Transcript SAT Math Prep

SAT MATH PREP
Introduction
• Danny Pham
• [email protected]
• Graduated from California State University in Los
Angeles
• Hobbies include watching Professional League of
Legends, reading Japanese Comics, and
Cooking
Class Schedule
• Weekly
Week 1 – Numbers and Operation
Week 2 – Algebra and Geometry I
Week 3 – Algebra and Geometry II
Week 4 – Statistics and Probability
Week 5 – Review
• Daily Class Breakdown
Review Homework and Quiz (25 minutes)
Lecture (65 minutes)
Quiz (30 minutes)
[Everything here is tentative]
Class Set Up
• Power Points, Homework and Quiz Answers will all be
posted online
• Email will be sent out how to access all of these on
Monday
• Homework and Quizzes will be graded and there will be a
Final Grade at the end
• The more effort you put in, the more successful you’ll be
SAT Math Format
• 3 sections broken down into 20/18/16 questions to be
done in 25/25/20 minutes
• Total of 54 questions with only 44 having the ¼ penalty for
being incorrect
• 10 Student Response (Grid-in) Problems
Today’s Topic – Numbers and Operations
1. Basic Strategies
2. Definitions
3. Translating Word Problems
4. Basic Math
5. Logic
6. Sets
7. Counting Techniques
8. Sequence and Series
Basic Strategies I
• Read each Question Carefully
• Circle the Question and Underline Clues
• Example:
In a class of 78 students 41 are taking French, 22
are taking German. Of the students taking French or
German, 9 are taking both courses. How many students
are not enrolled in either course?
Basic Strategies II
• Use Calculator whenever possible
• Process of Elimination/Brute Force
• Example:
Of the following, which is greater than ½ ?
A. 2/5
B. 4/7
C. 4/9
D. 5/11
E. 6/13
Definitions I
• Numbers
Definition II
• Prime numbers – 2,3,5,7,11,13,…
• Even numbers – 0, 2, 4, 6, …. , 2k, ….
• Odd numbers – 1, 3, 5, 7, ….. , 2k + 1, ….
• Consecutive Numbers –
1,2,3,4 or 7,8,9,10 or 2,4,6,8 or 5,10,15,20
• Factors – Numbers that can be divided into another
number; Factors of 12 : 1,2,3,4,6,12
• Multiples – Multiples of a given number are numbers that
can be divided by that number with remainder;
Multiple of 7: 7, 14, 21, 28, 35, …
• Numerator/Denominator – Top and Bottom of a Fraction
Reciprocal – Flip the Top and Bottom of a Fraction.
Division Algorithm
• When “a” is divided by “b”, it can be written as
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𝑎 = 𝑏𝑥 + 𝑟
where x is the quotient, r is the remainder and 0≤ 𝑟 ≤ 𝑏
Example:
If k is divided by 7, the remainder is 6. What is the
remainder if k+2 is divided by 7?
𝑘 = 7𝑥 + 6
𝑘 + 2 = 7𝑥 + 6 + 2
𝑘 + 2 = 7𝑥 + 7 + 1
𝑘 + 2 = 7(𝑥 + 1) + 1
Translating Word Problems
Add
Subtract
Multiply
Division
Equal
Sum
Difference
Product
Quotient
Is
Increased by
Decreased by
Times
Each
Was
More Than
Less Than
Double/Twice
Share Equally Has
Total
Fewer Than
Each
Every
Are
Together
How Many
More
In all
Per
Will Be
Combined
How Much is
Left
Of
Out of
Gives
Percent
Yields
Change
Basic Math I
• Fractions?
• Ratio – Relationship between two entities expressed
using “to”, :, and as a fraction
• Example:
3 dogs to 4 cats, 6 dimes : 1 marble, $3/lbs
• Unit Conversion Example:
Michael is making wind chimes for a fundraiser. He can
make one wind chime every 10 minutes. How many wind
chimes can he make from now until the fundraiser starts
in 3 hours?
1 𝑤𝑖𝑛𝑑 𝐶ℎ𝑖𝑚𝑒
•
10 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
×
60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
1 ℎ𝑜𝑢𝑟
×
3 ℎ𝑜𝑢𝑟𝑠
1
= 18 𝑤𝑖𝑛𝑑 𝑐ℎ𝑖𝑚𝑒𝑠
Basic Math II
• Proportion – Equation relating two Fractions/Ratios
• Example:
The ratio between the length of a box to its width is 5:2. If
the length of the box is 15 inches, what is its width?
• 𝐿𝑒𝑡 𝑤 = 𝑖𝑡𝑠 𝑤𝑖𝑑𝑡ℎ
•
𝐿𝑒𝑛𝑔𝑡ℎ 5
:
𝑊𝑖𝑑𝑡ℎ 2
• Use Cross Multiplication
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5𝑤 = 30
𝑤=6
=
15
𝑤
Basic Math III
• Decimal – Decimals can be obtained by dividing the
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numerator by denominator in fractions
Percent – Per hundred which means divide by hundred
Example: 30% means 30/100 or .30
Scientific Notation – A way to write big numbers as a
multiple of 10
Example: 12345 could be written as 1.2345 × 104
When re-writing from Scientific Notation to Standard,
positive exponent means move the decimal that many
times to the right and negative means move that many
times to the left.
