Geo_SE_FULL_01.pdf - High School Math Solution Geometry Student Edition Volume 1 Sandy Bartle Finocchi and Amy Jones Lewis with Josh Fisher Janet

Geo_SE_FULL_01.pdf - High School Math Solution Geometry...

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Unformatted text preview: High School Math Solution Geometry Student Edition Volume 1 Sandy Bartle Finocchi and Amy Jones Lewis with Josh Fisher, Janet Sinopoli, and Victoria Fisher GEO_SE_FM_Vol 1.indd FM-1 5/26/18 10:24 AM 501 Grant St., Suite 1075 Pittsburgh, PA 15219 Phone 888.851.7094 Customer Service Phone 412.690.2444 Fax 412.690.2444 © Carnegie Learning, Inc. Cover Design by Anne Milliron Copyright © 2018 by Carnegie Learning, Inc. All rights reserved. Carnegie Learning and MATHia are registered marks of Carnegie Learning, Inc. All other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. Permission is granted for photocopying rights within licensed sites only. Any other usage or reproduction in any form is prohibited without the expressed consent of the publisher. ISBN: 978-1-60972-415-3 Student Edition, Volume 1 Printed in the United States of America 1 2 3 4 5 6 7 8 9 CC 21 20 19 18 17 GEO_SE_FM_Vol 1.indd FM-2 5/26/18 10:24 AM © Carnegie Learning, Inc. Manifesto • FM-3 GEO_SE_FM_Vol 1.indd FM-3 5/26/18 10:24 AM ACKNOWLEDGMENTS High School Math Solution Authors • • • • • Sandy Bartle Finocchi, Senior Academic Officer Amy Jones Lewis, Director of Instructional Design Josh Fisher, Instructional Designer Victoria Fisher, Instructional Designer Janet Sinopoli, Instructional Designer Foundational Authors • • • William S. Hadley, Co-Founder David Dengler Mary Lou Metz Vendors • • Lumina Datamatics, Ltd. Mathematical Expressions, LLC Images • Alison Huettner for project management and editorial review. • Jacyln Snyder for her contributions to the Teacher’s Implementation Guide facilitation notes. • Harry Lynch for his contributions and review of the Statistics and Probability strand. • The members of Carnegie Learning Cognitive Scientist Team—Brendon Towle, John Connelly, Bob Hausmann, Chas Murray, and Martina Pavelko— for their insight in learning science and collaboration on MATHia® Software. • John Jorgenson, Chief Marketing Officer, for all his insight and messaging. • Carnegie Learning Education Services Team for content review and providing customer feedback. • The entire Carnegie Learning staff for their hard work and dedication to transforming math education. • The families of the authoring team for their continued support. © Carnegie Learning, Inc. Special Thanks FM-4 • Acknowledgments GEO_SE_FM_Vol 1.indd FM-4 5/26/18 10:24 AM Mathematics is so much more than memorizing rules. It is learning to reason, to make connections, and to make sense of the world. We believe in Learning by Doing(TM)—you need to actively engage with the content if you are to benefi t from it. The lessons were designed to take you from your intuitive understanding of the world and build on your prior experiences to then learn new concepts. My hope is that these instructional materials help you build a deep understanding of math. Sandy Bartle Finocchi, Senior Academic Officer You have been learning math for a very long time—both in school and in your interactions in the world. You know a lot of math! In this course, there’s nothing brand new. It all builds on what you already know. So, as you approach each activity, use all of your knowledge to solve problems, to ask questions, to fix mistakes, and to think creatively. © Carnegie Learning, Inc. Amy Jones Lewis, Director of Instructional Design At Carnegie Learning we have created an organization whose mission and culture is defined by your success. Our passion is creating products that make sense of the world of mathematics and ignite a passion in you. Our hope is that you will enjoy our resources as much as we enjoyed creating them. Barry Malkin, CEO, Carnegie Learning Acknowledgments • FM-5 5 GEO_SE_FM_Vol 1.indd FM-5 5/26/18 10:24 AM TABLE OF CONTENTS Volume 1 Student Edition Module 1: Reasoning with Shapes Topic 1: Using a Rectangular Coordinate System 1.1  The Squariest Square From Informal to Formal Geometric Thinking ..........................................................M1-7 1.2  Hip to Be Square Constructing a Coordinate Plane ............................................................................. M1-17 1.3  Ts and Train Tracks Parallel and Perpendicular Lines ............................................................................. M1-33 1.4  Where Has Polly Gone? Classifying Shapes on the Coordinate Plane .......................................................... M1-51 1.5  In and Out and All About Area and Perimeter on the Coordinate Plane ........................................................ M1-69 Topic 2: Composing and Decomposing Shapes 2.1  Running Circles Around Geometry Using Circles to Make Conjectures ........................................................................ M1-111 2.2  The Quad Squad 2.3  Into the Ring Constructing an Inscribed Regular Polygon ......................................................... M1-145 2.4  Tri- Tri- Tri- and Separate Them Conjectures About Triangles .................................................................................. M1-161 2.5  What’s the Point? Points of Concurrency ............................................................................................. M1-175 © Carnegie Learning, Inc. Conjectures About Quadrilaterals ......................................................................... M1-127 Topic 3: Rigid Motions on a Plane 3.1  Put Your Input In, Take Your Output Out Geometric Components of Rigid Motions ............................................................ M1-205 3.2  Bow Thai Translations as Functions ....................................................................................... M1-217 FM-6 • Table of Contents GEO_SE_FM_Vol 1.indd FM-6 5/29/18 8:01 PM 3.3  Staring Back at Me Reflections as Functions ......................................................................................... M1-229 3.4  Turn Yourself Around Rotations as Functions ............................................................................................ M1-243 3.5  OKEECHOBEE Reflectional and Rotational Symmetry ................................................................. M1-257 Module 2: Establishing Congruence Topic 1: Congruence Through Transformations 1.1  The Elements Formal Reasoning in Euclidean Geometry ................................................................M2-7 1.2  ASA, SAS, and SSS Proving Triangle Congruence Theorems................................................................. M2-23 1.3  I Never Forget a Face Using Triangle Congruence to Solve Problems ...................................................... M2-39 Topic 2: Justifying Line and Angle Relationships 2.1  Proof Positive Forms of Proof ........................................................................................................... M2-61 2.2  A Parallel Universe Proving Parallel Line Theorems ............................................................................... M2-83 © Carnegie Learning, Inc. 2.3  Ins and Outs Interior and Exterior Angles of Polygons .............................................................. M2-103 2.4  Identical Twins Perpendicular Bisector and Isosceles Triangle Theorems .................................. M2-119 2.5  Corners in a Round Room Angle Relationships Inside and Outside Circles ................................................... M2-141 Topic 3: Using Congruence Theorems 3.1  SSS, SAS, AAS, . . . S.O.S! Using Triangle Congruence to Determine Relationships Between Segments .... M2-185 3.2  Props To You Properties of Quadrilaterals ................................................................................... M2-197 3.3  Three-Chord Song Relationships Between Chords .............................................................................. M2-225 Table of Contents • FM-7 GEO_SE_FM_Vol 1.indd FM-7 5/29/18 8:01 PM Volume 2 Student Edition Module 3: Investigating Proportionality Topic 1: Similarity 1.1  Big, Little, Big, Little Dilating Figures to Create Similar Figures ..................................................................M3-7 1.2  Similar Triangles or Not? Establishing Triangle Similarity Criteria .................................................................. M3-23 1.3  Keep It in Proportion Theorems About Proportionality ............................................................................. M3-37 1.4  This Isn’t Your Average Mean More Similar Triangles .............................................................................................. M3-65 1.5  Run It Up the Flagpole Application of Similar Triangles ............................................................................... M3-79 1.6  Jack’s Spare Key Partitioning Segments in Given Ratios .................................................................... M3-95 Topic 2: Trigonometry 2.1  Three Angle Measure Introduction to Trigonometry ................................................................................ M3-121 2.2  The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent ........................................ M3-137 2.3  The Sine Ratio 2.4  The Cosine Ratio Cosine Ratio, Secant Ratio, and Inverse Cosine ................................................... M3-171 2.5  We Complement Each Other Complement Angle Relationships ......................................................................... M3-187 2.6  A Deriving Force Deriving the Triangle Area Formula, the Law of Sines, © Carnegie Learning, Inc. Sine Ratio, Cosecant Ratio, and Inverse Sine ....................................................... M3-155 and the Law of Cosines ........................................................................................... M3-199 FM-8 • Table of Contents GEO_SE_FM_Vol 1.indd FM-8 5/29/18 8:01 PM Module 4: Connecting Geometric and Algebraic Descriptions Topic 1: Circles and Volume 1.1  All Circles Great and Small Similarity Relationships in Circles ...............................................................................M4-7 1.2  A Slice of Pi Sectors and Segments of a Circle ............................................................................ M4-25 1.3  Do Me a Solid Building Three-Dimensional Figures ....................................................................... M4-45 1.4  Get to the Point Building Volume and Surface Area Formulas for Pyramids, Cones, and Spheres ................................................................................................... M4-65 Topic 2: Conic Sections 2.1  Any Way You Slice It Cross-Sections .......................................................................................................... M4-101 2.2  X 2 Plus Y 2 Equals Radius2 Deriving the Equation for a Circle .......................................................................... M4-119 2.3  A Blip on the Radar Determining Points on a Circle .............................................................................. M4-133 2.4  Sin2 θ Plus Cos2 θ Equals 12 The Pythagorean Identity ....................................................................................... M4-149 © Carnegie Learning, Inc. 2.5  Going the Equidistance Equation of a Parabola............................................................................................ M4-159 2.6  It’s a Stretch Ellipses ...................................................................................................................... M4-187 2.7  More Asymptotes Hyperbolas ............................................................................................................... M4-211 Table of Contents • FM-9 GEO_SE_FM_Vol 1.indd FM-9 5/26/18 10:25 AM Module 5: Making Informed Decisions Topic 1: Independence and Conditional Probability 1.1  What Are the Chances? Compound Sample Spaces..........................................................................................M5-7 1.2  And? Compound Probability with And .............................................................................. M5-27 1.3  Or? Compound Probability with Or ................................................................................ M5-41 1.4  And, Or, and More! Calculating Compound Probability .......................................................................... M5-57 Topic 2: Computing Probabilities 2.1  Table Talk Compound Probability for Data Displayed in Two-Way Tables............................ M5-81 2.2  It All Depends Conditional Probability ............................................................................................. M5-99 2.3  Give Me 5! Permutations and Combinations .......................................................................... M5-113 2.4  A Different Kind of Court Trial Independent Trials................................................................................................... M5-135 2.5  What Do You Expect? Glossary ............................................................................................................................... G-1 Index ......................................................................................................................................... I-1 © Carnegie Learning, Inc. Expected Value ......................................................................................................... M5-149 FM-10 • Table of Contents GEO_SE_FM_Vol 1.indd FM-10 5/26/18 10:25 AM LESSON STRUCTURE Each lesson has the same structure. Key features are noted. 1. Learning Goals The Quad Squad Learning goals are stated for each lesson to help you take ownership of the learning objectives. 2 2. Connection Each lesson begins with a statement connecting what you have learned with a question to ponder. Conjectures About Quadrilaterals Learning Goals Classify each figure using as many names as possible. • Use diagonals to draw quadrilaterals. • Make conjectures about the diagonals of special quadrilaterals. • Make conjectures about the angle relationships of special quadrilaterals. • Categorize quadrilaterals based upon their properties. • Make conjectures about the midsegments of quadrilaterals. • Understand that the vertices of cyclic quadrilaterals lie on the same circle. 1. © Carnegie Learning, Inc. © Carnegie Learning, Inc. 2. 2 1 Warm Up Return to this question at the end of this lesson to gauge your understanding. 