Contents Test-Taking Tips... 8 Unit 1 Number and Number Relations... 9 Lesson 1: Number Concepts...10 Computing with Real Numbers 2 Effects of Computations on Real Numbers 2 Evaluating Radical Expressions 1 Simplifying Radical Expressions 1 EOC Practice...18 Lesson 2: Ratios and Proportions...20 Ratios 4 Proportions 4 Rates 4 Percent 4 Scale 4 EOC Practice...31 Unit 2 Algebra...33 Lesson 3: Linear Equations...34 Slope 6 Linear Equations and Their Graphs 6 Using the Point-Slope Formula 6 EOC Practice...48 Unit 3 Patterns, Relations, and Functions...49 Lesson 4: Number Patterns...50 Rules for Patterns 26 Sequences 26 Arithmetic Sequences 26 Geometric Sequences 26 Linear and Non-Linear Patterns 20 EOC Practice...61 Lesson 5: Applications of Functions...63 Linear Functions 27 Quadratic Functions 27 Exponential Functions 27 EOC Practice...71 4
Unit 4 Geometry...73 Lesson 6: Geometric Figures...74 Points, Lines, and Angles 10, 11 Polygons 9, 10 Triangles 9, 10 Triangle Inequality 9, 10 Congruent Triangles 18 Constructing Congruent Triangles 9 Proving Triangles Are Congruent Using Postulates 18 Similar Triangles 4, 18 Applications of Similar Triangles 4, 18 Triangle Proportionality Theorem 4 Quadrilaterals 9, 10 Interior and Exterior Angles 9, 10 Solids 9 EOC Practice...110 Lesson 7: Trigonometry...114 Pythagorean Theorem 12 Special Right Triangles 18 Ratios of a Right Triangle 3, 18 Using Trigonometric Ratios 3, 18 Applications of Trigonometric Ratios 8 EOC Practice...132 Lesson 8: Circles...135 Parts of a Circle 13 Central Angles 13 Inscribed Angles 13 Finding Arc Lengths 13 Finding Areas of Sectors 13 Finding the Lengths of Chord Segments 13 Finding Measures of Angles Formed by Chords, Secants, and Tangents 13 EOC Practice...150 Lesson 9: Coordinate Geometry...153 Distance on a Number Line 16 Distance on a Coordinate Grid 16 Translations 14 Reflections 14 Rotations 16 Dilations 15 EOC Practice...165 5
Lesson 10: Logic, Reasoning, and Proof...168 Deductive Reasoning: Syllogisms 17, 23 Deductive Reasoning: Conditional Statements 23 Critiquing for Validity 17, 23 Converse Statements 23 Inverse Statements 23 Contrapositive Statements 23 Geometric Deduction 10, 19 Making Deductive Proofs 10, 19 Indirect Proofs 17, 19 Using Inductive Reasoning 17 Deductive and Inductive Reasoning 17 Validating a Conjecture 10, 17 EOC Practice...192 Unit 5 Measurement...195 Lesson 11: Geometric Measurement...196 Surface Area of a Cone 7 Surface Area of a Sphere 7 Surface Area of a Pyramid 7 Volume of a Cone 7 Volume of a Sphere 7 Volume of a Pyramid 7 EOC Practice... 205 Unit 6 Data Analysis, Probability, and Discrete Mathematics... 207 Lesson 12: Data Analysis... 208 Frequency Charts 22 Bar Graphs 22 Circle Graphs 22 Applications of Matrices 22 Line Graphs 22 Scatter Plots and Lines of Best Fit 5, 20, 22 Finding the Line of Best Fit Using Technology 5, 20 Classifying Data Sets 20 EOC Practice... 235 6 Table of Contents
Standards: A1, B2, C3 Lesson 13: Probability...238 Counting Techniques 24 Combinations 24 Permutations 24 Theoretical Probability 21 Probability of Independent and Dependent Events 21 Probability of Mutually Exclusive Events 21 Probability of Non-Mutually Exclusive Events 21 Conditional Probability 21 EOC Practice...255 Lesson 14: Discrete Mathematics...257 Fair Games 25 Elections 25 EOC Practice...264 100 To the Teacher: Grade-Level Expectation numbers for each lesson section are listed in the table of contents and at the top of each page in the section. (See example at the left.) 7
Lesson 1 Number Concepts GLE: 2 In this lesson, you will compute and predict the effect of computations on real numbers. You will also simplify and determine the value of radical expressions. Computing with Real Numbers When adding, subtracting, multiplying, or dividing positive and negative real numbers, the result can be either positive or negative. Here are the rules to follow. Addition The sum of two positive numbers is positive. ( ) ( ) 25 7 32 The sum of two negative numbers is negative. ( ) ( ) 4 ( 16) 20 The sum of one positive and one negative number will have the sign of the number with the greater absolute value. ( ) ( ) or 12 ( 8) 4 3 ( 17) 14 ( ) ( ) or 9 28 19 20 15 5 Subtraction The difference of two positive numbers, two negative numbers, or one positive and one negative number can be either positive or negative. ( ) ( ) or 26 3 23 11 12 1 ( ) ( ) or 5 ( 11) 6 23 ( 8) 15 ( ) ( ) 19 ( 2) 21 ( ) ( ) 8 32 40 Subtracting a number is the same as adding its opposite. 10 Unit 1 Number and Number Relations
