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The Ultimate SAT Math Strategies Guide Created by Sherman Snyder Fox Chapel Tutoring Pittsburgh, PA Welcome to The Ultimate SAT Math Strategies Guide. Unlike the study guides available on the shelves of your local book store, this unique guide focuses on math strategies rather than math content . Navigation through the strategy guide is entirely up to you! Feel free to choose your own path…….just click on the links at the bottom of each page to advance to the next topic or example. The choice is yours to make. The recommended starting point is the student success model. From there you will be able chose one of three paths to navigate: Test Tips, Definitions, and Math Strategies. Go to Success Model

Math SAT Success Model Test Taking Tips Math Definitions & Concepts Math Strategies Student Success Return to Introduction

SAT Test Taking Tips Two Rules Back to Success Model

Math Definitions and Concepts The Top 25
Return to Table of Contents Math Definitions and Concepts The Top 25 Back to Success Model

Math Strategies Math Topics Back to Success Model

Absolute Value Back to Top 25 Definition: How far a number is from zero. An alternative definition is the numeric value of a quantity without regards to its sign. The absolute value of a number is always positive or zero. The symbol “|….|” is used to denote absolute value of a quantity. Applications: Values: |6.5| = 6.5; |- 3.2| = 3.2; |0| = 0 Solving equations: |x - 5| = 3 Solving inequalities: See math strategy Graphs of functions: See math strategy

Arc Back to Top 25 Definition: An unbroken part of the circumference of a circle. An arc can be measured by its length or by its central angle. When measured by its central angle, the arc has the same degree measure as the central angle. arc central angle Applications: Finding the length of an arc Finding area of a sector Finding internal angles of an isosceles triangles with one vertex at the central angle isosceles triangle

Average (arithmetic mean)
Return to Table of Contents Average (arithmetic mean) Back to Top 25 Definition: The most commonly used type of average on the SAT sum of values number of values average (arithmetic mean) = Applications: Usually involves values expressed in terms of variables, not numerical values. See math strategy Note: You will never be asked to calculate the mean of a list of numbers. Such questions always ask for the median, not the mean of the list.

Average Speed Back to Top 25 Definition: The total distance traveled by an object divided by the total time traveled total distance traveled total time Average speed = Applications: Word problems that involve the motion of an object Caution: If a question involves the motion of an object at two different rates and asks for the overall average speed of the object, the correct answer will be the average of the two given rates if and only if each segment of motion occurs over the same time period. If the motion of each segment occurs over the same distance, the above definition of average speed must be applied.

Bisector Back to Math Definitions Definition: A line segment, line, or plane that divides a geometric figure into two congruent halves. Applications: Most common application involves angle bisectors. angle bisector

Central Angle Back to Top 25 Definition: An angle whose vertex is at the center of a circle. The measure of a central angle is also the measure of the arc that the angle encloses. Applications: Finding the length of an arc Finding area of a sector Finding internal angles of an isosceles triangles with one vertex at the central angle Note: You will never be asked questions about inscribed angles Applications: See isosceles triangle central angle inscribed angle

Diagonal Back to Top 25 Definition: A line segment joining two non-consecutive vertices of a polygon. In the figure, the three dashed lines are diagonals Applications: Finding the number of diagonals in a polygon of “n” sides (see example) Finding the number of possible triangles formed by all diagonals from one vertex of the polygon

Digit Back to Top 25 Definition: The set of integers from “0” to “9” in the decimal system that are used to form numbers. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Note: The number zero is contained in the set of digits Applications: Formation of integers

Directly Proportional
Return to Table of Contents Directly Proportional Back to Top 25 Definition: A relationship between two variables in which the ratio of the value of the dependent variable to the value of the independent variable is a constant. If y is proportional to x, then y/x is a constant. This can be written in equation form as y =kx where k is a proportionality constant. Applications: See math strategy Proportions, ratios, and probability are closely related in many applications. See math strategy

Distance Between Points
Return to Table of Contents Distance Between Points Back to Top 25 Definition: The distance between two points on a number line is the absolute value of the difference between the two points. The order of subtraction does not affect the result. 3 -4 Distance = |3 - (-4)| = |7| = 7 or Distance = |-4 - 3| = |-7| = 7 Distance = 7 Applications: Any question that contains the words distance, points, and number line requires the application of the above definition.

Divisor Back to Top 25 Definition: A number or quantity to be divided into another number or quantity (the dividend) A number that is a factor of another number Applications: Questions involving long division and remainders. See math strategy For some questions the word “divisor” can be replaced with the word “factor”.

Factor Back to Top 25 Definition: A factor of a number or expression, N, is a number or expression that can be multiplied by another number or expression to get N. When a number or expression is written as a product of its factors, it is said to be in factored form. Example: (2)(4)(15) = Example: (x + 1)(x + 2) = x2 + 3x +2 Factors Factors Applications: See math strategy

Function Back to Top 25 Definition: A special relationship between two quantities in which one quantity, the argument of the function, also known as the input, is associated with a unique value of the other quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The notation f(x) is said “F of X”. An example of a function is f(x) = 2x, a function which associates with every number twice as large. Applications: See math strategy

Inversely Proportional
Return to Table of Contents Inversely Proportional Back to Top 25 Definition: The product of the value of the independent variable and the value of the dependent variable is constant. Can be written as k = xy, or y = k/x. The relationship between “x” and “y” can be expressed graphically as Applications: Questions that begin with the words “If “y” is inversely proportional to “x” and…” Questions that contain a table of “x” and “y” values that have a constant product.

Median Back to Top 25 Definition: The middle number in a sorted list of numbers. Half the numbers are less and half the numbers are greater. If the sorted list contains an even number of values, the median is the average of the two numbers in the middle of the list. Example: 2, 3, 3, 6, 8, 9, 9 Example: 2, 3, 3, 3, 5, 6, 7, 9 Median = 4 Applications: See math strategy

Multiple Back to Top 25 Definition: The product of an integer by an integer. For example, the multiples of 4 are 4, 8, 12, 16, 20, …..For any positive integer there are an infinite number of multiples. Applications: Finding the value of a term in a repeating sequence. Variety of questions that require understanding of the multiple definition

Percent Back to Top 25 Definition: A ratio that compares a number to 100. Percent means “out of one hundred”. For example: 10% means 10/100, 750% means 750/100, “k%” means k/100 Applications: See math strategy

Percent Change Back to Top 25 Definition: The amount of change in a quantity divided by the original amount of the quantity times 100%. % change = amount of change original amount x 100% Applications: See math strategy

Probability Back to Top 25 Definition: The likelihood of the occurrence of an event. The probability of an event is a number between 0 and 1, inclusive. If an event is certain, it has a probability of 1. If an event is impossible, it has a probability of 0. Applications: Elementary probability Probability of independent/dependent events Geometric probability

Proportional Back to Top 25 Definition: An equation showing that two ratios are equal Two variables are proportional if their ratio is constant. If a is proportional to b, then a/b is a constant. Can be written in equation form as a = kb where k is a proportionality constant. Applications: Proportions, ratios, and probability are closely related in many applications. See math strategy

Rate Back to Top 25 Definition: A rate is a ratio that compares two quantities measured with different units. For example, the speed of a car is a rate that compares distance and time. Applications: When the word “rate” is contained in a question, create a ratio of the two given quantities identified in the question. Such questions usually vary the value of one of the given quantities and ask for the value of the second quantity that will maintain the given rate. To solve efficiently, create a proportion of the two ratios and solve for the unknown quantity. See math strategy

Sector Back to Top 25 Definition: A sector of a circle is the portion of a circle bounded by two radii and their intersected arc. Sector Applications: Area of sector Length of arc AB

Sequences Back to Top 25 Definition: A sequence is an ordered set of numbers. Four types of sequences on the SAT. Arithmetic sequence: A sequence of numbers that has a common difference between each number. 3,7,11,15,19,23 Geometric sequence: A sequence of numbers that has a common ratio between each number. 3, 6,12, 24, 48, 96 Repeating sequence: A sequence of numbers that form a repeating pattern. See math strategy “Other” sequence: A sequence that does not fit any of the above three categories. A formula is usually provided that can be used to determine each value of the sequence. Applications: Any or all of the above types of sequences will be found on every SAT. However, the sequence names used above will never be found in any SAT questions. Instead, a description of the sequence is used. Bottom line….know the sequence definitions.

Similar Triangles Back to Top 25 Definition: Two triangles are similar if and only if all pairs of corresponding angles are congruent and all pairs of corresponding sides are proportional. 7 6 4 3.5 3 2 Applications: (See figure at right) When a smaller triangle is completely inside a larger triangle such that corresponding angles are congruent or one pair of corresponding sides are parallel, the two triangles are similar. Congruent angles

Slope of a Line Back to Top 25 Definition: Slope is a measure of the tilt or steepness of a line. Slope is calculated as the vertical distance divided by the horizontal distance between two points. Slope is also a measure of the amount that the dependent variable (often “y”) changes as the independent variable (often “x”) changes by one unit. Slope = ∆y ∆x = y2 - y1 x2 - x1 rise run Applications: Slope of a line when two points are known Identification of the “x” or “y” value of a point when the coordinates of a second point are known and the slope of the line is given. Slope of a line parallel or perpendicular to another line Linear relationships or functions that ask for the change in the value of a quantity as the independent variable is changed.

Venn Diagram Back to Top 25 Definition: A diagram (usually made of circles) that shows all possible relations between sets. Applications: Venn diagrams (2 sets): See math strategy Venn diagram (3 sets): See math strategy

Zero Back to Top 25 Definition: Zero is an even integer (thus it is divisible by 2) that is neither positive nor negative. As a result, zero is the smallest non-negative number. Zero is also the smallest of 10 digits. Zero is a whole number, a rational number, and a real number. Division by zero results in an undefined value. Applications: Questions that ask for the number of integers, the number of even integers, or the number of positive integers that are contained in a solution set. Questions that ask for a specific value of “x” for which a function is not defined.

Questions You Can Count On
Return to Table of Contents Questions You Can Count On Back to Tips Formation of even/odd numbers: See strategy Equation of a line or slope of a line perpendicular to another line: See strategy The “If…..then what is the value?” question: See strategy Tangent lines to a circle: See strategy Parallel lines cut by one or more transversals: See strategy A figure that is rotated, flipped, reflected, taken apart, unfolded is usually either question 3, 4, or 5 in the 20 multiple choice section of math. Learn More

Return to Table of Contents Questions You Can Count On Back to Tips Questions that contain an inequality: See strategy System of equations: See strategy Rules of exponents: See strategy Rate and or ratio questions: See strategy Two types of definition questions Substitution into an expression: See strategy Words in quotations: See strategy Sequence questions Average (arithmetic mean) questions: See strategies Previous Learn More

Return to Table of Contents Questions You Can Count On Back to Tips Long division and remainder questions: See strategies Percentage questions: See strategies Use of function notation and function translations or reflections: See strategy Probability of events occurring: See strategy Geometric probability: See strategy Counting problems including the number of ways to pair objects: See strategy Area of irregular shapes and area of sectors Previous Learn More

Return to Table of Contents Questions You Can Count On Back to Tips Creation of a cost equation for the purchase of an item or service Median of a list of numbers: See strategy Absolute value equation or inequality: See strategy Directly or indirectly proportion questions Similar shapes (usually triangles): See strategy Patterns of number or shapes/objects: See strategy Overlap of data sets (Venn diagram applications): See strategy Previous

The Two Test Taking Rules
Return to Table of Contents The Two Test Taking Rules Back to Tips Keep it simple. View each question through the lens of simplicity, not the lens of complexity. The math portion of the SAT is not a two headed monster. With good reasoning skills and an understanding of basic math definitions and content, every question can be solved with little difficulty. Having this mindset will often lead to increased confidence. Answer the question. Make sure you answer the question being asked, not the question being assumed. Before choosing an answer, read the last half of the last sentence. If the questions asks for the cost of three pounds of bananas, do not choose the per pound cost. If a question asks for the value of the “y” variable, do not choose the value of the “x” variable. If the question asks for the value of the largest of three consecutive integers, do not choose the smallest integer. If the questions asks for the value of “4x”, do not choose the value of “x”. Answer the question being asked!

The Three Questions Back to Tips What piece of information do I need? This is a crucial question to ask. SAT questions are asked in ways that are more abstract than a typical math question. The answer to this question will ensure you are heading down the correct path toward the answer. What is the strategy for finding this information? This is where most students have difficulty. A good strategy is usually needed at this point. If none can be identified, students will go to Plan B (substitution of answers, elimination and guess), or skip the question. What do I do with the information? This is the math step that usually requires using a formula.

Test Day Tips Back to Tips Replace calculator batteries. Replace the batteries in your calculator (usually four AAA batteries) with fresh, out of the package batteries. Do not replace with the batteries that are rolling around in your desk drawer…..the ones that should have been tossed out the last time you replaced batteries. Prepare a survival kit. In a lunch bag, pack bottled water and many snacks. Include one chocolate bar to be consumed between sections seven and eight of the ten section test. Fatigue will be high at this point during the test. Eat the chocolate bar for a burst of energy and tough it out until the end. Have your admission ticket and photo ID. This a common sense issue. Take a watch to the testing center. You do not have control over the amount of time for each test section. However, with a watch, you are in a position to control the use of your time. If the testing room has a clock on the wall, your watch may not be needed. Learn More

Test Day Tips Back to Tips Take plenty of No. 2 wood pencils. Mechanical pencils are not permitted apparently due to cheating issues. Bubble in the student-generated response answers: Some students forget to bubble the answers. Four math sections…do not panic. The SAT is comprised of ten sections: three writing, three reading, three math, and one “experimental section”. The experimental section will be an additional writing, reading, or math section that will not be part of your final score. The experimental section is not identified. Do your best on all sections! Proctors are not your friend. The test proctor is there to make a few bucks on a Saturday morning. They are not there to help you in anyway. They are prone to making mistakes with the timing of sections, have been observed talking on the phone causing noise issues, and often have a nasty disposition. They are not your friend! Previous

What Study Guides Will Never Reveal
Return to Table of Contents What Study Guides Will Never Reveal Back to Tips Be prepared to reason: Math content is plentiful in study guides, however math strategies are virtually nonexistent. To be successful on the SAT, reasoning skills are as important as having basic math content knowledge and basic computational skills. Complex computational skills not required: The SAT is a test of quantitative reasoning skills, not computational skills. With strong reasoning ability, only basic calculations are needed to answer most questions. Basic calculations should be done without a calculator: Calculators are absolutely, positively not needed for the SAT, however, you should absolutely, positively use one…..sparingly. Avoid using the calculator for basic addition and multiplication operations, especially those involving negative numbers. Student calculator input errors often lead to costly mistakes that are absolutely avoidable. Answer the easy questions first. All questions are equally weighted. Do not try the hard questions first. Attempt the questions in the order they are presented. Learn More

Return to Table of Contents What Study Guides Will Never Reveal Back to Tips No need to memorize formulas: There is no need to memorize formulas….they are all provided. If a formula is needed and is not contained on the list of formulas at the beginning of each math section, then the formula will be provided in the text of the question. The bottom line is this….if you believe a formula is needed to solve a specific problem and the formula is not provided, look for an alternative way (and often more efficient way) to solve the problem. Need to know math definitions: Definitions are not provided. You are expected to know all math definitions. Examples include slope of a line, average (arithmetic mean), percent, percent change, average speed, etc. Cross multiplication is your best friend: The solution to many questions is made easier by using cross multiplication. Look for opportunities to use it. Never enter a value for “pi” into your calculator: Entering “pi’ into your calculator will often result in a close approximation to the correct answer, not the exact answer. Solve questions in terms of “pi”, especially the student-produced response questions that require exact answers. Previous Learn More

Return to Table of Contents What Study Guides Will Never Reveal Back to Tips The words “arithmetic” and “geometric” sequence are not used: Students are not expected to know the definition of these sequences, as suggested by study guides. Instead of using the words “arithmetic” and “geometric” sequences, SAT questions describe the characteristics of these sequences. Never asked to find the domain of a function: This topic is discussed in study guides, however, it is not found on the SAT reasoning test. More likely to find this topic on the SAT math subject test. Never asked to calculate the average of a list of numbers: When a list of values is provided, analysis of the median (sometimes mode) is always asked. Do not be fooled into making a lengthy calculation of the mean of a list of numbers….it is never asked for. Inscribed shape questions: When a shape is inscribed inside a second shape, their centers always coincide. This is often useful when developing a strategy to solve this class of questions. Do not need to use permutations or combinations: Although both topics are discussed in most study guides, you can always use Fundamental Counting Principles to solve counting problems. Previous

The Usual Study Guide Tips
Return to Table of Contents The Usual Study Guide Tips Back to Tips Use the figure when figuring. All figures are drawn to scale unless stated otherwise. Use this to your advantage. If there is a note stating the figure is not drawn to scale, you must stick to the facts when drawing conclusions about the answer. To guess or not to guess. There is a ¼ point penalty for each missed multiple choice question. The conventional wisdom is to guess if one answer choice can be eliminated. My recommendation is to guess if two of five choices can be eliminated. Guess on student generated response questions. No penalty is given for missing a student produced response question. If the answer is not known, take a guess. Student produced response answers must be non-negative rational numbers. All non-negative integers (including zero) and all fractions are acceptable answers.

