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# Mathematics

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### Overview

The SAT Mathematics test includes both multiple-choice questions and questions that require a written numerical answer. Students’ mathematical reasoning is assessed in separate areas of number and operations; algebra and functions; geometry and measurement; data analysis; probability and statistics. These areas follow the guidelines set by the National Council for Teachers of Mathematics, and are part of the Mathematics Common Core in many states.

### Why Mathematics?

Required math classes and testing are familiar to most high school students, especially those who are preparing to attend college. Students are cleared to take college math courses by their performance on standardized tests and their high school classwork. In addition, many colleges and universities require college-level math courses as a General University Requirement for graduation, so students in all majors take some form of math.

### But I’m not in STEM!

Students preparing to go into STEM fields after graduation — science, technology, engineering, and mathematics studies– already know a lot of math is in their future. Mathematics is an essential component, because the language of mathematics is the only way to explain physical relationships. However, other fields require the discipline and logical thinking gained from learning mathematical problems. For example, students who excel in constructing the sorts of arguments to solve proofs and equations can apply those skills to law and liberal arts. Social science majors in psychology, sociology, and economics use statistical thinking as a basis for experimentation. The arts, including music, are firmly rooted in mathematical concepts of proportion, symmetry, and balance.

### Review Basic Concepts

Number and operations questions on the SAT require a review of basic and familiar concepts such as the properties of integers, word problems, fractions, ratios, and sequences. Be familiar with mathematical operations to find square roots and squares of numbers. Practice word problems, equations, place value and scientific notation, prime numbers, and ratios, proportions and percent.

### Strategies for Test Preparation

As with the other sections of the SAT, students who have a lot of experience with solving similar types of questions have an advantage. Teachers and tutors can guide individuals to improve weaknesses and supplement strengths during practice sessions. Remember that students can use approved calculators on the math tests. Questions are arranged from easy to hard, so work the most familiar problems first. Check answers carefully if there is extra time after the test.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Boulder, CO: visit: Tutoring in Boulder, CO

150 150 Deborah

### Overview

Students taking the SAT Mathematics test are assessed on areas such as number and operations, algebra and functions, geometry and measurement, data analysis, and probability and statistics. The Algebra and Functions strand includes operations on algebraic expressions, solving equations and inequalities, quadratic equations, rational systems of linear equations and inequalities, direct and inverse variation, and basic types of functions. Test preparation for SAT Mathematics includes review of specific types of questions, as well as tips and tricks to answer multiple-choice and response grid questions correctly.

### Review of Basic Algebra

Basic algebraic concepts that are often covered in high school introductory courses are tested on the SAT Mathematics portion. Students are expected to be able to perform operations on algebraic expressions such as factoring, evaluate exponents and roots, and solve equations and inequalities correctly. Other questions test knowledge of absolute value, translating words into mathematical expressions, and understanding inequalities. During test preparation, students will learn to solve problems that incorporate concepts and use the multiple-choice format to their advantage.

### Review of Systems of Equations

In order to solve systems of equations or inequalities, students must be able to solve for one variable that is correct for both sentences in the system. They can then use that value to solve for the other variable. Suppose the equations are 4x + 2y = 14 and x – 2y = 11. Adding the equations together results in 4x + x + 2y -2y = 25 or 5x = 25. If x = 5, then what does y equal? 20 +2y = 14, or 2y = -6, or y = -3 in the first equation, and 5 + 6 =11 in the second equation. A closer look at the second equation reveals that 5 + [-2(-3)] = 5 + 6, because a negative number times a negative number is a positive number.

### Review of Functions

Mathematical functions involve relationships between data points, and require students to know and apply specialized vocabulary such as domain and range. Topics such as basic function notation, functions as models, and graphs of linear and quadratic functions are covered in algebra and other courses throughout high school and college.

### Strategies for Test Preparation

Teachers usually require students to show every step of their work, as well as a correct solution, in order to get good scores on homework assignments and tests. The SAT requires students to work quickly and recognize the correct answer from a number of alternatives. Often, students can estimate the answer and eliminate distractors that are obviously incorrect through mathematical logic and reasoning.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in New Canaan, CT: visit: Tutoring in New Canaan, CT

150 150 Deborah

### Overview

The SAT Mathematics test questions assess different areas of mathematical knowledge and application such as number and operations, algebra and functions, geometry and measurement, probability and statistics, and data analysis. Data analysis questions address students’ ability to interpret information presented in tables, graphs, and charts, recognize change and trends, and analyze change by performing calculations.

