Teacher s guide Table of Contents 066792 Teacher s Guide Introduction................................................................................... TG1 Pacing Guide... TG5 Standards Correlations......................................................................... TG31 Graphic Organizers............................................................................ TG33 Unit 1: Quadratics and Complex Numbers Lesson 1: Analyzing Quadratics in Vertex Form f (x) = (x h) 2........................................... 1 Lesson 2: Finding the y-intercept in Standard Form and Analyzing Graphs.............................. 24 Lesson 3: Factoring ax 2 + bx + c = 0.................................................................. 29 Lesson 4: Finding the Vertex of Quadratics........................................................... 44 Lesson 5: Analyzing Quadratics in Vertex Form f (x) = a(x h) 2 + k...55 Lesson 6: Quadratic Inequalities and Finding x-intercepts from Vertex Form... 65 Lesson 7: The Quadratic Formula and the Discriminant... 76 Lesson 8: Imaginary Numbers...................................................................... 90 Lesson 9: Complex Numbers and Operations with Complex Numbers................................... 95 Lesson 10: Graphing Complex Numbers........................................................... 110 Lesson 11: Sequences... 121 Lesson 12: Sums in Sequences.................................................................... 125 Lesson 13: Series... 130 Answer Key.................................................................................... 135 Unit 2: Right Triangle Trigonometry Lesson 1: Right Triangles........................................................................ 141............................................... 169 Answer Key.................................................................................... 181 Unit 3: Circles and Spheres Lesson 1: Secants, Tangents, Chords, and Arcs... 183 Lesson 2: Congruent Chords and Arcs, Sectors... 192 Lesson 3: Central, Inscribed, and Other Angles..................................................... 200 Lesson 3 Alternative: Central Angle and Arc Properties.............................................. 221 Lesson 4: Two Tangents......................................................................... 230 Lesson 5: Arc Lengths and Areas of Sectors... 234 Lesson 6: Spheres... 239 Answer Key.................................................................................... 245 Unit 4: Statistics Lesson 1: Mean and Standard Deviation........................................................... 249 Lesson 2: Empirical Rule... 265 Lesson 3: Collecting and Organizing Data... 272 Lesson 4: Sample Data from One Population....................................................... 277 iii Georgia Mathematics 2 Teacher Resource Binder
TEACHER S GUIDE Table of Contents Lesson 5: Comparing Distributions... 283 Lesson 6: Posing Questions... 287 Lesson 7: Random Sampling..................................................................... 296 Lesson 8: Sampling from Non-Normal Distributions................................................ 301 Answer Key.................................................................................... 307 Unit 5: Step and Piecewise Functions Lesson 1: Piecewise Functions.................................................................... 309 Lesson 2: Exponential Functions.................................................................. 328 Lesson 3: Base e... 342 Answer Key.................................................................................... 351 Unit 6: Linear and Quadratic Regression Lesson 1: Scatter Plots and Linear Regression... 353 Lesson 2: Piecewise and Absolute Value Functions.................................................. 386 Lesson 3: Modeling Data... 405 Lesson 4: Collecting, Analyzing, and Modeling Data... 421 Answer Key.................................................................................... 437 iv Georgia Mathematics 2 Teacher Resource Binder
Teacher s Guide Introduction Introduction The Georgia Math 2 Program is a complete set of materials developed in concert with the Georgia Math 2 program to support success in Math 2 and achievement of Georgia s Performance Standards, as well as to prepare students for the EOCT and further studies in mathematics. Topics are built around accessible core curricula, ensuring that the Georgia Math 2 Program is useful for striving students and diverse classrooms. This program realizes the importance of mastering prerequisite skills, of pre-teaching new concepts, and of providing ongoing support to Math 2 students. It employs a variety of instructional models to meet the learning needs of a range of students. The Georgia Math 2 Program includes components that support inquiry-based learning, instruction, and practice. The scope and sequence address the prerequisite skills and preview topics necessary to successfully solve the Leaning