ADVANCED PREPARATION FOR THE
NEW
SAT EXAM
The Math Part
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A comprehensive review of the salient elements of the math section of the SAT
General strategies for maximizing the level of success
Specific strategies to deploy during the SAT exam
A repertoire of effective tactics and strategies for solving problems
Guided practice, topic by topic, on the specific material and problem types in the SAT
A collection of 4 highly representative Practice Tests and Solutions
Elliott M. Rosenberg
La Jolla Country Day School
La Jolla, CA
ISBN 978-1-934048-00-9
Copyright by Author
i
Advanced Preparation for the NEW SAT Exam – The Math Part
Table of Contents . . . . . . . . . . . . . . . . . . . . .
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About the Author / Acknowledgments . . . .
i
ii
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MODULE ONE – The Basics . . . . . . . . . . .
Getting Started . . . . . . . . . . . . . . . . . . . . . . . .
Arithmetic Operations . . . . . . . . . . . . . . .
Algebraic Concepts . . . . . . . . . . . . . . . . . .
Geometric Figures and Relationships . . .
Rules and Formulas . . . . . . . . . . . . . . . . . . . .
Fractions . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponential Arithmetic . . . . . . . . . . . . . .
Distribution . . . . . . . . . . . . . . . . . . . . . . .
Factoring . . . . . . . . . . . . . . . . . . . . . . . . .
Solving Inequalities . . . . . . . . . . . . . . . . .
Simplifying Radicals . . . . . . . . . . . . . . . .
Arithmetic Operations with Radicals . .
Geometric Figures – Perimeter and Area
Volume . . . . . . . . . . . . . . . . . . . . . . . . . . .
Surface Area . . . . . . . . . . . . . . . . . . . . . .
Coordinate Geometry . . . . . . . . . . . . . . .
Advanced Concepts . . . . . . . . . . . . . . . . . . . .
Difference of Two Squares . . . . . . . . . . . .
Squaring Numbers Ending with 5 . . . . . .
Fun with Pythagorean Triples . . . . . . . . .
Advanced Functions and Their Graphs .
1
2
2
4
5
12
12
12
13
13
14
14
15
16
17
18
18
22
22
22
22
23
MODULE TWO – The SAT . . . . . . . . . . . . . . 27
The SAT I Exam . . . . . . . . . . . . . . . . . . . . . . . . 28
Long and Short Term Pretest Strategies . . . . 34
Taking the Test . . . . . . . . . . . . . . . . . . . . . . . . . 35
Solution Strategies . . . . . . . . . . . . . . . . . . . . . . 36
MODULE THREE – Building Skills . . . . . 42
Preliminary Assessment Test . . . . . . . . . . . . 43
Guided Practice . . . . . . . . . . . . . . . . . . . . . . . 74
Arithmetic and Algebraic Operations . . 74
Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Proportions . . . . . . . . . . . . . . . . . . . . . . . . 82
Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Variation and Task Completion . . . . . . . 90
Measures of Central Tendency . . . . . . . . 94
Probability . . . . . . . . . . . . . . . . . . . . . . . . . 98
Multiple Variable Problems . . . . . . . . . . . 102
Coordinate Geometry . . . . . . . . . . . . . . . . 106
Geometric Formulas . . . . . . . . . . . . . . . . . 111
Advanced Functions . . . . . . . . . . . . . . . . . 116
Module FOUR – Four Practice Tests . . . . .
Practice Test 1 . . . . . . . . . . . . . . . . . . . . .
Answer Sheet 1 . . . . . . . . . . . . . . . . . . . .
Answer Key 1 . . . . . . . . . . . . . . . . . . . . . .
Solutions Test 1 . . . . . . . . . . . . . . . . . . . .
Practice Test 2 . . . . . . . . . . . . . . . . . . . . .
Answer Sheet 2 . . . . . . . . . . . . . . . . . . . .
Answer Key 2 . . . . . . . . . . . . . . . . . . . . . .
Solutions Test 2 . . . . . . . . . . . . . . . . . . . .
Practice Test 3 . . . . . . . . . . . . . . . . . . . . .
Answer Sheet 3 . . . . . . . . . . . . . . . . . . . . .
Answer Key 3 . . . . . . . . . . . . . . . . . . . . . .
Solutions Test 3 . . . . . . . . . . . . . . . . . . . .
Practice Test 4 . . . . . . . . . . . . . . . . . . . . .
Answer Sheet 4 . . . . . . . . . . . . . . . . . . . . .
Answer Key 4 . . . . . . . . . . . . . . . . . . . . . .
Solutions Test 4 . . . . . . . . . . . . . . . . . . . . .
