Download rubric – implementing standards for mathematical

High School
CCSS
Mathematics II
Curriculum
Guide
-Quarter 1Columbus City
Schools
Page 1 of 162
Table of Contents
RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE ....................... 11
Mathematical Practices: A Walk-Through Protocol .............................................................................. 16
Curriculum Timeline .............................................................................................................................. 19
Scope and Sequence ............................................................................................................................... 20
Similarity 1, 1a, 1b, 2, 3, 4, 5 ................................................................................................................. 29
Teacher Notes .......................................................................................................................................... 30
Are You Golden? ................................................................................................................................. 43
The Gumps ........................................................................................................................................... 47
The Gumps and Similar Figures .......................................................................................................... 55
Draw Similar Triangles ........................................................................................................................ 61
Similar Quilt Blocks............................................................................................................................. 63
Quilt Calculations ................................................................................................................................ 64
Investigating Triangles with Two Pairs of Congruent Angles ............................................................. 67
Similar Triangles Application .............................................................................................................. 71
Find the Scale Factor............................................................................................................................ 72
Let’s Prove the Pythagorean Theorem ................................................................................................. 76
Proving the Pythagorean Theorem, Again!.......................................................................................... 80
Trigonometric Ratios G-SRT 6, 7, 8 ...................................................................................................... 82
Teacher Notes .......................................................................................................................................... 83
Exploring Special Right Triangles (45-45-90)..................................................................................... 97
Exploring Special Right Triangles (30-60-90)..................................................................................... 99
Discovering Trigonometric Ratios ..................................................................................................... 104
Make a Model: Trigonometric Ratios ................................................................................................ 108
Let’s Measure the Height of the Flagpole .......................................................................................... 112
Applications of Trigonometry Using Indirect Measurement ............................................................. 114
Find the Missing Side or Angle ......................................................................................................... 122
Between the Uprights ......................................................................................................................... 124
Solve the Triangle .............................................................................................................................. 129
Right Triangle Park ............................................................................................................................ 135
Find the Height................................................................................................................................... 136
Find the Height Data Sheet ................................................................................................................ 137
Applications of the Pythagorean Theorem ......................................................................................... 138
Memory Match – Up .......................................................................................................................... 140
Memory Match – Up Cards ............................................................................................................... 141
Similar Right Triangles and Trigonometric Ratios ............................................................................ 147
Similar Right Triangles and Trigonometric Ratios ............................................................................ 149
Hey, All These Formulas Look Alike!............................................................................................... 155
Problem Solving: Trigonometric Ratios ............................................................................................ 157
Grids and Graphics Addendum ............................................................................................................ 162
Page 2 of 162
Math Practices Rationale
CCSSM Practice 1: Make sense of problems and persevere in solving them.
Why is this practice important?



What does this practice look like when students are
doing it?


What can a teacher do to model this practice?


What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?



Helps students to develop critical thinking
skills.
Teaches students to “think for themselves”.
Helps students to see there are multiple
approaches to solving a problem.
Students immediately begin looking for
methods to solve a problem based on previous
knowledge instead of waiting for teacher to
show them the process/algorithm.
Students can explain what problem is asking as
well as explain, using correct mathematical
terms, the process used to solve the problem.
Frame mathematical questions/challenges so
they are clear and explicit.
Check with students repeatedly to help them
clarify their thinking and processes.
“How would you go about solving this
problem?”
“What do you need to know in order to solve
this problem?”
What methods have we studied that you can
use to find the information you need?
Students can explain the relationships
between equations, verbal descriptions,
tables, and graphs.
Students check their answer using a different
method and continually ask themselves, “Does
this make sense?”
They understand others approaches to solving
complex problems and can see the similarities
between different approaches.
Showing the students shortcuts/tricks to solve
problems (without making sure the students
understand why they work).
Not giving students an adequate amount of
think time to come up with solutions or
processes to solve a problem.
Giving students the answer to their questions
instead of asking guiding questions to lead
them to the discovery of their own question.
Page 3 of 162
CCSSM Practice 2: Reason abstractly and quantitatively.
Why is this practice important?


What does this practice look like when students are
doing it?



What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?