Tasks in the Georgia Math 2 Program. The 3-ring binders include: More than 150 hours of lessons and activities Essential Questions for each instructional topic Vocabulary Examples Step-by-step graphing calculator instructions for the TI-Nspire Prerequisite Skills Practice Previews of Math 2 topics Additional practice opportunities Purpose of Materials The Georgia Math 2 Program has been organized to coordinate with the Georgia Math 2 Program, the Frameworks, and the Learning Tasks. Each lesson includes examples of the prerequisite skills for the upcoming lessons in the Georgia Math 2 Program, offering step-by-step procedures for solving the problems. These examples incorporate concept and skill development and guided practice, and then move on to the application of prerequisite skills and concepts in problem-solving situations. This program includes all the prerequisite skills needed for the topics included in the Georgia Math 2 Program. These include: TG1 Georgia Mathematics 2 Teacher Resource Binder
Teacher s guide Introduction Quadratics and Complex Numbers Right Triangle Trigonometry Circles and Spheres Statistics Step and Piecewise Functions Linear and Quadratic Regression Problem solving, reasoning and proof, communication, connections, and representations are infused throughout. In addition to re-teaching and reinforcing prerequisite skills and concepts, the Math 2 materials suggest strategies for previewing or pre-teaching the Learning Tasks from Math 2. Further, they offer additional assistance and practice for students who need extra support in meeting the expectations and standards of Georgia Math 2. Structure of the Binder The Georgia Math 2 Teacher Resource Binder is provided, for your convenience, in binder format. The materials are completely reproducible. Tabs allow you to access the sections of the binder quickly and easily. The Teacher s Guide is the first section. Written for you, this section helps you to navigate the materials with the pacing guide, offers a collection of graphic organizers and suggested strategies for their use, and shows how the lessons correlate to the Math 2 Program and the Georgia Performance Standards. The next sections focus on prerequisite skills and preview concepts for each of the topics in the six units of the Georgia Math 2 curriculum: Quadratics and Complex Numbers; Right Triangle Trigonometry; Circles and Spheres; Statistics; Step and Piecewise Functions; and Linear and Quadratic Regression. The lessons in the Georgia Math 2 Program can be implemented as prescribed in one of the pacing guides, yet the design is flexible so that you can mix and match activities as the needs of your students and your instructional style dictate. Note: Although not included in the Math 2 materials, the Station Activities found in the Georgia Math 2 Program correspond to the content in the Georgia Math 2 units and provide students with the opportunity to apply concepts and skills. These hands-on activities could be used in the Math 2 class, providing students another opportunity to develop concepts and skills related to the unit, while you have a chance to circulate, observe, speak to individuals and small groups, and informally assess and plan. TG2 Georgia Mathematics 2 Teacher Resource Binder
Teacher s guide Introduction Structure of Units Nearly all of the instructional units have some common features. Each lesson begins with Essential Questions, vocabulary titled Words to Know, and a list of the prerequisite skills to be covered in the lesson. The instruction is followed by a few guided practice examples that address the prerequisite skills to be developed and/or reinforced. Then, students have an opportunity to apply the skills and concepts learned in the lesson through practice sheets. The instructional portion of the lesson ends with a preview of the coming lesson and Learning Task, where students will be applying the prerequisite skills just practiced. As necessary and with timepermitting, students can work with you to complete assignments from the Math 2 classroom. 1. Georgia Performance Standards for the Corresponding Math 2 Lesson All Georgia performance standards that are addressed in the corresponding Math 2 lesson are listed. 2. Essential Questions These are intended to guide students thinking as they proceed through the investigation of the learning task. By the end of each lesson, students should be able to answer the questions. 3. Words to Know Vocabulary terms and formulas are provided as background information for instruction or to review key concepts that are addressed in the lesson. 4. List of Prerequisite Skills This section lists all of the prerequisite skills, based on the Math 2 lesson and Learning Task, that will be addressed in the lesson. Depending on students needs, you might cover all or just one of the prerequisite skills listed. 5. Written for you, this section gives some suggestions about the skills presented in the Learning Tasks and helps you to guide students in developing these skills. al strategies include investigation, direct instruction, modeling, and discussion. 6. Guided Practice This section provides examples of concepts necessary for developing prerequisite skills. The examples are solved through step-by-step instructions. TG3 Georgia Mathematics 2 Teacher Resource Binder
