Download Lesson 8-3 – Special Right Triangles

Curriculum and Instruction – Office of Mathematics
3rd Quarter
GEOMETRY
DRAFT
Introduction
In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District
is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,
 80% of our students will graduate from high school college or career ready
 90% of students will graduate on time
 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity
In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, College and Career Ready standardsaligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a
grade. College and Career Ready Standards are rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN
State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor.
Focus
•
•
•
•
The Standards call for a greater focus in
mathematics. Rather than racing to cover topics
in a mile-wide, inch-deep curriculum, the
Standards require us to significantly narrow and
deepen the way time and energy is spent in the
math classroom. We focus deeply on the major
work of each grade so that students can gain
strong foundations: solid conceptual
understanding, a high degree of procedural skill
and fluency, and the ability to apply the math
they know to solve problems inside and outside
the math classroom.
For geometry, the major clusters, account for
53% of time spent on instruction.
Supporting Content - information that
supports the understanding and implementation
of the major work of the grade.
Additional Content - content that does not
explicitly connect to the major work of the grade
yet it is required for proficiency.
Subject to revision
Coherence
Rigor
Thinking across grades:
•
The Standards are designed around coherent
progressions from grade to grade. Learning is
carefully connected across grades so that
students can build new understanding onto
foundations built in previous years. Each
standard is not a new event, but an extension of
previous learning.
Conceptual understanding:
•
The Standards call for conceptual understanding
of key concepts, such as place value and ratios.
Students must be able to access concepts from a
number of perspectives so that they are able to
see math as more than a set of mnemonics or
discrete procedures.
•
Procedural skill and fluency:
•
The Standards call for speed and accuracy in
calculation. While the high school standards for
math do not list high school fluencies, there are
suggested fluency standards for algebra 1,
geometry and algebra 2.
Linking to major topics:
•
Instead of allowing additional or supporting
topics to detract from course, these concepts
serve the course focus. For example, instead of
data displays as an end in themselves, they are
an opportunity to do grade-level word
problems.
Application:
•
The Standards call for students to use math
flexibly for applications in problem-solving
contexts. In content areas outside of math,
particularly science, students are given the
opportunity to use math to make meaning of and
access content.
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
GEOMETRY
Our collective goal is to ensure our students graduate ready for
college and career. The Standards for Mathematical Practice
describe varieties of expertise that mathematics educators at all
levels should seek to develop in their students. These practices
rest on important “processes and proficiencies” with longstanding
importance in mathematics education. The first of these are the
NCTM process standards of problem solving, reasoning and proof,
communication, representation and connections.
Look for and
express
regularity in
repeated
reasoning
Look for and
make use of
structure
Problem
Solving
Connection
Reasoning and
Proof
Make sense of
problems and
persevere in
solving them
Reason
abstractly and
quatitatively
Mathematical
Practices
Attend to
precision
Construct viable
arguments and
crituqe the
reasoning of
others
Model with
mathematics
Use appropriate
tools
strategically
Subject to revision
DRAFT
Representation
Communication
The second are the strands of mathematical proficiency specified in
the National Research Council’s report Adding It Up: adaptive
reasoning, strategic competence, conceptual understanding
(comprehension of mathematical concepts, operations and relations)
procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition
(habitual inclination to see mathematics and sensible, useful and
worthwhile, coupled with a belief in diligence and one’s own efficacy).
Throughout the year, students should continue to develop proficiency
with the eight Standards for Mathematical Practice.
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
GEOMETRY
DRAFT
How to Use the Mathematic Curriculum Maps
This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that ultimately our
students can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their
instructional practice in alignment with the three College and Career Ready shifts in instruction for Mathematics. We should see these shifts
in all classrooms:
1) Focus
2) Coherence
3) Rigor
Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to
reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage
resources around each of the three shifts that teachers should consistently access:
The TNCore Mathematics Standards
The Tennessee Mathematics Standards:
Teachers can access the Tennessee State standards, which
https://www.tn.gov/education/article/mathematics- are featured throughout this curriculum map and
standards
represent college and career ready learning at reach
respective grade level.
Mathematical Shifts
Focus
The standards are focused on fewer topics so students can
http://achievethecore.org/shifts-mathematics
learn more
Coherence
http://achievethecore.org/shifts-mathematics
Topics within a grade are connected to support focus, and
learning is built on understandings from previous grades
Rigor
http://achievethecore.org/shifts-mathematics
The standards set expectations for a balanced approach to
pursuing conceptual understanding, procedural fluency,
and application and modeling
Subject to revision
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
GEOMETRY
DRAFT
Curriculum Maps:







