The Ultimate SAT Math Strategies Guide

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Introduction toThe Ultimate SAT Math

Strategies Guide

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•Unique math study guide that focuses on math strategies rather than math content•Study guide is designed to provide step-by-step development of math strategies in an easy-to-use format•Each 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 Topic

Number and OperationsLinear ProportionalityVenn DiagramsRatios and their MultiplesRatios, Proportion, ProbabilityCounting Problems The Handshake ProblemLong Division and RemaindersPercent ChangePercentagesRepeating Sequences

AlgebraUsing 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 Lengths

Data Analysis, Statistics, and Probability Arithmetic Mean

FunctionsUsing Function NotationReflections - x axisReflections - y axisReflections - Absolute ValueTranslations - Horizontal ShiftTranslations - Vertical ShiftTranslations - Vertical StretchTranslations - Vertical Shrink

Venn Diagram

Strategy: 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

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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.

Application: Used when members of two or more groups (sets) have common members.

18 22 10

Total number of students = 50

Number of students that study math only:

40 – 22 = 18

Number of students that study history only:

32 – 22 = 10

Number of students that study history = 32

Number of students that study math = 40

Number of students that study math and history = 22

Step 1 40 + 32 = 72

Step 2 72 – 50 = 22

Math History

18 + 22 + 10 = 50

The Handshake Problem

Strategy: The total number of handshakes that can be exchanged within a group of people of size “n” is equal to ½n(n -1).

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Reasoning: 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.

Alternative Solution: Total number of handshakes can be found by addition of the number of handshakes exchanged by each individual person.

5 + 4 + 3 + 2 + 1 + 0 = 15 handshakes

½n(n -1) = ½(6)(5) = 15 total handshakes shared by a group of 6 people

n = 6 people

n - 1 = 5 handshakes

Line Segment or Diagonal Length in a Rectangular Solid

Strategy: 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.

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Reasoning: 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.

a

b

c

Right Triangle

Line Segment

c2 = a2 + b2

Pythagorean Theorem

Interval Spacing

Strategy: The interval spacing on a number line is found by a two-step process:1.Determine the distance between two known points on the number line2.Divide the distance by the number of intervals separating the two known points

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Reasoning: By design, the number line has equal distance between each tick mark on the line

Application: Used to identify an unknown coordinate on number line. Also used to identify the value of specific term in an arithmetic sequence.

3 18 23

What is this value?

2.5

(18 - 3)6

= 2.5

18 + 2(2.5) = 23

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

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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

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

6

3 < x < 15

9 6

15

9

63

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)

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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.

Function notation such as f(x), g(x), and h(x) is a useful way 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.

Introduction

Important Note: Function notation is not a mathematical operation. See example of commonly made mistake.

Function TranslationsHorizontal 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)

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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.

y = f(x)

y = f(x-2)

y = f(x+4)

2

2

Venn Diagram Example 1

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?

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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.

What is the strategy for identifying essential information?:Use the properties of Venn diagrams and ratios to find the value of “n”

Solution Steps

Football Baseball

28 14n

1) Create a proportion of the number of football players to baseball players

n + 28n + 14

53

=

2) Solve for “n” using cross multiplication: 5n + 70 = 3n + 84

2n = 14 n = 7

Venn Diagram 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?

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What 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.

Solution Steps

Math Music

175 m50

1) Create an appropriate Venn diagram to help visualize the given information.

2) Find the value of m, the number of students that take music only

175 + 50 + m = 350 m = 125

3) Find the value of m + 50, the number of students that take music

m + 50 = 125 + 50 = 175

The Handshake Problem Example 1

Question: 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?

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What 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.

Solution Steps

1) Find the number of games played between the 8 teams

½(8)(7) = 28 individual games played without repeats

2) Multiply by 4 to account for the fact that each team plays exactly four games with each of the other 7 teams

Total number of games played: 28 x 4 = 112 games

The Handshake Problem Example 2

Question: How many diagonals can be drawn inside a regular polygon with 6 congruent sides.

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What 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 ½n(n - 3).

Solution Steps

n = 6 sides

n -3 = 3 diagonals

½n(n - 3) = ½(6)(6 - 3) = 9 diagonals can be drawn in a regular polygon

with 6 sides

Line Segment Length in Solid Example 1

Question: What is the volume of a cube that has a diagonal length of 4√3?