Appendix – Solution Strategy Summary
121
122
137
139
140
152
167
169
170
182
196
198
199
211
225
227
228
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Advanced Preparation for the NEW SAT Exam – The Math Part
Preface
Each year, more than a million students
throughout the United States undergo the
daunting experience of applying to colleges. It is
often approached with a mixture of excitement,
anxiety, and trepidation. For many, this choice is
viewed as a major step in shaping their future,
and positioning themselves to pursue, and
hopefully then realize their dreams. For others, it
is an important step in finding themselves, and
learning what they really want to do with the rest
of their lives. Either way, the choice of course of
study, and the endeavor to find a school of
higher learning that can unlock the opportunities
must be viewed as a process – the accumulation
of knowledge and skills, and the compilation and
refinement of this personal package to represent
an attractive match between a college’s goals for
assembling a Freshman class, and the attributes
offered by each student candidate.
One rather significant component of the
admissions process is the assessment of scores
achieved by each student on standardized tests.
For many years, the most commonly accepted
benchmark of all College Entrance Exams has
been the SAT. While the specifics of the testing
format, and even the definition of what the letters
S A T stand for has changed over the years, the
test is still used widely by colleges as a way of
comparing the academic preparation and the
abilities of students who apply for admission.
Just as the whole college application experience
can be viewed as a process, preparation for the
SAT must also be approached methodically as a
process. The exam is designed to provide an
assessment of the knowledge, aptitude, and
reasoning abilities of each student. So in a way,
every aspiring applicant, has been preparing for
this test for many years. However, the
comprehensiveness of the examination, and the
manner in which the test is structured, worded,
and scored, positions it as a type of test where
serious attention to preparation can provide a
student with a strong competitive edge.
This Advanced Preparation for the NEW SAT
Exam – The Math Part publication is designed
to leverage the preparation process by providing
the student with the following:
1) A comprehensive review of the salient
elements of the math section of the SAT.
2) General strategies for maximizing the level
of success by providing insights to prudent steps
to follow throughout the preparation period
leading right up to taking the actual SAT.
3) Specific strategies to deploy during the SAT
exam to ensure that a solid preparation effort is
then complemented with strong execution
during the test.
4) A repertoire of effective tactics and strategies
for breaking down a problem, recognizing the
key concepts, and developing an efficient
solution approach.
5) Guided practice, topic by topic, on the
specific material and problem types typically
included in the test.
6) A collection of four highly representative
Practice Tests that are modeled after the actual
SAT exams. These model tests provide students
with the opportunity to gain highly relevant
experience for taking the SAT, and a means of
assessing their progress and readiness for this
challenge.
It should be understood that this book offers no
tricks or other methods for divining the correct
answers to the problems in the math section of
the SAT. It does, however, provide a methodical
approach to building the mathematical analysis
and processing skills needed to maximize a
student’s performance on the exam. This text
can be used effectively by any student working
independently on his or her own. It is also a
useful resource for those who engage the
support of a tutor to assist them with the
preparation. The pace at which you choose to
work with this guide can be flexible. However,
the amount of material covered in the SAT is
significant, so adequate lead time must be
factored into your equation for success.
Advanced Preparation for the NEW SAT Exam – The Math Part
About the Author
Elliott Rosenberg has a BS in Electrical
Engineering from Pennsylvania State University
(1968) and a MS in Computer Science from
Rensselaer Polytechnic Institute (1971). He had a
35-year career in the Business Information
Systems field with United Technologies
Corporation. Subsequently, he served for 9 years
as a Learning Skills Instructor at Watkinson
School in Hartford, Connecticut. He is currently a
Substitute Math Teacher at La Jolla Country Day
School in La Jolla, California. Mr. Rosenberg has
also been tutoring students in mathematics and
SAT Preparation for 40 years.
The material in this book reflects the author’s
many years of experience using and teaching
mathematics. The specific emphasis on
knowledge
requirements
and
preparation
strategies for the Math Part of the NEW SAT I
Exam is designed to provide both strong and
challenged students with the understanding of
math and effective test taking tactics that will help
boost their scores in the College Entrance Exam.
Acknowledgments
Sincere appreciation is due to Joel E. Rosenberg
and Douglas A. Levy for their invaluable
support in reviewing, editing, and validating the
accuracy, appropriateness, and balance of the
material in this book.
This book is a key resource for helping students
prepare for the SAT College Entrance
Examination. SAT is a registered trademark of
the College Entrance Examination Board. While
references are made to test content and scoring
procedures documented on the College Board
website (www.collegeboard.com), and in other
publications of this organization, the College
Entrance Examination Board was not involved
in producing this text.
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ADVANCED PREPARATION FOR THE
NEW
SAT EXAM
The Math Part
Elliott Rosenberg has assembled everything a student might need for thorough SAT math
preparation. He includes thoughtful explanations of content, user-friendly testing
strategies, and sample tests with complete solutions. He has a gift for finding the simplest
route to a solution without sacrificing understanding. His brevity and clarity are amazing.
This is a go-to resource for SAT preparation.
Paige A. Kenausis
Mathematics Teacher and SAT tutor