Locate the TDOE Standards in the left column. Analyze the language of the standards and match each standard to a learning target in
the second column.
Consult your McGraw-Hill/Glencoe Teachers’ Edition (TE) and other cited references to map out your week(s) of instruction.
Plan your weekly and daily objectives, using the standards' explanations provided in the second column. Best practices tell us that
making objectives measureable increases student mastery.
Carefully review the web-based resources provided in the 'Content and Tasks' column and use them as you introduce or assess a
particular standard or set of standards.
Review the CLIP Connections found in the right column. Make plans to address the content vocabulary, utilizing the suggested
literacy strategies, in your instruction.
Examine the other standards and skills you will need to address in order to ensure mastery of the indicated standard.
Using your McGraw-Hill/Glencoe TE and other resources cited in the curriculum map, plan your week using the SCS lesson plan
template. Remember to include differentiated activities for small-group instruction and math stations.
Subject to revision
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
GEOMETRY
Essential Understandings
Content & Tasks
DRAFT
CLIP Connections
Unit 4 Similarity
Using Similar Triangles ( 8 days)
G.SRT Similarity, Right Triangles, and
Trigonometry
Prove theorems using similarity
G.SRT.B.4 Prove theorems about triangles.
G.SRT.B.5 Use congruence and similarity
criteria for triangles to solve problems and to
prove relationships in geometric figures.
Lesson 7.3 Similar Triangles
G.SRT Similarity, Right Triangles, and
Trigonometry
Prove theorems using similarity
G.SRT.B.4 Prove theorems about triangles.
G.SRT.B.5 Use congruence and similarity
criteria for triangles to solve problems and to
prove relationships in geometric figures.
Lesson 7-4 Parallel Lines and Proportional
Parts (midsegments was previously covered in
unit 2)
G.SRT Similarity, Right Triangles, and
Trigonometry
Prove theorems using similarity
G.SRT.B.4 Prove theorems about triangles.
G.SRT.B.5 Use congruence and similarity
criteria for triangles to solve problems and to
prove relationships in geometric figures.
Lesson 8-1 Geometric Mean
Find the geometric mean between
two numbers
Solve problems involving relationships
between parts of a right triangle and the
altitude to its hypotenuse
G.SRT Similarity, Right Triangles, and
Lesson 7.5 Parts of Similar Triangles
Subject to revision

Identify similar triangles using
the AA Similarity Postulate and
the SSS and SAS similarity
Theorems

Use similar triangles to solve
problems

Use proportional parts within
triangles

Use proportional parts with parallel
lines

Lesson 7.3 pp. 474-483
Similarity
AA Similarity
SSS & SAS Similarity
Applying Similar Triangles
Contrast and compare the triangle
congruence theorems with the triangle
similarity theorems.
Lesson 7-4 pp. 484 - 492
Triangle Side Splitter Theorem
Dividing a Line Segment into Equal Parts
Parallel Lines and Proportional Segments See
Mathematics, Instructional Resources,
Geometry, Task Arc: Investigating Coordinate
Geometry
Partitioning See Mathematics, Instructional
Resources, Geometry, Task Arc: Investigating
Coordinate Geometry
However You Want to Slice It
Comparing Shapes tncore.org/
Use multiple representations to explore
angle bisectors and proportions. See p.
492, #47
Lesson 8-1 pp. 531 - 539
Special Relationships Within Right Triangles
Prove the Pythagorean Theorem Using
Similarity
Lesson 7-5 pp. 495 -503
Utilize Tasks to include the Standards
for Mathematical Practice where
students have to reason, justify,
explain, construct & model their
thinking.
What is an arithmetic mean and a geometric
mean of two numbers? Are they ever equal?
Justify your answer.
Find a counterexample: If the measure
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
Trigonometry
Prove theorems using similarity
G.SRT.B.4 Prove theorems about triangles.
G.SRT.B.5 Use congruence and similarity
criteria for triangles to solve problems and to
prove relationships in geometric figures.
GEOMETRY
Essential Understandings

