**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
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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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