 + 6 6 - 2 x ? = Order of Operations
We have just discussed most the basic arithmetic operations:
Addition: +
Subtraction: -
Multiplication: (), [ ], { } , x
Division: ÷, /
Exponentiation: 22
3 + 6 x
6 - 2 = ?

41. 3 + 6 x 6 – 2 = ? Order of Operations
What are some of the issues we face in solving this problem?
Where do we start?
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 ( )

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 PEMDAS Examples: Simplify the following
Simple examples:
x 10 65
Class Examples:
5 + 8 ÷ 2 9
x 10 54
PEMDAS

44. Order of Operations PEMDAS “Difficult” examples (More operations)
3 (2 + 1) – ÷ 2 7
Class Examples:
64 ÷ (8 – 4)2 x 9 – 52 11
5 x (8 – 4)2 ÷ 4 – 2 18
PEMDAS

45. Prime Numbers and FactoRing
Math Skills – Week 1
Prime Numbers and FactoRing

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 Class Example:
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!!!
Do 42 example on the board, 1:42 , 2:21 , 3:14 , 6:7 , 7:6

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, 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)
Not so clear for larger numbers, so a good place to start is using the following rules.

49. Factoring Together: Class Examples: 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:
Divide the number by 2
If 2 divides the number evenly, it is the first prime factor
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.
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” 60
First few prime numbers:
2, 3, 5, 7, 11, 13, 17 2 30 2 15
The prime factorization of 60 = 2 x 2 x 3 x 5
Factor the number much in the same way as previously discussed, except this time we 3 5 5 1

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
315
3 x 3 x 5 x 7
106
2 x 53
106 = 2 x 53…we stop after 8 since 8^2 = 64
315 = 3 x 3 x 5 x 7….we stop because we get 1
201 = 3 x 67…we stop after 11 because 11^2 > 67

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