Students develop reasoning skills that help
them to understand if their answers make
sense and if they need to adjust the answer to
a different format (i.e. rounding)
Students develop different ways of seeing a
problem and methods of solving it.
Students are able to translate a problem
situation into a number sentence or algebraic
expression.
Students can use symbols to represent
problems.
Students can visualize what a problem is
asking.
Ask students questions about the types of
answers they should get.
Use appropriate terminology when discussing
types of numbers/answers.
Provide story problems and real world
problems for students to solve.
Monitor the thinking of students.
“What is your unknown in this problem?
“What patterns do you see in this problem and
how might that help you to solve it?”
Students can recognize the connections
between the elements in their mathematical
sentence/expression and the original problem.
Students can explain what their answer
means, as well as how they arrived at it.
Giving students the equation for a word or
visual problem instead of letting them “figure
it out” on their own.
Page 4 of 162
CCSSM Practice 3: Construct viable arguments and critique the reasoning of others
Why is this practice important?


What does this practice look like when students are
doing it?


What can a teacher do to model this practice?






What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?


Students better understand and remember
concepts when they can defend and explain
it to others.
Students are better able to apply the
concept to other situations when they
understand how it works.
Communicate and justify their solutions
Listen to the reasoning of others and ask
clarifying questions.
Compare two arguments or solutions
Question the reasoning of other students
Explain flaws in arguments
Provide an environment that encourages
discussion and risk taking.
Listen to students and question the clarity of
arguments.
Model effective questioning and appropriate
ways to discuss and critique a mathematical
statement.
How could you prove this is always true?
What parts of “Johnny’s “ solution confuses
you?
Can you think of an example to disprove
your classmates theory?
Students are able to make a mathematical
statement and justify it.
Students can listen, critique and compare
the mathematical arguments of others.
Students can analyze answers to problems
by determining what answers make sense.
Explain flaws in arguments of others.
Not listening to students justify their
solutions or giving adequate time to critique
flaws in their thinking or reasoning.
Page 5 of 162
CCSSM Practice 4: Model with mathematics
Why is this practice important?

What does this practice look like when students are
doing it?


What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?

Helps students to see the connections
between math symbols and real world
problems.
Write equations to go with a story problem.
Apply math concepts to real world problems.
Use problems that occur in everyday life and
have students apply mathematics to create
solutions.
Connect the equation that matches the real
world problem. Have students explain what
different numbers and variables represent in
the problem situation.
Require students to make sense of the
problems and determine if the solution is
reasonable.
How could you represent what the problem
was asking?
How does your equation relate to the
problems?
How does your strategy help you to solve
the problem?
Students can write an equation to represent
a problem.
Students can analyze their solutions and
determine if their answer makes sense.
Students can use assumptions and
approximations to simplify complex
situations.
Not give students any problem with real
world applications.
Page 6 of 162
CCSSM Practice 5: Use appropriate tools strategically
Why is this practice important?

What does this practice look like when students are
doing it?



What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?


What actions might the teacher make that inhibit
the students’ use of this practice?


Helps students to understand the uses and
limitations of different mathematical and
technological tools as well as which ones can
be applied to different problem situations.
Students select from a variety of tools that
are available without being told which to
use.
Students know which tools are helpful and
which are not.
Students understand the effects and
limitations of chosen tools.
Provide students with a variety of tools
Facilitate discussion regarding the
appropriateness of different tools.
Allow students to decide which tools they
will use.
How is this tool helping you to understand
and solve the problem?
What tools have we used that might help
you organize the information given in this
problem?
Is there a different tool that could be used to
help you solve the problem?
Students are sufficiently familiar with tools
appropriate for their grade or course and
make sound decisions about when each of
these tools might be helpful.
Students recognize both the insight to be
gained from the use of the selected tool and
their limitations.
Only allowing students to solve the problem
using one method.
Telling students that the solution is incorrect
because it was not solved “the way I showed
you”.
Page 7 of 162
CCSSM Practice 6: Attend to precision.
Why is this practice important?


What does this practice look like when students are
doing it?


What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?



What does proficiency look like in this practice?



What actions might the teacher make that inhibit
the students’ use of this practice?



