The Ultimate SAT Math Strategies Guide Demo Version

1. The Ultimate SAT Math StrategiesGuide Demo Version Created by Sherman Snyder Fox Chapel Tutoring Pittsburgh, PA 412-352-6596 Use in PowerPoint Slide Show Mode Go to Table of Contents Go to Introduction

2. Introduction to The Ultimate SAT Math Strategies Guide • 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 Go to Table of Contents Go to first strategy

3. Table of Contents Click on Highlighted Topic Number and Operations Linear Proportionality Venn Diagrams Ratios and their Multiples Ratios, Proportion, Probability Counting Problems The Handshake Problem Long Division and Remainders Percent Change Percentages Repeating Sequences Algebra Using New Definitions Elimination of Like Terms and Factors 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 Geometry and Measurement Dividing Irregular Shapes Line Segment Length in Solids Putting Shapes Together 3-4-5 Triangle 30-60-90 Triangle 45-45-90 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 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 Data Analysis, Statistics, and Probability Arithmetic Mean

4. Venn Diagram 18 + 22 + 10 = 50 Total number of students = 50 History Math 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 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 40 + 32 = 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

5. The Handshake Problem n - 1 = 5 handshakes Strategy: The total number of handshakes that can be exchanged within a group of people of size “n” is equal to ½n(n -1). n = 6 people 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. ½n(n -1) = ½(6)(5) = 15 total handshakes shared by a group of 6 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 Return to Table of Contents See example of strategy

6. 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. Line Segment a c Reasoning: By finding a right triangle within the solid, Pythagorean Theorem can be used to find the segment or diagonal length. b Right Triangle c2 = a2 + b2 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. Pythagorean Theorem Return to Table of Contents See example of strategy

7. 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 2.5 3 23 18 (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

8. Triangle Side Lengths Strategy: The 3rd side of any triangle is greaterthanthe difference and smallerthanthe sumof 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 6 9 9 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. Return to Table of Contents See example of strategy

9. 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) 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. 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

10. 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

11. Venn Diagram Example 1 Baseball Football 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 28 n 14 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 ratios 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

12. Venn Diagram Example 2 Music Math 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 175 50 m 3) Find the value of m + 50, the number of students that take music 2) Find the value of m, the number of students that take music only 1) Create an appropriate Venn diagram to help visualize the given information. 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. 175 + 50 + m = 350 m = 125 m + 50 = 125 + 50 = 175 Return to Table of Contents Return to strategy page Return to previous example

13. 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? Solution Steps 1) Find the number of games played between the 8 teams What essential information is needed? How many games are played between the eight teams. ½(8)(7) = 28 individual games played without repeats 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. 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 Return to Table of Contents Return to strategy page See another example of strategy

14. The Handshake Problem Example 2 Question: How many diagonals can be drawn inside a regular polygon with 6 congruent sides. Solution Steps n = 6 sides n -3 = 3 diagonals 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). 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

15. 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 • 45-45-90 triangle a a a√2 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. 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

16. 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? E Solution Steps A B D 1) Diagonal BD is the hypotenuse of right triangle BCD. Find the length of BD. C 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 3-4-5 triangle. The length of BD is 20. (12-16-20) E What is the strategy for identifying essential information? Half the length of diagonal BD is equivalent to the desired distance. Use Pythagorean theorem. A B 24 2) Half the length of diagonal BC is 20/2 = 10 (shown in white on diagram) 12 D 16 C Return to Table of Contents Return to strategy page Return to previous example

17. 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? Solution Steps 1) Find the common value. 16.5 - 4 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 the value to the eighth term to find value of tenth term. Tenth term = 16.5 + 2.5 + 2.5 Tenth term = 21.5 Return to Table of Contents Return to strategy page See another example of strategy

18. 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+1 e) 3·2n+2 Solution Steps 2n+1 P 2n+2 1) Find the interval spacing 2n+2 - 2n+1 Expand the powers 2n ·22 - 2n ·21 Common factor is 2n 2n (22 - 21) Simplify 22 - 21 What essential information is needed? The interval spacing can be used to find the value of “P”. 2n (2) Divide by six intervals Interval spacing 2n (2) 6 2n 3 = 3 2) 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 3 Expand the powers and factor 2n+1 + (3) = 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

19. 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) 16 d) 30 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 > 23 - 8 Length of third side > 15 2) Determine the smallest integer length of third side of 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. Smallest integer length is 16 Return to Table of Contents Return to strategy page See another example of strategy

20. 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 yes a) 5 - 2 < x < 5 + 2 3 < x < 7 b) 7 - 3 < x < 7 + 3 4 < x < 10 yes c) 8 - 3 < x < 8 + 3 no 5 < x < 11 What essential information is needed? The range of possible triangle side lengths for each answer choice. d) 6 - 5 < x < 6 + 5 yes 1 < x < 11 yes e) 6 - 6 < x < 6 + 6 0 < x < 12 2)Test third number of 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) (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

21. 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”? Correct Solution Steps 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 = 100 + b b = 5900 (incorrect answer) 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 = 100 + b b = 200 (correct answer) Return to Table of Contents Return to strategy page See example of strategy

22. 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 + 7 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(a) = a + 7 = 8 + 7 f(a) = 15 Return to Table of Contents Return to strategy page See another example of strategy

23. Using Function Notation Example 2 y = f(x) 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)? Solution Steps 2 -2 2 -2 What essential information is needed? The math expression g(1) from which the value of g(1) can be determined 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). 1) Find the expression for g(1) 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 Return to Table of Contents Return to strategy page See another example of strategy

24. Using Function Notation Example 3 Question: Using the table to the right, if f(3) = k, what is the value of g(k)? Solution Steps 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. 1) Find the value of “k” using table. 2) Find the value of g(5) using table. f(3) = k f(3) = 5 g(5) = 4 Return to Table of Contents Return to strategy page Return to example 1