Conquering SAT MathBook

402 Pages·2016·3.42 MB·English

McGRAW-HILL’s
CONQUERING
SAT MATH
SECOND EDITION
Robert Postman
Professor of Mathematics and Education
Mercy College, New York
Ryan Postman
Mathematics Department
Pascack Hills High School, New Jersey
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DOI: 10.1036/0071493417
ACKNOWLEDGMENTS
This book shows you how to get your highest possible score on the New SAT:
Mathematics Test. Hundreds of high school students, just like you, field test-
ed this book. Much of what is here reflects these students’ opinions about
what was most helpful to them. The book has also been reviewed by many
high school teachers and college faculty, and it incorporates many of their
suggestions about what will be most helpful to you.
This book has over 600 realistic SAT problems with explained answers.
Many of those problems are in five SAT Math Practice Tests. The rest of the
problems are in the chapters that review all of the mathematics concepts on
the new SAT. These review chapters have hundreds more examples and prac-
tice exercises to help you sharpen your mathematics skills.
Special thanks go to Judy Brendel, Mathematics Supervisor, who sup-
ported the development of this book. Particular thanks go to John Uhl, Janet
Telesmanich, Charleen Martinelli, Kevin Killian, and Michelle Gaeta. These
talented mathematics teachers reviewed the SAT Math Practice Tests.
Thanks are due also to the hundreds of high school students who reviewed
every part of this book. Thanks are also due to Texas Instruments for per-
mission to use images of their calculators.
Professional
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CONTENTS
CHAPTER 1: INTRODUCTION / 1
CHAPTER 2: NUMBERS AND OPERATIONS ON NUMBERS / 8
CHAPTER 3: FACTORS AND MULTIPLES / 20
CHAPTER 4: RATIOS AND PROPORTIONS / 32
CHAPTER 5: PERCENTS / 43
CHAPTER 6: MEAN, MEDIAN, AND MODE / 54
CHAPTER 7: POWERS AND RADICALS / 65
CHAPTER 8: BASIC ALGEBRA / 76
CHAPTER 9: COORDINATE GEOMETRY / 91
CHAPTER 10: FUNCTIONS AND MATHEMATICAL MODELS / 103
CHAPTER 11: TRIANGLES / 122
CHAPTER 12: QUADRILATERALS AND OTHER POLYGONS / 148
CHAPTER 13: CIRCLES / 164
CHAPTER 14: INTERSECTING LINES / 179
CHAPTER 15: SOLIDS / 193
CHAPTER 16: COUNTING, PROBABILITY, AND LOGIC PROBLEMS / 210
CHAPTER 17: DATA INTERPRETATION / 224
iv CONTENTS
CHAPTER 18: SAT WORD PROBLEMS / 245
CHAPTER 19: SAT MATH PRACTICE TEST 1 / 255
CHAPTER 20: SAT MATH PRACTICE TEST 2 / 285
CHAPTER 21: SAT MATH PRACTICE TEST 3 / 313
CHAPTER 22: SAT MATH PRACTICE TEST 4 / 343
CHAPTER 23: SAT MATH PRACTICE TEST 5 / 369
CHAPTER 1
INTRODUCTION
Conquering the New SAT Math covers all the mathematics topics and all the
problem types on the new version of the SAT. It is the SAT you will take.
Every page in this book was field tested by high school students and reviewed
by experienced high school math teachers. We listened very carefully to stu-
dent and teacher recommendations to create a book that will work for you.
You learn mathematics by doing mathematics. The review sections of this
book contain hundreds of examples and practice problems. The review sec-
tions also feature worked-out model SAT problems—more than 300 practice
SAT problems with explained answers. There are another 270 SAT problems
with explained answers in five SAT Mathematics Practice Tests. That’s more
than 600 worked-out SAT practice problems in all.
We review each of the 17 major areas on the SAT in 17 separate chapters.
Each chapter begins with a clear review and practice with answers. Next
come several worked-out SAT problems to show you how to apply the math-
ematics concepts to the SAT format. Then comes a whole set of practice SAT
problems in each chapter, with explained answers. The problems generally
increase in difficulty.
The five mathematics practice tests follow the review chapters. Each test
reflects the mix of mathematics problems you will find on the real SAT. The
tests are deliberately a little more difficult than the actual SAT. You will get
an estimated mathematics score range for each test you complete.
We show you strategies for answering SAT multiple choice and “grid-in”
math questions. However, the most important strategy is to think mathe-
matically, just like the people who make up the test do. These test writers al-
ways have particular mathematics skills and concepts in mind as they write
an item. Knowing how to spot those concepts is usually the secret to getting
a high score. You’ll learn that from this book.