Math Strategies Table of Contents Geometry and Measurement Dividing Irregular Shapes Line Segment Length in Solids Putting Shapes Together 3-4-5 Triangle Triangle Triangle Distance Between Two Points Midpoint Determination in x-y Coordinate Midpoint Determination on Number Line Exterior Angle of a Triangle Perpendicular Lines Interval Spacing - Number Line Triangle Side Lengths Similar Triangle Properties The Slippery Slope Parallel Lines and Transversals Tangent Line to a Circle Number and Operations Ordering of Negative Numbers Linear Proportionality Venn Diagrams (2 sets) Venn Diagrams (3 sets) Ratios and their Multiples Ratios, Proportion, Probability Rate Counting Problems The Pairing Strategy Long Division and Remainders Percent Change Dealing With Percentages Repeating Sequences Consecutive Integers Even/Odd Integer Creation Algebra Using New Definitions: Type 1 Using New Definitions: Type 2 Solving Simple Inequalities Equivalent Strategy System of Equations Matching Game Factoring Strategy Word problems Basic Rules of Exponents Additional Rules of Exponents Absolute Value Inequalities Creation of Math Statements Parabolas Single Term Denominators Making Connections Data Analysis, Statistics, and Probability Average (Arithmetic Mean) Median of Large Lists Elementary Probability Probability of Independent Events Geometric Probability The Unit Cell It’s Absolutely Easy! Functions Using Function Notation Reflections - x axis Reflections - y axis Reflections - Absolute Value Translations - Horizontal Shift Translations - Vertical Shift Translations - Vertical Stretch Translations - Vertical Shrink The Basics All the Equations You Need! The Important Definitions You Need! Back to Success Model

Math Strategies Table of Contents Lesson 1 Algebra Strategies Using New Definitions: Type 1 Using New Definitions: Type 2 Solving Simple Inequalities Equivalent Strategy System of Equations Matching Game Factoring Strategy Word problems Basic Rules of Exponents Additional Rules of Exponents Absolute Value Inequalities Creation of Math Statements Parabolas Single Term Denominators Making Connections Back to Success Model Back to Math Topics

Math Strategies Table of Contents Lesson 2 Geometry and Measurement Strategies Dividing Irregular Shapes Line Segment Length in Solids Putting Shapes Together 3-4-5 Triangle Triangle Triangle Distance Between Two Points Midpoint Determination in x-y Coordinate Midpoint Determination on Number Line Exterior Angle of a Triangle Perpendicular Lines Interval Spacing - Number Line Triangle Side Lengths Similar Triangle Properties The Slippery Slope Parallel Lines and Transversals Tangent Line to a Circle Back to Success Model Back to Math Topics

Math Strategies Table of Contents Lesson 3 Number and Operations Strategies Ordering of Negative Numbers Directly Proportional Venn Diagrams (2 sets) Venn Diagrams (3 sets) Ratios and their Multiples Ratios, Proportion, Probability Rate Counting Problems The Pairing Strategy Long Division and Remainders Percent Change Dealing With Percentages Repeating Sequences Consecutive Integers Even/Odd Integer Creation Back to Success Model Back to Math Topics

Math Strategies Table of Contents Lesson 4 Functions Strategy Using Function Notation Reflections - x axis Reflections - y axis Reflections - Absolute Value Translations - Horizontal Shift Translations - Vertical Shift Translations - Vertical Stretch Translations - Vertical Shrink Back to Success Model Back to Math Topics

Math Strategies Table of Contents Lesson 5 Data Analysis, Statistics, and Probability Strategies Average (Arithmetic Mean) Median of Large Lists Elementary Probability Probability of Independent Events Geometric Probability The Unit Cell It’s Absolutely Easy! Back to Success Model Back to Math Topics

All The Equations You Need! Strategy: Great news! The equations on this page are the only ones you need to be successful on the SAT. Area of Circle = π r2 Circumference of Circle = 2π r Area of rectangle = lw Reasoning: If the equation is not on this page, you do not need to use it. Hooray! Examples include quadratic formula, combinations, permutations, equation of a line or circle, surface area and volume of a cone, pyramid, or sphere. If one of these equations is needed to solve a problem, it will be provided. Area of triangle = ½ bh Volume of rectangular solid = lwh Volume of cylinder = π r2h Pythagorean theorem c2 = a2 + b2 Application: There are plenty of questions on the SAT for which these formulas are used. To save time when taking the SAT, it is recommended that you memorize these basic formulas. Triangle Click for details Triangle Click for details Return to Table of Contents See example of strategy

All The Equations You Need! Example 1 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

All The Equations You Need! Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

The Important Definitions You Need! Strategy: These definitions are extremely important for you to memorize. Unlike formulas, definitions are not provided on the SAT. Average speed = total distance traveled total time Distance between two points = Reasoning: Students often consider these definitions to be formulas. They are not formulas! Formulas are derived in geometry using proofs. Slope = ∆y ∆x = y2 - y1 x2 - x1 rise run Click for more details Application: These definitions are extremely valuable resources when solving a variety of problems on the SAT. The definition of empty set, integer, positive and negative numbers, even and odd numbers, digits, and percentages are also important to know. Average (arithmetic mean) = sum of values number of values Click for more details Percent change = amount of change original amount X 100% Click for more details Return to Table of Contents See example of strategy

The Important Definitions Example 1 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

The Important Definitions Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Ordering of Negative Numbers Strategy: Visualize the position of a single negative value or a list of negative values as they would appear on a number line On the number line shown below, which letter best represents the location of the value -2/5? Click to see answer A -⅜ -¼ -⅛ -½ B C D E Reasoning: As you move left on a real number line, the values get smaller. This property is especially useful when ranking negative numbers. On the number line shown below, which letter best represents the location of the value -5/2? Click to see answer Application: Any question that requires you to rank the values of negative values from smallest to largest or vice-versa. Also useful when assigning values to positions on a number line. -7 -4 -1 -10 A B C D E Return to Table of Contents See example of strategy

Ordering of Negative Numbers Example 1 Question: If a < 0, which of the four numbers is the greatest? a B) 2a + 2 4a D) 8a + 2 E) It cannot be determined from the information given Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Ordering of Negative Numbers Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Directly Proportional “y” is directly proportional to “x” Strategy: Often given values for “x1” and “y1” and asked to find value for “x2” when given “y2”. Use the following proportion: y x y = kx (x2, y2) (x1, y1) Reasoning: The ratio of y:x is constant for any two points. Click to see properties of directly proportional Properties of a directly proportional include the following: Graph of “y” versus “x” is linear and passes through the origin. Has the form of y = kx. Slope of line is the ratio of y:x for any point on the line Slope of line is equal to proportionality constant “k”. Application: Any relationship that can be expressed as ratios. In addition to points on a line, examples include amount of ingredients in recipes, number of marble colors in a container, and segment lengths of a number line. Return to Table of Contents See example of strategy

Directly Proportional Example 1 Question: A machine can produce 80 computer hard drives in 2 hours. At this rate, how many computer hard drives can the machine produce in 6.5 hours? Solution Steps 1) Create a ratio representing rate of computer hard drive production: What essential information is needed? Rate of computer hard drives per hour. 80 hard drives 2 hours 2) Create a linear proportion to solve for number of hard drives produced in 6.5 hours: What is the strategy for identifying essential information?: Ratio the number of computer hard drives to the number of hours required to produce them. With this ratio, create a linear proportion to answer the question. 80 hard drives 2 hours “n” hard drives 6.5 hours = 3) Solve for ‘n”: 2n = (80)(6.5) n = 260 Return to Table of Contents Return to strategy page See another example of strategy

Directly Proportional Example 2 Question: If y varies directly as x, and if y = 10 when x = n and y = 15 when x = n + 5, what is the value of n? Solution Steps 1) Create a linear proportion to solve for n. What essential information is needed? A link between y and x that can be used to solve for n. 10 n 15 n + 5 = 2) Solve for n using cross multiplication: What is the strategy for identifying essential information? The ratio y/x is a constant. Create a proportion and solve for n. 15n = 10(n + 5) 15n = 10n + 50 5n = 50 n = 10 Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Back to Definition Venn Diagram (2 sets) = 50 Total number of students = 50 Strategy: To determine the overlap (intersection) of members in two groups (sets), use the following approach: Step 1: add the total number of members from both groups Step 2: subtract the sum consisting of the total number of members in one group only and both groups from the number of members in step 1 Math History 18 22 10 Number of students that study math = 40 Number of students that study history = 32 Reasoning: By eliminating the overlap of members, the sum of three numbers in the Venn diagram will equal the total number of members being counted. Step 1 = 72 Step 2 72 – 50 = 22 Number of students that study math and history = 22 Application: Used when members of two or more groups (sets) have common members. Number of students that study math only: 40 – 22 = 18 Number of students that study history only: 32 – 22 = 10 Return to Table of Contents See example of strategy

Venn Diagram (2 sets) Example 1 Football Baseball 28 14 n Question: The Venn diagram to the right shows the distribution of students who play football, baseball, or both. If the ratio of the number of football players to the number of baseball players is 5:3, what is the value of n? Solution Steps What essential information is needed? Connection between the number of players in each sport to “n”, the number of players that participate in both sports. 1) Create a proportion of the number of football players to baseball players n + 28 n + 14 5 3 = What is the strategy for identifying essential information?:Use the properties of Venn diagrams and proportions to find the value of “n” 2) Solve for “n” using cross multiplication: 5n + 70 = 3n + 84 2n = 14 n = 7 Return to Table of Contents Return to strategy page See another example of strategy

Venn Diagram (2 sets) Example 2 Question: The 350 students at a local high school take either math, music, or both. If 225 students take math and 50 take both math and music, how many students take music? Solution Steps 1) Create an appropriate Venn diagram to help visualize the given information. Math Music 175 m 50 What essential information is needed? Connection between the multitude of given information and the unknown quantity. 2) Find the value of m, the number of students that take music only What is the strategy for identifying essential information? Use the properties of Venn diagrams to help “visualize” the given information. m = m = 125 3) Find the value of m + 50, the number of students that take music m + 50 = = 175 Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Venn Diagrams (3 sets) Football Baseball Strategy: When analyzing the overlap of three data sets, it is important to understand the meaning of each section of the resulting Venn diagram (see example) 6 7 8 3 4 5 9 Reasoning: The interpretation of data in each section is determined by the rules of logic Soccer The number of athletes that play all three sports = 3 The number of athletes that play two sports. Example: football and baseball = 10 Application: Data sets in which there is overlap of members of two or more sets. Applications include student choices of school classes or sport activities, and overlapping properties of various real numbers The number of athletes that play one sport only = 23 The number of athletes that play two sports only = 16 The number of athletes that play football only (6), baseball only (8), or soccer only ( 9) Return to Table of Contents See example of strategy

Venn Diagrams (3 sets) Example 1 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Venn Diagrams (3 sets) Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Ratios and Their Multiples Strategy: When given the total number of several different objects and a ratio that describes their distribution, create an equation to find the exact number of each object. (click to see example) A jar contains a total of 30 red, yellow, and blue marbles. The number of each marble color in the jar follows the ratio 3 red: 2 yellow: 1 blue. How many of each color are there in the jar.? Reasoning: For discrete objects like marbles, bowling balls, and people, the total number of each object in the group must be a multiple of their respective ratio value. 3x + 2x + x = 30 marbles 6x = 30 marbles x = 5 blue marbles 2x = 10 yellow marbles Application: Questions that ask for the distribution of angles in a triangle or the distribution of objects among containers. 3x = 15 red marbles Total = 30 marbles Return to Table of Contents See example of strategy

Ratios and Their Multiples Example 1 Question: The measures of the interior angles in a triangle are in the ratio 9:4:2. What is the measure of the largest angle in the triangle? Solution Steps 1) Create equation using ratio values 9x + 4x + 2x = 180 degrees What essential information is needed? The measure of each individual angle. 2) Solve equation for “x”. Multiply by nine to find measure of largest angle. What is the strategy for identifying essential information? Create and solve an equation using the angle ratios and the fact that the sum of the interior angles is 180 degrees in a triangle. 9x + 4x + 2x = 180 degrees 15x = 180 degrees x = 12 degrees 9x = 108 degrees Return to Table of Contents Return to strategy page See another example of strategy

Ratios and Their Multiples Example 2 Question: Cookies are distributed within four separate jars in the ratio of 7:5:3:1. The total number of cookies contained in the four jars is 48. How many cookies are contained in the jar with the greatest number of cookies? Solution Steps 1) Create equation using ratio values 7x + 5x + 3x + x = 48 cookies What essential information is needed? The number of cookies in each jar. 2) Solve equation for “x”. Multiply by seven to find measure of largest angle. What is the strategy for identifying essential information? Create and solve an equation using the given ratios and the fact that the total number of cookies contained in the four jars is 48. 7x + 5x + 3x + x = 48 cookies 16x = 48 cookies x = 3 cookies 7x = 21 cookies Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Ratios, Proportions, Probability Connections Strategy: When the whole consists of two parts and the parts are expressed as a ratio of each other, there are several connections between ratios, proportions, and probability that are useful to solve a variety of problems. The ratio of red to blue marbles is 1 to 3. Connection 1: The total number of marbles in the can must be a multiple of four marbles (1 + 3 = 4). Reasoning: For the example shown to the right, three out of every four marbles in the can are blue. To maintain this ratio, the total number of marbles in the can must remain a multiple of four. As a result, the probability of selecting a blue marble is ¾. Connection 2: The probability of randomly selecting a blue marble from the can is ¾. Connection 3: To maintain this ratio when adding to or removing marbles from the can, a proportion should be used. Application: Problems involving lengths of line segments, rate/time, areas and perimeters, sizes of angles Return to Table of Contents See example of strategy

Ratios, Proportions, Probability Example 1 Question: During the month of February (28 days) the city of Pittsburgh had two days on which it snowed for every five days on which it did not snow. For the month of February, the number of days on which it did not snow was how much greater than the number of days on which it snowed? Solution Steps 1) Set up a proportion using the following strategy: For every seven days (2 + 5 = 7) during the month of February, it snowed 2 days. Find the number of days it snowed. 2 7 n 28 = n = 8 days of snow What essential information is needed? Need to determine the number of days in which it snowed and the number of days in which it did not snow. 2) Find the number of days in which it did not snow. 28 days - 8 days = 20 days What is the strategy for identifying essential information?: Use proportions to determine essential information. 3) Subtract the result of Step 1 from the result of Step 2 20 days – 8 days = 12 days greater Return to Table of Contents Return to strategy page See another example of strategy

Ratios, Proportions, Probability Example 2 Question: The ratio of almonds to cashews in a mixture is 2:3. How many pounds of almonds are there in a seven pound mixture of almonds and cashews. Solution Steps 1) Set up ratio of almonds to mixture. 2 pounds almonds + 3 pounds cashews = 5 pounds mixture What essential information is needed? The number of pounds of almonds required to maintain proper mixture ratio. 2 pounds of almonds 5 pounds of mixture Ratio: 2) Create proportion to solve problem. What is the strategy for identifying essential information? Use proportions to determine essential information. 2 5 n pounds of almonds 7 pounds of mixture = 5n = 14 Cross multiply 14 5 n = pounds of almonds Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Rate Strategy Strategy: For all questions that require the rate of two quantities to be held constant, create a proportion to solve for the new value of one quantity when the value of a second quantity is changed a given amount. Definition: A rate is a ratio that compares two quantities measured with different units. For example, the speed of a car is a rate that compares distance and time. Reasoning: A proportion is an equation stating that two ratios are equivalent. Note: When you read the word rate in a question, think ratio! Application: Any question that requires the rate to be held constant. Examples of constant rate include speed of an object, rate of work, rate of flow of a liquid, rate of growth of money, etc. Return to Table of Contents See example of strategy