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What 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.

Solution Steps

1) Establish relationships between cube diagonal length and side length using properties of a cube

a

a√2

a

a

•Let “a” be the side length of cube•The longer side length of right triangle found using properties of 45-45-90 triangle

4√3

2) Apply Pythagorean theorem to find side length a2 + (a√2)2 = (4√3)2

a = 4

Volume = a3 = 43 = 64

Line Segment Length in Solid Example 2

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What 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.

Solution Steps

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

1) Diagonal BD is the hypotenuse of right triangle BCD. Find the length of BD.

A

C

B

D

E

24

12

16

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)

2) Half the length of diagonal BC is 20/2 = 10 (shown in white on diagram)

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 16.5. What is the value of the tenth term?

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What 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.

Solution Steps

1) Find the common value.

16.5 - 45 intervals

= 12.5 5 intervals

= 2.5

2) Add twice the common value of 2.5 to the eighth term value of 16.5.

Tenth term = 16.5 + 2.5 + 2.5

Tenth term = 21.5

Interval Spacing Example 2

Question: On the number line above, what is the value of point P? a)2n+½ b) 2n+¾ c) 3·2n

d) 3·2n+1 e) 3·2n+2

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What 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.

Solution Steps2n+1 2n+2 P

1) Find the interval spacing

2) Find the value of “P”

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 (2)6

= 2n 3

Interval spacing

2n+1 + (3)2n 3

= 2n+1 + 2n Expand the powers and factor2n ·21 + 2n = 2n (21 + 1)

3∙ 2n Value of point “P”

3

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) 14 b) 15 c) 16d) 30 e) 31

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What essential information is needed? The smallest possible length of the third side of the triangle

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.

Solution Steps

1) Find the smallest possible length of the third side

2) Determine the smallest integer length of third side of triangle

Length of third side > 23 - 8

Length of third side > 15

Smallest integer length is 16

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)

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What 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.

Solution Steps

1) Determine range of possible side lengths using first two numbers

2)Test third number of each answer choice

a) 5 - 2 < x < 5 + 2 3 < x < 7b) 7 - 3 < x < 7 + 3

c) 8 - 3 < x < 8 + 3

d) 6 - 5 < x < 6 + 5

e) 6 - 6 < x < 6 + 6

4 < x < 10

5 < x < 11

1 < x < 110 < x < 12

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”

yes

yes

no

yes

yes

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”?

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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.

Correct use of function notation: C(n) is replaced with 300 when n is replaced with 20 in the function equation.

Solution Steps for Commonly Made Mistake

1) Replace “C” with 300 and replace “n” with 20

C(n) = 5n + b

300(20) = 5(20) + b

6000 = 100 + b

b = 5900 (incorrect answer)

C(n) = 5n + b

Correct Solution Steps

300 = 5(20) + b 300 = 100 + b

b = 200 (correct answer)

Using Function Notation Example 1

Question: If f(x) = x + 7 and 5f(a) =15, what is the value of f(-2a)?

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What 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).

Solution Steps

1) Find the value of “a”

Given 5f(a) = 15 Divide both sides by 5

Result f(a) = 3

Given f(x) = x + 7 Evaluate f(a)

f(a) = a + 7 = 3

Result: a = -4

2) Use a = -4 to find f(-2a)

f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8)

f(a) = a + 7 = 8 + 7

f(a) = 15

Using Function Notation Example 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)?

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What essential information is needed? The math expression g(1) from which the value of g(1) can be determined

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).

Solution Steps

y = f(x)2

2

-2

-2

1) Find the expression for g(1)

g(x) = f(2x) + 2

g(1) = f(2) + 2

2) Find the value of f(2) from the graph of y = f(x)

f(2) = 2

g(1) = 2 + 2 g(1) = 4

Using Function Notation Example 3

Question: Using the table to the right, if f(3) = k, what is the value of g(k)?

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What 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.

Solution Steps

x f(x) g(x)

1 3 8

2 4 10

3 5 8

4 6 6

5 7 4

1) Find the value of “k” using table.

f(3) = k

2) Find the value of g(5) using table.

f(3) = 5

g(5) = 4