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Welcome Welco me to the SAT SAT Teaching Teaching Systems We’ve developed our educational package to integrate you, your students, the video component, We’ve and the supplemental materials into an effective learning system.
The program delivers information in a clear clear,, concise, example-lled manner that teaches with the perspective of the learner in mind. The supplemental material allows students structured opportunities to practice and enhance their knowledge of basic and advanced concepts. Each module contains the following items: a lesson plan, worksheets, and various testing components, and a practice exam.
The Lesson Plan has three parts:
• Pre-viewing reviews the basic elements of the SA SAT T test. • Viewing the program offers a fun fast-paced fast-paced way to teach important concepts. • Post-viewing provides worksheets to reinforce the concepts taught in the video. Testing components consist o:
• Worksheets that have your students practice the material to reinforce reinforce the concepts concepts and topics introduced.
• PracticeTestwhichcoversallthelearning PracticeTestwhichcoversallthelearningobjectivesandcanbe objectivesandcanbeusedeitherasahome usedeitherasahomework work assignmentorasapracticetestinclass.
We hope that you and your students nd Teaching Systems benecial and enjoyable. Be sure to check out Cerebellum.com for special offers, new subjects, and other great resources!
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SAT MATH
Welcome Welco me to the SAT SAT Teaching Teaching Systems We’ve developed our educational package to integrate you, your students, the video component, We’ve and the supplemental materials into an effective learning system.
The program delivers information in a clear clear,, concise, example-lled manner that teaches with the perspective of the learner in mind. The supplemental material allows students structured opportunities to practice and enhance their knowledge of basic and advanced concepts. Each module contains the following items: a lesson plan, worksheets, and various testing components, and a practice exam.
The Lesson Plan has three parts:
• Pre-viewing reviews the basic elements of the SA SAT T test. • Viewing the program offers a fun fast-paced fast-paced way to teach important concepts. • Post-viewing provides worksheets to reinforce the concepts taught in the video. Testing components consist o:
• Worksheets that have your students practice the material to reinforce reinforce the concepts concepts and topics introduced.
• PracticeTestwhichcoversallthelearning PracticeTestwhichcoversallthelearningobjectivesandcanbe objectivesandcanbeusedeitherasahome usedeitherasahomework work assignmentorasapracticetestinclass.
We hope that you and your students nd Teaching Systems benecial and enjoyable. Be sure to check out Cerebellum.com for special offers, new subjects, and other great resources!
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Lesson Plan Video: 150 minutes
Lesson: 3 days
Pre-Viewing
• :00 Warm Warm Up: The Math section covers arithmetic, percentages, decimals, order of operations, fractions, averages,
ratios, statistics, probability, geometry, geometry, functions, and algebra (including higher level “Algebra II”). • :00 Test-Prep: In each math section of the SAT, SAT, the questions are arranged in order of difculty. To To help you allocate your time on the SAT, SAT, we like to label the questions with three t hree degrees of difculty: the Good, the Bad and the Ugly. The student-produced response questions are the only part of the SA SAT T Math Section that are not multiple choice. These questions require you to ll in your own answer by marking the ovals on your answer grid. Viewing
• :04 Playing Video: Since this workbook and the SA SAT T Math module cover the same material, you can watch one whole module then do the relevant workbook part, part, watch part of the module and work on that part of the workbook, try the workbook rst rst and then watch the DVD–it’s all up to you! The great thing about the DVD is that you can always go back and review any sections or subjects that are giving you trouble. The workbook and the video are an unbeatable tagteam combo. • :04 Wrap Up: When you’re ready, ready, you can have students take the Practice Tests Tests provided on the t he CD-ROM. The idea i dea is that if you take these t hese tests in a setting similar to the real tests your students will be better prepared come test day.
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SAT Math Section Contents Drill 1: The Good, the Bad, and the Ugly Drill 2: Student-Produced Response Questions Drill 3: Denitions Drill 4: Percentages Drill 5: Percent Increase & Decrease Drill 6: Decimals Drill 7: Fractions Drill 8: Average Questions Drill 9: Median & Mode Drill 10: Square Roots Drill 11: Exponents Drill 12: Ratios Drill 13: Proportions Drill 14: Algebraic Manipulation Drill 15: Inequalities Drill 16: Simultaneous Equations Drill 17: Absolute Value, Direct & Inverse Variation Drill 18: Quadratic Equations Drill 19: Functions Drill 20: Domain and Range Drill 21: Functions as Models Drill 22: Algebra: Experiments Drill 23: Algebra: Using Actual Numbers Drill 24: Algebra: Working Backwards With the Answers Drill 25: Probability Drill 26: Geometry: Angles Drill 27: Geometry: Triangles Drill 28: Geometry: Perimeter Perimeter,, Area, Parallel Lines Drill 29: Geometry: Circles and Volume Drill 30: Coordinate Plane and Slope
7 8 9 9 10 11 11 13 14 15 16 17 19 20 22 23 24 26 26 28 30 33 34 36 37 38 40 43 45 47
The math portion of this workbook needs to be completed with a calculator, the same one that you will use on the actual test. It is also important that you practice your SAT work under quiet, test-like conditions to create the same kind of environment that you will experience on the day of the test. ™
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Drill 1: The Good, the Bad, and the Ugly In each math section of the SAT, the questions are arranged in order of difculty. To help you allocate your time on the SAT, we like to label the questions with three degrees of difculty — the Good, the Bad and the Ugly. Every student needs to do all the questions listed as “Good” and “Bad.” Picking up points on these questions is crucial. Do not rush through the Good and Bad questions to get the Ugly ones. On the 25-minute, 20-question multiple-choice math section:
The Good…Questions 1 to 8. The Bad…Questions 9 to 17. The Ugly…Questions 18 to 20. Everyone needs to do at least 1–17. On the 20-minute, 16-question multiple-choice math section:
The Good…Questions 1 to 5. The Bad…Questions 6 to 12. The Ugly…Questions 13 to 16. Everyone needs to do at least 1–11. As you attempt each question, you need to know if it is Good, Bad, or Ugly. The expectation of how difcult the question is will help you avoid traps. If you are shooting for a 500… you need to do all the Good and Bad questions. Do not worry about the Ugly questions. If you are shooting for a 600… you need to do 18 questions on both of the 20-question sections and 14 on the 16-question section. If you are shooting for a 700… you need to do all the questions. Everyone wants to score as high as possible on the SAT. However, you can’t realistically shoot for 700 until you can get to 600. Likewise, you can’t shoot for 600 until you can get 500. Improvements come in steps. Increasing the number of questions you attempt in a section leaves less time for the Good questions. Thus, doing more questions before you are ready can actually lower your score.
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Drill 2: Student-Produced Response Questions The student-produced response questions are the only part of the SAT Math Section that are not multiple choice. These questions require you to ll in your own answer by marking the ovals on your answer grid. Your answer will be graded as correct whether it is entered as a fraction or a decimal, as long as the answer ts into the grid. If your answer to a question is 100 over 200, this would need to be reduced because it does not t into the four slots available to grid-in your numbers.
Practice entering in your answers. Hint: Ifittsintheanswergrid,youhaveyouranswer.Don’treduceorroundoff
ifyouareabletotinyourresponse.Savethetime.