3. Key Terms 4. • coincident • interior angle of a polygon • kite • isosceles trapezoid • midsegment • cyclic quadrilateral You have classified quadrilaterals by their side measurements and side relationships. What conjectures can you make about different properties of quadrilaterals? LESSON 2: The Quad Squad • M1-127 GEO_SE_M01_T02_L02.indd 127 18/05/18 10:48 AM Lesson Structure GEO_SE_FM_Vol 1.indd FM-11 • FM-11 5/26/18 10:25 AM 3. Getting Started Each lesson begins with a Getting Started. When working on the Getting Started, use what you know about the world, what you have learned previously, or your intuition. The goal is just to get you thinking and ready for what's to come. 3 GETTING STARTED Cattywampus Think about: Why can a concave quadrilateral have only one angle greater than 180°? A quadrilateral may be convex or concave. The quadrilaterals you are most familiar with—trapezoids, parallelograms, rectangles, rhombi, and squares—are convex. For a polygon to be convex it contains all of the line segments connecting any pair of points. It is concave if and only if at least one pair of its interior angles is greater than 180°. Consider the two quadrilaterals shown. A quadrilateral has exactly two GLDJRQDOV. Convex Concave 1. Draw the diagonals in the two quadrilaterals shown. What do you notice? © Carnegie Learning, Inc. The diagonals of any convex quadrilateral create two pairs of vertical angles and four linear pairs of angles. © Carnegie Learning, Inc. 2. Make a conjecture about the diagonals of a convex quadrilateral and about the diagonals of a concave quadrilateral. 3. Label the vertices of the convex quadrilateral as well as the point of intersection of the diagonals. Identify each pair of vertical angles and each linear pair of angles. M1-128 • TOPIC 2: Composing and Decomposing Shapes GEO_SE_M01_T02_L02.indd 128 18/05/18 10:48 AM FM-12 • Lesson Structure GEO_SE_FM_Vol 1.indd FM-12 5/26/18 10:26 AM 4. Activities AC T I V I T Y 4 2.1 You are going to build a deep understanding of mathematics through a variety of activities in an environment where collaboration and conversations are important and expected. Quadrilaterals Formed Using Concentric Circles Let’s explore the diagonals of different convex quadrilaterals. Consider a pair of concentric circles with center A. Diameter ¯ BC is shown. AC T I V I T Y 2.2 Quadrilaterals Formed Using a Circle Use a new piece of patty paper for each quadrilateral. For precision, it is important to use a straightedge when tracing or drawing line segments. B C A In the previous activity, you drew quadrilaterals using a pair of 1. Draw quadrilateral BDCE by following concentric circles. Now let’s draw quadrilaterals using only one these steps. circle. Circle P with diameter ¯ QR is shown. ¯ • Construct the perpendicular bisector to BC through AC T I V I T Y point A of the concentric circles. Use a new piece of patty paper for each quadrilateral. For • Use patty paper to trace ¯ BC. precision, it is important to use a straightedge when tracing or • Without moving the patty paper, draw diameter ¯ DE of the inner Q R drawing line segments. P that it is not perpendicular or coincident to ¯ circle in such a way BC. Two line segments are coincident if they lie In theofprevious two 1. activities, you used the BDCE. properties of the diagonals • Connect the endpoints the diameters to drawquadrilateral quadrilateral Draw QSRT by following these steps. exactly on top of the to discover each member of the quadrilateral family. You investigated • Construct the perpendicular bisector to ¯ QR through each other. ¯between ¯?the diagonals of quadrilaterals. relationships a. What do you know about BC and DE point P. 2.3 You will learn how to solve new problems, but you will also learn why those strategies work and how they are connected to other strategies you already know. Making Conjectures About Quadrilaterals • Use patty paper to trace ¯ QR. 1. Make a conjecture• about the diagonals of the described Without moving the patty paper, draw diameter ¯ ST of quadrilaterals. your reasoning using b. Use a ruler to measure the lengthsExplain of each side of the circle P in such a way that it examples. is not perpendicular or quadrilateral. What do you notice? ¯ coincident to QR. Remember: ¯? ¯ and DC d. What do you know about BE b. Use a ruler to measure the lengths of each side of the quadrilateral. What do you notice? e. What names can be used to describe quadrilateral BDCE? Explain your reasoning. Write the name of the quadrilateral b. rectangles Keep this quadrilateral on the patty paper. to compare with other c. Use a protractor to determine the measure of each interior quadrilaterals angle of the quadrilateral. Which conjecture thatthat you made in you will create in the previous lesson does this measurement support? this lesson. GEO_SE_M01_T02_L02.indd 129 © Carnegie Learning, Inc. a polygon is an angle inside the polygon between two adjacent sides. © Carnegie Learning, Inc. © Carnegie Learning, Inc. © Carnegie Learning, Inc. a. parallelograms• Connect the endpoints of the diameters to draw quadrilateral QSRT. c. Use a protractor to determine the measure of each interior An interior angle of angle of the quadrilateral. What do you ¯ and ST ¯? a. What do younotice? know about QR LESSON 2: The Quad Squad • M1-129 d. What names can be used to describe quadrilateral QSRT? Explain your reasoning. Write the most specific name for the quadrilateral on the patty paper. c. quadrilaterals with pairs of adjacent congruent sides • Making mistakes are a critical part of learning, so take risks. • There...
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