The Ultimate SAT Math Strategies GuideDemo VersionGo to Table of ContentsGo to IntroductionCreated by Sherman SnyderFox Chapel TutoringPittsburgh, PA412-352-6596Use in PowerPoint Slide Show Mode

Introduction toThe Ultimate SAT Math Strategies Guide Go to Table of ContentsGo to first strategyUnique math study guide that focuses on math strategies rather than math contentStudy guide is designed to provide step-by-step development of math strategies in an easy-to-use formatEach math strategy is accompanied by examples that provide opportunity to apply strategy to a variety of question formats Study guide to be used in conjunction with traditional paperback study guides available in bookstores

Table of ContentsClick on Highlighted TopicNumber and OperationsLinear ProportionalityVenn DiagramsRatios and their MultiplesRatios, Proportion, ProbabilityCounting Problems The Handshake ProblemLong Division and RemaindersPercent ChangePercentagesRepeating SequencesAlgebraUsing New DefinitionsElimination of Like Terms and FactorsEquivalent StrategySystem of EquationsMatching GameFactoring StrategyWord problemsBasic Rules of ExponentsAdditional Rules of ExponentsAbsolute Value InequalitiesCreation of Math Statements Geometry and MeasurementDividing Irregular ShapesLine Segment Length in SolidsPutting Shapes Together3-4-5 Triangle30-60-90 Triangle45-45-90 TriangleDistance Between Two PointsMidpoint Determination in x-y CoordinateMidpoint Determination on Number Line Exterior Angle of a TrianglePerpendicular LinesInterval Spacing - Number LineTriangle Side LengthsData Analysis, Statistics, and Probability Arithmetic MeanFunctionsUsing Function NotationReflections - x axisReflections - y axisReflections - Absolute ValueTranslations - Horizontal ShiftTranslations - Vertical ShiftTranslations - Vertical StretchTranslations - Vertical Shrink

Venn DiagramStrategy: To determine the overlap (intersection) of members in two groups (sets), use the following approach:Step 1: add the number of members of each group Step 2: subtract the total number of members that are in either group or both groups from the result of step 1 Return to Table of ContentsSee example of strategyReasoning: By eliminating the overlap of members, the sum of three numbers in the Venn diagram will equal the total number of members being counted. Application: Used when members of two or more groups (sets) have common members. 182210Total number of students = 50Number of students that study math only: 40 22 = 18Number of students that study history only: 32 22 = 10Number of students that study history = 32Number of students that study math = 40Number of students that study math and history = 22Step 1 40 + 32 = 72Step 2 72 50 = 22MathHistory18 + 22 + 10 = 50

The Handshake ProblemStrategy: The total number of handshakes that can be exchanged within a group of people of size n is equal to n(n -1). Return to Table of ContentsSee example of strategyReasoning: For a total of n people, each person can shake hands with n -1 other people. However, each handshake is shared by two people. Application: Useful for determining the total number of games played in a sport league, or the number of lines that can be drawn between pairs of points on a plane when no more than two points are collinear. n(n -1) = (6)(5) = 15 total handshakes shared by a group of 6 peoplen = 6 peoplen - 1 = 5 handshakes

Line Segment or Diagonal Length in a Rectangular SolidStrategy: To find the length of a diagonal or a line segment that connects two edges of a rectangular solid, create a right triangle within the solid that uses the unknown segment as the hypotenuse. Return to Table of ContentsSee example of strategyReasoning: By finding a right triangle within the solid, Pythagorean Theorem can be used to find the segment or diagonal length. Application: Any question that asks for the length of a line segment or diagonal in a rectangular solid. The information provided in the question will be sufficient to apply Pythagorean Theorem. abcRight TriangleLine Segment

Interval SpacingStrategy: The interval spacing on a number line is found by a two-step process:Determine the distance between two known points on the number lineDivide the distance by the number of intervals separating the two known pointsReturn to Table of ContentsSee example of strategyReasoning: By design, the number line has equal distance between each tick mark on the lineApplication: Used to identify an unknown coordinate on number line. Also used to identify the value of specific term in an arithmetic sequence.23

Triangle Side LengthsStrategy: The 3rd side of any triangle is greater than the difference and smaller than the sum of the other two sidesReturn to Table of ContentsSee example of strategyReasoning: 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 triangleApplication: 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. 963 < x < 151563

Using Function NotationStrategy: 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) Return to Table of ContentsSee example of strategyReasoning: 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.