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(Answersareonpage48-50)
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Drill 3: Defnitions All numbers except fractions and decimals. For example: -7, 0, 2 are all integers. Even: Divisible by 2. -4 is even. 8 is even. And don’t forget that 0 (zero) is even, too. Odd: Not divisible by 2. Positive: Greater than 0. 1/2 is positive. So is 0.4 and 100. Negative: Less than 0. Prime numbers: A prime number is only divisible by itself and 1. Whole numbers: Any number except for fractions and negatives. Digits: The numbers 0 through 9. Consecutive numbers: Numbers that are in order. 2, 3, 4, 5, etc. Distinct: Numbers that are different. 4 and 3 are distinct. 4 and 4 are not distinct. Order of Operations: PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Divisibility: Dividing so that there is nothing left over. For example, 8 is divisible by 4 since 4 divides perfectly into 8. 9 is not divisible by 4 since 4 does not go perfectly into 9. Remainder: The part left over when you divide. When 9 is divided by 4 the remainder is 1. Multiples: Numbers that your original number divides into perfectly. Multiples of 4 would be 4, 8, 12, 16, 20, 24, etc. Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, etc. Factors: All the numbers that divide perfectly into your original number. The factors of 16 are 1, 2, 4, 8, 16. The factors of 50 are 1, 2, 5, 10, 25, 50. Zero: Zero is even and an integer. However, 0 (zero) is neither positive nor negative. Integers:
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Drill 4: Percentages REMEMBER: You can convert a percent to a decimal
by moving the decimal point two places to the left. (35% = .35) Or you can convert a percent to a fraction by placing it over 100. (35% = 35/100) Remember, a percent is simply a part over a whole, times 100. (Answers are on pages 40-43.) 1. What is 40% of 70? Problems:
2. 80% of 25 = 3. Which is greater, 30% of 45, or 45% of 30? 4. 60 percent of 40 percent of 300 is equal to which of the following? (A) 12 percent of 300 (B) 18 percent of 300 (C) 20 percent of 300 (D) 24 percent of 300 (E) 30 percent of 300 5. In a class of 24 students, 9 students scored between 80% and 90% on a test, 3 scored over 90%, and 4 scored between 70% and 80%. What percentage of students scored below 70% on the test? (A) 66% (B) 50% (C) 33% (D) 24% (E) 13%
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Drill 5: Percent Increase & Decrease Here is the formula: Percent increase or decrease = number increase or decrease / original whole
(Answers are on pages 40-43.) 1. A student was able to read 30 pages in an hour. After taking a speed reading course, the student was able to read 45 pages an hour. By what percent did the student’s reading ability increase? (A) 15% (B) 30% (C) 45% (D) 50% (E) 75% Problems:
2. During the rst semester at law school, there were 350 students enrolled. At the start of the second semester, there were 270 students. By approximately what percent did the rst-year student body decrease? (A) 15% (B) 23% (C) 31% (D) 37% (E) 45% 3. After a stern memo was circulated at the ofce, monthly production levels of new computers went up 25%. If 232 computers a month were being produced before the memo, how many were being produced a month after the memo? (A) 240 (B) 258 (C) 290 (D) 312 (E) 348
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Drill 6: Decimals Don’t be fooled into wasting time on decimal problems like these below. Pull out your calculator and start pushing buttons. When practicing on these problems, be sure to use the same calculator you intend to use when you take the SAT. Problems: (Answers are on pages 40-43.) 1. 4.02 + 6.679 = 2. 5.31 + 7.006 = 3. 4.9 - 6.23 = 4. 7.67 x 3.1 = 5. 9.24 ÷ 3.67 =
Drill 7: Fractions Here are the basics: To add or subtract fractions, nd a common denominator. To multiply fractions, just multiply the numerators by the numerators, and the denominators with the denominators. And to divide fractions, ip the second fraction over, and then multiply them. Problems:
(Answers are on pages 40-43.)
1. 2/3 + 4/5 =
5. 6/7 x 19/21 =
2. 3/4 + 7/12 =
6. 12/13 x (-5/8)
3. 8/9 – 7/3 =
7. 2/5 ÷ 3/7 =
4. 3/7 – 7/8 =
8. 9/25 ÷ 5/3 =
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Drill 8: Average Questions Average Questions (also known as “arithmetic mean”) will always involve three pieces of information:
the average, the number of items being averaged, and the total sum of all the things being averaged. A typical SAT question will give you two of these three pieces of information. Your job is to gure out the missing piece. There are 3 scenarios:
1) To solve for an average, divide the total sum by the number of items. 2) To nd the total sum of all the items being averaged, multiply the average by the number of items being averaged. 3) To nd the number of items being averaged, divide the total sum by the average. Problems:
(Answers are on pages 40-43.)
1. Bobby took 3 tests and scored an 87, 93, and 99. What was the average (arithmetic mean) of his three test scores?
2. If Doug’s average phone bill for the year came out to 40 dollars per month, how much money did Doug spend on his phone bill for the entire year?
3. If 25 is the average of 14, x, and 40, what is the value of x?
4. The average (arithmetic mean) of three numbers is 29. If two of the numbers are 21 and 24, what is the third number? (A) 13 (B) 29 (C) 42 (D) 45 (E) 87
5. What is the average (arithmetic mean) of all even integers from 1 to 20 inclusive? (A) 8 (B) 10 (C) 11 (D) 12 (E) 20 6. The top three students at Tony Clifton High School averaged a 96 test score on the Spanish nal. If the average of two of the students was 94, what did the third student score on the test to bring their collective average up to 96? (A) 90 (B) 94 (C) 96 (D) 98 (E) 100
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7. An airline sold 60 coach tickets, each at a price of $200. This same airline also sold 20 rst-class tickets, each for $600. What was the average cost of a ticket on this ight? (A) 200 (B) 300 (C) 400 (D) 510 (E) 710 8. If a basketball player averaged 34 points a game during a 6-game series, and scored 54 during the sixth and nal game of this series, how many points did the player average over the rst 5 games? (A) 28 (B) 30 (C) 34 (D) 38 (E) 54
9. For the 5 months from January 1st until the end of May, a bus service that operates between Los Angeles and Las Vegas sold an average of 600 round trip tickets per month. If the company sold 900 tickets in January and 600 tickets in February, what is the average number of tickets that were sold in March, April and May? (A) 500 (B) 600 (C) 900 (D) 1500 (E) 3000
Drill 9: Median & Mode The median is dened as the middle number in a group of numbers.
In order to determine what the median is, it is important to put your numbers in ascending order. If your set has an even amount of numbers, take the average of the middle two numbers to nd your median. The mode is the number that appears most often in a set of numbers. One good way to remember what mode means is to think “the most” since mode and most sound alike. Also important to note is that there can be more than one mode in a set of numbers. Problems:
(Answers are on pages 40-43.)