Teacher s guide Introduction 7. Student Practice Sheets Each sub-lesson includes practice problems to support students achievement of the prerequisite skills. These sheets are written for the student. They can be used in any combination of teacher-led instruction, cooperative learning, or independent application of knowledge. 8. Preview Each sub-lesson ends in a section designed to help students prepare for the upcoming coordinated lesson in the Georgia Math 2 Program. This section gives an overview of the skills and concepts required for completing the Learning Tasks. 9. Answer Key Answers for all of the Prerequisite Practice problems are provided at the end of each unit. 10. Technology Most instructional topics list an EasiTeach RM keyword for computer-based exploration that can be used as available in your school. Additionally, step-by-step instructions for using the TI-Nspire are provided whenever graphing calculators are referenced. TG4 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Problem Solving with Trigonometric Ratios Georgia Performance Standard MM2G2: Students will define and apply sine, cosine, and tangent ratios to right triangles. c. Solve application problems using the trigonometric ratios. Essential Questions 1. How can trigonometric ratios be used to solve problems (such as finding area and or perimeter) of other geometric shapes? 2. How can right triangles and trigonometric ratios be used to indirectly measure heights and distances? 3. In what ways can trigonometric ratios be used in engineering and buildings? WORDS TO KNOW adjacent side the side of a triangle that is formed by one of the sides of the angle angle of elevation the angle formed by an imaginary horizontal line from an observer to the line of sight to an object up from the horizontal from the same observer angle of depression the angle formed by an imaginary horizontal line from an observer to the line of sight to an object down from the horizontal from the same observer cosine the ratio of the length of the side adjacent to the angle to the length of the hypotenuse hypotenuse longest side of a right triangle; the side opposite the 90 angle opposite side the side of a triangle that is NOT formed by one of the sides of the angle sine the ratio of the length of the side opposite to the angle to the length of the hypotenuse tangent the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle 169 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2.2.1: Angles of Elevation and Depression Prerequisite Skills Identifying sine, cosine, and tangent ratios Using technology to find sine, cosine, and tangent ratios Identifying Sine, Cosine, and Tangent Ratios Students sometimes have difficulty remembering and applying trigonometric ratios in right triangles. Remind them of the mnemonic SOHCAHTOA, which stands for: Sine = Opposite Hypotenuse = O H Cosine = Adjacent Hypotenuse = A H Tangent = Opposite Adjacent = O A When applying the trigonometric ratios to each angle, it may help students to first label the sides of the triangle opposite, adjacent, and hypotenuse. Walk students through the following example on the whiteboard or overhead. Example In right triangle ABC below, C is a right angle. What is the length of the side opposite angle A? What is the length of the side adjacent to angle A? What is the length of the hypotenuse? What are the values of sin A, cos A, and tan A? 170 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY By looking at the diagram, we see that: The side opposite angle A is BC, which has length 35. The side adjacent to angle A is AC, which has length 12. The hypotenuse of the triangle is AB, which has length 37. Using this information, we can calculate the trigonometric ratios of angle A: sin A = cos A = tan A = length of side opposite angle A hypotenuse length of side adjacent to angle A hypotenuse = 35 37 = 12 37 length of side opposite angle A length of side adjacent to angle A = 35 12 171 Georgia Mathematics 2 Teacher Resource Binder