Rate Strategy Example 1 Question: The rate of motion of a baseball is k feet per 2 seconds. In terms of k, how many seconds will it take a baseball to move k + 50 feet if the rate of motion is constant? A) B) C) D) E) Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Rate Strategy Example 2 Question: The rate of flow of water from a hose is 4 gallons per 20 seconds. At this rate, how many gallons of water can the hose provide in 5 minutes? Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Counting Problems Strategy: Use “Fundamental Counting Principles” (FCP) and reasoning to solve many counting problems that do not involve pairing of objects. For pairing problems, see Handshake/ Pairing strategy. Fundamental Counting Principles: If one event can happen in n ways, and a second, independent event can happen in m ways, the total number of ways in which two events can happen is n times m. Reasoning: FCP represent a broad class of counting principles that include permutations and combinations. Some counting problems will have constraints. Such problems, along with reasoning, can be solved using these principles. A restaurant uniform consists of a hat, shirt, and pants. If a worker has two hats, four shirts, and three pair of pants to choose from, how many uniforms can the worker create? Step 1: Choice of a hat, shirt, or pants is independent of each other . Step 2: Multiply the number of each together to find the total number of uniforms. Application: Any problem asking you to figure the number of ways to select or arrange members of a group. Examples include numbers, letters of the alphabet, or officers of a club. 2 x 4 x 3 = 24 uniforms Return to Table of Contents See example of strategy

Counting Problems Example 1 Question: Five individual pictures of the Jones family consists of the Jones parents and each of the four Jones children. The individual pictures are to be arranged vertically on a living room wall. How many arrangements of pictures can be made if the parent picture must be placed at the top of the arrangement? Solution Steps 1) Determine the number of arrangements of pictures. Take into account there is a constraint: the top picture must be the Jones parents. Top position → 1 picture to choose Second position → 4 pictures to choose What essential information is needed? The number of ways the five pictures can be arranged vertically on the wall. Third position → 3 pictures to choose Fourth position → 2 pictures to choose Fifth position → 1 picture to choose What is the strategy for identifying essential information?: Use fundamental counting principles. 2) Multiply each number together to find the total number of arrangements 1 x 4 x 3 x 2 x 1 = 24 arrangements Return to Table of Contents Return to strategy page See another example of strategy

Counting Problems Example 2 Question: A certain restaurant offers ice cream specials that consist of two scoops of ice cream and one topping. If there are four toppings to choose from and four flavors of ice cream, how many different ice cream specials can be created if the two scoops of ice cream must be different flavors? Solution Steps 1) Determine the number of ways to pair scoops of ice cream if there are four flavors to choose from. Vanilla Chocolate Strawberry Peach 2 3 1 4 5 6 What essential information is needed? A special consists of two groups → the number of toppings and the number of ways to pair up four flavors of ice-cream. 2) Multiply the number of toppings (4) and number of pairs of flavors (6) to find the total number of ice cream specials What is the strategy for identifying essential information? Use fundamental counting principles to identify the number specials. 4 x 6 = 24 specials Return to Table of Contents Return to strategy page Return to previous example

The Pairing Strategy n - 1 = 5 handshakes per person Strategy: The total number of ways to pair “n” objects is equal to ½n(n -1). n = 6 people Reasoning: For a total of “n” objects, each object can be paired with “n -1” other objects. However, each pair is shared by two objects. Click to see an example of the total number of handshakes exchanged by 6 people. ½n(n -1) = ½(6)(5) = 15 total handshakes shared by a group of 6 people Application: Examples include determining the total number of games played in a sport league, or the number of ways a two scoop ice cream cone can be created from a known number of available flavors. Alternative Solution: Total number of handshakes can be found by addition of the number of handshakes exchanged by each individual person. = 15 handshakes Return to Table of Contents See example of strategy

The Pairing Strategy Example 1 Question: In a baseball league with 6 teams, each team plays exactly 4 games with each of the other 5 teams in the league. What is the total number of games played in the league? Solution Steps 1) Find the number of games played between the 6 teams What essential information is needed? How many games are played between the eight teams. ½(6)(5) = 15 individual games played without repeats What is the strategy for identifying essential information?: Find the number of games played between the 6 teams using the handshake problem strategy. Multiply the result by 4 to account for the fact that each team plays exactly 4 games with each of the other 5 teams. 2) Multiply by 4 to account for the fact that each team plays exactly four games with each of the other 5 teams Total number of games played: 15 x 4 = 60 games Return to Table of Contents Return to strategy page See another example of strategy

Back to Diagonal Definition The Pairing Strategy Example 2 Question: How many diagonals can be drawn inside a regular polygon with 6 congruent sides. Solution Steps n = 6 sides What essential information is needed? The total number of diagonals drawn from the 6 vertices of the polygon. n -3 = 3 diagonals What is the strategy for identifying essential information? Use the pairing strategy with modifications. Polygons have sides that do not require lines connecting adjacent vertices. To account for this, multiply the total number of vertices “n” by “n - 3” rather than “n - 1”. Total number of diagonals is ½n(n - 3). ½n(n - 3) = ½(6)(6 - 3) = 9 diagonals can be drawn in a regular polygon with 6 sides Return to Table of Contents Return to strategy page Return to previous example

Back to Divisor Definition Back to Frequent Questions Long Division and Remainders Strategy: Find a value for the unknown variable k by adding the given divisor to the given remainder. Process the value found for k as specified in the question. Divide this result by the new divisor to find the desired remainder. Click to see a review of long division. Example: When the positive integer k is divided by 7, the remainder is 6. What is the remainder when k + 8 is divided by 7 ? = x dividend = divisor x quotient + remainder Reasoning: Long division questions always involve analysis of the remainder, not the quotient. All long division questions provide a value for the divisor and remainder. By choosing a value of 1 for the quotient, a value for the dividend (unknown variable k) can be easily and quickly found. 1 7 13 -07 6 Application: Any long division question that expresses the dividend as a variable rather than a numerical value. Return to Table of Contents See example of strategy

Long Division and Remainders Example 1 Question: When d is divided by 9, the remainder is 7. What is the remainder when d + 4 is divided by 9? Solution Steps Find a possible value for d by adding the remainder to divisor: d = = 16 What essential information is needed? Find a number for d that satisfies the requirements. Add 4 to d, divide by 9, and find the remainder. Divide 20 by 9: 20 / 9 = 2 with remainder 2 Add d = 16 to 4: d + 4 = 20 What is the strategy for identifying essential information? Add remainder to the divisor. This will quickly provide a possible value for d. The new remainder is 2 Return to Table of Contents Return to strategy page See another example of strategy

Long Division and Remainders Example 2 Question: When n is divided by 7, the remainder is 5. What is the remainder when 3n is divided by 7? Solution Steps Find a possible value for n by adding the remainder to divisor: n = = 12 What essential information is needed? Find a number for n that satisfies the requirements. Multiply n by 3, divide by 7, and find new remainder. Divide 36 by 7: 36 / 7 = 5 with remainder 1 Multiply n = 12 by 3: 3n = 36 What is the strategy for identifying essential information? Add remainder to the divisor. This will quickly provide a possible value for n. The new remainder is 1 Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Dealing With Percentages Strategy: When a percentage is quoted as a number or variable, express the percentage as a ratio with the percentage in the numerator and the number 100 in the denominator. 10 % should be written as k % should be written as Reasoning: Percentages are expressed as a ratio of a number over 100 in mathematics. This strategy will avoid issues related to expressing a percentage as a decimal when the given percentage is a variable rather than a numerical value. Note: If a question expresses percentages as a numerical value only, it is okay to use the decimal form of a percentage. Application: Any question that contains a percentage expressed as a variable. Return to Table of Contents See example of strategy

Dealing With Percentages Example 1 Question: If k% of 60% of 180 is 54, what is the value of k? Solution Steps 1) Create a mathematical statement that properly expresses k% What essential information is needed? A mathematical statement is needed that properly describes the given information and provides a way to solve for the value of “k”. k% should be expressed as k 100 Math statement is: k 100 60 x 180 = 54 2) Solve for “k” using algebra What is the strategy for identifying essential information?: Two strategies are required: Creation of Mathematical Statements Percentages Strategy Eliminate zero’s k 100 60 x 180 = 54 Multiply by 100 k(6)(18) = 5400 k = 50 Return to Table of Contents Return to strategy page See another example of strategy

Dealing With Percentages Example 2 Question: If the length of a rectangle is increased 40% and the width is decreased 40%, how does the new area compare to the original area? Solution Steps 1) Choose convenient values for length and width Note: A square is a rectangle. Great shape to use for area calculations Convenient original area is 100. Use length of 10 and width of 10 What essential information is needed? Rectangle lengths and widths that meet the percent change requirements. 2) Apply percentage changes What is the strategy for identifying essential information? Start with convenient length and width values. Apply the required percentage changes to each value. Calculate new rectangle area and compare to original value. New length = = 14 New width = = 6 3) Calculate new area and compare New area = (14)(6) = 84 Area is reduced by 16% Return to Table of Contents Return to strategy page See another example of strategy

Dealing With Percentages Example 3 Question: What is ½ percent of 8? Solution Steps 1) Express percentage in proper form What essential information is needed? Need to convert ½ percent into an appropriate form to answer question. 100 = 1 200 Recommended form is: 2) Determine answer to question What is the strategy for identifying essential information? Use percentage strategy. Express percentage as a fraction over 100 rather than decimal form. Multiply recommended form by 8 1 1 200 x 8 = 1 25 = .04 25 Return to Table of Contents Return to strategy page Return to example 1

Back to Definition Percent Change Strategy: Percent change is defined as the amount of change in the quantity divided by the original amount of the quantity times 100%. % change = amount of change original amount x 100% Reasoning: This a well known definition in mathematics. Mostly used in chemistry and physics. Caution: Do not divide amount of change by the final amount Application: Can be used for any question involving percent increase or decrease. Return to Table of Contents See example of strategy

Percent Change Example 1 Question: Elliot’s height at the end of third grade was 48 inches. His height at the end of sixth grade was 60 inches. What was the percent change in Elliot’s height? a) b) c) 20 d) e) 30 Solution Steps 1) Determine the change in Elliot’s height Change in height = height at end of 6th grade - height at end of 3rd grade What essential information is needed? The change in height is essential to determining percent change. Change in height = 60 inches - 48 inches Change in height = 12 inches 2) Determine the percent change in Elliot’s height What is the strategy for identifying essential information?: Determine the change in height from the end of third grade to the end of sixth grade using subtraction. 12 inches 48 inches x 100% Percent change = Percent change = 25% Return to Table of Contents Return to strategy page See another example of strategy

Percent Change Example 2 Question: For the years 1990 to 2005, the function above expresses the projected population of Mathville. What is the projected percent increase in population of Mathville from 1990 to 2005? P(t) = 500t + 25,000 Solution Steps 1) Determine the population in 1990 and 2005 using function equation. P(t) = 500t + 25,000 Note: t = 0 for 1990 and t = 15 for 2005 P(0) = 500(0) + 25,000 = 25,000 What essential information is needed? The change in projected population is essential to determining percent change. P(15) = 500(15) + 25,000 = 32,500 Change in population = 7,500 people 2) Determine the percent change in population from 1990 and 2005. What is the strategy for identifying essential information? Using the function equation, determine the population in 1990 and Subtract the two values to determine the change in population. 7,500 25,000 x 100% Percent change = Percent change = 30% Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Repeating Sequence 2nd term 6th term 10th term Strategy: For any sequence that repeats, the value of the last term before the sequence repeats will always be repeated for any multiple of its term number. A C F T A C F T A C……. 4th term 8th term Reasoning: The letter “T” is the last letter before the sequence repeats. “T” appears as the 4th, 8th, 12th,.. 20th,….40th term value. Term numbers that are a multiple of 4 will always have the letter “T” as its value for this sequence. The term number of letter “C” will always be the following: 4n + 2 where “n” is an integer value and 2 is the remainder when the term number is divided by the multiple 4 Application: Used when any sequence of numbers or objects repeat. Examples include numbers or letters, days of the week, hours on the clock, remainders from long division. Return to Table of Contents See example of strategy

Repeating Sequence Example 1 Question: If the day of the week is Friday and it is assigned the value of one, what day of the week would be assigned the value one hundred? Solution Steps 1) Identify the day at end of cycle If Friday is day one of the cycle, Thursday is the end of the weekly cycle and is assigned the value of seven Apply multiple of seven to Thursday What essential information is needed? Identify the appropriate multiple number for the repeating sequence. 2) Find the remainder when one hundred is divided by the value seven What is the strategy for identifying essential information?: Identify the day of week at end of cycle, apply the multiple of seven to this day, identify the day assigned the value of one hundred. 100 7 = 14 with a remainder of 2 3) Identify day assigned the value of one hundred For remainder of two, day one hundred is two days beyond Thursday → Saturday Return to Table of Contents Return to strategy page See another example of strategy

Repeating Sequence Example 2 Question: A pattern consisting of three red circles, two blue circles, three yellow circles, and three green circles was painted side by side along the perimeter of a rectangular box. If the color of the last painted circle was blue, which of the following could be the total number of circles painted on the box? a) b) c) 86 d) e) 92 Solution Steps 1) Identify multiple number for sequence Add total number of circles in pattern: 3 red + 2 blue + 3 yellow + 3 green = 11 Multiple number is 11 for sequence Conclusion: Third green circle is always a multiple of 11 in sequence. What essential information is needed? Multiple number for sequence and possible remainders for a blue circle 2) Identify possible remainders for blue circle Blue circles are located at positions four and five in sequence. Correct choice is a value that is 4 or 5 greater than a multiple of 11 Correct choice is (11)(8) + 4 = 92 What is the strategy for identifying essential information? Use repeating sequence principles to identify essential information Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Using New Definitions Type 1 Example: For all positive integers x, let be defined to be (x+1)(x+2). What is the value of ? Strategy: Read and apply the new definition carefully before choosing answers. What does mean? How do I determine a value? Apply the definition in given form = (4+1)(4+2) Reasoning: The new definition will typically break down to a simple application involving basic math operations. Operation is easy to apply for any value of “x” = (5)(6) = 30 Final answer Caution: Do not foil (x+1)(x+2). More efficient to apply definition in factored form. Return to Table of Contents See example of strategy

Back to Frequent Questions Using New Definitions Type 2 Example: A positive integer is said to be “bi-factorable” if it is the product of two consecutive integers. How many positive integers less than 100 are bi-factorable? Strategy: Read and apply the new definition carefully before choosing answers. Note the defined word is in “quotations” and there is no math expression as in Type 1. What does the definition “bi-factorable” mean? How do I determine a value? 1 x 2 = 2 Smallest integer less than 100 that is “bi-factorable” 2 x 3 = 6 Reasoning: Requires reasoning to apply the intended meaning due to lack of a math expression as in Type 1. Type 2 “New Definition” questions are usually more difficult to solve than Type 1. 8 x 9 = 72 9 x 10 = 90 Largest integer less than 100 that is “bi-factorable” Result: There are nine positive integers less than 100 that are “bi-factorable” Return to Table of Contents See example of strategy

Using New Definitions Example 1 Question: Let <x> be defined as the sum of the integers from 1 to x, inclusive. What is the value of <53> - <50>? Solution Steps Apply the definition to each quantity: <53> = …+1 <50> = …+1 What essential information is needed? Find the value of each quantity and perform the subtraction operation. Look for cancellation opportunities: <53> - <50> = ( …) – (50+49+…) <53> - <50> = <53> - <50> = 156 What is the strategy for identifying essential information?: Carefully apply the definition of <x> to each quantity. Look for opportunities to simplify the solution process through cancellation of like terms. Note: No calculator needed due to cancellation of like terms. Without cancellation strategy, problem would be consume too much time. Return to Table of Contents Return to strategy page See another example of strategy

Using New Definitions Example 2 Question: Let ©(x) be defined as ©(x) = (10-x) for all values of x. If ©(b) = ©(2b-2) what is the value of b? Solution Steps Apply definitions to each expression: ©(b) = 10-b ©(2b-2) = 10-(2b-2) What essential information is needed? Find the value of b that satisfies the given equation. Set both expressions equal to each other and solve: 10-b = 10-(2b-2) Distribute (-) 10-b = 10-2b+2 Subtract 10 -b = -2b+2 Add 2b b = 2 What is the strategy for identifying essential information? Carefully apply given definition to the expressions on each side of the equation. Set both expressions equal to each other and solve for b using simple math operations. Return to Table of Contents Return to strategy page See another example of strategy