You Decide How Much Review You Need
This book is designed to help you no matter how much SAT mathematics
preparation you need. Which type of student are you?
✓I need mathematics review.
✓I need to practice SAT problems.
✓I need to take practice SAT Tests.
Work through the review chapters and then take the practice tests.
✓I need to practice SAT problems.
✓I need to take practice SAT Tests.
Skip the teaching and practice in the review chapters and go directly to the
Model Problems in each chapter. Complete the practice problems in each
chapter and take the practice tests.
✓I just need to take practice SAT Tests.
Skip the review chapters and go to the practice tests.
1
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2 CONQUERING SAT MATH
MATHEMATICS TEST OVERVIEW
There are two types of SAT mathematics problems, multiple choice and stu-
dent response (grid-in). Here is an overview of the Math sections of the SAT
Two 25-minute test sections
One 20-minute test section
Total 70 minutes
Multiple-choice items 44 items
Grid-ins 10 items
Total 54 items
Math Is Not my Favorite Subject
OK, we understand that. Most of the other parts of the SAT reflect things you
might do in a normal week—reading, understanding, and writing. The only
time most people do SAT mathematics problems is on the mathematics SAT.
But, that’s the way it is. If things ever change, we’ll be the first to tell you.
Until then, we’ll show you the way to your highest possible score.
MATHEMATICS TEST STRATEGIES
There are different strategies for each problem type. Let’s take them in order.
Multiple Choice
Remember, a multiple choice item always shows you the correct answer.
Each multiple choice item has five answer choices. SAT scoring deducts 1/4
point for each incorrect answer. That means that, on average, you’ll lose a
small fraction of a point if you just guess an answer. However, if you are sure
that one or more of the answers are wrong, on average, you will gain if you
guess. It’s just simple mathematics.
Eliminate and Guess
If you know the answer, fill in that oval on the answer sheet. If you don’t
know the answer, and you can definitely eliminate at least one answer, cross
off the answers you know are incorrect right in the test booklet. Then guess
from among the remaining answers.
Use the Test Booklet
The test booklet is yours. Do your work in the test booklet. Cross off incor-
rect answers in the test booklet. Draw diagrams—whatever you like.
Darken the Correct Oval
The machine that scores multiple choice tests does not know if you are right
or wrong. The machine just “knows” if you have filled in the correct oval, so
it is possible to know the correct answer but be marked wrong.
The fewer times your eyes go from the test sheet to the answer sheet, the
less likely you are to make a recording error. We recommend this practice.
Write all the answers next to the problem for a two-page spread of questions.
Then transfer the answer to the Answer Sheet.
CHAPTER 1 / INTRODUCTION 3
2. In which of the following choices would the symbol > create an incorrect
statement?
7 3
A.
10 4
A
2 1
B.X .66 .5
3 2
5 5
C.
7 9
7 3
D.
8 4
5 4
E.
6 9
Estimate
Estimate the answer. An estimate may lead you to the correct answer or it
may enable you to cross off some answers you know are incorrect.
Work From the Answers
You know that one of the answers must be correct. You may be able to find
the correct answer by substituting answers in the problem.
Grid-ins
The answer format for grid-in items is shown below. Write your answer in
the four spaces at the top of the grid. But the scoring machine just reads the
ovals you fill in. You must fill in the correct oval to receive credit.
/ /
. . . .
0 0 0
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
6 6 6 6
7 7 7 7
8 8 8 8
9 9 9 9
You may enter the digits and symbols shown in each column. The symbols
represent the fraction line and the decimal point. These choices put some
natural limits on the answers. You’ll read about those limits in this section.
How to Grid-in an Answer
Here are some examples of grid-in answers. Remember, you must fill in the
ovals to receive credit.
5
Here’s how to enter the fraction in the grid. Notice that we entered the
13
numerator and the denominator separated by the slash, and then we filled in
the corresponding ovals.
4 CONQUERING SAT MATH
5 / 1 3
/ /
. . . .
0 0 0
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
6 6 6 6
7 7 7 7
8 8 8 8
9 9 9 9
3
Say that your answer is . There are several ways to fill in the grid.
5
3
You can grid in . You can grid in the decimal equivalent. 0.6
5
3 / 5 0 . 6
/ / / / / /
. . . . . . . .
0 0 0 0 0 0
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9
If your answer is 197, you can fill in the grid two different ways.
1 9 7 1 9 7
/ / / / / /
. . . . . . . .
0 0 0 0 0 0
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9
2
The fraction is equivalent to the repeating decimal 0.666 ... You can grid-
3
2
in , or .666 or .667.
3