GLE: 2 Multiplication and Division The product or quotient of two positive numbers is positive. ( ) ( ) 4 7 28 ( ) ( ) 27 3 9 The product or quotient of two negative numbers is positive. ( ) ( ) 2 ( 11) 22 ( ) ( ) 64 ( 8) 8 The product or quotient of one positive and one negative number is negative. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( 8) 48 3 14 42 34 ( 2) 17 20 4 5 Practice For questions 1 through 8, add, subtract, multiply, or divide. 1. 12 ( 30) 5. 6 ( 13) 2. 50 ( 14) 6. 10 ( 22) 3. 45 54 7. 17 ( 43) 4. 64 8 8. 88 4 9. Evaluate: 11 9 A. 99 B. 20 10. Evaluate: 7 13 A. 20 B. 6 C. 20 C. 6 D. 99 D. 20 Lesson 1 Number Concepts 11
GLE: 2 Effects of Computations on Real Numbers You can use what you know about operations with real numbers to predict the results of computations, even if you do not know the actual numbers involved. Example What type of number will result from multiplying two numbers less than 1? First observe that, since each of the numbers is less than 1, they will both be negative numbers. Since the product of two negative numbers is positive, the result of this computation will be a positive number. Since each of the numbers is less than 1, it follows that their product will be a positive number that is greater than the absolute value of either of the original numbers. Therefore, the result of multiplying two negative numbers less than 1 is a positive number greater than the absolute value of either of the original numbers. Example What type of number will result from dividing a negative number by a positive number less than 1? First, observe that the quotient of a negative number and a positive number will be negative. Recall that the quotient of a negative number and a positive number less than 1 has an absolute value greater than the original numbers. But, since the number being divided is negative, this means that the actual quotient will be less than the original numbers. Therefore, the result of dividing a negative number by a positive number less than 1 will be a negative number that is less than either number. 12 Unit 1 Number and Number Relations
GLE: 2 Practice 1. What type of number will result from subtracting a larger positive number from a smaller positive number? 2. What type of number will result from multiplying two positive numbers less than 1? 3. What type of number will result from dividing a positive number by a positive number greater than 1? 4. What type of number will result from adding a number less than 5 to the number 5? 5. What type of number will result from dividing the number 18 by a number greater than 2? 6. If each value in the set 0 x 7 is multiplied by 3, what is the resulting range of values? Lesson 1 Number Concepts 13
Evaluating Radical Expressions The nth root of a number x is represented by the symbol n x. The is called the radical sign, the n is called the index (tells which root to take), and the x is called the radicand. Finding the nth root of a number is the inverse of raising a number to the nth power. Therefore, n x r if and only if r n x. When n is even, there are two roots of x, r and r, since in this case both r n x and ( r ) n x. However, n x usually represents the principal, or positive, nth root of x. When no index is shown, you need to fi nd the square root of the number (that is, n 2). Example What is 81? Since 9 2 81, 81 9. What is 7 78,125? Since 5 7 78,125, 7 78,125 5. GLE: 1 n If x r where r is an integer, then x is called a perfect nth power. In the previous examples, 81 is a perfect second power, or perfect square, and 78,125 is a perfect seventh power. Practice For questions 1 through 6, find the nth root of each number. 1. 144 4. 3 512 2. 4 81 5. 5 16,807 3. 6 4,096 6. 400 7. Evaluate: 4 6,561 A. 9 B. 27 C. 81 D. 1,640.25 8. Evaluate: 784 A. 16 B. 28 C. 56 D. 392 14 Unit 1 Number and Number Relations
GLE: 1 Simplifying Radical Expressions You may be asked to fi nd the square root of a number that is not a perfect square. In many of these cases, you can simplify the radical expression. Square roots of nonperfect squares are irrational numbers. The product property of radicals states that the square root of a product equals the product of the square roots of each factor. ab a b, when a and b are positive. 36 9 4 6 3 2 6 6 When you are asked to fi nd the square root of a nonperfect square number, simplify the expression by looking for perfect-square factors. The expression is in simplest form when there are no perfect-square factors other than 1 in the radicand. Example Simplify the expression 45. 45 9 5 9 5 3 5 3 5 is in simplest form because 5 has no perfect-square factors other than 1. Example Simplify the expression 14. 14 7 2 7 2 Neither 7 nor 2 is a perfect square. Since 14 does not have a perfect-square factor other than 1, 14 cannot be simplifi ed. Lesson 1 Number Concepts 15