Using New Definitions Example 3 Question: For positive integers a and b, let a b be defined by a b = ba . Which of the following is equal to 243. C) E) D) Solution Steps Apply definitions to each expression: What essential information is needed? Find the value of b that satisfies the given equation. Set both expressions… What is the strategy for identifying essential information? Carefully apply the given definition using the values in each answer choice. Return to Table of Contents Return to strategy page Return to example 1

Back to Frequent Questions Solving Simple Inequalities Example: For all values of x, what must be true about the value of “n” in the inequality k – n < k + 2? Strategy: Always solve the inequality directly by eliminating like terms and/or factors before analyzing answer choices. Recommended Solution Reasoning: By eliminating like terms or factors, the inequality often simplifies to one of the answer choices. Without simplification, each answer choice typically requires time consuming analysis to determine correct choice. Step 1: Eliminate like terms by subtraction k – n < k + 2 n > - 2 Step 2: Solve for “n” Caution: Do not choose values for “k” and use guess and check methods. Can be time consuming. Return to Table of Contents See example of strategy

Solving Simple Inequalities Example 1 Question: For all values of x, what is a possible value of x that satisfies the inequality x + 5 > x + 7? Solution Steps 1) Cancel x term from both sides of inequality: x + 5 > x + 7 What essential information is needed? All possible values of x that will make the left expression greater than the right expression. 2) Evaluate remaining terms of inequality: 5 > 7 This result is impossible The correct answer is the empty set. What is the strategy for identifying essential information?: Look for like term cancellation opportunities that eliminate the need to do time consuming guess and check steps. Note: Cancellation of like terms by subtraction provides a clear result to analyze. Return to Table of Contents Return to strategy page See another example of strategy

Solving Simple Inequalities Example 2 Question: If a + b > a - b, which of the following statements must be true? a) b < a b) a < b c) a = b d) b > e) a > 0 Solution Steps 1) Simplify inequality by elimination and consolidation of like terms What essential information is needed? From answer choices it is clear a method is needed to condense the number of variables to one on each side of the inequality. Eliminate “a” from both sides a + b > a - b Add “b” to both sides b > -b Divide “b” from both sides 2b > 0 What is the strategy for identifying essential information?: Look for like term cancellation opportunities that reduce the number of variables and eliminate the need to do time consuming guess and check steps. b > 0 Correct answer choice is “d” Return to Table of Contents Return to strategy page Return to previous example

Equivalent Strategy Example: What is equivalent to the following equation? Strategy: When a question asks for an equivalent form of an equation or expression, review all answer choices for guidance on ways to process the given equation/expression. Equivalent Forms Reasoning: Equations or expressions can be expressed in an infinite number of equivalent forms. The answer choices often provide valuable guidance on how to transform the equation or expression into the correct answer choice. Click to see equivalent forms All of the above are equivalent forms of the original equation. Answer choices on the SAT will typically include one of the above equivalent forms and four incorrect choices. Return to Table of Contents See example of strategy

Equivalent Strategy Example 1 Question: For x ≠ 0, which of the following is equivalent to a) 6x b) 12x c) 24x d) 6x2 e) 12x2 Solution Steps 1) Review answer choices for clues Conclusion: Answer choices suggest equivalent form requires elimination of fractions in numerator and denominator What essential information is needed? Guidance on how the expression should be transformed into “correct” equivalent form 2) Eliminate fractions by multiplying numerator by reciprocal of denominator 2 What is the strategy for identifying essential information?: Review answer choices for guidance on “correct” equivalent form. Return to Table of Contents Return to strategy page See another example of strategy

Equivalent Strategy Example 2 Question: If k is a positive integer, which of the following is equivalent to k + 2k + 2k + 2k ? a) 24k b) 4k c) 42k d) 2k e) 2k+4 Solution Steps 1) Review answer choices for clues Conclusion: Answer choices suggest equivalent form requires simplification of radical expression What essential information is needed? Need clues that better define equivalent form of expression. 2) Simplify radical using proper rules 2k + 2k + 2k + 2k = 4(2k) What is the strategy for identifying essential information? Review answer choices for guidance on correct solution path. 22(2k) 2k+2 Return to Table of Contents Return to strategy page See another example of strategy

Equivalent Strategy Example 3 Question: For all x > -2, which of the following expressions is equivalent to ? a) x + 2 = 10x b) x + 2 = 20x c) x + 2 = 10x d) x + 2 = 20x2 e) x(100x - 1) = 2 Solution Steps 1) Review answer choices for clues Conclusion: Answer choices suggest equivalent form requires squaring of radical What essential information is needed? Need clues that better define equivalent form of expression. 2) Square radical using proper rules What is the strategy for identifying essential information? Review answer choices for guidance on correct solution path. 3) Transform equation and factor Return to Table of Contents Return to strategy page Return to example 1

Back to Frequent Questions System of Equations Example: What is the value of “w” in the following system of equations? 3w = x – y + 4 w = z – x – 9 2w = y – z + 11 3w = x – y + 4 w = z – x – 9 2w = y – z + 11 Strategy: Solve a system of equations using elimination method or by reasoning. Do not use substitution . 6w = Reasoning: A system of three or more equations takes considerable time to solve using substitution methods. The questions are typically designed to be quickly solved by reasoning or by elimination of unwanted variables by the elimination method. w = 1 Strategy: Use elimination method. Reasoning method not practical without more information about the values of or relationships between the variables. Return to Table of Contents See example of strategy

System of Equations Example 1 Question: At a used book sale, Hillary paid $5.25 for 2 paperback books and 3 hardback books, while Ally paid $6.75 for 4 paperback books and 3 hardback books. At these prices, what is the cost, in dollars, for 3 paperback books? Solution Steps 1) Solution using reasoning skills The only difference between Hillary’s book order and Ally’s book order is the number of paperback books purchased. What essential information is needed? The unit price for a paperback book. Ally spent $1.50 more than Hillary to purchase 2 additional paperback books. 2) Find the unit cost for paperback books What is the strategy for identifying essential information?: Can use system of equations to develop two cost equations. An alternative method is to apply reasoning skills. Unit cost = $1.50/2 paperback books Unit cost = $0.75 3) Find the cost for 3 paperback books Total cost = $2.25 Return to Table of Contents Return to strategy page See another example of strategy

System of Equations Example 2 Question: In the system of equations below, what is the value of x + y? x + y - 4z = 400 x + y + 6z = 1200 Solution Steps 1) Subtract second equation from first equation and solve for the value of z: x + y - 4z = 400 x + y + 6z = 1200 What essential information is needed? Need a value for the expression x + y or separate values of x and y. -10z = -800 z = 80 What is the strategy for identifying essential information? Use elimination to determine value of expression x + y. 2) Substitute the value of z into first equation and solve for x + y: x + y -4(80) = 400 x + y -320 = 400 x + y = 720 Return to Table of Contents Return to strategy page Return to previous example

The Matching Game for Equalities Example: If k is a constant and 2(kx + 4) = 6x + 8 for all values of x, what is the value of k? Equivalent expressions Strategy: When two equivalent expressions are set equal to each other, match corresponding terms and solve for the unknown constant. “k” is unknown constant 2(kx + 4) = 6x + 8 2kx + 8 = 6x + 8 Distribute Reasoning: Terms on each side of the equal sign can be easily matched and common factors and/or terms can often be eliminated. This will allow the possibility of quickly identifying the value of the unknown constant. Match corresponding terms 2kx + 8 = 6x + 8 Set equal and solve for “k” 2kx 2x 6x = 2kx = 6x k = 3 Return to Table of Contents See example of strategy

The Matching Game Example 1 Question: If xy2 + 5 = xy + 5, which of the following values of y are solutions to the equation? I II) III) 1 a) I only b) II only c) III only d) II and III only e) I, II, and III Solution Steps 1) Cancel like terms from both sides of equation. xy2 + 5 = xy + 5 Subtract 5 2) Cancel common factors from both sides of equation. What essential information is needed? All possible values of “y” that make the left side of equation equal to the right side. xy2 = xy Divide out “x” What is the strategy for identifying essential information? Look for like term and common factor cancellation opportunities that eliminate the need to do time consuming guess and check steps. 3) Evaluate y2 = y for possible solutions Solutions are 0 and 1. Correct answer choice is d Return to Table of Contents Return to strategy page See another example of strategy

The Matching Game Example 2 Question: In the equation below, k and m are constants. If the equation is true for all values of x, what is the value of m? (x + 6)(x – k) = x2 - 4x + m Solution Steps 1) Convert expression on left side to trinomial form by distributing: x2 - kx + 6x - 6k = x2 - 4x + m x2 - (k – 6)x - 6k = x2 - 4x + m What essential information is needed? Need value of “m” that will make expression on right side of equal sign equivalent to the expression on left side. 2) Match like terms on each side: x2 - (k – 6)x - 6k = x2 - 4x + m What is the strategy for identifying essential information? Match terms in expression on left side of equal sign to corresponding terms in expression on right side. m = - 6k -(k – 6) = -4 3) To solve for “m” need value of “k” -(k – 6) = -4 -k + 6 = -4 k = 10 m = - 6k m = - 6(10) m = - 60 Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Factoring Strategy Example: In factored form Strategy For the following expression, what is the largest integer value for which the expression is positive? (4a - 2)(4 - a) Given equation is in factored form. Reason through problem in this form. Strategy: If an expression is in factored form, generally leave it that way. If an expression can be factored, it is always to your advantage to factor it. Reasoning: Working in factored form provides opportunities to quickly reason through problems with little computation. Example: Can be factored Strategy What conditions must be true for the following expression to be odd? a is odd b is odd a + b is odd a2 +ab Reason through problem with the expression in factored form a(a + b) Return to Table of Contents See example of strategy

Factoring Strategy Example 1 Question: If x2 – y2 = 92 and x + y = 23, what is the value of x – y? Solution Steps x2 – y2 = (x + y)(x – y) 1) Write in factored form What is the essential information needed?: Need values for x and y. Better approach is to directly find a value for the expression x – y. 92 23 ? 2) Solve for x - y What is the strategy for identifying essential information?: x + y and x – y are factors of x2 – y2 . Write x2 – y2 in factored form. Divide the value of x2 – y2 by the value of x + y. x – y = 92 23 x - y = 4 Return to Table of Contents Return to strategy page See another example of strategy

Factoring Strategy Example 2 Question: If (x + 2)(x – 5) < 0, how many integer values of x are possible? Solution Steps When considered a parabola, two properties are useful to answer question: The parabola opens upward The parabola has roots at x = -2 and x = 5 What is the essential information needed?: Need to identify specific integer values of x that result in a value less than zero for the left side of the inequality. -2 5 What is the strategy for identifying essential information?: It can be reasoned that the two linear binomial factors on the left side of the inequality describe a parabola. Use the properties of parabolas to determine answer. {-1, 0, 1, 2, 3, 4} There are six integer values between -2 and 5 that result in a value less than zero Return to Table of Contents Return to strategy page Return to previous example

Word Problems How many days are there in h hours and m minutes? Strategy: Use a two-step strategy to solve most word problems: 1) Eliminate choices that do not properly model the situation (often obvious). 2) Eliminate choices that do not provide proper units (dimensions)for the solution. No Step 1: Both minutes and hours should be smaller than days, not greater. Likely need to divide both terms in answer by a number or variable. No No Reasoning: By reasoning, some choices will obviously not appear to be proper solutions. Of those remaining, some will likely have wrong or inconsistent units. Step 2: To end with units of days, divide hours by 24 hours per day. Also, divide minutes by 1440 minutes per day. No This choice properly converts hours and minutes into days. Yes Return to Table of Contents See example of strategy

Word Problems Example 1 Question: Water from a leaking roof is collected in a bucket. If n ounces of water are collected every m minutes, how many ounces of water are collected in z minutes? Solution Steps 1) Determine units of correct answer Final answer represents quantity of water collected Units of final answer should be ounces What essential information is needed? Need to establish relationships between the given variables that provide dimensionally correct answer. 2) Arrange the three variables in proper way that provides correct units What is the strategy for identifying essential information? Determine the proper units of final expression that are consistent with question being asked. Create an expression that is consistent with the required units. Units of minutes cancel - ounces remain ounces minute x minutes n m (z) = nz Replace units with corresponding variables Return to Table of Contents Return to strategy page See another example of strategy

Word Problems Example 2 Question: In a certain grocery store, there are b stockcases with c shelves in each stockcase. If a total of d cans is to be stored on each of the shelves, what is the number of cans per shelf? Solution Steps 1) Determine units of correct answer Final units should be cans per shelf What essential information is needed? Need to establish relationships between the given variables that provide dimensionally correct answer. 2) Divide the total number of cans (d) by the total number of shelves What is the strategy for identifying essential information? Determine the proper units of final expression that are consistent with question being asked. Create an expression that is consistent with the required units. c shelves stockcase b stockcases x = bc shelves Number of cans per shelf = d bc Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Rules of Exponents Basic Rules Example: If x and y are positive integers and (23x )(23y) = 64, what is the value of x + y? Strategy: When the bases of two powers are the same in an equation, use these three basic rules to combine the two powers into a power with a single base. The value on the right hand side of the equation should be converted into a power with the same base as the power on the left hand side of the equation. Product of Two Powers Rule: Power of a Power Rule: Quotient of Two Powers Rule: Reasoning: The three basic rules of exponents evolve from the fundamental definition of “exponentiation” that states: xa means “x” multiplied “a” times. Caution: The product and power rules are often confused for one another. Return to Table of Contents See example of strategy

Basic Rules of Exponents Example 1 Question: If p and n are positive integers, and 32p = 2n , what is the value of p/n? Solution Steps 1) Convert 32p to a power with base 2 What essential information is needed? Need to establish a relationship between expressions on left side and right side of equal sign that clarify the relationship between p and n. 32p = 2n (25)p = 2n 25p = 2n 2) Set exponents equal to each other and solve for p/n. What is the strategy for identifying essential information?:Use rules of exponents to covert 32 to a power with a base of 2. 5p = n p n 1 5 = Return to Table of Contents Return to strategy page See another example of strategy

Basic Rules of Exponents Example 2 Question: If 28x+2 = 643 , what is the value of 4x? Solution Steps 1) Convert 643 to a power with base 2. What essential information is needed? Need to establish a relationship between expressions on left side and right side of equal sign that clarify the relationship between the two exponents. 28x+2 = 643 28x+2 = (26)3 28x+2 = 218 2) Set exponents equal to each other to solve for the value of “4x” What is the strategy for identifying essential information? Use rules of exponents to convert 64 to a power with base 2. 8x + 2 = 18 8x = 16 4x = 8 Note: No need to solve for “x”. Can solve directly for the value of 4x. Return to Table of Contents Return to strategy page Return to previous example

Rules of Exponents Additional Rules Strategy: Use these additional rules of exponents when needed. Of the four additional rules, the negative exponent and rational (fractional) exponent rules are utilized most. Power of a Product Rule: Negative Exponent Rule: Reasoning: When an equation contains a variable with a negative exponent and rational exponent, follow a two step process to isolate variable: Zero Exponent Rule: 1) Convert the negative exponent to a positive exponent using rule 2) Raise both sides of equation to the reciprocal of the rational exponent. Rational (fractional) Exponent Rule: Application: Questions with expressions that contain negative exponents and/or rational exponents. Return to Table of Contents See example of strategy

Additional Rules of Exponents Example 1 Question: Positive integers a, b, and c satisfy the equations a-½ = ¼ and b-¾ = ⅛. What is the value of a + b? Solution Steps 1) Apply negative exponent rule to each equation What essential information is needed? The values of a and b are needed. a-½ = ¼ b-¾ = ⅛ 1 a½ = ¼ 1 b¾ = ⅛ a½ = 4 b¾ = 8 2) Raise both sides of each equation to the reciprocal of the rational exponent What is the strategy for identifying essential information?: Use negative exponent rule and raise both sides of each equation to the reciprocal of the rational exponent. (a½ )2 = 42 (b¾ )4/3 = 84/3 a = 16 b = 16 a + b = 32 Return to Table of Contents Return to strategy page See another example of strategy