Basic Math IV
• Exponents – The number of times by which a number
multiplies itself
Example 35 = 3 × 3 × 3 × 3 × 3
• Rules for Multiplying and Dividing numbers with the same
base and different exponent
• Multiply – Add the exponent
Divide – Subtract the exponent
• Example:
24 × 26 = 210
𝑦3
1
−3
= 𝑦 = 3
6
𝑦
𝑦
Basic Math V
• Zero exponent for any base except 0 is equal to 1
Example:
50 = 1
• When a number with an exponent is raised to another
exponent, the two exponent are multiplied
• Example:
33 2 = 36 = 729
• Use Calculator
Sets I
• Sets – Sets are list of numbers that are primarily
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organized in brackets
Example : A{2, 4, 6, 8} , B{4, 8, 12, 16}
Union – A set collecting all units of each set
Example : {2, 4 , 6, 8, 12, 16}
Intersection – A set collecting only the common units
Example : {4, 8}
Subset – A smaller set made from a bigger set
Example C{4, 8} C is a subset of A
Complement – If a subset contains some units from a
bigger set, the complement would be a set that contains
the rest.
Sets II
• Example:
In a class of 78 students 41 are taking French, 22 are
taking German. Of the students taking French or German, 9
are taking both courses. How many students are not
enrolled in either course?
A. 6
B. 15
C. 24
D. 33
E. 54
• 41 + 22 − 9 = 54
• 78 − 54 = 24
Logic
• Definitions:
• And – Must meet both Conditions
• Or – Must have at least one Condition
• Exclusive Or – One or the other
• Counter-Example – One example to prove any statement
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to be false
If—Then Statements – Includes a Hypothesis and
Conclusion: If P then Q.
Converse – If Q then P
Inverse – If not P then not Q
Contrapositive – If not Q then not P.
Logic II
• Example:
All of Kay’s brothers can swim.
If the statement above is true, which of the following must
also be true?
(A) If Fred cannot swim, then he is not Kay’s brother.
(B) If Dave can swim, then he is not Kay’s brother.
(C) If Walt can swim, then he is Kay’s brother.
(D) If Pete is Kay’s brother, then he cannot swim.
(E) If Mark is not Kay’s brother, then he cannot swim.
Logic III
• Example:
The set consists of all multiples of 6. Which of the following
sets are contained within ?
I.The set of all multiples of 3
II.The set of all multiples of 9
III.The set of all multiples of 12
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
Counting Techniques I
• Permutation – A way by which a set of things can be
arranged. Order matters so the number of thing in each
set is decreased by 1 upon selection.
• Example 1:
Five people are running in a race. How many different
ways can the people be placed at the end of the race?
5 × 4 × 3 × 2 × 1 = 120
• Example 2:
• Five people are running in a race. How many different
ways can first, second, and third be placed?
5 × 4 × 3 = 60
Counting Techniques II
• Combination – A way to select things out of another group
and order doesn’t matter.
• Example –
• 12 kids are working on a project. 2 kids are designated to
buy materials for the project. How many different pairs of
kids can be chosen to buy the materials?
12 × 11 = 132
132 ÷ 2 = 66
• 𝑛𝑃𝑟 =
• Calculator
𝑛!
,
𝑛−𝑟 !
𝑛𝐶𝑟 =
𝑛!
𝑟! 𝑛−𝑟 !
Sequence and Series I
• A Sequence is an Ordered List of Numbers defined by a
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Function 𝑎𝑛 with Domain in Natural Numbers
Example: 5,12,19,26,….
𝑎𝑛 = 7𝑛 − 2
Arithmetic Sequence – Sequence that have a common
difference “d” between each term
General Formula: 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
Geometric Sequence – Sequence that have a common
ratio “r” between each term
Example: 4,16,64,256,…
General Formula: 𝑎𝑛 = 𝑎1 𝑟 𝑛−1
Sequence and Series II
• Example Problem:
-20, -16, -12, -8…
In the sequence above, each term after the first
is 4 greater than the previous term. Which of the
following could NOT be a term in the sequence?
(A) 0
(B) 200
(C) 440
(D) 668
(E) 762
Sequence and Series III
• Series – Summation of all terms in a Sequence or up until
a certain point
• Arithmetic Series :
𝑛
𝑖=1 𝑎𝑖
=
𝑛(𝑎𝑛 +𝑎1 )
2
• Example – Add 1 – 100
𝑛)
𝑎
(1−𝑟
1
𝑛
• Finite Geometric Series:
𝑖=0 𝑎𝑖 =
1−𝑟
𝑎1
𝑛
• Infinite Geometric Series:
𝑖=0 𝑎𝑖 = 1−𝑟