Recognize and use proportional
relationships of corresponding
angle bisectors, altitudes, and
medians of similar triangles

Use the Triangle Angle Bisector
Theorem
Content & Tasks
Triangle Angle Bisector Theorem
DRAFT
CLIP Connections
of an altitude and side of a triangle are
proportional to the corresponding
altitude and corresponding side of
another triangle, then the triangles are
similar.
Unit 5 –Trigonometry
Trigonometry with Right Triangles (8 days)
G-SRT Similarity, Right Triangles, and
Trigonometry
Define trigonometric ratios and solve
problems involving right triangles
Lesson 8-3 – Special Right Triangles

Identify and apply side ratios in
45-45-90 right triangles.

Identify and apply side ratios in
30-60-90 right triangles
Define trigonometric ratios and solve
problems involving right triangles
G-SRT.C.7 Explain and use the relationship
between the sine and cosine of
complementary angles.
G-SRT.C.8 Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles
in applied problems.★
G-SRT.C.6
Subject to revision
Discovering Special Right Triangles
Learning Task
Finding Right Triangles in your Environment
Learning Task
H.O.T. Problems
p.559 #50
Explain how you can find the lengths of two
legs of a 30-60-90 triangle in radical form if
you are given the length of the hypotenuse.
Create your own triangles Learning Task
G-SRT.C.6 Understand that by similarity, side
ratios in right triangles are properties of the
angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
G-SRT Similarity, Right Triangles, and
Trigonometry
Lesson 8.3 pp. 552-559
Utilize Tasks to include the Standards for
Mathematical Practice where students
have to reason, justify, explain, construct
& model their thinking.
Lesson 8-4 – Trigonometry


Define trigonometric ratios for acute
angles in right triangles
Use trigonometric rations and
Pythagorean Theorem to solve right
triangles
Use the relationship between the sine
and cosine of complementary angles.
Lesson 8.4 pp. 562-571
Discovering Trigonometric Ratio Relationships
learning task p.22
H.O.T. Problems
p.570 #65
Explain how you can use ratios of the side
lengths to find the angle measures of the
acute angles in a right triangle.
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
G-SRT Similarity, Right Triangles, and
Trigonometry
Define trigonometric ratios and solve
problems involving right triangles
G-SRT.C.8
GEOMETRY
Essential Understandings
Lesson 8-5 – Angles of Elevation and
Depression