Additional Rules of Exponents Example 2 Question: If 4-y/2 = 16-1 , then y = ? Solution Steps 1) Apply negative exponent rule to both sides of equation What essential information is needed? Need to directly solve for the value of “y” 4-y/2 = 16-1 1 4y/2 = 16 4y/2 = 16 What is the strategy for identifying essential information? Use negative exponent rule first. Solve for value of “y” by converting powers on both sides of equation to the same base. 2) Convert to same powers 4y/2 = 16 4y/2 = 42 y 2 = 2 y = 4 Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Absolute Value Inequalities Example: A manufacturer produces picture frames between 28 and 42 inches in width. If x represents the size, in inches, of the picture frames produced by the manufacturer, which of the following represents all possible values of x ? Strategy: To solve absolute value inequalities quickly, use a three step approach: Using given information eliminate choices representing the wrong solution type Remove absolute value, evaluate positive solution, and eliminate choices With remaining choices evaluate negative solution and choose correct answer Possible Solution Types: a < x < b x < a or x > b Example of Solution: 28 < x < 42 x < 28 or x > 42 Possible Inequality: | x – 35 | < 7 | x – 35 | > 7 Solution details for | x – 35 | < 7 Remove absolute value: +/-(x – 35) < 7 Reasoning: Absolute value inequalities have properties that can be used to eliminate wrong choices. Positive solution: x – 35 < 7 x < 42 Negative solution: x – 35 > -7 x > 28 Overall solution: 28 < x < 42 Return to Table of Contents See example of strategy

Absolute Value Inequalities Example 1 Question: For a certain airline company, the weight of pilots must be between 140 and 200 pounds. If w pounds is the acceptable weight of a pilot for this airline company, which of the following represents all possible values of w? a) │w - 170│= b) │w + 140│< 60 c) │w - 170│> d) │w -170│< 30 e) │w - 140│< 60 Solution Steps 1 & 2) Remove absolute value sign, solve positive solution, eliminate choices that do not meet the solution w < 200. a) w = Not a solution b) w < Not a solution c) w > Not a solution d) w < Possible solution e) w < Possible solution What essential information is needed? The correct answer must be the solution to 140 < w < 200. 3) Evaluate negative solution What is the strategy for identifying essential information?: Use the absolute value strategy to identify answer d) w > -30 w > Solution e) w > -60 w > 110 Not a solution Return to Table of Contents Return to strategy page See another example of strategy

Absolute Value Inequalities Example 2 Question: A certain manufacturer of pencils requires all pencils to meet a length specification between 6.9 and 7.0 inches inclusive. If x is the length of a pencil that meets the specification, which of the following represents the length of pencils that do not meet the specification? a) │x - 6.0│< b) │x - 6.0│> .05 c) │x - 6.0│> d) │x │> .05 e) │x + 6.0│> 13.0 Solution Steps 1 & 2) Remove absolute value sign, solve positive solution, eliminate choices that do not meet the solution x > 7.0 a) x < Not a solution b) x > Not a solution c) x > Possible solution d) x > Possible solution e) x > Possible solution What essential information is needed? The correct answer will be the solution to x < 6.9 or x > 7.0 3) Evaluate negative solution x < 5.0 Not a solution c) x < -1.0 What is the strategy for identifying essential information? Use the absolute value strategy to identify answer d) x < -.05 x < Solution x < -7.0 Not a solution e) x < -13.0 Return to Table of Contents Return to strategy page Return to previous example

Creation of Math Statements from Words Example: If three times a number x is twelve less than x, what is x ? Words Symbol Translation Is, the same as, is equal to = Equals Sum of, more than, greater than + Addition Less than, difference, fewer - Subtraction Of, product, times × Multiplication For, per ÷ Division Strategy: Use the information in the table to the right to translate words into mathematical expressions and equations. Reasoning: These are common words that are utilized in questions. When properly translated, the solution to a question is usually straightforward. Translation: 3x = x – 12 Solution: x = -6 Return to Table of Contents See example of strategy

Creation of Math Statements from Words Example 1 Question: If ¾ of 3x is 15, what is ½ of 6x? Solution Steps 1) Create the proper math statement What essential information is needed? Create a math statement that properly describes the given information. ¾ of 3x is 15 times equals ¾ · 3x = 15 2) Solve for the value of “3x” What is the strategy for identifying essential information?: Use the table of words to convert the given information into the proper math statement. Recognize that ½ of 6x equals 3x. Solving for the value of 3x will provide correct answer to question. ¾ · 3x = 15 multiply by 4 3 5 [¾ · 3x] = [15] 4 3 Correct answer 3x = 20 Return to Table of Contents Return to strategy page See another example of strategy

Creation of Math Statements from Words Example 2 Question: Which of the following expresses the number that is 15 less than the product of 4 and x + 1? -4x + 14 -4x + 16 4x - 11 4x - 13 4x - 15 Solution Steps 1) Create the proper math statement from given information Product of 4 and x + 1 4(x + 1) 15 less than product of 4 and x + 1 4(x + 1) - 15 What essential information is needed? Create a math statement that properly describes the given information. 2) Simplify the math statement to match answer choices What is the strategy for identifying essential information? Use the table of words to convert the given information into the proper math statement. 4(x + 1) Distribute 4 4x Subtract 15 4x Correct answer Return to Table of Contents Return to strategy page Return to previous example

The Parabola Example: The quadratic function f is given by f(x) = ax2 + bx + c, where “a” is a positive constant and “c” is a negative constant. Which of the figures could be the graph of f? Standard form of a parabola f(x) = ax2 +bx + c Strategy: Many questions about the parabola (sometimes called “the quadratic function”) require an understanding of the impact of constants “a”, “b”, and ‘c” on the graph of f(x). “a” positive opens up “a” negative opens down Reasoning: 1) The coefficient or constant “a” directly influences the x2 term of the function. When f(x) = x2, the parabola opens up. When f(x) = -x2, the parabola opens in the opposite direction or down. 2) The constant “c” is the function value for f(0) = “c”. This is the definition the y-intercept. 3) The impact of “b” is more complicated and usually not important. Click to show answer “c” positive positive “y” intercept “c” negative negative “y” intercept Return to Table of Contents See example of strategy

The Parabola Example 1 Solution Steps Question: The quadratic function f is given by f(x) = ax2 + bx + c, where “a” is a positive constant and “c” is equal to zero. Which of the figures could be the graph of f? “a” positive “c” positive “a” positive “c” negative “a” positive “c” zero A B C What essential information is needed? Need to know the effects of constants “a” and “c” on the graph of a parabola. D E What is the strategy for identifying essential information?: Use parabola strategy to determine effects of “a” and “c”. “a” negative “c” zero “a” negative “c” zero What is the correct choice? (click to verify choice) Return to Table of Contents Return to strategy page See another example of strategy

The Parabola Example 2 Solution Steps Question: The quadratic function f is given by f(x) = ax2 + bx + c, where the product “ac“ is a positive constant. Which of the figures could be the graph of f? “a” positive “c” zero “a” positive “c” negative “a” positive “c” zero A B C What essential information is needed? Need to know the effects of constants “a” and “c” on the graph of a parabola. D E What is the strategy for identifying essential information? Use parabola strategy to determine effects of “a” and “c”. “a” negative “c” zero “a” negative “c” negative “ac” = positive What is the correct choice? (click to verify choice) Return to Table of Contents Return to strategy page Return to previous example

Single Term Denominator Equations Example: If , what is the value of ? 5x + y x 26 5 = y 5x + y x 26 5 = Strategy: When an expression contains two or more variable terms in the numerator and a single variable term in the denominator, expand the expression by placing each term in the numerator over the variable in the denominator 5x x y 26 5 = + 5x x y 26 5 = - y x 1 5 = Reasoning: The expression will often easily simplify into the form required to directly answer the question. Alternative Solution: This problem can also be solved using cross multiplication. Although the algebra is straightforward, students often struggle to isolate the answer when a ratio is required. Try it! Return to Table of Contents See example of strategy

Single Term Denominator Example 1 Question: What is the value of if and ? 7x + y + z y x = 14 z = 5 Solution Steps 1) Expand the expression 7x y z + What essential information is needed? Need values of each variable or find way to simplify the expression using the given ratio values. Note: The value of each ratio is given 2) Substitute given ratio information and simplify What is the strategy for identifying essential information? Use single term denominator strategy to simplify the expression without the need to identify values of each variable. x y 1 14 = y 1 = z y 5 = 7[ ] 1 14 1 2 6.5 Return to Table of Contents Return to strategy page See another example of strategy

Single Term Denominator Example 2 Question: If , what is the value of ? Solution Steps What essential information is needed? Need values of each variable or find way to simplify the expression using the given ratio values. What is the strategy for identifying essential information? Use single term denominator strategy to simplify the expression without the need to identify values of each variable. Return to Table of Contents Return to strategy page Return to previous example

Dividing Irregular Shapes in Polygon Shapes Example: Which of the following represents the area of the five-sided figure shown to the right? Correct strategy x y 5 450 Rectangle Strategy: Always divide irregular polygon shapes into rectangles (or squares) and right triangles. Do not divide the shape into trapezoids or parallelograms. Click to see the animation. Triangle Incorrect strategy Reasoning: The area and perimeter of rectangles and right triangles are usually easy to determine from the given information. In particular, right triangles can be solved using Pythagorean theorem or properties of and triangles. Trapezoid Triangle Note: You are setting a “trap” when a shape is divided into a trapezoid. Return to Table of Contents See example of strategy

Dividing Irregular Shapes Example 1 Question: In the figure above, what is the perimeter of triangle ABC? A B C 4 3 8 6 Figure not drawn to scale Solution Steps 4 9 A C 1) Divide the shape into a rectangle and right triangle (see original figure) 4 9 What essential information is needed? Sides AB and BC easy to determine. Need to divide figure into shapes that will provide an efficient way to find the length of segment AC 2) Determine the length of each side of triangle ABC Determine length of sides AB and BC from properties of triangle AB = 5 and BC = 10 Determine length of side AC from Pythagorean Theorem What is the strategy for identifying essential information?: Divide the shape into a rectangle and right triangle. Return to Table of Contents Return to strategy page See another example of strategy

Dividing Irregular Shapes Example 2 Question: In the rectangle above, the sum of the areas of the shaded region is 14. What is the area of the unshaded region? Solution Steps x y What essential information is needed? What is the strategy for identifying essential information? Divide the shape into a rectangle and right triangle. Return to Table of Contents Return to strategy page Return to previous example

Line Segment or Diagonal Length in a Geometric Solid Example: In the figure shown to the right, the endpoints of the line segment are midpoints of two edges of a cube of volume 64cm3. What is the length of the line segment? Strategy: To find the length of a diagonal or a line segment that connects two edges of a geometric solid, create a right triangle within the solid that uses the unknown segment as the hypotenuse. Click to see the animation. Line Segment a c b Right Triangle c2 = a2 + b2 Pythagorean Theorem Reasoning: By finding a right triangle within the solid, Pythagorean Theorem can be used to find the segment or diagonal length. Helpful Hint: The diagonal of any cube is equal to the cube side length times √3 Caution: Does not apply for rectangular solids (shoe box shape) Return to Table of Contents See example of strategy

Line Segment Length in Solid Example 1 Question: What is the volume of a cube that has a diagonal length of 4√3? Solution Steps 1) Establish relationships between cube diagonal length and side length using properties of a cube What essential information is needed? Side length of the cube is needed to find the volume. Let “a” be the side length of cube a 4√3 The longer side length of right triangle found using properties of triangle a a What is the strategy for identifying essential information?: Use the properties of a cube, the diagonal length, and Pythagorean theorem to find the side length. a√2 2) Apply Pythagorean theorem to find side length a2 + (a√2)2 = (4√3)2 a = 4 Volume = a3 = 43 = 64 Return to Table of Contents Return to strategy page See another example of strategy

Line Segment Length in Solid Example 2 Question: In the figure above, if AB = 24, BC = 12, and CD = 16, what is the distance from the center of the rectangular solid to the midpoint of AB? A C B D E Solution Steps 1) Diagonal BD is the hypotenuse of right triangle BCD. Find the length of BD. A C B D E 24 12 16 What essential information is needed? A connection between given side lengths, the center of solid, and the midpoint of AB Can easily find the length of BD by recognizing that triangle BCD is a multiple of the triangle. The length of BD is 20. ( ) What is the strategy for identifying essential information? Half the length of diagonal BD is equivalent to the desired distance. Use Pythagorean theorem. 2) Half the length of diagonal BC is 20/2= 10 (shown in white on diagram) Return to Table of Contents Return to strategy page Return to previous example

Total area of three shapes = 10 Putting Shapes Together Area = 5 Area = 3 Area = 2 Strategy: When asked to piece together several regular shapes into one shape, sum together the areas of individual pieces. The final shape will have the same area as the sum of the individual pieces. Unit Area = 1 block Which of the shapes below could be made from the three individual shapes shown above? Reasoning: The area must be conserved provided there is no overlap when the individual pieces are combined into one shape. Click to see the animation of the correct choice. Area = 9 Area = 10 Area = 8 Return to Table of Contents See example of strategy

Putting Shapes Together Example 1 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Putting Shapes Together Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

3-4-5 Triangle Strategy: Recognizing the right triangle in a figure can save time and reduce the possibility of error when determining side lengths of a triangle. 5 4 Reasoning: Recognizing triangles as do not require calculation of the third side using Pythagorean Theorem. Triangles with common multiple lengths of a are similar to the 3 Application: Look for right triangles with side lengths that are multiples of Common examples include , , , and triangles. Use similar triangle properties to determine unknown side lengths, not Pythagorean Theorem. 10 15 8 12 6 9 Return to Table of Contents See example of strategy

3-4-5 Triangle Example 1 Question: In the figure above, what is the area of ∆ABC? 100 80 A B C Solution Steps 1) Use properties of triangle to find length of BC Side CA has a length of 80. This is a multiple of four (4 x 20 = 80) Side AB (hypotenuse) has a length of 100. This is a multiple of five (5 x 20 = 100) Conclusion: Side BC is a multiple of 3 and will have a length of 60. (3 x 20 = 60) What essential information is needed? Side length BC is needed to find the triangle area. What is the strategy for identifying essential information?: Can use Pythagorean theorem, however, more efficient to use properties of triangle. 2) Calculate the area of ∆ABC Area = ½(base)(height) = ½(80)(60) Area = 2400 Return to Table of Contents Return to strategy page See another example of strategy

3-4-5 Triangle Example 2 Question: In the figure above, what is the perimeter of ∆XYZ? x y z 55 33 Solution Steps 1) Use properties of triangle to find length of XZ Side YZ has a length that is a multiple of three (3 x 11 = 33) Side XY has a length that is a multiple of five (5 x 11 = 55) Conclusion: Side XZ is a multiple of four and will have a length of 44. (4 x 11 = 44) What essential information is needed? The length of side XZ is needed to find perimeter. What is the strategy for identifying essential information? ∆XYZ is a right triangle. Can use Pythagorean theorem, however, it is easier and more efficient to use triangle relationships. 2) Calculate the perimeter of ∆XYZ Perimeter = XY + YZ + XZ Perimeter = Perimeter = 132 Return to Table of Contents Return to strategy page Return to previous example

Triangle Strategy: If the leg of a right triangle is expressed in terms of , the triangle is likely a The coefficient associated with the is the length of the shorter leg. The hypotenuse is twice the length of the shorter leg. 600 300 5√3 5 10 Reasoning: This relationship is derived by splitting an equilateral triangle into two congruent triangles. The relationships between sides are derived using Pythagorean Theorem. The formula for this relationship is found on the SAT formula sheet. Coefficient Note: The triangle is not a triangle Application: Consider using for any triangle that has a 300 or 600 angle. Also, use for any right triangle that has a 300 or 600 angle. Return to Table of Contents See example of strategy

Triangle Example 1 Question: In triangle ABC shown above, the length of side BC is half the length of side AB. The length of side AC is 4√3. What is the length of side AB? C A B Solution Steps 1) Identify connection to right triangle Triangle side BC = ½ side AB Triangle side AC has length 4√3 Conclusion: ∆ABC is a triangle What essential information is needed? A connection between side lengths that justifies calling triangle ABC a right triangle. 2) Use properties of triangle to find length of AB Side BC is short leg of triangle Side AC is long leg of triangle Side AB is hypotenuse of triangle What is the strategy for identifying essential information?: Use properties of triangle or triangle to establish connection to right triangle. 3) Determine length of side AB AC = 4√3 BC = 4 AB = 2 x 4 = 8 Return to Table of Contents Return to strategy page See another example of strategy