Find the median in each of the following sets of numbers: 1. (6, 9, 10, 2, 5)
Find the mode in each of the following sets of numbers: 4. (3, 2, 4, 5, 2, 4, 4)
2. (2, 3, 4, 5, 1)
5. (1, 1, 2, 3, 4, 5, 6, 6, 7)
3. (8, 2, 4, 1)
6. (12, 15, 15, 15, 16, 17, 17)
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7. What is the median of the rst 3 positive multiples of 7? (A) 3 (B) 7 (C) 14 (D) 21 (E) 28
8. What is the average (arithmetic mean) of the mode of set A and the median of set B? (A) 12 (B) 23 (C) 24 (D) 25 (E) 31
Set A = (12, 13, 15, 19, 23, 23, 24, 30) Set B = (12, 8, 27, 25, 31)
Drill 10: Square Roots The square root of a number is the number that needs to be multiplied by itself, or squared, to get to that
number. Rules:
1) You CAN multiply or divide square roots. For example: √2 × √5 = √10 2) You CANNOT add or subtract square roots. For example: √2 + √5 ≠ √7 Hints:
1) The square root of zero is still equal to zero. √0 = 0 2) The square root of one equals one. √1 = 1 3) The square root of any fraction between zero and one gets LARGER. √1/4 > 1/4 Thus, when taking the quantitative comparison section of the SAT, don’t automatically assume that a square root makes a number smaller. It can be equal or even larger. Problems:
(Answers are on pages 48-50.)
1. 3√16 – 3√9 = (A) 3 (B) 4 (C) 5 (D) 3√7 (E) 7
2. √10 × √5 = (A) 5 (B) 5√2 (C) 2√25 (D) 25√2 (E) 50√5
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3. √1/4 + √1/25 = (A) 2/29 (B) 1/5 (C) 1/4 (D) 1/2 (E) 7/10
Drill 11: Exponents An exponent is the number of times a base is raised to a power. Rules:
1) When you multiply exponents together, you actually add them. When you divide exponents, you subtract. x 2 multiplied by x3 equals x5. 2) As was the case with square roots, you CANNOT ADD or SUBTRACT exponents. X 2 + X3 does not equal X5. Hints:
1) Zero squared equals zero and one squared still equals one. A number squared can still equal itself. Also squaring a fraction or decimal actually makes the number smaller. 1 /4 = 1/16 These are both important concepts when attacking quantitative comparison questions. 2) Please note that -10 2 = -100 since a negative times a negatives gives a positive. So when you are solving an exponent question the answer can be the positive or negative version of itself. Thus if x 2 = 100, x can equal 10 or -10. This concept is very important for quantitative comparison questions. Problems:
(Answers are on pages 40-43.)
1. (7) (104 ) + (2) (103 ) + 4 = (A) 7,204 (B) 70,204 (C) 72,004 (D) 72,040 (E) 72,404
3. If y4 = 81, then 2y = (A) 2 (B) 3 (C) 4 (D) 8 (E) 16
2. x2 = 8, then x4 = (A) 64 (B) 32 (C) 16 (D) 8 (E) 4 ™
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Drill 12: Ratios A ratio compares how many parts you have of two or more things. If a 1st grade class trip has 1 parent for every 5 children, then the ratio of parents to children is 1 to 5. This can be written as 1:5 or 1/5; they both mean the same thing. The important thing to note is that these expressions are comparing the ratio of parents to children. These numbers are not necessarily the actual number of parents and children on the trip, but for every 1 parent on this trip, there will be 5 children as well. This relationship will remain constant.
Most SAT ratio questions are designed to have you add up the parts in your ratio. 1 parent for every 5 children means that 6 is your total number of parts, or in this case, people. This simple addition step is usually the rst step for solving any ratio question. Now we’re in a position to answer some basic ratio questions. What fraction of the trip’s participants are parents? 1/6 (1 out of every 6 people). What fraction of the trip’s participants are children? 5/6 (5 out of every 6 people). In order to answer either of these questions, we rst needed to determine that 6 is the total number of people from which these fractions would be judged. So the rst thing we do when we come to a ratio question is to add up the parts to nd the total. 1 parent for every 5 children dealt with 6 people at a time. This is what we call the “Before.” If we are told that there are 30 people on the trip, this is what we call the “After.” In order to get your numbers to the “After,” you must gure out something that is known as the “ratio jumper.” In order to solve this problem you must ask yourself, “how do I get from the Before to the After” or specically, “What number do I multiply 6 by to get to 30?” Or you can go in reverse and gure out what you need to divide 30 by to get to 6. Either way, this work needs to be done to determine the ratio jumper. In this case the ratio jumper is 5. Once you establish this number, the rest of this question becomes calculator work because you will multiply all of the “Before” numbers by the same ratio jumper number. Parents
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(Answers are on pages 48-50.) 1. The ratio of attendance at a college basketball game was recorded as 14 students for every 1 professor. If there were 3000 people at the game, how many of them were professors? (A) 1 (B) 14 (C) 200 (D) 256 (E) 2800 Problems:
2. If the ratio of dogs to cats at an animal shelter is 7 to 5, and dogs and cats are the only animals at the shelter, what fractional part of the animals at this shelter are cats? (A) 7/5 (B) 5/7 (C) 7/12 (D) 5/12 (E) 12/35
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3. If x:y:z = 1:3:9 and z = 27, then x + y = (A) 4 (B) 12 (C) 13 (D) 18 (E) 39 4. In a room containing only children, the ratio of boys to girls is 2:3. Boys are what fractional part of the total children in the room? (A) 3/2 (B) 2/3 (C) 3/5 (D) 2/5 (E) 2/9
5. The instructions for sewing a sweater suggest using 6 feet of red yarn for every 3 feet of white yarn and 1 foot of blue yarn. If the total sweater uses 15 feet of white yarn, how much yarn is used on the entire sweater? (A) 15 (B) 50 (C) 90 (D) 150 (E) 300 6. A college basketball team has a win-to-loss ratio of 4 to 3. If the team has played a total of 35 games, how many more games has the team won than lost? (A) 1 (B) 4 (C) 5 (D) 7 (E) 20
Drill 13: Proportions A typical proportion question gives you two sets of fractions with one of the four numbers missing. Your job is to cross-multiply to solve for the missing variable. These are GREAT calculator questions. The key is to keep your numbers consistent. Problems: (Answersareonpages40-43.)
1. A weight of 3 pounds is equal to 48 ounces. A weight of 1/2 pound is equal to how many ounces? (A) 48 (B) 32 (C) 16 (D) 8 (E) 4
2. A wheel turns 60 times every 3 minutes. At this rate, how many times will the wheel turn in 4 minutes? (A) 20 (B) 40 (C) 60 (D) 80 (E) 100
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3. If a recipe that feeds 4 people uses 6 ounces of avoring, how many ounces of avoring are needed to feed 6 people? (A) 6 (B) 8 (C) 9 (D) 12 (E) 24 4. At a kindergarten lunch, each child will eat one slice of pizza. If each pizza contains 8 slices, and there are 256 children at the kindergarten, how many pizzas are needed to ensure that each child has one slice? (A) 256 (B) 128 (C) 32 (D 16
(E) 8 5. Jeff requires 7 hours of sleep per night during the ve-day school week, and 9 hours of sleep per night over his two-day weekends. How many hours of sleep does Jeff get during the course of a 16-week semester? (A) 35 (B) 53 (C) 256 (D) 716 (E) 848 Student-produced Response Question:
6. There are 22 students for every 1 teacher at an elementary school. If 14 teachers work at the school, how many students go to the school?
