NEW YORK ALGEBRA TABLE OF CONTENTS

NEW YORK ALGEBRA TABLE OF CONTENTS CHAPTER 1 NUMBER SENSE & OPERATIONS TOPIC A: Number Theory: Properties of Real Numbers {A.N.1} PART 1: Closure...1 PART 2: Commutative Property...2 PART 3: Associative Property...3 PART 4: Distributive Property...4 PART 5: Identity Element...5 PART 6: Inverse Property...7 TOPIC B: Meanings of Operations and Procedures PART 1: Simplifying Radical Terms (no variable in the radicand) {A.N.2}...9 PART 2: Operations on Radicals {A.N.3} Section a: Addition and Subtraction of Radicals...10 Section b: Multiplication of Radicals...13 Section c: Division of Radicals...16 PART 3: Scientific Notation {A.N.4} Section a: Evaluating Scientific Notation...18 Section b: Multiplying and Dividing Scientific Notation...20 PART 4: Solving Algebraic Problems {A.N.5} Section a: Algebraic Problems Involving Fractions Section 1: Addition...22 Section 2: Subtraction...25 Section 3: Multiplication...27 Section 4: Division...29 Section b: Algebraic Problems Involving Decimals...31 Section c: Algebraic Problems Involving Percents (Percent Decrease, Percent Increase, and Discount)...32 Section d: Ratio, Proportion, and Direct Variation Section 1: Proportion Equations...37 Section 2: Direct Variation...40 Section 3: Proportionality Constants...42 Section 4: Ratio and Proportion Word Problems (See Chapter 12A)...43 PART 5: Factorials, Absolute Values, and Exponential Expressions {A.N.6} Section a: Evaluating Factorial Expressions...46 Section b: Evaluating Absolute Value Expressions...47 Section c: Evaluating Exponential Expressions...49 PART 6: Fundamental Counting Principle {A.N.7}...52 PART 7: Permutations {A.N.8}...54 CHAPTER 2 ALGEBRAIC VARIABLES & EXPRESSIONS TOPIC A: Translating Verbal Phrases into Algebraic Expressions and Equations {A.A.1} PART 1: Literal Translations of Algebraic Expressions and Equations...59 PART 2: Determining Expressions for Algebraic Word Problems...60 TOPIC B: Translating Algebraic Expressions into Verbal Phrases {A.A.2}...63 TOPIC C: Operations on Monomial Expressions {A.A.12} PART 1: Multiplication of Monomials...64 PART 2: Division of Monomials...67 TOPIC D: Adding, Subtracting, and Multiplying Monomials and Polynomials {A.A.13} PART 1: Addition of Monomials and Polynomials Section a: Monomials...71 Section b: Polynomials...75 PART 2: Subtraction of Monomials and Polynomials Section a: Monomials...81 Section b: Polynomials...85

PART 3: Multiplication of Binomials and Polynomials Section a: Binomials or Polynomials by a Monomial...92 Section b: Binomials and Polynomials...95 TOPIC E: Dividing a Polynomial by a Monomial or Binomial {A.A.14} PART 1: Binomial or Polynomial by a Monomial...100 PART 2: Binomial or Polynomial by a Binomial...104 CHAPTER 3 FACTORING TOPIC A: Identifying and Factoring the Difference of Two Perfect Squares {A.A.19} PART 1: Different Types of Factoring...107 PART 2: Factoring the Difference of Two Perfect Squares...108 TOPIC B: Factoring Algebraic Expressions {A.A.20} PART 1: Factoring the Greatest Common Factor (GCF)...112 PART 2: Factoring Trinomials...117 PART 3: Factoring Completely...120 CHAPTER 4 ALGEBRAIC FRACTIONS (Rational Expressions) TOPIC A: Undefined Algebraic Fractions {A.A.15}...125 TOPIC B: Simplifying Algebraic Fractions {A.A.16}...128 TOPIC C: Operations on Algebraic Fractions PART 1: Addition of Algebraic Fractions {A.A.17}...132 PART 2: Subtraction of Algebraic Fractions {A.A.17}...136 