Solve problems involving angles of
elevation.
Solve problems involving angles of
depression.
Content & Tasks
Lesson 8.5 pp. 574-581
Find that Side or Angle Task
Interstate
DRAFT
CLIP Connections
H.O.T. Problems
p.580 #25
Classify the statement below as true or false.
Explain. “As a person moves closer to an
object he or she is sighting, the angle of
elevation increases”
Unit 7 – Measurement and Modeling in Two and Three Dimensions
Volume Formulas (6 days)
G-GMD Geometric Measurement and
Dimension
Explain volume formulas and use them to
solve problems
G-GMD.A.1. Give an informal argument for the
formulas for the circumference of a circle, area
of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s
principle, and informal limit arguments.
G.GMD.A.3. Use volume formulas for
cylinders, pyramids, cones, and spheres to
solve problems.★
Lesson 12.4 – Volumes of Prisms and
Cylinders
 Find volumes of prisms
Find volumes of cylinders
H.O.T. Problems
p.853 #44
Write a helpful response to the following
questions posted on an Internet garden forum.
“I am new to gardening. The nursery will
deliver a truckload of soil, which they say is 4
yards. I know that a yard is 3 feet, but what is
a yard of soil? How do I know what to order?”
Utilize Tasks to include the Standards for
Mathematical Practice where students
have to reason, justify, explain, construct
& model their thinking.
G-GMD Geometric Measurement and
Dimension
Explain volume formulas and use them to
solve problems
G-GMD.A.1
G.GMD.A.3
Lesson 12.5 – Volumes of Pyramids and
Cones
G-GMD Geometric Measurement and
Dimension
Explain volume formulas and use them to
solve problems
Lesson 12.6 – Volume of Spheres
Find volumes of spheres
Subject to revision
Lesson 12.4 pp. 847-854
How much money is that? (prism)
Centerpiece (cylinder)
 Find volumes of pyramids
Find volumes of cones
Lesson 12.5 pp. 857-863
Doctors Appointment (cone)
Great Egyptian Pyramids (pyramid)
Lesson 12.6 pp. 873-878
Guessing Gumballs Task
H.O.T. Problems
p.862 #40
Compare and contrast finding volumes of
pyramids and cones with finding volumes of
prisms and cylinders.
H.O.T. Problems
p.870 #48
Write a ratio comparing the volume of a
sphere with radius r to the volume of a
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
GEOMETRY
Essential Understandings
Content & Tasks
G-GMD.A.1
G.GMD.A.3
DRAFT
CLIP Connections
cylinder with radius r and height 2r. Then
describe what the ratio means.
Unit 7 – Measurement and Modeling in Two and Three dimensions
Modeling and Problem Solving (5 days)
G-MG Modeling with Geometry
Apply geometric concepts in modeling
situations
G.MG.A.1 Use geometric shapes, their
measures, and their properties to describe
objects (e.g., modeling a tree trunk or a human
torso as a cylinder).★
G-MG Modeling with Geometry
Apply geometric concepts in modeling
situations
No Glencoe Lesson
Use three-dimensional figures to
describe objects
Subject to revision
A plastic beach ball has a diameter of 15
inches. About how much plastic is used to
make the beach ball? Describe what
processes you used to find the answer.
Utilize Tasks to include the Standards for
Mathematical Practice where students
have to reason, justify, explain, construct
& model their thinking.
No Glencoe Lesson
Model and solve problems using figures
and their measurements
Glencoe Lesson CCSS Lab 14: Geometry
Lab: Population Density (Supplement)
Illustrative mathematics G-MG.A.2 multiple
lessons and tasks that can be used
No Glencoe Lesson
Model and solve problems using figures
and their measurements
Illustrative mathematics G.MG.A.3 multiple
lessons and tasks that can be used
G-MG.A.2 Apply concepts of density based on
area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot).
★
G-MG Modeling with Geometry
Apply geometric concepts in modeling
situations
G-MG.A.3 Apply geometric methods to solve
design problems (e.g., designing an object or
structure to satisfy physical constraints or
minimize cost; working with typographic grid
systems based on ratios).★
Illustrative mathematics G.MG.A.1 multiple
lessons and tasks that can be used
A candy factory makes peanut butter
chocolate balls with a 1 inch diameter using a
chocolate shell and a peanut ball center. If the
diameter of the peanut butter ball is .5 inch,
find the volume of chocolate used to make
each ball. What percentage of each ball is
peanut butter? Justify your answer.
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Revised 12/15/15
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
GEOMETRY
Essential Understandings
Content & Tasks
DRAFT
CLIP Connections
Unit 6: Properties of Circles
Angles and Segments in circles (5 days)
G - C Circles
Understand and apply theorems about
circles
G.C.A.1 Prove that all circles are silimar
Lesson 10-1 – Circles and Circumference