Drill 14: Algebraic Manipulation Most basic Algebra problems are designed to make you solve for a missing variable. In order to do these questions your job will be to isolate that missing variable on one side of the equation. You can add, subtract, multiply or divide, but remember to do it to both sides of the equation. That is the key step in algebra–you can do whatever you want, as long as you do the same thing to both sides of the equation. One good way to attack these questions is to think in terms of opposites. If you are solving for x, and one side of the equation contains 2x (which is 2 times x), you will want to divide both sides of the equation by 2. If one side of the equation contains x - 3, you will want to add 3 to both sides of the equation. Doing the opposite will often help get rid of the unwanted parts and isolate the variable on one side of the equation. Problems:
(Answers are on pages 40-43.)
1. Solve for x. 3x + 10 = 34 2. Solve for y. 2y – 5 = 19 3. If a = 4 then (2 – a)/2 = (A) -2 (B) -1 (C) 0 (D) 1 (E) 2
4. If 14 - y = 3y – 2, then y = (A) 0 (B) 2 (C) 4 (D) 6 (E) 8
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5. If (t + 3) (4 + 22) = 40, then t = (A) 1 (B) 2 (C) 3 (D) 4 (E) 8
7. If 14/a = 42/9, then a = (A) 2 (B) 3 (C) 4 (D) 6 (E) 9
6. If 4x + 20y = 88, then x + 5y = (A) 4 (B) 8 (C) 11 (D) 22 (E) 44
8. If 5x-7 = 28, then 3x = (A) 7 (B) 14 (C) 21 (D) 28 (E) 35
Drill 15: Inequalities < Means less than > Means greater than
The rule to memorize… WHEN YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER, THE LESS THAN OR GREATER THAN SYMBOL FLIPS.
If you forget to switch the sign, you will get this question wrong. Problems:
(Answers are on pages 40-43.)
1. If -2x + 10 < 20, then (A) x < -5 (B) x > -5 (C) x < 5 (D) x > 5 (E) x < 10
2. If 4x – 6 < 18 + 6x, then (A) x < 12 (B) x > 12 (C) x < -12 (D) x > -12 (E) x < 24
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3. If 2 < x < 5 and 3 < y < 8, which of the following must be true for x + y? (A) 1 < x + y < 8 (B) 2 < x + y < 8 (C) 3 < x + y < 8 (D) 3 < x + y < 13 (E) 5 < x + y < 13
4. If 3 < A < 7 and 4 < B < 10, which of the following must be true for b – a? (A) -3 < b – a < 7 (B) 1 < b – a < 3 (C) 4 < b – a < 6 (D) 7 < b – a < 17 (E) 10 < b – a < 14
Drill 16: Simultaneous Equations When dealing with simultaneous equations, rst line them up and then combine the two equations by either adding or subtracting. The most common mistake is to try to deal with the two equations separately. When you combine the equations by adding or subtracting, one of the variables will drop out and you will be able to solve for the other variable. Once you have one variable, simply substitute it back into one of the original equations to solve for the other variable. (Answers are on pages 40-43.) 1. If 2a + 5b = 20 and 3a – 5b = 30, then a = ? Problems:
2. If 2x + 4y = 10 and 3x + 5y = 20, then 5x + 9y = ?
3. If 3x + 4y = 24 and 4x + 3y = 25, then x + y = ?
4. If a = 4 + b and 3a = 12 – 2b, what is the value of a? (A) 24 (B) 12 (C) 8 (D) 4 (E) 3
5. If a + b = 16, b + c = 20, and a + c = 40, then a + b + c = ? (A) 30 (B) 32 (C) 34 (D) 36 (E) 38 6. If 3b + 4c = 30, then 12b + 16c = ? (A) 15 (B) 30 (C) 60 (D) 90 (E) 120 7. Three roses and two tulips cost $10.00 and four roses and ve tulips cost $18.00. How much do one rose and one tulip cost? (A) 2 (B) 4 (C) 7 (D) 14 (E) 28 ™
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Drill 17: Absolute Value, Direct & Inverse Variation Absolute Value Simply put, the absolute value of a number is the distance between that number and 0 (zero) on the number line. Very important point! Absolute value is a distance, so it’s ALWAYS POSITIVE. Again, because the absolute value of a number is a distance–its distance from 0 on the number line–it is always positive. What is the absolute value of 8? 8. What is the absolute value of negative 8? 8 again. Absolute value = always a positive number. Here’s how to write an absolute value: l8l = 8. This says that the absolute value of 8 is 8. l-8l = 8. This says that the absolute value of -8 is 8. (Notice that two lines are on either side of the number we’re trying to nd the absolute value of.)
Here’s another way to think of absolute value that will help you out. Take the number 8 again. 8 is the absolute value of what 2 numbers? That’s right, 8 and -8. Every positive number is the absolute value of two numbers. That number itself and its negative.
Direct Variation The equation for direct variation is: x = ky
where k is a constant, and x and y are variables.
This seems like a tricky equation, but all it’s saying is that y changes directly as x does. That means, when x changes, y changes in the same way. If x doubles, y doubles. If x triples, y triples. And so on. Just remember the equation x = ky.
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Inverse Variation The equation for Indirect Variation is: xy = k
where k is a constant, and x and y are variables.
All this is saying is that y changes inversely as x changes. When x changes, y changes in an opposite way. If x doubles, y gets cut by half. For any inverse variation question, just remember the equation xy = k. Then just plug in the numbers they give you, and use your algebra knowledge to solve for x or y, whichever the question asks for. Problems
(Answers are on pages 40-43.)
1. Give the absolute value of the following expressions: A) -4 B) -1/2 C) –√64 D) 3/4 E) -12
Give the formulas for: 2. Direct variation 3. Inverse variation
Drill 18: Quadratic Equations This is still the best way to convert an unfactored equation into a factored one. The most common quadratic equations used on the SAT are: (x + y) 2 = x 2 + 2x y + y 2 (x – y) 2 = x 2 – 2x y + y 2 (x + y) (x – y) = x 2 – y 2 FOIL... stands for First, Outside, Inside, Last.
Memorizing the above 3 equations will save you time on any SAT question that involves factoring. Problems:
(Answers are on pages 40-43.)