PART 3: Multiplication of Algebraic Fractions {A.A.18}...139 PART 4: Division of Algebraic Fractions {A.A.18}...143 CHAPTER 5 LINEAR EQUATIONS & INEQUALITIES TOPIC A: Identifying the Difference Between Algebraic Expressions and Algebraic Equations {A.A.3}...147 TOPIC B: Solving Linear Equations in One Variable {A.A.22} PART 1: Single Operation...147 PART 2: Two or More Operations...150 PART 3: Using the Distributive Property...154 TOPIC C: Solving Literal Equations for a Given Variable {A.A.23}...157 TOPIC D: Solving Linear Inequalities in One Variable {A.A.24}...159 TOPIC E: Solving Equations Involving Fractional Expressions {A.A.25}...166 TOPIC F: Solving Algebraic Proportions in One Variable {A.A.26}...169 TOPIC G: Translating Verbal Sentences {A.A.4} PART 1: Into Mathematical Equations...172 PART 2: Into Mathematical Inequalities...175 TOPIC H: Writing Algebraic Equations or Inequalities to Represent a Situation {A.A.5} PART 1: Equations...176 PART 2: Inequalities...180 TOPIC I: Analyzing and Solving Verbal Problems {A.A.6} PART 1: Linear Equations in One Variable...182 PART 2: Linear Inequalities in One Variable...190 TOPIC J: Verifying Solutions to Equations and Inequalities {A.A.21} PART 1: Linear Equations in One Variable...191 PART 2: Linear Inequalities in One Variable...192 CHAPTER 6 SOLVING QUADRATIC, EXPONENTIAL, & SYSTEMS OF EQUATIONS TOPIC A: Solving Systems of Linear Equations Algebraically {A.A.10} (See Chapter 11E) PART 1: Using Addition or Subtraction...193 PART 2: Using Substitution...194 PART 3: Using Any Method...196 TOPIC B: Using Systems of Linear Equations to Solve Verbal Problems {A.A.7}...199 TOPIC C: Using the Multiplication Property of Zero to Solve Quadratic Equations {A.A.27}...202 TOPIC D: Using Quadratic Equations to Solve Verbal Problems {A.A.8}...208 TOPIC E: Verbal Problems Involving Exponential Growth and Decay {A.A.9}...211 TOPIC F: Solving Systems of One Linear and One Quadratic Equation {A.A.11} PART 1: Algebraic Solutions...212 PART 2: Graphic Solutions...214

TOPIC G: Solving Algebraic Proportions Resulting in Quadratic Equations {A.A.26}...218 TOPIC H: Roots of Quadratic Equations and Factors of Quadratic Expressions {A.A.28} PART 1: Factoring Quadratic Expressions and Finding the Roots...219 PART 2: Determining the x-intercepts of Quadratic Equations...220 PART 3: Writing Quadratic Equations Given the Roots...221 CHAPTER 7 PATTERNS, RELATIONS, & FUNCTIONS TOPIC A: Elements of a Set {A.A.29} PART 1: Set-builder Notation...223 PART 2: Inequality and Interval Notation...224 TOPIC B: Finding the Complement of a Subset {A.A.30}...230 TOPIC C: Finding the Intersection and Union of Sets {A.A.31} PART 1: Intersection of Sets...231 PART 2: Union of Sets...234 PART 3: Venn Diagrams...235 CHAPTER 8 COORDINATE GEOMETRY -- PROPERTIES OF EQUATIONS TOPIC A: Understanding Slope as a Rate of Change Between Dependent and Independent Variables {A.A.32}..243 TOPIC B: Determining the Slope of a Line Given Two Points on the Line {A.A.33}...245 TOPIC C: Finding a Point Given a Point and Slope (edification)...247 TOPIC D: Writing Equations of Lines PART 1: Given Slope and a Point on the Line {A.A.34}...248 PART 2: Given Two Points on the Line {A.A.35}...250 PART 3: Parallel to the Axes {A.A.36}...251 TOPIC E: Determining the Slope of a Line From its Equation {A.A.37}...252 TOPIC F: Determining the y-intercept of a Line From its Equation (edification)...253 TOPIC G: Determining the