Download Geometry - CPM Student Guidebook

Geometry
Connections
Table of Contents
Student Edition
Chapter 1 Shapes and Transformations
1
Section 1.1 Course Introduction Problems
1.1.1
1.1.2
1.1.3
1.1.4
1.1.5
Creating a Quilt Using Symmetry
Making Predictions and Investigating Results
Perimeter and Area of Enlarging Tile Patterns
Logical Arguments
Building a Kaleidoscope
3
7
12
16
21
Section 1.2 Transformations and Symmetry
1.2.1
1.2.2
1.2.3
1.2.4
1.2.5
Spatial Visualization and Reflections
Rigid Transformations: Rotations and Translations
Using Transformations
Using Transformations to Create Shapes
Symmetry
26
31
36
40
44
Section 1.3 Shapes and Probability
1.3.1 Attributes and Characteristics of Shapes
1.3.2 More Characteristics of Shapes
1.3.3 Introduction to Probability
Chapter Closure
Chapter 2 Angles and Measurement
49
53
57
62
71
Section 2.1 Angle Relationships
2.1.1
2.1.2
2.1.3
2.1.4
2.1.5
Complementary, Supplementary, and Vertical Angles
Angles Formed by Transversals
More Angles Formed by Transversals
Angles in a Triangle
Applying Angle Relationships
73
78
83
89
93
Section 2.2 Area
2.2.1
2.2.2
2.2.3
2.2.4
Units of Measure
Areas of Triangles and Composite Shapes
Areas of Parallelograms and Trapezoids
Heights and Area
98
102
105
110
Section 2.3 The Pythagorean Theorem
2.3.1 Squares and Square Roots
2.3.2 Triangle Inequality
2.3.3 The Pythagorean Theorem
113
117
121
Chapter Closure
125
Chapter 3 Justification and Similarity
133
Section 3.1 Introduction to Similarity
3.1.1
3.1.2
3.1.3
3.1.4
Similarity
Proportional Growth and Ratios
Using Ratios of Similarity
Applications and Notation
135
139
144
147
Section 3.2 Triangle Similarity
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
Conditions for Triangle Similarity
Creating a Flowchart
Triangle Similarity and Congruence
More Conditions for Triangle Similarity
Determining Similarity
Applying Similarity
Chapter Closure
Chapter 4 Trigonometry and Probability
152
157
161
165
169
173
176
185
Section 4.1 The Tangent Ratio
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
Constant Ratios in Right Triangles
Connecting Slope Ratios to Specific Angles
Expanding the Trig Table
The Tangent Ratio
Applying the Tangent Ratio
187
192
195
198
202
Section 4.2 Probability Models
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
Introduction to Probability Models
Theoretical and Experimental Probability
Using an Area Model
Choosing a Probability Model
Optional: Applications of Probability Methods
Chapter Closure
204
209
212
216
221
224
Chapter 5 Completing the Triangle Toolkit
233
Section 5.1 Trigonometry
5.1.1
5.1.2
5.1.3
5.1.4
Sine and Cosine Ratios
Selecting a Trig Tool
Inverse Trigonometry
Trigonometric Applications
235
239
243
246
Section 5.2 Special Right Triangles
5.2.1 Special Right Triangles
5.2.2 Pythagorean Triples
250
254
Section 5.3 Completing the Triangle Toolkit
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
Finding Missing Parts of Triangles
Law of Sines
Law of Cosines
Ambiguous Triangles
Choosing a Tool
Chapter Closure
Chapter 6 Congruent Triangles
257
262
265
269
273
278
287
Section 6.1 Congruent Triangles
6.1.1
6.1.2
6.1.3
6.1.4
Congruent Triangles
Conditions for Triangle Congruence
Flowcharts for Congruence
Converses
287
293
297
301
Section 6.2 Closure Activities
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
Angles on a Pool Table