1. If x – y =5 and x2 – y 2 = 15, then x + y =?
2. If a/b + b/a = 8 what is the value of (a + b) (1/a + 1/b)? (A) 6 (B) 10 (C) 16 (D) 32 (E) 64
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Drill 19: Functions Functions are one of the most intimidating-looking types of problems on the math sections of the SAT. However, these questions are usually more bark than bite. Function questions will give you a strange looking symbol along with a formula next to it. Your job is to run some numbers through this formula to come up with an answer. Remember, the symbols in these problems have no mathematical value other than what the problem assigns to them. Don’t worry if you don’t recognize the symbols–no one else will either. (Answers are on pages 40-43.) Questions 1, 2, and 3 refer to the following function: a and b are distinct integers. a Ψ b equals the larger of the numbers a and b. a ΨΨ b equals the smaller of the two numbers a and b. Problems:
1. What is the value of (3Ψ2)? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
3. What is the value of (4ΨΨ6) ΨΨ (2Ψ1)? (A) 1 (B) 2 (C) 4 (D) 5 (E) 6
2. What is the value of (2Ψ3) Ψ (5Ψ4)? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Questions 4, 5, and 6 refer to the following function: For all positive integers x greater than 1, let x ♠ be the product of all positive integers less than x. For example, 4♠ = 3 × 2 × 1 = 6 4. What is the value of 3♠ × 3♠? (A) 3 (B) 4 (C) 6 (D) 9 (E) 81
5. What is the value of 5♠ - 4♠? (A) 1 (B) 2 (C) 6 (D) 18 (E) 24
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6. What is the value of 4♠♠? (A) 6 (B) 16 (C) 24 (D) 64 (E) 120 7. a and b are non-zero integers and a $ b = 5 a / b. What is the value of 4$10? (A) 10$4 (B) 7$4 (C) 6$5 (D) 6$15 (E) 2$$10
Function Notation On test day, you’ll see “ofcial” function notation. Don’t worry, it’s not difcult! F(x) This is read as “F of x”. Here’s a sample function problem. F(x) = 10x + 2 This is read as “F of x equals 10 times x plus 2. F(3) Your task is to solve this function when x = 3. [We get that from “F(3).] All we do is plug in 3 for x. 10 times 3 equals 30. 30 plus 2 equals 32. That’s it. Just be aware that they might throw in another letter other than F, but it’s all the same. G(x) This is read as “G of x.” No real difference! H(x) This is read as “H of x.” Again, no real difference. Just be prepared!
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Drill 20: Domain and Range Other concepts about functions that you’ll need to know are domain and range. The domain of a function is the set of values for which the function is dened. The range of a function is the set of the results of the function.
What does all that mean? For domain, it’s just the set of all input values for x. That is, domain describes what makes sense to plug in for x. For range, that’s just the results or the output of the function. Let’s look at a function and determine its domain. F(x) = X+2 X–3 Now, here’s something the SAT will check to see if you know; you can’t have a 0 (zero) in the denominator of a fraction. So, for this function, what numbers for x will result in a zero in the denominator? 3. If we plug in 3 for x, in the denominator we have: 3 – 3 = 0. We can’t have that! -3 is ne for x, because -3 – 3 = -6. The domain of our function is the set of all number except 3. We write that as F(x) = X+2 X–3 x ≠ 3. Here’s another thing to remember that will help you out with the domains of functions: you can’t get the square root of a negative number–it just doesn’t exist. The √-1? Doesn’t exist. √-2? Doesn’t exist. So, if in a function problem you see x (your domain) under a square root, you have to make sure the numbers you plug in for x won’t result in a negative number. Example: F(x) = √x - 2 If we plug in anything less than 2 for x, we get a negative number under the square root. For instance, let’s plug in 1 for x. 1 – 2 = -1 We can’t take the square root of that. So the domain of this function is all numbers greater than or equal to 2. We write this as: F(x) = √x - 2 Domain = x ≥ 2.
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Let’s gure out the range of function. Remember, the range of a function is the set of all the possible values that are the result of applying the function. What is the range of this function? F(x) = x2 Well, look at x 2. We know that any number squared won’t end up negative, so x won’t end up a negative number. So, x has to be a number greater than or equal to zero. The range is all POSSIBLE values, so The range of F(x) = x 2 > 0 (Answers are on pages 48-50.) 1. If the function F is dened by f(x) = x2– 6, then f(a – b) is equivalent to (A) a2 – 2ab + b 2 – 6 (B) a2 – 2ab + b2 + 6 (C) a2 + b2 – 36 (D) a2 + b2 + 36 (E) 2ab + b2 + 6 Problems:
2. Let the function K be dened by k(x) = 2 – 4x. If the domain of the function k is -2 < x < 4, what is the smallest value in the range of the function? (A) -20 (B) -14 (C) 7 (D) 14 (E) 20 3. Let the function F be dened by f(x) = 12 –x 2. If the domain of function f is -12 < x < 1, what is the largest value in the range of the function? (A) -12 (B) -6 (C) 0 (D) 6 (E) 12
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Drill 21: Functions as Models The SAT will ask you to consider some real-life situations involving functions. These require a little bit of thought, but are not particularly difcult. The graph above tells us about Suzy’s lemonade stand. It shows us Suzy’s Lemonade Stand the number of cups of lemonade she sold at different prices. 400 On the horizontal axis, you can see she sold cups of lemonade for 10 300 cents, 20 cents, 30 cents, and 40 cents. The vertical axis shows us how many cups sold–100, 200, 300, 400. 200 The graph shows us the function of how the price affects the number of cups sold. 100 d l o S s p u C
0 $0.10
$0.20
$0.30
$0.40 $0.50
Price
The question might ask something like this:
If Suzy wants to sell the maximum number of cups of lemonade, what price should she set for a cup?” Looking at the graph, you see that the line peaks at around 300 cups. Looking down, we see the price for those cups was 20 cents. So 20 cents is our answer.
Linear Functions Let’s look at linear functions. A linear function is just an equation whose graph is a straight line. Like this. You’ll need to know this formula:
Y
y = mx + b The values of x and y can vary. m is the slope of a line. X
b is the y-intercept. This is where the line
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Take a look at this graph of a linear function below:
4
For this line, m (the slope) = -2. b (the y-intercept) = 4.
3
2
1 0 -4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
Let’s look at a problem with a graph. 10 9 8 7 6 5 4 3 2 1 0 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 -1 -2 -3 -4 -5 -6 -7 -8
1
2
3
4
5
6
7
8
9 10
If the line above, line g, has a slope of -3, what is the y-intercept of line g? Let’s look at our formula: y = mx + b First off, we know that the y-intercept is b, so we’ll be solving for b. Also, they give us the slope, -3. y = -3x + b Now we need to nd x and y. We can tell from the graph above that the point (2, 4) is on line g. Let’s plug in these numbers. 4 = -3(2) + b 4 = -6 + b
-9 -10
Let’s get b by itself by adding 6 to each side. 4+6=b 10 = b Our y-intercept is 10. Now we know that line g hits the y axis at 10.