Slope, y-intercept, and Equation of Graphed Lines (edification)...255 TOPIC H: Identifying Equations of Parallel and Perpendicular Lines {A.A.38}...263 TOPIC I: Determining Whether a Point is on the Equation of a Line {A.A.39}...268 TOPIC J: Determining Whether a Point is in the Solution Set of a System of Linear Inequalities {A.A.40}...270 TOPIC K: Determining the Vertex and Axis of Symmetry of a Parabola From its Equation {A.A.41} (See Chapter 11H) PART 1: Vertex (Turning Point)...272 PART 2: Axis of Symmetry...274 CHAPTER 9 TRIGONOMETRIC FUNCTIONS & THE PYTHAGOREAN THEOREM TOPIC A: Finding Sine, Cosine, and Tangent Ratios {A.A.42}...279 TOPIC B: Evaluating Trigonometric Ratios {A.A.42} PART 1: Using the Table of Trigonometric Values...285 PART 2: Using a Calculator...285 PART 3: Using Any Method...285 TOPIC C: Determining the Measure of an Angle of a Right Triangle Given Two Side Measures {A.A.43}...288 TOPIC D: Finding the Measure of a Side of a Right Triangle Given an Angle and a Side Measure {A.A.44}...290 TOPIC E: Trigonometry Word Problems {A.A.44} PART 1: Sine Problems...293 PART 2: Cosine Problems...296 PART 3: Tangent Problems...297 PART 4: Multi-step Problems...300 TOPIC F: Determining the Measure of a Third Side of a Right Triangle Using the Pythagorean Theorem {A.A.45} PART 1: Pythagorean Theorem...308 PART 2: Converse of the Pythagorean Theorem...316 CHAPTER 10 GEOMETRY TOPIC A: Finding the Area of 2-Dimensional Figures {A.G.1} PART 1: Area of Squares...319 PART 2: Area of Rectangles...321 PART 3: Area of Triangles...325 PART 4: Area of Parallelograms...328 PART 5: Area of Circles...330

PART 6: Area of Polygons...333 PART 7: Area of Sectors of Circles...339 TOPIC B: Finding the Perimeter or Circumference of 2-Dimensional Figures {A.G.1} PART 1: Perimeter of Squares...343 PART 2: Perimeter of Rectangles...344 PART 3: Perimeter of Triangles...346 PART 4: Perimeter of Parallelograms...347 PART 5: Circumference of Circles...348 PART 6: Perimeter of Regular Polygons...350 TOPIC C: Perimeter, Circumference, and Area Combination Problems {A.G.1}...351 TOPIC D: Calculating Volume of 3-Dimensional Figures {A.G.2} PART 1: Volume of Cubes...360 PART 2: Volume of Rectangular Solids...362 PART 3: Volume of Cylinders...368 TOPIC E: Calculating Surface Area of 3-Dimensional Figures {A.G.2} PART 1: Surface Area of Cubes...372 PART 2: Surface Area of Rectangular Solids...577 PART 3: Surface Area of Cylinders...374 CHAPTER 11 COORDINATE GEOMETRY -- GRAPHS OF EQUATIONS TOPIC A: Determining if a Relation is a Function {A.G.3} PART 1: Using Ordered Pairs...381 PART 2: Using Graphs...384 TOPIC B: Graphing Linear, Quadratic (Parabolic), Absolute Value, and Exponential Functions {A.G.4} PART 1: Types of Functions...390 PART 2: Graphs of Functions Section a: Linear Functions...391 Section b: Quadratic Functions...398 Section c: Absolute Value Functions...400 Section d: Exponential Functions...401 TOPIC C: Affect of Changing the Coefficients of a Function on its Graph {A.G.5} PART 1: Linear Functions...401 PART 2: Quadratic Functions...404 PART 3: Absolute Value Functions...409 PART 4: Exponential Functions...411 TOPIC D: Graphing Linear Inequalities {A.G.6} PART 1: Linear Inequalities in Two Variables...413 PART 2: Inequalities on a Coordinate Grid...416 TOPIC E: Graphing and Solving Systems of Linear Equations and Inequalities {A.G.7} (See Chapter 6A) PART 1: Systems of Linear Equations with Rational Coefficients in Two