Investigating a Triangle
Creating a Mathematical Model
Analyzing a Game
Using Transformations and Symmetry to Create a Snowflake
Chapter Closure
306
309
311
315
318
324
Chapter 7 Proof and Quadrilaterals
333
Section 7.1 Introduction to Chapters 7 - 12
7.1.1
7.1.2
7.1.3
7.1.4
Properties of a Circle
Building a Tetrahedron
Shortest Distance Problems
Using Symmetry to Study Polygons
335
339
343
349
Section 7.2 Proof and Quadrilaterals
7.2.1
7.2.2
7.2.3
7.2.4
7.2.5
7.2.6
Special Quadrilaterals and Proof
Properties of Rhombi
More Proof with Congruent Triangles
More Properties of Quadrilaterals
Two-Column Proofs
Explore-Conjecture-Prove
352
356
359
363
365
369
Section 7.3 Coordinate Geometry
7.3.1 Studying Quadrilaterals on a Coordinate Grid
7.3.2 Coordinate Geometry and Midpoints
7.3.3 Quadrilaterals on a Coordinate Plane
Chapter Closure
Chapter 8 Polygons and Circles
373
376
380
383
391
Section 8.1 Angles and Area of a Polygon
8.1.1
8.1.2
8.1.3
8.1.4
8.1.5
Pinwheels and Polygons
Interior Angles of a Polygon
Angles of Regular Polygons
Regular Polygon Angle Connections
Finding the Area of Regular Polygons
393
398
402
405
408
Section 8.2 Ratio of the Area of Similar Figures
8.2.1 Area Ratios of Similar Figures
8.2.2 Ratios of Similarity
413
417
Section 8.3 Area and Circumference of a Circle
8.3.1 A Special Ratio
8.3.2 Area and Circumference of Circles
8.3.3 Circles in Context
Chapter Closure
421
424
428
434
Chapter 9 Solids and Constructions
441
Section 9.1 Solids
9.1.1
9.1.2
9.1.3
9.1.4
9.1.5
Three-Dimensional Solids
Volume and Surface Area of Prisms
Prisms and Cylinders
Volumes of Similar Solids
Ratios of Similarity
443
446
450
453
456
Section 9.2 Construction
9.2.1
9.2.2
9.2.3
9.2.4
Introduction to Construction
Constructing Bisectors
More Exploration with Constructions
Finding a Centroid
Chapter Closure
Chapter 10 Circles and Expected Value
459
463
466
470
473
481
Section 10.1 Relationships Within Circles
10.1.1
10.1.2
10.1.3
10.1.4
10.1.5
Introduction to Chords
Angles and Arcs
Chords and Angles
Tangents and Chords
Problem Solving with Circles
483
487
492
496
499
Section 10.2 Expected Value
10.2.1 Designing Spinners
10.2.2 Expected Value
10.2.3 More Expected Value
502
507
510
Section 10.3 Equation of a Circle
10.3.1 The Equation of a Circle
Chapter Closure
514
518
Chapter 11 Solids and Circles
525
Section 11.1 Pyramids, Cones, Spheres
11.1.1
11.1.2
11.1.3
11.1.4
11.1.5
Platonic Solids
Pyramids
Volume of a Pyramid
Surface Area and Volume of a Cone
Surface Area and Volume of a Sphere
527
533
536
541
545
Section 11.2 More Circle Relationships
11.2.1 Coordinates on a Sphere
11.2.2 Tangents and Arcs
11.2.3 Secant and Tangent Relationships
Chapter Closure
Chapter 12 Conics and Closure
548
553
556
564
571
Section 12.1 Conic Sections
12.1.1
12.1.2
12.1.3
12.1.4
12.1.5
Introduction to Conic Sections
Graphing Parabolas Using the Focus and Directrix
Circles and Ellipses
The Hyperbola
Conic Equations and Graphs
573
576
580
583
587
Section 12.2 Closure Activities
12.2.1
12.2.2
12.2.3
12.2.4
Using Coordinate Geometry and Construction to Explore Shapes
Euler’s Formula for Polyhedra
The Golden Ratio
Using Geometry to Find Probability
Appendices
A
B
C
Points, Lines, and Planes
Euclidean Geometry
Non-Euclidean Geometry
591
594
597
601
607
607
612
616
Glossary
621
Index
647
Index to List of Symbols and Equations
658