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(Answer is on page 40-43.) 1. If an Alaskan dog sled team starts a race and travels 800 miles to the nish, as shown in the graph above, between what two points did the team reach its greatest average speed? (A) A to B 10 (B) B to C 9 (C) C to D 8 7 (D) D to E 6 (E) E to F Problem:
F
5 E
4 D
3 C
2 B
1 A
0
200 300 400 500 600 700 800 900 1000
Drill 22: Algebra: Experiments Anytime you see a problem with variables, like x or y, you can avoid doing the algebra by experimenting with your own numbers. Put in your own trial numbers for each of the variables to set up your experiment. After you assign numbers for all the variables, answer the question with those numbers. Whatever answer you come up with will be YOUR answer to the question. Now go down to the answer choices and substitute in the numbers that you have invented. The answer choice that gives you YOUR answer will be the correct answer to the question. Do you need to try it again with different numbers? No. Unlike the Quantitative Comparisons, there is nothing to mess up here. Run a set of numbers through the problem, come up with your answer, and nd it in the answer choices. This will help you answer even the toughest-looking algebra questions. It is important to know that YOU WILL STILL GET THE ANSWER TO THE PROBLEM NO MATTER WHAT NUMBERS YOU USE AS LONG AS YOU ARE CONSISTENT. Since you can use any numbers you want, our recommendation is to use numbers that make the math easy to do. So 10 is probably a better number to pick than 167. (Answers are on pages 40-43.) 1. Chip can do x pushups every minute. How many pushups can Chip do in one hour? (A) x (B) 3x (C) 6x (D) 30x (E) 60x Problems:
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2. Charles is 4 years older than Alex and 2 years older than Bob. If Alex is a years old, then in terms of Alex, the sum of their ages = (A) 3 a – 4 (B) 3 a – 2 (C) 3 a + 2 (D) 3 a + 4 (E) 3 a + 6
5. The sum of three positive consecutive even integers
3. Howard is now 5 years older than John was 2 years ago. John is now j years old. In terms of j, how many years old is Howard now? (A) j – 5 (B) j – 3 (C) j – 2 (D) j + 2 (E) j + 3
6. If x is an odd integer, which of the following must also be an odd integer? (A) x – 1 (B) x + 1 (C) 2 x (D) 2 x + 1 (E) 2 x + 2
is x. What is the value of the smallest of the three
integers? (A) (x – 6)/3 (B) (x + 6)/3 (C) x /3 – 6 (D) x /3 + 6 (E) 3 x – 6
4 If x/4, x /5, and x /6 are integers, which of the following is NOT necessarily an integer? (A) x /60 (B) x /30 (C) x /20 (D) x /12 (E) x /8
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Drill 23: Algebra: Using Actual Numbers Ifaquestionasksyoutondwhatfractionorpercentsomethingis,youcanuseyourownnumberstospeedup theprocess.ThisworksforanytimetheSATtriestoget“hypothetical”withyou.
Once again, it is better to think in terms of actual numbers as opposed to variables. In everyday life you don’t do algebra, you use actual numbers. Whenever you go shopping. Whenever you go to eat. This is what you do. So, it makes sense to use actual numbers wherever possible on the SAT. Your job is to pick numbers that meet the requirements of the question. (Answers are on pages 40-43.) 1. If a and b are two consecutive odd integers then b–a= (A) 0 (B) 1 (C) 2 (D) 3 (E) 5 Problems:
2. If a (b + c) is a positive number, which of the following must be positive? I. a II. b + c III. a + b + c (A) None (B) I only (C) II only (D) III only (E) I, II, and III 3. As part of a Christmas sale, an electronics store reduces its stereo prices by a 20% discount. Then looking to spark even more business, this same store reduces its discounted price by another 25% on New Years Day. By what overall percent has the stereo been reduced in price? (A) 50% (B) 45% (C) 40% (D) 25% (E) 20%
4. Charlie does 1/3 of his homework during his lunch break and 1/2 of what remains on his ride home on the school bus. What fractional part of his homework remains? (A) 1/6 (B) 1/3 (C) 1/2 (D) 2/3 (E) 5/6 5. A clothing designer discounts last year’s merchandise by 50% of the original price. After nding no increase in sales, the designer discounts the new sales price by an additional 20%. By what overall percent has the merchandise been reduced in price? (A) 20% (B) 30% (C) 60% (D) 70% (E) 75% 6. On Monday, Joey read 1/4 of a novel for his English class. On Tuesday he read 1/3 of what was left of the book. What fraction of the book did Joey read on Monday and Tuesday? (A) 1/4 (B) 2/7 (C) 1/3 (D) 1/2 (E) 2/3 ™
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Drill 24: Algebra: Working Backwards With the Answers When you are attacking a word problem and are having a hard time setting up an equation, all is not lost. Very often you can work your way out of this predicament by going down to the answer choices. The great thing about a multiple-choice test is that one of the ve answers MUST work. It may take a few tries, but you are guaranteed to nd the answer. Answer choices on the SAT always go in increasing or decreasing order. This means that when experimenting with your answer choices, you want to start in the middle with answer choice C. This way, even if it’s not the correct answer, it can still help you determine if you need a bigger or smaller number. Thus, if C gives you an answer that is too big, you certainly don’t need to try the two answer choices that will be larger (probably D and E). The answer has to be A or B. Try either one. If it works, you’ve got an answer. If it doesn’t, you still have your answer as it will have to be the one you didn’t try. This elimination technique is a good way to speed up and not have to try all ve answer choices. This technique will come in handy on Algebra problems as well as any time you nd yourself stuck on the test. Take advantage of the test being multiple-choice. (Answers are on pages 40-43.) 1. When x is divided by 9, the remainder is 6, and when x is divided by 6, the remainder is 0. Which of the following numbers could be x? (A) 36 (B) 100 (C) 106 (D) 108 (E) 114 Problems:
2. If t + 3 is an even positive integer then t could be which of the following? (A) -3 (B) -2 (C) -1 (D) 0 (E) 2
3. If 3a + 2 = 94 − a, what is the value of a? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 4. To celebrate his SAT Math score, Jesse ordered himself a set of personalized pencils. If Jesse lost 1/4 of the pencils the rst week he used them and 1/2 of the pencils that were left the second week, and Jesse now has 3 pencils remaining, how many pencils did Jesse order originally? (A) 20 (B) 16 (C) 12 (D) 8 (E) 6
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5. Frank weighs twice as much as George but 30 pounds less than Harry. If Harry weighs 110 pounds more than George, how much does Harry weigh? (A) 110 (B) 160 (C) 190 (D) 200 (E) 220
Drill 25: Probability Probability might only be the topic of one or two questions on the entire test. To solve a probability question, you need to gure out the total number of occurrences and divide it by the number of times that the event specied in the question can happen. For example: What is the probability of rolling a die and getting a 4? If you roll a die, there are 6 different numbers you could get, 1 through 6. This means that 6 is the total number of occurrences. Getting a 4 (which is the requirement of the question) would be 1 of 6 possible things that can happen. Thus there is a 1/6 probability of rolling a die and getting a 4. In probability situations involving more than one event, gure out the individual probability of each occurrence and multiply the results together.
For example: If there is a 1/6 chance of rolling a 4, the probability of rolling two 4s would be 1/6 × 1/6, which equals 1/36. (Answers are on pages 40-43.) 1. A two-sided coin is ipped twice. What is the probability that the coin will come up tails on both ips? Problems:
2. What is the probability that it will come up heads on both ips?
3. Someone rolls two dice with faces ranging from 1 to 6. What is the probability that the dice will add up to 7?
4. A bubble gum machine contains gum balls colored red, white and blue. If blue gum balls are 1/6 of the total and there are 21 red gum balls and 14 white gum balls, what are the odds of choosing at random a white gum ball? (A) 1/6 (B) 1/3 (C) 1/2 (D) 2/3 (E) 5/6
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Drill 26: Geometry: Angles The geometry on the SAT is pretty straightforward. Basically, you need to know a little bit about circles, triangles, rectangles, and parallel lines. You won’t have to memorize any complex formulas just some basic ones. And get this–even if you forget one of the formulas, don’t worry about it. All the formulas you need are provided for you at the beginning of each math section. ALL DIAGRAMS, UNLESS OTHERWISE STATED, ARE DRAWN TO SCALE!