### Why Graphs?

College students will encounter graphs in many textbooks and journal articles in the humanities, social sciences, physical and biological sciences and other disciplines. Part of the information is presented graphically because it is more economical, as well as adding interest and focus to the presentation. For example, a great deal of numerical data can be summarized by plotting one variable against another on a line graph. Graphic presentations are more accessible to those studying a relationship or phenomenon.

### Types of Graphs

Some of the most common graphs are pie or circle graphs, line graphs, bar graphs, and pictographs. Pie charts show relationships of facts to one another by presenting each percentage as a slice of an entire circle. Line graphs are useful as different relationships can be shown by presenting multiple lines. For example, a line graph show the high temperatures for an entire month and the low temperatures for the same month by displaying the dates along the x axis and temperatures on the y axis. Bar graphs present data along x and y axes with rectangular bars rather in lines. Sometimes bar and line graphs are combined on the same chart. Pictographs present data by using small icons to represent a quantity of data. They often accompany texts in economics, newspaper, and magazine articles.

### Data Interpretation

Many of the questions regarding graphs, charts, and tables on the SAT require more than recognizing the types of graphs. They require students to interpret the data that is presented. This requires more than looking at the chart. It is important to understand what type of information is being displayed on the graph and what the relationship is between data points. Read the labels to see what is being displayed, and be able to identify the relationships along different points in time. Read questions very carefully to make sure they are understood.

### More Strategies for Test Preparation

The same information that is presented in a graph can be presented in a chart. The chart or table can allow for precise figures to use in calculations. Many scientific journal articles require information to be presented in both charts and graphs. That way, students and other researchers can read them, do their own calculations, and draw their own conclusions.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Lauderhill, FL: visit: Tutoring in Lauderhill, FL

150 150 Deborah

### Overview

Probability and statistics concepts are tested as one of the strands of SAT Mathematics. Topics include measures of central tendency such as mean, median, and mode; elementary probability; and geometric probability. The test doesn’t contain long calculations of standard deviation or other statistical measurements.

### Why Statistics?

Probability and statistics concepts are often combined with data analysis, because some of the most common ways that data are measured mathematically involve statistics. College courses in education, the sciences, and humanities often include statistical concepts. In particular, advanced methods of statistical analysis are key to psychological and sociological research. Many articles in peer-reviewed journals demand an understanding of measurement, evaluation, and analysis to follow the arguments.

### Measures of Central Tendency

Common measures of central tendency include the mean, median, and mode. The arithmetic mean is simply the average; the median is the middle value in a list of scores when the numbers are arranged from largest to smallest, or from smallest to largest; and the mode is the most frequently occurring value in a set of scores. When a distribution is normal, such as the overall scores of a large group of students on the SAT, the mean, median, and mode are all the same. There may also be questions on the SAT regarding weighted averages, when groups do not have the same number, average of algebraic expressions, and using averages to find a particular value when one is missing.

### Elementary Probability and Geometric Probability

Students are often asked to determine the percentage of times an event may occur. If an event never happens, its probability is zero. If an event always happens, its probability is 1. Therefore, probabilities are expressed as a decimal or fraction between 0 and 1, or as a percentage. Events may be independent or dependent. Some probability questions may concern geometric figures. For example, a smaller circle may be inside a larger circle, and students may be asked to choose the probability that a point in within the area of the smaller circle. The radius of each circle is given, so the student compares the areas.

### Strategies for Test Preparation

Probability, statistics, and measurement questions are frequent within other math classes in middle school and high school, as well as any class that describes quantitative research. As in any other math strand of the SAT, familiarity and practice lead to confidence. Before the test, notice problems that involve statistical analysis in everyday life, such as in sports, weather, or politics. Use practice questions to determine strong and weak areas during preparation, and ask teachers and tutors to help you clarify any weak areas. During the test itself, use the test booklet to draw pictures, underline terms, and eliminate distractors.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Augusta, GA: visit: Tutoring in Augusta, GA

150 150 Deborah

### Overview

Students taking the SAT Mathematics test are assessed on areas such as number and operations, algebra and functions, geometry and measurement, data analysis, and probability and statistics. The Geometry and Measurement strand covers topics such as geometric notation, concepts such as points, lines, and angles in the plane. Geometric figures such as triangles, quadrilaterals, and other polygons; areas and perimeters, circles, basic solid geometry, and coordinate geometry are also discussed.