Triangle Example 2 Question: Equilateral triangle ABC has a side length of 4. If BD is an altitude of ∆ABC, what is the area of ∆ABD? A D B C 4 Solution Steps 1) Find the length of AD (base of ∆ABD) and length of BD (height of ∆ABD) Note: ABD is a triangle with angle BAD = 600 and angle ABD = 300 Conclusion: Side AD = 2; half the length of hypotenuse AB What essential information is needed? Need a connection between side length AB (value of 4), AD (base of ∆ABD), and BD (height of ∆ABD) Conclusion: Side BD = 2√3; √3 times the length of the short side AD What is the strategy for identifying essential information? The altitude of an equilibrium triangle divides the triangle into two triangles. Use properties of triangle to make connection. 2) Find the area of ∆ABD Area = ½(base)(height) Area = ½(2)(2√3) Area = 2√3 Return to Table of Contents Return to strategy page Return to previous example

Triangle Coefficient Strategy: If the hypotenuse of a right triangle is expressed in terms of √2 , the triangle is likely a The coefficient associated with the √2 is the length of each triangle leg. 450 5 5√2 Reasoning: This relationship is a property of the triangle. It can be derived using Pythagorean Theorem. The formula for this relationship is found on the SAT formula sheet. Application: Consider using for any triangle that has a 450 angle . Also, any right triangle that is isosceles will be a triangle. Return to Table of Contents See example of strategy

Triangle Example 1 Question: In the figure below, what is the area of the square? 10 Solution Steps 1) Use properties of triangle to find side length (Side length ) √2 = 10 What essential information is needed? Side length of square is needed to calculate area. Side length = 10 √2 2) Calculate area of square What is the strategy for identifying essential information?: Most efficient strategy is to recognize that the diagonal of a square divides the square into two congruent, isoceles triangles. Each triangle is a Area = (side length)2 (10) (√2) Area = = 100 2 Area = 50 Return to Table of Contents Return to strategy page See another example of strategy

Triangle Example 2 Question: In the figure above, if DC = 2√6, what is the value of BC? B C A D 450 300 Solution Steps 1) Find the length of AC using properties of triangle Note: AC is twice the length of AD and DC is √3 times the length of AD AD(√3) = DC = 2√6 AD = 2√6 √3 = 2√2 What essential information is needed? Need to make a connection between the value of DC and the value of BC. Conclusion: AC = 2(2√2) = 4√2 2) Find the length of BC using properties of triangle What is the strategy for identifying essential information? The two right triangles share a common side AC. Use properties of and triangles to make connection. Note: BC is √2 times the length of AC BC = (4√2)(√2) BC = 8 Return to Table of Contents Return to strategy page Return to previous example

Distance Between Two Points x-y Coordinate Plane Strategy: Draw the x-y coordinate, plot the points, and find a right triangle. Calculate the distance as shown. (3, 3.5) (-5, -2.5) d = 10 y2 - y1 Reasoning: As shown to the right, the distance formula is an outcome of applying Pythagorean Theorem in the x-y coordinate plane. The distance “formula” is not given on the SAT formula sheet. x2 - x1 Application: Multitude of problems involving lines and points in the x-y coordinate plane. See examples for specific applications. Return to Table of Contents See example of strategy

Distance Between Two Points Example 1 Question: If points A (6, 2), B(12, 2), and C(9, 9) are endpoints of triangle ABC, what is the perimeter of the triangle? Solution Steps 1) Find the length of side AB using distance formula for a number line What essential information is needed? Need to find the length of each side of triangle ABC. d = │12 - 6│ = 6 2) Find the length of congruent sides AC and BC using distance formula for x-y coordinate plane What is the strategy for identifying essential information?: A quick sketch of the triangle reveals an isosceles triangle with the non-congruent side AB parallel to the x-axis. The remaining two sides are congruent and require use of the distance formula to find side length. 3) Find the perimeter of triangle ABC Perimeter = 6 + √58 + √58 Perimeter = 6 + 2√58 Return to Table of Contents Return to strategy page See another example of strategy

Distance Between Two Points Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Midpoint Determination Number Line Midpoint Strategy: The midpoint (xm) between two endpoints on a number line is found by averaging the two endpoints. 5.5 5.5 4 x1 7 x2 xm Reasoning: The midpoint is equidistant from either endpoints. This is consistent with the properties of the average (mean) of two numbers. xm = 1.5 Application: Number line applications that requires the determination of midpoint or endpoint values. The midpoint “formula” is not given on the SAT formula sheet. Return to Table of Contents See example of strategy

Midpoint Determination Example 1 Question: If 3n and 3n+4 are end points on a number line, what is the midpoint? 3n+1 3n+2 3n+2.5 3n+3 41(3n) Solution Steps 1) Find the sum of the two endpoints 3n + 3n+4 Expand 3n+4 3n + 3n ·34 Factor 3n 3n ( ) = 3n (1 + 81) What essential information is needed? Find the point that is located midway between the two endpoints. 82(3n ) 2) Divide the sum by two to find midpoint What is the strategy for identifying essential information?: Use the midpoint determination strategy for finding midpoint on a number line 82(3n ) 2 = 41(3n ) Return to Table of Contents Return to strategy page See another example of strategy

Midpoint Determination Example 2 Question: If x - 2 and y are endpoints on a number line and x + 6 is the midpoint, which of the following expressions represents y? x x + 2 x + 12 x + 14 x + 16 Solution Steps 1) Apply the midpoint strategy to set up the solution. x + 6 = (x - 2) + y 2 2) Solve for the endpoint “y” What essential information is needed? Find the endpoint that has x + 6 as the midpoint when x - 2 is the other endpoint. x + 6 = (x - 2) + y 2 Cross multiply Simplify and solve for “y” 2(x + 6) = (x - 2) + y What is the strategy for identifying essential information? Apply the midpoint determination strategy to find the endpoint “y”. 2x + 12 = x y x +14 = y Return to Table of Contents Return to strategy page Return to previous example

Midpoint Determination x-y Coordinate Plane (6, 6) (-8, -4) Strategy: The midpoint (xm , ym ) between two endpoints on the x-y coordinate plane is found by averaging the x-coordinates and y-coordinates of the two endpoints. Midpoint (xm , ym ) y1 + y2 (-1, 1) Reasoning: The midpoint of each x-y coordinate point is equidistant from either endpoint. This is consistent with the properties of the average of two numbers x1 + x2 Application: In addition to the x-y coordinate, questions could ask for the midpoint on a number line. Some questions will give the midpoint and one end point and ask for the unknown end point. The midpoint “formula” is not given on the SAT formula sheet. Return to Table of Contents See example of strategy

Midpoint Determination Example 1 Question: In the x-y coordinate plane, the points (2, 8) and (12, 2) are on line m. The point (7, y) is also on line m. What is the value of y? Solution Steps 1) Midpoint analysis of “x” values The “x” value of 7 is the midpoint of 2 and 12 What essential information is needed? A method for determining the value of “y” Conclusion: The “y” value must be the midpoint of 2 and 8 What is the strategy for identifying essential information?: Can use two known points to find the equation of line m and use equation to find y. Equation of line not on SAT formula sheet. As a result, likely not the most efficient approach. As an alternative, midpoint analysis can be used. 2) Find the midpoint of 2 and 8 “y” value = 5 Note: Same result using equation of line……less efficient method. Return to Table of Contents Return to strategy page See another example of strategy

Midpoint Determination Example 2 Question: In the x-y coordinate plane, the midpoint of AB is (2, 3). If the coordinates of point A are (-1, 1), what are the coordinates of point B? Solution Steps 1) Find the endpoint by using the midpoint formula What essential information is needed? Need to connect coordinates of endpoint to the coordinates of midpoint. What is the strategy for identifying essential information? Use the midpoint formula to connect the coordinates of endpoints to the midpoint. Coordinates of endpoint are (5, 5) Return to Table of Contents Return to strategy page Return to previous example

Exterior Angle of a Triangle Strategy: Any exterior angle of a triangle is equal to the sum of the two remote interior angles Remote interior angles 450 750 x0 Reasoning: The sum of the two remote interior angles is supplementary to the third interior angle. Likewise, the exterior angle is supplementary to the third interior angle. Exterior angle Application: This strategy is a useful way to save time and potential calculation errors when an exterior angle of any triangle is needed. Return to Table of Contents See example of strategy

Exterior Angle of a Triangle Example 1 Question: In the figure above, line m is parallel to line k. What is the value of z? x0 y0 1100 m k z0 1000 Solution Steps 1) Find the value of x + y 1100 is an exterior angle; x and y are the remote interior angles Conclusion: x + y =1100 What essential information is needed? A strategy is needed to connect the known angle values to the unknown variables. 2) Find the value of y y is a linear pair with angle 1000 Conclusion: y = 800 and x = 300 What is the strategy for identifying essential information?: Can easily find the value of x + y using exterior angle of triangle strategy. Can also find the value of y. From alternate interior angles, x = z. 3) Find the value of z From alternate interior angles, z = x Conclusion: z = 300 Return to Table of Contents Return to strategy page See another example of strategy

Exterior Angle of a Triangle Example 2 Question: In the figure above, what is the sum of a + b + c + d? a0 b0 d0 c0 950 Solution Steps 1) Find the value of a + b 950 is an exterior angle; a and b are the remote interior angles Conclusion: a + b = 950 What essential information is needed? Need a connection between the given angle value of 950 and the unknown angle variables. 2) Find the value of c + d 950 is an exterior angle; c and d are the remote interior angles Conclusion: c + d = 950 What is the strategy for identifying essential information? The given angle of 950 is an exterior angle to both triangles. 3) Find the value of a + b + c + d a + b + c + d = 2(950 ) a + b + c + d = 1900 Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Perpendicular Lines Strategy: The slopes of perpendicular lines are opposite reciprocals of each other. q l Reasoning: This is a fundamental relationship developed in coordinate geometry Application: All questions involving perpendicular lines require comparison of slopes Return to Table of Contents See example of strategy

4 Perpendicular Lines Example 1 2 2 4 Question: In the xy-plane above, the equation of line m is 4x + 3y = 12. Which of the following is an equation of a line that is perpendicular to line m? a) y = x b) y = -4x + 3 c) y = 4x d) y = ¾x + 6 e) y = -¾x - 6 Solution Steps 1) Slope of line m Slope using figure Slope = ∆y ∆x 4 - 0 0 - 3 = = - 4 3 What essential information is needed? The slope of line m is needed to determine the slope of line perpendicular to line m Slope using equation of line m 4x + 3y = 12 3y = -4x + 12 y x + 12 = - 4 3 What is the strategy for identifying essential information?: Slope of line m can be determined from equation of line m or directly from figure. 2) Equation of line perpendicular to line m Line must have slope = ¾ Correct choice is y = ¾x + 6 Return to Table of Contents Return to strategy page See another example of strategy

Perpendicular Lines Example 2 Question: Line q is tangent to the circle at the point (4, -3). What is the slope of line q? (4, -3) q Solution Steps 1) Find slope of new line Slope of a line that passes through origin can be determined from the ratio of y/x for any point on the line. Slope of new line is -¾ What essential information is needed? Need to identify a line perpendicular to line q and determine the slope of the new line. 2) Find the slope of line q Slope of line q is the opposite reciprocal of slope of new line What is the strategy for identifying essential information? Draw a line from origin to point of tangency. This line is a radius and is perpendicular to line q. Slope of line q is 4 3 Return to Table of Contents Return to strategy page Return to previous example

Interval Spacing What is this value? Strategy: The interval spacing on a number line is found by a two-step process: Determine the distance between two known points on the number line Divide the distance by the number of intervals separating the two known points (Caution: Do not divide by the number of tick marks) 2.5 3 18 23 |18 - 3| 6 = 2.5 Reasoning: By design, the number line has equal distance between each tick mark on the line 18 + 2(2.5) = 23 Application: Used to identify an unknown coordinate on number line. Also used to identify the value of specific term in an arithmetic sequence. Return to Table of Contents See example of strategy

Interval Spacing Example 1 Question: The value of each term of a sequence is determined by adding the same number to the term immediately preceding it. The value of the third term of a sequence is 4 and the value of the eighth term is What is the value of the tenth term? Solution Steps 1) Find the common value. 5 intervals = 12.5 5 intervals = 2.5 What essential information is needed? The common value added to each term of the sequence. 2) Add twice the common value of 2.5 to the eighth term value of 16.5. What is the strategy for identifying essential information? Use interval spacing strategy to identify the common value. Add twice this value to the eighth term to find value of tenth term. Tenth term = Tenth term = 21.5 Return to Table of Contents Return to strategy page See another example of strategy

Interval Spacing Example 2 Question: On the number line above, what is the value of point P? 2n+½ b) 2n+¾ c) 3·2n d) 3·2n e) 3·2n+2 Solution Steps 2n+1 2n+2 P 1) Find the interval spacing 2n+2 - 2n+1 Expand the powers 2n ·22 - 2n ·21 Common factor is 2n 2n ( ) Simplify What essential information is needed? The interval spacing can be used to find the value of “P”. 2n (2) Divide by six intervals 2n (2) 6 = 2n 3 Interval spacing 3 What is the strategy for identifying essential information? Find the interval spacing by dividing the difference of the two endpoints by the number of intervals (six). Multiply the interval spacing by three and add to the value of the left endpoint. 2) Find the value of “P” 2n+1 + (3) 2n 3 Expand the powers and factor = 2n+1 + 2n 2n ·21 + 2n = 2n (21 + 1) 3∙ 2n Value of point “P” Return to Table of Contents Return to strategy page Return to previous example

Triangle Side Lengths Strategy: The 3rd side of any triangle is greater than the difference and smaller than the sum of the other two sides 3 < x < 15 6 9 Reasoning: A side length of 15 would require the formation of a line, not a triangle. A side length of 3 would also require the formation of a line, not a triangle 15 9 6 3 6 Application: Given two sides, choose the smallest or greatest integer value of third side. Given three sides as answer choices, which will not form a triangle. 9 Return to Table of Contents See example of strategy

Triangle Side Lengths Example 1 Question: If the side lengths of a triangle are 8 and 23, what is the smallest integer length of the third side? a) b) c) 16 d) e) 31 Solution Steps 1) Find the smallest possible length of the third side What essential information is needed? The smallest possible length of the third side of the triangle Length of third side > Length of third side > 15 What is the strategy for identifying essential information?: The third side of a triangle must be greater than the difference of the given two sides of the triangle. 2) Determine the smallest integer length of third side of triangle Smallest integer length is 16 Return to Table of Contents Return to strategy page See another example of strategy

Triangle Side Lengths Example 2 Question: Each choice below represents three suggested side lengths for a triangle. Which of the following suggested choices will not result in a triangle? a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12) d) (5, 6, 7) e) (6, 6, 11) Solution Steps 1) Determine range of possible side lengths using first two numbers a) < x < 5 + 2 3 < x < 7 yes b) < x < 7 + 3 4 < x < 10 yes c) < x < 8 + 3 5 < x < 11 no What essential information is needed? The range of possible triangle side lengths for each answer choice. d) < x < 6 + 5 1 < x < 11 yes e) < x < 6 + 6 0 < x < 12 yes What is the strategy for identifying essential information? Evaluate the first two numbers of each answer choice using triangle side length strategy. Test the third number of each answer choice by comparing to range of possibilities based on first two numbers. 2)Test third number of each answer choice a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12) d) (5, 6, 7) e) (6, 6, 11) Correct answer choice is “c” Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Similar Triangle Properties Strategy: Under construction Reasoning: Application: Return to Table of Contents See example of strategy

Similar Triangle Properties Example 1 x a 4 3 8 Question: In the figure to the right, what is the value of “a” ? Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Similar Triangle Properties Example 2 E D C B A Question: In the figure to the right, , , , and What is the length of ? What essential information is needed? Solution Steps What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

The Slippery Slope Strategy: When given linear equations as answer choices and a question about the amount of change in the “y” variable as the “x” variable is changed a given amount, use the properties of slope to quickly select the correct choice. If d represents the distance measured in meters from a particular coffee shop and t is time measured in minutes, which of the following equations describes the greatest increase in distance from the coffee shop during the period from t = 5 minutes to t = 8 minutes? Reasoning: Slope is a measure of the amount of change in the “y” value when the “x” value is changed by one unit. The constant in the equation has no impact on the amount of change in the dependent variable value. a) d = 50t - 100 b) d = 40t c) d = 40t + 100 d) d = -50t e) d = -500t Caution: Do not calculate distance values by direct substitution into each equation. Use properties of slope to quickly determine answer. Click for correct choice. Application: Any question Return to Table of Contents See example of strategy