Function TranslationsHorizontal ShiftStrategy: 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) Return to Table of ContentsReasoning: 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. y = f(x)y = f(x-2)y = f(x+4)2

Venn Diagram Example 1Question: 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? Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed? Connection between the number of players in each sport to n, the number of players that participate in both sports. What is the strategy for identifying essential information?:Use the properties of Venn diagrams and ratios to find the value of n 5n + 70 = 3n + 84

Venn Diagram Example 2Question: 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? Return to Table of ContentsReturn to strategy pageReturn to previous exampleWhat essential information is needed? Connection between the multitude of given information and the unknown quantity. What is the strategy for identifying essential information? Use the properties of Venn diagrams to help visualize the given information.175 + 50 + m = 350 m = 125

The Handshake Problem Example 1Question: In a baseball league with 8 teams, each team plays exactly 4 games with each of the other 7 teams in the league. What is the total number of games played in the league? Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed? How many games are played between the eight teams.What is the strategy for identifying essential information?: Find the number of games played between the 8 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 7 teams. (8)(7) = 28 individual games played without repeats

The Handshake Problem Example 2Question: How many diagonals can be drawn inside a regular polygon with 6 congruent sides.Return to Table of ContentsReturn to strategy pageReturn to previous exampleWhat essential information is needed? The total number of diagonals drawn from the 6 vertices of the polygon. What is the strategy for identifying essential information? Use the handshake problem 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 n(n - 3). n = 6 sidesn -3 = 3 diagonals

Line Segment Length in Solid Example 1Question: What is the volume of a cube that has a diagonal length of 43? Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed? Side length of the cube is needed to find the volume.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.aa2aaLet a be the side length of cubeThe longer side length of right triangle found using properties of 45-45-90 triangle43a2 + (a2)2 = (43)2 a = 4

Line Segment Length in Solid Example 2Return to Table of ContentsReturn to strategy pageReturn to previous exampleWhat essential information is needed? A connection between given side lengths, the center of solid, and the midpoint of AB What is the strategy for identifying essential information? Half the length of diagonal BD is equivalent to the desired distance. Use Pythagorean theorem. Can easily find the length of BD by recognizing that triangle BCD is a multiple of the 3-4-5 triangle. The length of BD is 20. (12-16-20)

Interval Spacing Example 1Question: 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 16.5. What is the value of the tenth term? Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed?The common value added to each term of the sequence.What is the strategy for identifying essential information? Use interval spacing strategy to identify the common value. Add twice the value to the eighth term to find value of tenth term. 2.5Tenth term = 16.5 + 2.5 + 2.5

Interval Spacing Example 2

Question: On the number line above, what is the value of point P? 2n+ b) 2n+ c) 32nd) 32n+1 e) 32n+2 Return to Table of ContentsReturn to strategy pageReturn to previous exampleWhat essential information is needed? The interval spacing can be used to find the value of P. 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. 2n+2 - 2n+1 Expand the powers 2n 22 - 2n 21 Common factor is 2n 2n (22 - 21) Simplify 22 - 21 2n (2) Divide by six intervals = 2n+1 + 2n Expand the powers and factor2n 21 + 2n = 2n (21 + 1)3

Triangle Side Lengths Example 1Question: If the side lengths of a triangle are 8 and 23, what is the smallest integer length of the third side?a) 14 b) 15 c) 16d) 30 e) 31 Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed? The smallest possible length of the third side of the triangleWhat 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.Length of third side > 23 - 8Length of third side > 15

Triangle Side Lengths Example 2Question: 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) Return to Table of ContentsReturn to strategy pageReturn to previous exampleWhat essential information is needed? The range of possible triangle side lengths for each answer choice. 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. a) 5 - 2 < x < 5 + 23 < x < 7b) 7 - 3 < x < 7 + 3c) 8 - 3 < x < 8 + 3d) 6 - 5 < x < 6 + 5e) 6 - 6 < x < 6 + 64 < x < 105 < x < 111 < x < 110 < x < 12a) (2, 5, 6)b) (3, 7, 7)c) (3, 8, 12)d) (5, 6, 7)e) (6, 6, 11)yesyesnoyesyes

Using Function Notation Example of Common MistakeQuestion: 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? Return to Table of ContentsReturn to strategy pageSee example of strategyCommon 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.Correct use of function notation: C(n) is replaced with 300 when n is replaced with 20 in the function equation.300(20) = 5(20) + b 6000 = 100 + b b = 5900 (incorrect answer) 300 = 5(20) + b 300 = 100 + b b = 200 (correct answer)

Using Function Notation Example 1Question: If f(x) = x + 7 and 5f(a) =15, what is the value of f(-2a)? Return to Table of ContentsReturn to strategy pageSee another example of strategyWhat essential information is needed? The value of a is needed to determine the value of f(-2a).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).Given 5f(a) = 15 Divide both sides by 5Result f(a) = 3Given f(x) = x + 7 Evaluate f(a)f(a) = a + 7 = 3Result: a = -4f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8)f(a) = a + 7 = 8 + 7

Using Function Notation Example 2Question: 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)? Return to Table of ContentsReturn to strategy pageSee another example of strategy What essential information is needed? The math expression g(1) from which the value of g(1) can be determinedWhat 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(x) = f(2x) + 2g(1) = f(2) + 2f(2) = 2

Using Function Notation Example 3Question: Using the table to the right, if f(3) = k, what is the value of g(k)? Return to Table of ContentsReturn to strategy pageReturn to example 1What essential information is needed? The value of k is needed to find g(k). 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.f(3) = k f(3) = 5

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