Angles Review:
There are 360º in a circle. This means that there are 180º in a half circle.
360˚
180˚
a
b
The sum of the angles of any four-sided gure equals 360º.
c a d
c
There are also 180º in a straight line. The sum of the angles of any triangle equals 180º.
b
When two straight lines cross each other, they form vertical angles. Vertical angles are always equal. A right angle is an angle that equals ninety degrees. b d All the angles of a rectangle are right angles. Bisect means to divide something into two equal parts. You can bisect a line or an angle. a
c
(Answers are on pages 40-43.) 1. In the gure below, line l is an angle bisector forming the smaller angles y, z, w, and v. What is the value of v + x + y? (A) 45º (B) 90º y x z (C) 135º w v (D) 180º l (E) 270º Problems:
2. In the gure below, a + c = (A) 90º (B) 130º c a (C) 140º b d (D) 170º (E) 180º
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3. In the gure below, line l bisects an angle to form the smaller angles b and c. What is the value of a + b? (A) 45º (B) 80º y x z (C) 90º w v (D) 135º (E) 180º
5. In the gure below, angle f = 25. a+b+c+d+e+f+g+h=
c
d
e
b l
4. In the gure below, angle a is equal to angle h, and angle g = 35. a +d + f = (A) 145º (B) 180º a b e f (C) 290º c d g h (D) 325º (E) 435º
f g
a
h
(A) 90º (B) 135º (C) 180º (D) 210º (E) It cannot be determined
Drill 27: Geometry: Triangles Triangle Review: An isosceles triangle is any triangle with two equal sides. In an isosceles triangle, the angles opposite the equal sides are also equal.
IsoscelesTriangle
If a triangle has three equal sides, it is called an equilateral triangle. In an equilateral triangle, all angles equal 60º.
60˚
60˚
60˚
EquilateralTriangle
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Largest Side
If a triangle has three sides of varying lengths, then the longest side is across from the longest angle, and the smallest side is across from the smallest angle. This is known as a scalene triangle.
Largest Agnle
A right triangle has an angle of ninety degrees. If you know two sides of a right triangle, you can nd the length of the third side by using the Pythagorean theorem. 5
4
Pythagorean theorem:
a2 + b2 = c2 c
b
3
Pythagorean Triplets:
3, 4, 5 right triangle 6, 8, 10 right triangle 5, 12, 13 right triangle 7, 24, 25 right triangle
a
Special Right Triangles: 30˚
2x
x√3
60˚
There are two special right triangles that show up on the SAT. These are the 30º-60º90º right triangle, and the 45º-45º-90º right triangle. These triangles can have sides of any length, but the cool thing is those lengths will always be in the same relationship to each other. On a 30º-60º-90º right triangle if the shortest side has a length of x, then the hypotenuse will have a length of 2x, and the third side will have a length of x√3
x
45˚ s√2 s
45˚ s
For the 45º-45º-90º right triangle, if the length of a side is s, then the other side will have a length of s as well, and the hypotenuse will have a length of s√2. Both of these special right triangles are included with the formulas at the beginning of each Math Section on the SAT. Remember this: if you see a right triangle on the SAT, chances are it’s either a special right triangle or a Pythagorean triplet. By recognizing these on the test, you can save yourself a lot of time and work.
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Problems: (Answers are on pages 40-43.)
1. Triangle ABC is an isosceles triangle. If Angle B = 40, what is the value of angle C? b (A) 40º (B) 45º (C) 70º (D) 90º a c (E) It cannot be determined 2. In the gure below, what is the measure of angle B? b (A) 30º (B) 45º 8 4√3 (C) 60º (D) 75º a c (E) It cannot be determined 3. If, in the gure below, ABCD is a rectangle with diagonals BD and AC that bisect each other, then AD = (A) 5 b c (B) 6 e 6 (C) 5√2 5 (D) 8 a d (E) 10 4. In the gure below, square ABEF shares a side with rectangle BCDE. If AC = 17, than diagonal BD = (A) 7 c b a (B) 10 5 (C) 13 (D) 25 f e d (E) 26
5. In the gure below, square ABCD has two diagonals, AC and BD, that bisect each other. If AB = 4, than DE = a (A) √2 b (B) 2 e (C) 2√2 4 (D) 4 (E) 4√2 c d
6. What is the value of x in the gure below? (A) 50º (B) 60º (C) 110º (D) 145º x (E) 155º 35˚
95˚
25˚
7. In the gure below, what is the value of x? (A) 90º (B) 60º (C) 45º (D) 40º (E) 30º x 30˚
80˚
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Drill 28: Geometry: Perimeter, Area, Parallel Lines To nd the perimeter of any gure, simply add the lengths of the sides together. Area is a little trickier than perimeter. Here are the formulas: The formula for area of a rectangle is length times width, or l × w. The formula for the area of a square is simply side squared, or s 2. The formula for the area of a triangle is one half times base times height, or 1/2bh.
If you have an odd shaped quadrilateral, like this, just drop a line in and divide it into two recognizable shapes.
a b c d g
e
f h
Let’s check out parallel lines. When two parallel lines are cut by a third line, eight angles are created. If you know one of those angles, you can nd the measure of the other seven. For example, if angle a equals 120º, then angle b equals 60º, because together they form a straight angle. Angle d equals 120º, because a and d are vertical angles. And angle c equals 60º, because angle b and c are vertical. On the bottom, angle e equals 120º because angle e and angle c are opposite interior angles. Angle f equals 60º, because e and f form a straight angle, angle g equals 60º, because f and g are vertical angles, and angle h equals 120º, because e and h are vertical angles.
Problems: (Answers are on pages 40-43.)
1. In the gures below, triangle ABC and quadrilateral DEFG have the same perimeter. What is the value of x? b e (A) 4 6 (B) 5 8 7 (C) 6 x f (D) 7 3 a c (E) 8 d 4 g 5
2. The gure below is made up of twelve identical squares, each with side of length 2. What is the perimeter of the gure? (A) 96 (B) 48 (C) 36 (D) 32 (E) 24
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3. In the gure below, triangle ABD is inscribed in rectangle ABCE. What is the area of triangle ABD? 8 b (A) 49 a (B) 28 4 (C) 16 (D) 12 c e d (E) 6 4. A triangle with base 4 and height 7 has an area that is one-third the area of a rectangle with a width 6. What is the length of the rectangle? (A) 4 8 b a (B) 7 5 (C) 12 4 (D) 14 c d (E) 28 5. In the gure below, rectangle ABCD is cut by two parallel lines, each of length 5. What is the area of the shaded region? (A) 8 (B) 12 (C) 14 (D) 20 (E) 32 6. In the gure below, lines l and m are parallel, and they are intersected by two other lines. What is the value of x + y? (A) 70º 70˚ (B) 110º x 70˚ l (C) 140º y 110˚ m (D) 180º (E) 210º 7. If l1 is parallel to l2 in the gure above, than a + b = (A) 300º b (B) 270º l (C) 245º l (D) 135º 45˚ a (E) 110º ™
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Drill 29: Circles and Volume When dealing with circles and volume on the SAT, it’s important to know when to use which formula. The good thing is that all the formulas you’ll need are printed at the beginning of the math section on the test, so refer to them whenever you need to refresh your memory. If all else fails, circle problems are great for estimating. And remember, whenever you can, eliminate answer choices. Here are the formulas:
The radius, r, is the distance from the center of the circle to any point on the edge of the
d
circle. Once you know the radius, you can gure out everything else. r
c
The diameter, d, is the straight line that runs from one side of the circle to the other, passing
through the center. The diameter equals 2 times the radius. The circumference, C, is the distance around the outside of the circle, kind of like a perimeter. The formula for circumference is 2 πr or πd. The area, A, of a circle is the amount of space inside the circle. The formula for the area of a circle is πr2. The formula for calculating the volume of a rectangle is L x W x h. The formula for calculating the volume of a cylinder is πr2h. Problems: (Answers are on pages 40-43.)