The Slippery Slope Example 1 Investment Value at t Years A -30t + 50 B -10t - 50 C -10t + 50 D 10t - 50 E 30t - 50 Question: The table to the right gives the value in dollars of five different investments at t years after the investment was started. The value of which investment falls the greatest amount during the period t = 4 to t = 9 ? Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

The Slippery Slope Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Using Function Notation Strategy: Replace the variable in the function expression (right side of equal sign) with the value, letter, or expression that has replaced the variable (usually x) in the function notation (left hand side of equal sign) Introduction Function notation such as f(x), g(x), and h(x) are useful ways of representing the dependent variable “y” when working with functions. For example, the function y = 2x + 5 can be written as f(x) = 2x + 5, g(x) = 2x + 5, or h(x) = 2x + 5. Reasoning: Function notation is a road map or guide that directly connects the “x” value for a given function with one unique “y” value. Application: Function notation can be applied in many different ways on the SAT. See examples for details. Function notation is commonly used to describe translations and reflections of functions. See Table of Contents for additional strategies that use function notation. Important Note: Function notation is not a mathematical operation. See example of commonly made mistake. Return to Table of Contents See example of strategy

Using Function Notation Example of Common Mistake Question: At a certain factory, the cost of producing control units is given by the equation C(n) = 5n + b. If the cost of producing 20 control units is $300, what is the value of “b”? Solution Steps for Commonly Made Mistake 1) Replace “C” with 300 and replace “n” with 20 C(n) = 5n + b Common mistake: Function notation should not be used as a math operation. C(n) should be replaced with 300 when n = 20. Do not multiply 300 and 20 as in a math operation. 300(20) = 5(20) + b 6000 = b b = 5900 (incorrect answer) Correct Solution Steps Correct use of function notation: C(n) is replaced with 300 when n is replaced with 20 in the function equation. C(n) = 5n + b 300 = 5(20) + b 300 = b b = 200 (correct answer) Return to Table of Contents Return to strategy page See example of strategy

Using Function Notation Example 1 Question: If f(x) = x + 7 and 5f(a) =15, what is the value of f(-2a)? Solution Steps 1) Find the value of “a” Given 5f(a) = 15 Divide both sides by 5 What essential information is needed? The value of “a” is needed to determine the value of f(-2a). Result f(a) = 3 Given f(x) = x Evaluate f(a) f(a) = a + 7 = 3 Result: a = -4 What is the strategy for identifying essential information?: Use the given information and properties of function notation to identify the value of “a”. Use this value to evaluate f(-2a). 2) Use a = -4 to find f(-2a) f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8) f(8) = 8 + 7 f(-2a) = 15 Return to Table of Contents Return to strategy page See another example of strategy

Using Function Notation Example 2 y = f(x) 2 -2 Question: The graph of y = f(x) is shown to the right. If the function y = g(x) is related to f(x) by the formula g(x) = f(2x) + 2, what is the value of g(1)? What essential information is needed? The math expression g(1) from which the value of g(1) can be determined Solution Steps 1) Find the expression for g(1) g(x) = f(2x) + 2 What is the strategy for identifying essential information? Find the expression for g(1) by substitution and the value of g(1) using the graph of y = f(x). g(1) = f(2) + 2 2) Find the value of f(2) from the graph of y = f(x) f(2) = 2 g(1) = g(1) = 4 Return to Table of Contents Return to strategy page See another example of strategy

Using Function Notation Example 3 x f(x) g(x) 1 3 8 2 4 10 5 6 7 Question: Using the table to the right, if f(3) = k, what is the value of g(k)? What essential information is needed? The value of “k” is needed to find g(k). Solution Steps What is the strategy for identifying essential information? Use the table of function values to find “k”. Once known, find g(k) using the table of function values. 1) Find the value of “k” using table. f(3) = k f(3) = 5 2) Find the value of g(5) using table. g(5) = 4 Return to Table of Contents Return to strategy page See another example of strategy

Using Function Notation Example 4 Question: If f(x) = x + 8, for what value of x does f(4x) = 4? Solution Steps 1) Determine an expression for f(4x) What essential information is needed? Need to determine the value of “x” that satisfies f(4x) = 4. f(x) = x + 8 f(4x) = 4x + 8 2) Set the expression for f(4x) equal to 4 and solve for the value of “x” What is the strategy for identifying essential information? Use function notation principles to determine an expression for f(4x). Set the expression equal to the value of 4. f(4x) = 4x + 8 = 4 4x + 8 = 4 4x = -4 x = -1 Return to Table of Contents Return to strategy page Return to example 1

Function Reflections x - Axis Strategy: The reflection of a function y = f(x) around the x-axis is easily performed by graphing the opposite (negative) of each y-value. Using function notation, this can be communicated as y = - f(x). y = f(x) Reasoning: The reflection of a function around the x-axis can be viewed as a mirror image of the original reflection. Imagine the x-axis as a flat mirror that reflects and produces an image of the original function on the opposite side of the x-axis. y = - f(x) Reflection of f(x) Application: x-axis reflections can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Reflections: x - Axis Example 1 Question: If point (a, b) is reflected over the x-axis, what are the coordinates of the point after the reflection? Solution Steps A reflection over the x-axis is described by y = -f(x). To accomplish the reflection, change the sign of the y-coordinate only. What essential information is needed? Must determine which, if any, coordinate signs will be affected. Correct answer is (a,-b) What is the strategy for identifying essential information?: For an x-axis reflection, use the function notation y = -f(x) as a guide. Note: Do not get confused by the original sign of the y-coordinate. If the original sign is “-y”, the reflected point will have the sign “+y”. Return to Table of Contents Return to strategy page See another example of strategy

Function Reflections: x - Axis Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Function Reflections y - Axis Strategy: The reflection of a function y = f(x) around the y-axis is easily performed by graphing the opposite (negative) of each x-value. Using function notation, this can be communicated as y = f(-x). y = f(-x) Reflection of f(x) y = f(x) Reasoning: The reflection of a function around the y-axis can be viewed as a mirror image of the original reflection. Imagine the y-axis as a flat mirror that reflects and produces an image of the original function on the opposite side of the y-axis. Application: y-axis reflections can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Reflections: y - Axis Example 1 Question: For the graph of the function f shown above, for what point does f(x) = f(-x)? (-1, 0) (0, 1) (2, 2) (5, 0) Solution Steps 1) Reflect f(x) about the y - axis ( click to show reflection) What essential information is needed? Must determine which, if any, coordinate signs will be affected. 2) Identify the point for which f(x) = f(-x) The only point that remains the same after reflection is the y intercept f(0) = 1 and f(-0) = 1 Correct choice is (0, 1) What is the strategy for identifying essential information?: Helps to recognize that f(x) = f(-x) describes a reflection about the y - axis. Return to Table of Contents Return to strategy page See another example of strategy

Function Reflections: y - Axis Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Function Reflections Absolute Value Strategy: The absolute value of function y = f(x) is easily created by graphing the opposite (negative) of each y-value that is negative on the original function. Using function notation, this can be communicated as y = |f(x)|. y = |f(x)| y = f(x) Reasoning: The absolute value of a function is a reflection of y = f(x) around the x-axis for those intervals of x that have negative y values. Application: Absolute value can be created for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Reflections: Absolute Value Example 1 Question: The graph of y = f(x) is shown above. Which of the choices could be the graph of y = │f(x)│? A B C D E What essential information is needed? Need to determine the effect of absolute value on the graph of f(x) Solution Steps What is the strategy for identifying essential information?: The absolute value strategy should be used. The absolute value reflects the graph of y = f(x) about the x- axis for intervals of “x” where f(x) < 0. Correct answer is choice C Return to Table of Contents Return to strategy page See another example of strategy

Function Reflections: Absolute Value Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Function Translations Horizontal Shift Strategy: A horizontal shift of a function y = f(x) is easily performed by sliding the function right or left parallel to the x-axis a specified distance. Using function notation, a shift to the right of 2 units can be communicated as y = f(x-2). A shift to the left of 4 units can be communicated as y = f(x+4) y = f(x) 2 y = f(x-2) 2 y = f(x+4) Reasoning: A horizontal shift described by y = f(x-2) has the same y-value at x = 2 as the original function f(x) at x = 0. Application: Horizontal shifts can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

-1 2 y = f(x) Function Horizontal Shift Example 1 Solution Steps Question: The graph of y = f(x) is shown to the right. Which of the following could be the graph of y = -f(x+1) ? Click to see answer choices Horizontal shift left y = f(x+1) x-axis reflection y = -f(x) Horizontal shift right y = f(x-1) A B C What essential information is needed? Need to interpret the impact of -f(x+1) on the original function y = f(x). Horizontal shift right x-axis reflection y = -f(x-1) Horizontal shift left x-axis reflection y = -f(x+1) What is the strategy for identifying essential information? Use the function notation strategy and the properties of function reflections and translations to choose the correct answer. D E What is the correct choice? (click to verify choice) Return to Table of Contents Return to strategy page See another example of strategy

Function Horizontal Shift Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Function Translations Vertical Shift y = f(x)+4 Strategy: A vertical shift of the function y = f(x) is easily performed by sliding the function up or down parallel to the y-axis a specified distance. Using function notation, a shift down of 2 units can be communicated as y = f(x)-2. A shift up of 4 units can be communicated as y = f(x)+4 y = f(x) 2 y = f(x)- 2 Reasoning: A vertical shift described by y = f(x)-2 decreases the y-value by 2 units for each value of x on the original function y = f(x). Application: Vertical shifts can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Vertical Shift Example 1 5 Question: The figure to the right shows the graph of function f(x) in the x-y coordinate plane. If the area between f(x) and x-axis is 10, what is the area between the function f(x)+2 and x-axis ? y = f(x) Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Function Vertical Shift Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Function Translations Vertical Stretch y = 2f(x) Strategy: A vertical stretch of the function y = f(x) is easily performed by multiplying each y-value by a specified amount greater than one. Using function notation, a vertical stretch of 2 units can be communicated as y = 2f(x). Multiply each y-value by 2 y = f(x) Reasoning: A vertical stretch described by y = 2f(x) multiplies each y-value by 2 units for each value of x on the original function y = f(x). Application: Vertical stretches can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Vertical Stretch Example 1 Question: The graph of y = f(x) is shown above. Which of the choices could be y = 2f(x)? A B C D E What essential information is needed? Need to understand the impact on y = f(x) when f(x) is multiplied by 2. Solution Steps What is the strategy for identifying essential information?: y = 2f(x) describes a vertical stretch. Apply the properties of a vertical stretch to y = f(x). A vertical stretch multiplies each “y” value on f(x) by two. As a result, the x-intercepts remain the same on y = 2f(x). The correct answer choice is E Return to Table of Contents Return to strategy page See another example of strategy

Function Vertical Stretch Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Function Translations Vertical Shrink Strategy: A vertical shrink of the function y = f(x) is easily performed by multiplying each y-value by a specified amount between zero and one. Using function notation, a vertical shrink of ½ units can be communicated as y = ½f(x). Multiply each y-value by ½ y = f(x) y = ½f(x) Reasoning: A vertical shrink described by y = ½f( x) multiplies each y-value by ½ units for each value of x on the original function y = f(x). Application: Vertical shrinks can be performed for any function using the strategy described above. Return to Table of Contents See example of strategy

Function Vertical Shrink Example 1 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Function Vertical Shrink Example 2 Question: Page under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Average (Arithmetic Mean) Problems Strategy: Apply the basic definition of average (arithmetic mean) to solve this class of problems. sum of values number of values average = Reasoning: Information will typically be given for the average and the number of values. The sum of values will be always be needed to reason through question and will typically consist of an expression with unknown variable(s). Often used form: sum of values = (average)( number of values) Application: 1) Problems that ask for an unknown value when given remaining values in the list and the average value of the list. 2) Problems that provide the average of a list of numbers, removes a number from the list, gives the new average, and asks for the value of the removed number. Caution: You will rarely be asked to find the average of a list of values. Instead, you will typically be asked to find the median of a list of values. Return to Table of Contents See example of strategy

Average (Arithmetic Mean) Example 1 Question: If the average of 6 and x is 12, and the average of 5 and y is 13, what is the average of x and y? Solution Steps 1) Determine the values of x and y: 6 + x 2 = 12 5 + y = 13 What essential information is needed? Need values of x and y to determine average value. 6 + x = 24 5 + y = 26 x = 18 y = 21 What is the strategy for identifying essential information?: Apply basic definition of average to find values of x and y separately. 2) Find average of x and y using basic definition of average: 2 Average = Average = 19.5 Return to Table of Contents Return to strategy page See another example of strategy

Average (Arithmetic Mean) Example 2 Question: The average of five positive odd integers is 15. If n is the greatest of these integers, what is the greatest possible value of n? Solution Steps 1) Find the sum of the five positive odd integers. What essential information is needed? The sum of the five positive odd integers is needed and a strategy to determine the greatest possible value of “n” Sum of values = (15)(5) = 75 2) Determine the greatest possible value of “n” using reasoning skills What is the strategy for identifying essential information? Apply the definition of average to find sum. Use reasoning skills to determine greatest possible value of “n” The four smallest positive integers are 1, 1, 1, 1 with a sum of four. The greatest possible value of “n” is n = = 71 Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Median of Large Lists Strategy: The middle value in a list of ascending or descending ordered values is the median. Large lists of values (more than 7 values) are usually structured in table form or bar chart form. Either form will not require rewriting of the order by the student. Additional Helpful Hints 1) For an ordered list with an odd number of values, the median is the middle value. 2) For an ordered list with an even number of values, the median is the average of the two middle values. Reasoning: Values provided in table form are similar to values provided in histogram form. In both forms it is easy to determine the cumulative total number of values starting with the lowest value. Caution: Do not confuse median with mean. When presented a table of values or a list of values, the question typically requires determination of the median, not the mean. Application: When values are organized in tables, questions will generally ask for the median directly or will give the median and ask for the value of an unknown variable. Return to Table of Contents See example of strategy

Median of Large Lists Example 1 Question: The scores on a recent physics test for 20 students are shown in the table to the right. What is the median score for the test? Solution Steps Score Number of Students 100 95 1 90 85 2 80 3 75 4 70 65 60 1 2 3 Sum = 7 1 2 3 4 Sum = 11 What essential information is needed? When the test scores are ordered from largest to smallest, find the middle score for the list. What is the strategy for identifying essential information?: With the test scores in table form, no additional ordering is needed. With 20 students, the median is the average of the scores of the 10th and 11th students. The 5th , 6th , and 7th students each received a score of 80 on the test The 8th ,9th ,10th ,and 11th students each received a score of 75 on the test Median score is 75 Return to Table of Contents Return to strategy page See another example of strategy

Median of Large Lists Example 2 Question: If the median of 10 consecutive odd integers is 40, what is the smallest integer among these integers? Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Elementary Probability Strategy: Divide the number of values that meet the given criteria by the total number of values in the set. Given information: {10, 12, 13, 18, 21, 23, 25, 29} Question: What is the probability of choosing a prime number at random from the above set? Reasoning: This is the basic definition of probability. The probability of an event is a number between 0 and 1, inclusive. If an event is certain, the probability is 1. If an event is impossible, the probability is 0. Essential information: 1)The number of values meeting the question criteria is 3 2)The total number of values in the set is 8 Application: Additional applications include finding the probability of choosing a particular object (marbles, cookies, coins) from a container with more than one type of object. Solution: Probability = ⅜ Return to Table of Contents See example of strategy

Elementary Probability Example 1 Question: A jar contains red, blue, and yellow marbles in the ratio 9:4:2. If a marble is selected at random, what is the probability of selecting a blue marble? Solution Steps 1) Determine the ratio of blue marbles to total number of marbles For every 15 total marbles in the jar ( = 15) there are 4 blue marbles What essential information is needed? The ratio of number of blue marbles to the total number of marbles. 2) Determine the probability The probability can be found by using the ratio of blue marbles to total marbles. What is the strategy for identifying essential information?: Use the properties of ratios to determine the essential information. Use the ratio to determine the probability. Probability = 4 15 Note: It is not necessary to know the exact number of each marble in the jar. Ratios are sufficient for probability. Return to Table of Contents Return to strategy page See another example of strategy