### Geometric Notation

Some of the multiple-choice questions on the SAT will refer to basic geometric notation, such as lines parallel to each other ∥, the definitions of point, line, and plane, and measuring the lengths of line segments according to their endpoints. Students will be required to read and interpret figures and diagrams in order to choose the correct answer. Angles are presented with both rays on the plane, so some questions will refer to supplementary and vertical angles, perpendicular lines, and complementary angles.

### Triangles

Students may be required to apply the definitions of isosceles and equilateral triangles, as well as know the properties of all types of right triangles and be able to apply the Pythagorean Theorem. Some questions may refer to congruent and similar triangles. Some may refer to the triangle inequality; that the sum of the lengths of any two sides of a triangle is greater than the length of a third side.

### Perimeter, Area, and Other Measurements

The area and perimeter of many figures can be calculated by knowing the relationships between their parts. For example, a square is a parallelogram with 4 equal sides and 4 right angles. The diagonals of the square form two special 45-45-90 triangles. The measurement can be calculated by the lengths of the side or by the length of the diagonal. Although formulas to calculate measurements are presented at the beginning of the test, practice before the test is necessary to save time and use them correctly.

### Coordinate Geometry

Some of the questions will present diagrams that are located in the Cartesian coordinate plane, and students will be required to read and interpret those diagrams. Watch carefully for questions involving the slope of a line, finding midpoints of a line segment, and finding the distance between and two points by applying the Pythagorean Theorem. Draw pictures in the test booklet if there are not diagrams available.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Smyrna, DE: visit: Tutoring in Smyrna, DE

#### Math Review of Binomial Distributions

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### Overview

A binomial experiment  has a fixed number of independent trials, and each trial has only two possible outcomes.  Each of the trials has the same probability of success.  The probability distribution is called a binomial distribution.

### What Types of Problems Are Binomial?

All the characteristics of a binomial experiment are present, then the distribution will be binomial.  A quiz has 8 questions, and each question has 4 alternatives.  If a student guesses randomly on every question, what is the probability of getting 5 or more correct?  It is a binomial experiment because it has a fixed number of trials (8), and the student is guessing randomly, so each question is independent of every other.  Each question can either be answered correctly or incorrectly.  In addition, each question has the same probability of success.

### What Types of Problems Are Not Binomial?

In a problem that is not a binomial problem, all the characteristics are not present.  For example, suppose a standard deck of cards is used, and the number of aces in 5 trials are recorded, but the cards are not replaced after each trial.  It is not a binomial distribution, because every trial depends on one another.  During the first trial, the population of cards is 52, the second, 51, the third, 50, and the fourth 49, so the trials are not independent.  Similarly, if there are three possible outcomes, rather than two, the problem will not be binomial.

### What Does a Binomial Distribution Look Like?

A binomial distribution is symmetrical, with the smallest values on either side of the mean, when it is graphed with the probability held constant, the number of trials on the x axis and the function on the y axis.  The more trials are done, the more the distribution spreads out, and the more the distribution flattens.

### How Is the Binomial Distribution Used?

Many types of real-world situations have a fixed number of  independent trials with only two outcomes.  For example, treatments often have outcomes of success or failure, basketball players either hit or miss free throws, and potential voters either cast their votes or not in a particular election.  Binomial distributions can be used to determine the probabilities of those events and others like them.

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#### Properties of Probability

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### Overview:

Probability is the measure of the likelihood of an event.  The basic mathematics of probability theory started with games of chance, but it can be applied to many situations, from weather forecasting to politics.  Probabilities range from 0 (no likelihood) to 1 (certainty), and are expressed as rational numbers.

### What Is the Sample Space?

The sample space is the set of all possible outcomes of an event.  For a coin toss, the coin will either result in heads or tails.  For the roll of one die, the sample space is all the values on the faces of the die, or a set of {1, 2, 3, 4, 5, 6}.  For the roll of a pair of dice, the sums will be in a set from 2 (both dice give you a 1), the smallest sum possible, to 12 (both dice give you 6), the largest sum possible.