1. A circle has an area of 16π. What is its diameter? (A) 16 (B) 8 (C) 6 (D) 4 (E) 2 2. In the gure to the below, a right triangle is inscribed in a circle, with the hypotenuse of the triangle passing through the center of the circle. What is the area of the
shaded region? 6
8
(A) 100π (B) 100π – 48 (C) 25π – 48 (D) 25π – 24 (E) 16π – 24
3. In the gure below, two identical circles are inscribed in a rectangle. If the area of the rectangle is 72, then what is the area of one of the circles? (A) 6 π (B) 9 π 6 (C) 12 π (D) 18 π (E) 36 π 4. Kevin rolls a tire with a diameter of one foot down the street. If he rolls the tire 41 feet, approximately how many revolutions has the tire made? (A) 25 (B) 20 (C) 16 (D) 13 (E) 8
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5. Jennifer purchases a box at a garage sale. If the box measures 4 feet by 8 feet by 3 feet, what is the volume of the box? (A) 24 (B) 32 (C) 56 (D) 96 (E) 120
6. The base of a tin can has a radius of 4 and the can has a height of 7. What is the volume of the can? (A) 28 π (B) 49 π (C) 112 π (D) 128 π (E) 142 π
Drill 30: Coordinate Plane and Slope Problems:
(Answers are on pages 40-43.)
1. In the gure below, a line is to be drawn through point P so that it never crosses the y-axis. Through which of the following points will the line never pass? (A) (3, 0) y (B) (3, -1) x P(3,2) (C) (-3, -2) (D) (3, -2) (E) (3, 4) 2. In the gure below, what is the distance from point P to the origin? (A) 3 y P(4,3) (B) 4 x (C) 4.5 (D) 5 (E) 7
3. If l1 contains points A (4, -2) and B (-7, -2), what is the slope of the line? (A) –3/4 (B) –4/7 (C) 0 (D) 2/3 (E) 3/2 4. If l1 has a slope of 3/5 and contains points (3, 4) and (a, 7), what is the value of a? (A) –8 (B) –4 (C) –3/4 (D) 3 (E) 8
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MATH DRILL ANSWERS Drill 2: Student-Produced Response Questions 1. 5/10 okay as is 2. 2.5 okay as is 3. 15/35 = 3/7 need to reduce 4. 15.5 = 31/2 need to change to improper fraction
5. .5767 = .576 or .577 need to round off or drop last number (either is acceptable)
Drill 4: Percentages 1. 28 2. 20 3. They are equal; both equal 13.5% 4. D. 24 percent of 300 (take 60 percent of 40 to get 24) 5. C. 33 percent (9 + 3 + 4 = 16, so 8 scored below 70%. 8 is 33% of 24.) Drill 5: Percent Increase and Decrease 1. D. 50% (15/30 = 50%) 2. B. 23% (350 – 270 = 80, 80/350 = 22.857% and question asks for approximate answer.) 3. C. 290 (x/232 = 25%, x = 58, 58 + 232 = 290)
Drill 6: Decimals 1. 10.699 2. 12.316 3. -1.33 4. 23.777 5. 2.5177 Drill 7: Fractions 1. 22/15 2. 4/3 3. -13/19 4. -25/56 5. 114/147 6. -15/26 7. 14/15 8. 27/125 Drill 8: Average Questions 1. 93 2. 480 3. x = 21 4. C. 42 5. C. 11 6. E. 100 7. B. 300 8. B. 30 9. A. 500
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MATH DRILL ANSWERS Drill 9: Median and Mode 1. 6 2. 3 3. 3 4. 4 5. 1, 6 (two modes) 6. 15 7. C. 14 8. C. 24 (mode of set A = 23, median of set B = 25) Drill 10: Square Roots 1. A. 3 2. B. 5√2 3. E. 7/10 Drill 11: Exponents 1. C. 72,004 2. A. 64 3. D. 8 4. B. 9/1021 Drill 12: Ratios 1. C. 200 (ratio jumper = 200) 2. D. 5/12 3. B. 12 (ratio jumper = 3) 4. D. 2/5 5. B. 50 (ratio jumper = 5) 6. C. 5 (ratio jumper = 5)
Drill 13: Proportions 1. D. 8 2. D. 80 3. C. 9 4. C. 32 5. E. 848 6. 308 Drill 14: Algebraic Manipulation 1. x = 8 2. y = 12 3. B. -1 4. C. 4 5. B. 2 6. D. 22 (divide the rst equation by 4) 7. B. 3 8. C. 3x = 21 Drill 15: Inequalities 1. B. x > -5 2. D. x > -12 3. E. 5 < x + y < 13 4. A. -3 ≤ b – a ≤ 7 (test the range by making b – a as big and as small as possible)
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MATH DRILL ANSWERS Drill 16: Simultaneous Equations 1. a = 10 2. 5x + 9y = 30 (add the equations together and you have the answer) 3. x + y = 7 (add the equations together, divide by 7, and you have the answer) 4. D. 4 5. E. 38 (add all 3 equations together and divide by 2) 6. E. 120 (multiply the rst equation by 4 to get the answer) 7. B. 4 Drill 17: Absolute Value, Direct & Inverse Variation 1. A) 4 B) 1/2 C) √64 or 8 D) 3/4 E) 12 2. Direct variation: x = ky 3. Inverse variation: xy = k Drill 18: Quadratic Equations 1. x + y = 3 2. B. 10 Drill 19: Functions 1. C. 3 2. E. 5 3. B. 2 4. B. 4 5. D. 18 6. E. 120 7. D. 6Ψ15
Drill 20: Domain and Range 1. A 2. B 3. E Drill 21: Functions as Models 1. E Drill 22: Algebra: Experiments 1. E. 60x 2. E. 3a + 6 3. E. j+3 4. E. x/8 (do an experiment where x = 60) 5. A. (x – 6)/3 6. D. 2x + 1 Drill 23: Algebra: Using Actual Numbers 1. C. 2 2. A. None 3. C. 40% 4. B. 1/3 5. C. 60% 6. D. 1/2 Drill 24: Algebra: Working Backwards with the Answers 1. E. 114 2. C. –1 3. C. 2 4. D. 8 5. C. 190
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