Elementary Probability Example 2 Question: A certain bowling center has two sizes of bowling balls, twelve pounds and sixteen pounds. For every 3 twelve pound bowling balls there are 4 sixteen pound bowling balls. If a bowling ball is chosen at random, what is the probability that a sixteen pound bowling ball will be selected? Solution Steps 1) Determine the ratio of blue marbles to total number of marbles For every 7 bowling balls (3 + 4 = 7), there are 4 sixteen pound bowling balls 2) Determine the probability What essential information is needed? The ratio of the number of sixteen pound bowling balls to the total number of bowling balls. The probability can be found by using the ratio of sixteen pound bowling balls to the total number of bowling balls Probability = 4 7 What is the strategy for identifying essential information? Use the properties of ratios to determine the essential information. Use the ratio to determine the probability. Note: The strategy for this problem is identical to the previous example. The questions are slightly different, however both involve ratios Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Probability of Independent Events Strategy: Multiply the probabilities of the individual events together to find the overall probability. Definition: Two events are independent if the outcome of the first event has no effect on the outcome of the second event Reasoning: Each individual first event must be paired with each individual second event. To account for the total number of outcomes meeting the given criteria (value in numerator) and the total number of possible outcomes (value in denominator), the individual probabilities must be multiplied together. Example: David has a red, yellow, blue, and green hat. He also has a red and blue shirt. If an outfit consists of a hat and shirt, what is the probability that David will wear an all red outfit? Solution: The probability of choosing a red hat is ¼ and the probability of choosing a red shirt is ½. Application: Popular applications include the probability of an outcome when a coin is flipped multiple times and the probability of passing multiple academic courses The overall probability is (¼)(½) = ⅛ Return to Table of Contents See example of strategy

Probability of Independent Events Example 1 Question: Adam has a 90% chance of passing history and a 60% chance of passing calculus. What is the probability that Adam will pass calculus and not pass history? Solution Steps 1) Determine the probability that Adam will pass calculus Probability = 60 100 6 10 = What essential information is needed? Are these events independent of each other? 2) Determine the probability that Adam will not pass history Probability = 10 100 = 1 What is the strategy for identifying essential information?: It can be assumed that passing history is independent of passing calculus. The two events are independent and the individual probabilities can be multiplied together. 3) Determine the probability that Adam will pass calculus and not pass history Overall probability = 1 10 6 x = 6 100 = 3 50 Return to Table of Contents Return to strategy page See another example of strategy

Probability of Independent Events Example 2 Question: The three cards shown to the right were taken from a box of ten cards, each with a different integer from 0 to 9. What is the probability that the next two cards selected from the box will both have an even integer on it? 1 5 7 Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Definition Back to Frequent Questions Geometric Probability Strategy: Divide the area of the smaller geometric shape by the area of the larger geometric shape. Definition: Geometric probabilities involve the use of two or more geometric figures. Example: A small circle with radius 3 is completely inside a larger circle with radius 6. If a point is chosen at random from the large circle, what is the probability that the point will be in the small circle? Reasoning: For planar geometrical shapes, area is the proper quantity to compare when selecting a point inside the figure. Essential information: Application: Usually involve simple shapes such as circles, rectangles, and squares. In all cases there is a smaller shape inside the larger shape and the analysis requires calculation of shape area. 1) Area of small circle is π(3)2 = 9π 2) Area of large circle is π(6)2 = 36π Solution: Probability = ¼ Return to Table of Contents See example of strategy

Geometric Probability Example 1 Question: In the figure above, each of the small circles has a radius of 3 and the large circle has a radius of 9. If a point is chosen at random inside the larger circle, what is the probability that the point does not lie in the shaded area? Solution Steps 1) Find the area of each circle What essential information is needed? Need the area of the large circle and area of each of the smaller circles. Area of each small circle = π(3)2 = 9π Area of large circle = π(9)2 = 81π What is the strategy for identifying essential information?: Use the formula for area of a circle to find areas of each circle. To find probability, ratio the area of the shaded region to the area of the large circle. 2) Find the geometric probability Probability = 81π - 2(9π) 81π = 63π Probability = 7 9 Return to Table of Contents Return to strategy page See another example of strategy

Geometric Probability Example 2 Question: The rectangle above with side length 4 contains circle C that has a radius of 1. If a point is chosen at random inside the rectangle, what is the probability that the point will lie in triangle ABC? C 4 B A Solution Steps 1) Find the area of triangle ABC Triangle ABC is a triangle AB is twice the radius of circle C and has a length of 2 AC and CB are congruent and are each equal to √2 Area = ½(√2)(√2) = 1 What essential information is needed? Need to determine the area of triangle ABC and the area of the rectangle. The length of AB is needed to find both areas. 2) Find the probability Probability = area of triangle area of rectangle = 1 (2)(4) What is the strategy for identifying essential information? Side AB is twice the radius of circle C. Knowing AB, use Pythagorean theorem to find AC and CB. Probability = ⅛ Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions The Unit Cell Three “Unit Cells” shown Leftover section Not a unit cell Strategy: Divide the given end of the metal strip into a smaller shape, called a “unit cell”, that can used to easily and quickly answer the question. Click to show the unit cell! One end of a 30-inch long metal strip is shown in the figure above. The lower edge was formed by removing a 1-in square from the end of each 3-inch length on one edge of the metal strip. What is the total perimeter, in inches, of the 30-inch metal strip? Reasoning: The unit cell is a repeating shape that comprises the entire object shape. Ten unit cells comprise the entire metal strip. Click to see calculation. The top horizontal section and the bottom notched section of each unit cell contributes = 8 inches to the perimeter. 1 in 3 in 30-in strip 3-in unit cell = 10 unit cells The “Unit Cell” Application: Any question that provides, in the form of a figure, a representative section of a longer object. The total perimeter is equal to: 10 unit cells x 8-in/unit cell + 2 vertical sides x 3-in Perimeter = 86 inches Return to Table of Contents See example of strategy

The Unit Cell Example 1 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

The Unit Cell Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

It’s Absolutely Easy! Strategy: Under construction Reasoning: Application: Return to Table of Contents See example of strategy

It’s Absolutely Easy! Example 1 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

It’s Absolutely Easy! Example 2 Question: Under construction Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Making Connections The “if…” Statement Example 1: If 4x2 = 18y = 36, what is the value of 2x2y? Example 2: If 2x + 7y = y, which of the following must equal 4x + 12y ? Strategy: For questions that begin with “If…” and end with “what is the value of…”, or “which of the following must equal…”, find a straightforward connection that links the given information (usually an equation) to the desired answer (usually the value of an expression). Example 1 4x2 = 18y = x2y? Connection #1: Set 4x2 = 36. Solve for 2x2 Connection #2: Set 18y = 36. Solve for y Connection? Example 2 2x + 7y = y x + 12y Connection: Subtract y from both sides of equation. Result is 2x + 6y = 0. Multiply both sides of equation by 2. Connection? Reasoning: The questions are designed to be solved in a straightforward way, provided the connection between the given information and the desired answer is made. To find the connection typically requires out of the box thinking. Return to Table of Contents See example of strategy

Making Connections Example 1 Question: If x is positive and x(x-1) = 30, what is the value of x(x+1) ? Solution Steps Identify the factors of 30 that are consecutive integers: 6(6-1) = 6(5) = 30 x = 6 What essential information is needed? Need to find a connection between the factored form of the expression on the left side of the equal sign and the value of 30 on the right side. What is the strategy for identifying essential information?: The factors on the left side are consecutive integers. Determine if the value 30 has factors that are consecutive positive integers. Note: Not necessary to foil the expression and solve as a quadratic equation x2 - x - 30 = 0 2) Find the value of x(x+1) for x = 6 6(6+1) = 6(7) = 42 Return to Table of Contents Return to strategy page See another example of strategy

Making Connections Example 2 Question: If x and y are positive numbers and , then what is the value of ? Solution Steps Solve directly for What essential information is needed? Need to find a connection between the equation and the expression. 2) Substitute result into expression What is the strategy for identifying essential information? Solve directly for and substitute the result into the expression Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Parallel Lines and Transversals Strategy: If uncertain of parallel line properties, use the diagram appearance to determine the relationship between pairs of angles. Note: This strategy is valid if and only if the figure is drawn to scale. Parallel Lines Reasoning: Any pair of angles will either be congruent (equal measure) or supplementary (sum to 180 degrees). Using the figure given in a question, it is usually obvious when angles are congruent. If they do not appear congruent, they are supplementary. In the figure shown above, pairs of red or pairs of blue angles are congruent. A pair consisting of a red and blue angle are supplementary. Application: Many questions contain parallel lines with two transversals (see example 2). Return to Table of Contents See example of strategy

Parallel Lines and Transversals Example 1 115o 50o xo Question: In the figure to the right, if m is parallel to n, what is the value of x ? 50o 65o n m q p 115o What essential information is needed? Determine the measures of the two remaining angles inside the triangle that contains angle x. Solution Steps 1) The two remaining angles inside the triangle are 50o (congruent to the 50o angle) and 65o (supplementary to the 115o angle). Click again to see animation of the angles. What is the strategy for identifying essential information?: Use the properties of parallel lines and transversals to determine the measures of the two angles. 2) Calculate the measure of angle x: x = ( ) x = 65o Return to Table of Contents Return to strategy page See another example of strategy

Parallel Lines and Transversals Example 2 yo xo 55o 180 - xo 180 - yo Question: In the figure to the right, if m is parallel to n, what is the value of x + y ? m 180 - xo 180 - yo n What essential information is needed? Need to define the two remaining angles inside the triangle in terms of x and y. Solution Steps 1) The two remaining angles inside the triangle are x (supplementary to angle x) and y (supplementary to angle y). Click again to see animation of the angles. What is the strategy for identifying essential information? Use the properties of parallel lines and transversals to define the measures of the two angles in terms of x and y. 2) Calculate the measure of angle x: (180 - x) + (180 - y) + 55 = 180 x + y = 235o Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Even/Odd Integers Strategy: Use the table of properties to the right to determine if an operation between two integers will result in an even or odd integer. Addition or Subtraction Multiplication odd + odd = even odd - odd = even odd x odd = odd even + even = even even - even = even even x even = even odd + even = odd odd - even = odd odd x even = even Reasoning: These integer formation properties eliminate the need to use the “plug in a number” strategy that is often more time consuming than applying the integer properties. Application: There is always at least one question that can be easily solved using these integer formation properties. Return to Table of Contents See example of strategy

Even/Odd Integers Example 1 Question: If a + b is an even integer, which of the following must be even? a) 2a + b b) 2a - b c) ab d) (a + 1)(b + 1) e) a2 - b2 Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Even/Odd Integers Example 2 Question: If 2a + b is an odd integer, which of the following must be true? a is odd b is odd 2a2 - b2 is odd Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

Consecutive Integers Strategy: Express the sum of three consecutive integers, consecutive odd integers, or consecutive even integers as the sum of the expressions shown to the right. Consecutive Integers n, n + 1, n + 2 Where n is any integer Consecutive Odd Integers n, n + 2, n + 4 Where n is an odd integer Reasoning: When you count by one’s from any number in the set of integers, consecutive integers are obtained. If you count by two’s beginning with any even/odd integer, consecutive even/odd integers are obtained. Consecutive Even Integers n, n + 2, n + 4 Where n is an even integer Application: Questions that ask for the smallest of three consecutive integers or consecutive odd/even integers when their sum is a specified value. Any question that begins with the phrase “Given three consecutive integers”. Return to Table of Contents See example of strategy

Consecutive Integers Example 1 Question: The average of a set of 5 consecutive even integers is 20. What is the smallest of these 5 integers? Solution Steps 1) Find the sum of the 5 integers using the definition of average What essential information is needed? Find the sum of the 5 consecutive even integers. Use the sum to find the smallest integer. 2) Find the smallest integer using consecutive even integer strategy What is the strategy for identifying essential information?: Use the definition of average to find the sum. Use the sum and the consecutive integer strategy to find the smallest integer. Return to Table of Contents Return to strategy page See another example of strategy

Consecutive Integers Example 2 Question: What is the median of 7 consecutive integers if their sum is 42? Solution Steps Find the smallest integer in a list of seven integers. What essential information is needed? The fourth value in a list of seven consecutive integers. What is the strategy for identifying essential information? Use the consecutive integer strategy to find the smallest integer. Add three to the smallest integer to find the value of the fourth integer. This will be the median value. 2) Find the median value by adding three to the smallest integer. Return to Table of Contents Return to strategy page Return to previous example

Back to Frequent Questions Tangent To A Circle Tangent line Strategy: If a line is drawn tangent to a circle, draw the radius of the circle to the point of tangency with the line. (Click again to draw radius) Reasoning: A tangent line and the radius always form a right angle at the point of tangency. The right angle relationship will be used in all applications involving tangent lines to circles. Application: Find the slope of the tangent line when given the coordinates of the point of tangency with the circle and the center of the circle. Find the perimeter of a shape when a circle is inscribed inside the given shape. Return to Table of Contents See example of strategy

Tangent To A Circle Example 1 Question: In the figure to the right, a circle is centered at the origin and is tangent to the line at point P. If the radius of the circle is 15, what is the slope of line? 9 12 15 P(9, b) P(9, -12) What essential information is needed? The radius and line are perpendicular to each other. Find the radius slope and use the relationship that the slope of perpendicular lines are opposite reciprocals of each other. Solution Steps 1) Using Pythagorean Theorem, the y-coordinate, b, has a value of -12. The slope of the radius is: What is the strategy for identifying essential information?:Use the radius length and the x-coordinate of point P to find b, the y-coordinate of point P. This is accomplished using Pythagorean Theorem. 2) Find the slope of line using the relationship between the slopes of perpendicular lines. Slope of line is Return to Table of Contents Return to strategy page See another example of strategy

Tangent To A Circle Example 2 x y z 30 60 Question: In the figure to the right, a circle is tangent to the side of equilateral triangle xyz and the radius equals 5. What is the perimeter of triangle xyz ? Radius What essential information is needed? The length of a side of the triangle. Solution Steps 1) Using properties of the triangle, the length of half the triangle side is 2) The triangle side length is What is the strategy for identifying essential information? The circle radius and the equilateral triangle side are perpendicular at the tangent point. Draw a right triangle and use the properties of the triangle to find the side length. Click again to show the right triangle 3) The perimeter is three times the triangle side length: Return to Table of Contents Return to strategy page Return to previous example

Strategy Section Concluded This is the end of the Strategy section. Please select one of the options at the bottom of this page Return to Table of Contents Return to first strategy Return to Introduction

Sample Strategy Strategy: The 3rd side of any triangle is greater than the difference and smaller than the sum of the other two sides Reasoning: A side length of 15 would require the formation of a line, not a triangle Application: A side length of 3 would also require the formation of a line, not a triangle Return to Table of Contents See example of strategy

Sample Strategy 1) The total cost of 4 equally priced notebooks is $5.00. If the price is increased by $0.75, how much will 6 of these notebooks cost at the new rate? $7.50 $8.00 $10.00 $12.00 $14.00 What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 2) If Jim traveled 20 miles in 2 hours and Sue traveled twice as far in twice the time, what was Sue’s average speed, in miles per hour? 5 10 20 30 40 What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 3) In the figure below, if CD is a line, what is the value x ? 45 60 90 100 120 C D x0 y0 Note: Figure not drawn to scale. What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 4) For which of the following functions is f(-2) > f(2) ? 3x2 3 3/x2 x2 + 2 3 - x3 What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 5) The energy required to stretch a spring beyond its natural length is proportional to the square of how far the spring is being stretched. If an energy of 20 joules stretches a spring 4 centimeters beyond its natural length, what energy, in joules, is needed to stretch this spring 8 centimeters beyond its natural length? 10 40 80 100 120 What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 6) The average (arithmetic mean) of x and y is 10 and the average of x, y, and z is 12. What is the value of z ? 2 4 12 16 26 What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 7) If Z is the midpoint of XY and M is the midpoint of XZ, what is the length of ZY if the length of MZ is 2 ? 2 4 6 8 More information is needed to answer question What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 8) In the figure below, line L is parallel to line m. What is the value of x ? 110 120 130 140 150 x0 600 1100 M L What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Strategy 9) If a and b are odd integers, which of the following must also be an odd integer? I only II only III only I and II II and III (a + b)b (a + b) +b ab +b What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents See example of strategy

Sample Factoring Strategy Example 1 Question: Solution Steps What essential information is needed? What is the strategy for identifying essential information?: Return to Table of Contents Return to strategy page See another example of strategy

Sample Factoring Strategy Example 2 Question: Solution Steps What essential information is needed? What is the strategy for identifying essential information? Return to Table of Contents Return to strategy page Return to previous example

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