### What Is A Fair Experiment?

In a fair experiment, all possible outcomes are equally likely.  The probability of any outcome is related to the total number of outcomes by a ratio of the number of outcomes in that event to the number of all possible outcomes of the event (the sample space).  Therefore, the probability that a coin will be heads is 1/2.  The probability that if one die is rolled, the number on top will be a 3 is 1/6.

### What If Events Are Not Equally Likely?

Sometimes, possible outcomes can be combined in such a way so that not all outcomes are equally likely.  Suppose two fair coins are tossed: there are 4 possibilities in the sample space {HH, HT, TH, TT}.  The probability of each event when order is important equals 1/4 for each possibility. However, if the question is merely “How many heads come up when two coins are tossed?”, there are only 3 possibilities in the sample space, 0 heads, 1 head, or 2 heads.  The event 0 heads is defined as {TT}, and the event 2 heads is defined as {HH}.  However, there are two possibilities for 1 head, either {HT} or {TH}, so the outcomes are not equally likely.

### What Are Mutually Exclusive Events?

If events are mutually exclusive, it means that neither sample space A or sample space B contain common elements.  Therefore, the probabilities can be added to form the probability of one event or the other occurring.  The sample space for tossing two fair dice and getting a sum of 7 consists of {(6,1), (1, 6), (2, 5), (5,2), (3,4), and (4,3)}.  The sample space for tossing two fair dice and getting a sum of 11 consists of {(6,5), (5,6)}.  The sample space for getting either a sum of 7 or a sum of 11 is the union of both sets.

TestPrep Academy is the premier SAT/ ACT services company for high school studies. We offer instructional programs and curriculum for students preparing for the PSAT, ACT and SAT.

#### Math Review of Random Numbers

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### Overview:  What Are Random Numbers?

In a set of random numbers, the numbers do not follow any pattern.  Each number has an equal probability of occurring, and each number event is independent of any others .  Most of the time, numbers that are close to random are generated by computer programs or calculator programs designed to do just that.

### What Are Some Examples of Random Events?

Many events are close to random. For example, individual molecules within a gas tend to move randomly, so that it cannot be predicted where an individual molecule will be.  Similarly,  the theory of radioactivity predicts that a percentage of atoms in a substance will decay into isotopes given an amount of time, but it does not predict precisely which particular atom will decay.

### How Are Random Numbers Found?

Numbers that are close to random can be found by consulting random number tables, as the result of computer programs to generate random numbers, and by using a calculator to generate random numbers.  In addition, games that depend on giving all the players a fair chance are often determined by using dice or a spinner divided into equal parts.  That way each number has an equal probability of being chosen.

### How Is Randomness Used In Statistics?

In scientific experiments, statistics are used as a tool to judge the results of an experimental treatment.  Subjects have an equal chance to be assigned to a treatment condition through random assignment.  Often a random number generator is used in order to assign subjects to treatment groups.  Not only does each subject have an equal chance of being assigned to any of the treatment groups, but the choice of any one subject is independent of all the others.  This minimizes errors that could occur if the assignment is not random, but confounded.

### What Are Monte Carlo Methods?

Monte Carlo methods are mathematical simulations that use random numbers to generate solutions.  They are called Monte Carlo methods because the earliest studies used equipment to generate random numbers similar to the methods in gambling casinos, such as decks of cards and roulette wheels.  Some simulations include the common practice of airlines overbooking seats on flights, because there are probabilities generated for how many passengers will arrive to take seats.  The computer simulation can be run for the number of seats on the airplane as the number of trials, using random numbers to suggest a possible outcome.

Test Prep Academy is the premier SAT/ACT services company for high school students. We offer instructional programs and curriculum for students preparing for the PSAT,  ACT, and SAT.

#### Measures of Central Tendency: Mean, Median, and Mode

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Overview:  What Are Measures of Central Tendency?

Measures of central tendency represent the most typical score or value in a group of scores on some measure in either a population or a sample of that population.  Along with measures of variability, they convey much information about a distribution.  The three most common measurements are the mean, median, and mode.

What is the Mean?