Give an argument to justify the
formula for the circumference of
a circle.
Prove that all circles are similar.
Lesson 10.1 pp.
Similar Circles Task
All Circles are Similar Task
Utilize Tasks to include the Standards for
Mathematical Practice where students
have to reason, justify, explain, construct
& model their thinking.
G-CO Congruence
Experiment with transformations in
the plane
G.CO.A.1 Know precise definitions of
angle, circle, perpendicular line, parallel
line, and line segment, based on the
undefined notions of point, line, distance
along a line, and distance around a
circular arc.
G-GMD Geometric Measurement and
Dimension
G.GMD.A.1
G - C Circles
Understand and apply theorems about
circles
G-C.A.2 Identify and describe relationships
among inscribed angles, radii, and chords.
Include the relationship between central,
inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the
radius of a circle is perpendicular to the
tangent where the radius intersects the circle.
Subject to revision
H.O.T. Problems
p.690 #54
Research and write about the history of pi and
its importance to the study of geometry.
Lesson 10-2 – Measuring Angles and
Arcs
Identify central angles, major arcs, minor
arcs, and semicircles and find their
measures.
Lesson 10.2 pp. 692-700
Circles and their Relationships among Central
Angles, Arcs and Chords
Investigating Angle Relationships in Circles
H.O.T. Problems
p.699 #62
Describe the three different types of arcs in a
circle and the method for finding the measure
of each one.
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
TN State Standards
G - C Circles
Understand and apply theorems about
circles
G-C.A.2
Subject to revision
GEOMETRY
Essential Understandings
Lesson 10-4 – Inscribed Angles

Identify and describe relationships
involving inscribed angles.
Prove properties of angles for a
quadrilateral inscribed in a circle.
Content & Tasks
Lesson 10.4 pp. 709-716
Opposite angles in a cyclic quadrilateral
DRAFT
CLIP Connections
H.O.T. Problems
p.715 #50
Compare and contrast inscribed angles and
central angles of a circle. If they intercept the
same arc, how are they related?
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Curriculum and Instruction – Office of Mathematics
3rd Quarter
Textbook Resources
ConnectED Site - Textbook and Resources
Glencoe Video Lessons
Hotmath - solutions to odd problems
Comprehensive Geometry Help:
Online Math Learning (Geometry)
I LOVE MATH
NCTM Illuminations
New Jersey Center for Teaching & Learning (Geometry)
Calculator
Finding Your Way Around TI-83+ & TI-84+ (mathbits.com)
Texas Instruments Calculator Activity Exchange
Texas Instruments Math Nspired
STEM Resources
Casio Education for Teachers
*Graphing Calculator Note: TI tutorials are available through
Atomic Learning and also at the following link: Math Bits graphing calculator steps Some activities require calculator
programs and/or applications.
Use the following link to access FREE software for your MAC.
This will enable your computer and TI Calculator to
communicate: Free TI calculator downloads
Subject to revision
GEOMETRY
RESOURCE TOOLBOX
CCSS
Common Core Standards - Mathematics
Common Core Standards - Mathematics Appendix A TN Core
CCSS Flip Book with Examples of each Standard
Geometry Model Curriculum
http://www.ccsstoolbox.org/
http://insidemathematics.org/index.php/high-school-geometry
http://www.azed.gov/azcommoncore/mathstandards/hsmath/
http://learnzillion.com/common_core/math/hs
http://www.livebinders.com/play/play/454480
https://www.livebinders.com/play/play?id=464831
http://www.livebinders.com/play/play?id=571735
North Carolina – Unpacking Common Core
http://thegeometryteacher.wordpress.com/the-geometry-course/
http://mathtermind.blogspot.com/2012/07/common-core- geometry.html
Utah Electronic School - Geometry
Ohio Common Core Resources
Chicago Public Schools Framework and Tasks
Mathy McMatherson Blog - Geometry in Common Core
Interactive Manipulatives
GeoGebra – Free software for dynamic math and science learning
NCTM Core Math Tools
http://www.keycurriculum.com/products/sketchpad (Not free) Any
activity using Geometer’s Sketchpad can also be done with any
software that allows construction of figures and measurement, such as
Cabri, Cabri Jr. on the TI-83 or 84 Plus,
TI-92 Plus, or TI-Nspire
DRAFT
Videos
Math TV Videos
The Teaching Channel
Teacher Tube
Khan Academy Videos (Geometry)
CLIP Resources
Literacy Skills and Strategies for Content Area
Teachers (Math, p. 22)
Glencoe Reading & Writing in the Mathematics Classroom
Graphic Organizers (9-12) (teachervision.com)
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