NAME: UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Prerequisite Practice 2.2.1: Identifying Sine, Cosine, and Tangent Ratios Use the diagram to answer the questions below. 1. What is the length of the side opposite angle X? 2. What is the length of the side adjacent to angle X? 3. What is the length of the hypotenuse of the triangle? 4. What is the value of sin X? 5. What is the value of cos X? 6. What is the value of tan X? 172 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Using Technology to Find Sine, Cosine, and Tangent Ratios Students can use technology to find the sine, cosine and tangent of any angle (rather than referring back to the triangles they created in Lesson 1.2). Students may have limited experience with this skill, so be careful when explaining the keystrokes. Make sure the students have the calculator set in Degree mode (rather than Radian mode). Review this skill by walking students through the following example on the whiteboard or overhead with graphing utility attached. Example Find sin 38, cos 17, and tan 65 using a calculator. 1. Locate the [SIN] key on your calculator. Type [SIN] then 38 followed by the ENTER key. The value of sin 38 is 0.615661475. 2. Locate the [COS] key on your calculator. Type [COS] then 17 followed by the ENTER key. The value of cos 17 is 0.956304756. 3. Locate the [TAN] key on your calculator. Type [TAN] then 65 followed by the ENTER key. The value of tan 65 is 2.144506921. 173 Georgia Mathematics 2 Teacher Resource Binder
NAME: UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Prerequisite Practice 2.2.1: Using Technology to Find Sine, Cosine, and Tangent Ratios Find the indicated quantity. Round to the nearest thousandth. 1. sin 42 2. sin 6 3. cos 22 4. cos 88 5. tan 2 6. tan 33 174 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2.2.1 Preview Explain to students that in the Problem Solving Learning Task they will need to be able to model real-world situations using triangles. They will also need to be able to use trigonometric ratios within those triangles to solve real-world problems. To model real-world situations using right triangles, students may need to first review the problems found in the Warm-Up. Ask students, How can we use a right triangle to represent the situation in Question 1? Appropriate answers will include a sketch of a right triangle with correctly labeled sides and angles. To challenge students, ask for other situations that can be modeled with right triangles. Appropriate answers will be any other application of right triangles to real-world situations. Follow this up by asking about how trigonometric ratios can be used with the right triangles to solve real-world problems. Students should respond by saying that the ratios can be used to find missing sides and angles in the right triangles that represent real-world situations. 175 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2.2.2: Trigonomic Identities Prerequisite Skills Identifying sine, cosine, and tangent ratios Identifying Sine, Cosine, and Tangent Ratios Students sometimes have difficulty remembering and applying trigonometric ratios in right triangles. Students may forget the sine, cosine and tangent ratios. Remind them of the mnemonic SOHCAHTOA, which stands for: Sine = Opposite Hypotenuse = O H Cosine = Adjacent Hypotenuse = A H Tangent = Opposite Adjacent = O A When applying the trigonometric ratios to each angle, it may help students to first label the sides of the triangle opposite, adjacent, and hypotenuse. Walk students through the following example on the whiteboard or overhead. Example In right triangle ABC below, C is a right angle. What is the length of the side opposite angle A? What is the length of the side adjacent to angle A? What is the length of the hypotenuse? What are the values of sin A, cos A, and tan A? 176 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY By looking at the diagram, we see that: The side opposite angle A is BC, which has a length of 24. The side adjacent to angle A is AC, which has a length of 10. The hypotenuse of the triangle is AB, which has a length of 26. Using this information we can calculate the trigonometric ratios of angle A: sin A = cos A = tan A = length of side opposite angle A hypotenuse length of side adjacent to angle A hypotenuse = 24 26 = 10 26 length of side opposite angle A length of side adjacent to angle A = 24 10 177 Georgia Mathematics 2 Teacher Resource Binder
NAME: UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Prerequisite Practice 2.2.2: Identifying Sine, Cosine, and Tangent Ratios Use the diagram to answer the questions below. 1. What is the length of the side opposite angle X? 2. What is the length of the side adjacent to angle X? 3. What is the length of the hypotenuse of the triangle? 4. What is the value of sin X? 5. What is the value of cos X? 6. What is the value of tan X? 178 Georgia Mathematics 2 Teacher Resource Binder
UNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2.2.2 Preview Explain to students that in the A Start at Trig Identities Learning Task, they will need to be able to identify the opposite, adjacent sides of a right triangle, as well as the hypotenuse. Students must be able to substitute these values into the Pythagorean Theorem. Once students write the Pythagorean Theorem written in terms of opposite, adjacent and hypotenuse, they will divide each side by the square of the hypotenuse. They then need to identify the sine and cosine ratios in the equation. This will help students find a relationship between the sine and cosine of any angle. 179 Georgia Mathematics 2 Teacher Resource Binder