The mean is the average value of all the numbers in the data set, and is a representation of the population mean.  It is computed by summing all values in the data set, and dividing by the number of values.  Many statistical calculations use the mean value to determine if treatment conditions have had an effect by comparing the mean of a pretreatment sample with a post treatment sample.

What Is the Median?

The median is the value in the middle if all the numbers in a data set are arranged from largest to smallest.  It is often reported rather than the mean value , if the values between the largest value and the smallest value are widely separated, such as with salaries or the price of housing in a community,  This is because the median is less affected by extreme values.  When calculating the median, first arrange the values from largest to smallest, listing all instances of a value as many times as it occurs.   Suppose the president of a company makes \$200,000 a year, the vice president makes \$50,000 a year, 2 supervisors make \$25,000 a year, 4 sales representatives make \$21,000 a year,  2 custodians and 2 warehouse workers make \$15,000 a year, and 3 clerical workers make \$12,000 a year.  The median salary is \$21,000.

What is the Mode?

The mode is the measure that occurs most frequently.  In the above example, there are 2 modes, the salary of the 4 sales representatives of \$21,000, and the salary of the 2 warehouse workers and 2 custodians, at \$15,000.  It is not as useful as a measure of central tendency because it can be affected by scores that are not the center of the data set, but are in other places in the data set.  If a third warehouse worker were hired at \$15,000, the new mode would be below the mean or the median.

Which Measure Is the Most Useful?

If a distribution has only one mode and is symmetric, such as the normal curve that reflects scores on the SAT and ACT, the mean will be the most useful.  If there is a large difference between the mean and the median, as when a distribution has many scores that are higher than the mean or lower than the mean, the median may be the most useful.  It is less affected by outlying scores, and may give a better picture than the mean.  However, the mean is most often used in many other statistical measures and is the most often reported.

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#### Review of Solid Geometry on the SAT and ACT

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Overview:  What Is Solid Geometry?

Solid geometry was developed after plane geometry as a way to describe the three-dimensional world and the objects in it.  In the ideal three-dimensional world, objects exist with faces and angles, depth and volume.  Objects are regular, or consist of a combination of regular objects, just as two-dimensional figures can be made from a combination of other two dimensional figures.  Solid geometry studies the properties, dimensions, and relationships of regular solids, as well as pyramids, cones, cylinders, prisms, spheres, and other solid objects.

What Are Regular Solids?

Regular solids have congruent faces, and the same number of faces meet at each vertex.  A solid with four regular triangular faces is called a tetrahedron, a cube has six square faces, an octahedron has eight triangular faces, a dodecahedron has 12 faces that are shaped like regular pentagons, and an icosahedron has 20 triangular faces.  They are symmetrical, and the only polygons that can fulfill all the requirements for sides, edges, and angles are triangles, squares, and pentagons.

Pyramids, Cones, Cylinders, Prisms

Some other solid figures do not fit the image of strict regularity.  For example, polygons other than triangles form the base of pyramids, yet the other faces are triangular.  Cones usually have a circular or elliptical base and an infinite number of faces tapering to a single point.  Cylinders are formed by the solid enclosed when two circles in parallel planes are joined. Prisms are a special class of solids with a polygonal base, a translation of that base in a parallel plane (similar to a cylinder) and sides that join them.

Spheres and Spheroids

Spheres are perfectly round in three dimensions.   They have many distinct properties, such as every point on the surface of the sphere is the same distance from the center, and that the width and the girth do not vary in a perfect sphere.  Objects that are close to spheres are spheroids and approximate many of the properties of spheres.  They may be elliptical rather than perfectly round, or polyhedrons that have a large number of sides.  Spherical geometry is the branch of trigonometry dealing with spheres and spheroids.

Applications of Solid Geometry

Many geometric solids have applications in nature and technology. Common crystal shapes include the tetrahedron, cube, and octahedron.  In addition, most viruses have the shape of a regular polyhedron because a basic unit protein can be repeated and packed into the smallest space possible.  The most common sphere is a ball or globe, and the most common spheroid is the Earth itself.

Do you need to know more about how test questions on the SAT or ACT are scored? Learn more about how we are assisting thousands of students each academic year.

Test Prep Academy is the premier test prep and private tutoring company for college-bound students. Our highly qualified test prep tutors deliver one-on-one personalized instruction that fit our student’s busy schedule.