Variables...422 PART 2: Systems of Linear Inequalities with Rational Coefficients in Two Variables...427 TOPIC F: Solving Systems of Linear and Quadratic Equations Graphically {A.G.9}...436 TOPIC G: Finding the Roots of a Parabolic Function Graphically {A.G.8}...440 TOPIC H: Determining the Vertex and Axis of Symmetry of a Parabola From its Graph {A.G.10} (See Chapter 8J)...441 CHAPTER 12 UNITS OF MEASUREMENT TOPIC A: Calculating Rates Using Appropriate Units {A.M.1}...445 TOPIC B: Converting Between Measurements {A.M.2}...447 TOPIC C: Calculating the Relative Error in Measuring Square and Cubic Units (A.M.3}...451 CHAPTER 13 COLLECTING, ORGANIZING, DISPLAYING, & ANALYZING STATISTICAL DATA TOPIC A: Organizing and Displaying Statistical Data PART 1: Qualitative or Quantitative Data {A.S.1}...453 PART 2: Univariate or Bivariate Data {A.S.2}...454 PART 3: Biased Collections or Displays of Data {A.S.3&15}...455 PART 4: Measures of Central Tendency {A.S.4} Section a: Mean...458 Section b: Median...460 Section c: Mode...461

Section d: Comparisons of Central Tendency...462 PART 5: Statistical Graphs -- Histograms, Cumulative Frequency Histograms, and Box-and-Whisker Plots (A.S.5) Section a: Frequency Tables and Histograms...464 Section b: Cumulative Frequency Tables and Histograms...470 Section c: Box-and-Whisker Plots...476 PART 6: Box-and-Whisker Plots Constructed From the Five Statistical Summaries {A.S.6}...477 PART 7: Scatter Plots of Bivariate Data {A.S.7}...479 PART 8: Scatter Plots -- Lines of Best Fit and Their Equations {A.S.8}...482 TOPIC B: Analyzing Statistical Data PART 1: Frequency and Cumulative Frequency Tables or Histograms and Box-and-Whisker Plots {A.S.9} Section a: Frequency Distribution Tables or Histograms...483 Section b: Cumulative Frequency Distribution Tables or Histograms...488 Section c: Box-and-Whisker Plots...495 PART 2: Published Statistical Reports and Graphs {A.S.10}...496 PART 3: Percentile Ranks of Items in Data Sets {A.S.11}...500 PART 4: Relationships Between Independent and Dependent Variables in Scatter Plots {A.S.12}...503 PART 5: Relationship Between Correlation and Causation {A.S.13&14}...506 TOPIC C: Making Predictions From Data PART 1: Recognizing the Effect of Linear Transformations of One-Variable on Statistical Data {A.S.16}...508 PART 2: Predictions Based on Interpolations or Extrapolations of Lines of Best Fit {A.S.17}...509 CHAPTER 14 PROBABILITY TOPIC A: Understanding Conditional Probability {A.S.18}...511 TOPIC B: Determining Numbers of Elements in Sample Spaces and Numbers of Favorable Events (A.S.19) PART 1: Numbers of Elements in Sample Spaces...515 PART 2: Numbers of Favorable Events...518 TOPIC C: Calculating the Probability of an Event and its Complement {A.S.20} PART 1: Probability of an Event...520 PART 2: Complement of an Event...525 TOPIC D: Determining Empirical Probabilities of Specific Sample Data {A.S.21}...528 TOPIC E: Calculating the Probability of a Set of Events {A.S.22} PART 1: Probability of Events that are More, Less, or Equally Likely to Occur...531 PART 2: Probability of Certain and Uncertain Events...533 TOPIC F: Calculating the Probability of Compound Events {A.S.23} PART 1: Probability of Independent Events...537 PART 2: Probability of Dependent Events...542 PART 3: Probability of Two Mutually Exclusive Events...546 PART 4: Probability of Two Non-Mutually Exclusive Events...549 APPENDIX HOW TO ADD MATH GRIDS & TABLES TO AN EXAM...555