Students are better able to understand new
math concepts when they are familiar with
the terminology that is being used.
Students can understand how to solve real
world problems.
Students can express themselves to the
teacher and to each other using the correct
math vocabulary.
Students use correct labels with word
problems.
Make sure to use correct vocabulary terms
when speaking with students.
Ask students to provide a label when
describing word problems.
Encourage discussions and explanations and
use probing questions.
How could you describe this problem in your
own words?
What are some non-examples of this word?
What mathematical term could be used to
describe this process.
Students are precise in their descriptions.
They use mathematical definitions in their
reasoning and in discussions.
They state the meaning of symbols
consistently and appropriately.
Teaching students “trick names” for symbols
(i.e. the alligator eats the big number)
Not using proper terminology in the
classroom.
Allowing students to use the word “it” to
describe symbols or other concepts.
Page 8 of 162
CCSSM Practice 7: Look for and make use of structure.
Why is this practice important?

What does this practice look like when students are
doing it?



What can a teacher do to model this practice?


What questions could a teacher ask to encourage
the use of this practice?
What does proficiency look like in this practice?
What actions might the teacher make that inhibit
the students’ use of this practice?




When students can see patterns or
connections, they are more easily able to
solve problems
Students look for connections between
properties.
Students look for patterns in numbers,
operations, attributes of figures, etc.
Students apply a variety of strategies to
solve the same problem.
Ask students to explain or show how they
solved a problem.
Ask students to describe how one repeated
operation relates to another (addition vs.
multiplication).
How could you solve the problem using a
different operation?
What pattern do you notice?
Students look closely to discern a pattern or
structure.
Provide students with pattern before
allowing them to discern it for themselves.
Page 9 of 162
CCSSM Practice 8: Look for and express regularity in repeated reasoning
Why is this practice important?

What does this practice look like when students are
doing it?


What can a teacher do to model this practice?



What questions could a teacher ask to encourage
the use of this practice?
What does proficiency look like in this practice?
What actions might the teacher make that inhibit
the students’ use of this practice?








When students discover connections or
algorithms on their own, they better
understand why they work and are more
likely to remember and be able to apply
them.
Students discover connections between
procedures and concepts
Students discover rules on their own
through repeated exposures of a concept.
Provide real world problems for students to
discover rules and procedures through
repeated exposure.
Design lessons for students to make
connections.
Allow time for students to discover the
concepts behind rules and procedures.
Pose a variety of similar type problems.
How would you describe your method? Why
does it work?
Does this method work all the time?
What do you notice when…?
Students notice repeated calculations.
Students look for general methods and
shortcuts.
Providing students with formulas or
algorithms instead of allowing them to
discover it on their own.
Not allowing students enough time to
discover patterns.
Page 10 of 162
RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE
Using the Rubric:
Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The
task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The
teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient”
describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more.
A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.
PRACTICE
Task:
Make sense of
problems and
persevere in
solving them.
EMERGING
NEEDS IMPROVEMENT


Is strictly procedural.
Does not require
students
to check solutions for
errors.
Teacher:



(teacher does thinking)
Task:
 Is overly scaffolded or
procedurally “obvious”.
 Requires students to
check answers by
plugging in numbers.
Teacher:
Does not allow for wait
time; asks leading
questions to rush
through
task.
Does not encourage
students to individually
process the tasks.
Is focused solely on
answers rather than
processes and
reasoning.




Allots too much or too
little time to complete
task.
Encourages students to
individually complete
tasks, but does not ask
them to evaluate the
processes used.
Explains the reasons
behind procedural steps.
Does not check errors
publicly.
PROFICIENT
(teacher mostly models)
Task:
 Is cognitively
(students take ownership)
Task:
 Allows for multiple entry
points and solution paths.
demanding.
 Requires students to
 Has more than one entry
defend and justify their
point.
solution by comparing
 Requires a balance of
multiple solution paths.
procedural fluency and
conceptual
Teacher:
understanding.
 Differentiates to keep
 Requires students to
advanced students
check solutions for
challenged during work
errors usingone other
time.
solution path.
 Integrates time for explicit
meta-cognition.
Teacher:
 Expects students to make
 Allows ample time for all
sense of the task and the
students to struggle with
proposed solution.
task.


Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions
EXEMPLARY
Expects students to
evaluate processes
implicitly.
Models making sense of
the task (given situation)
and the proposed
solution.
Summer 2011
Page 11 of 162
PRACTICE
Reason
abstractly and
quantitatively.
EMERGING
NEEDS IMPROVEMENT

Task:
Task:


Lacks context.
Does not make use of
multiple representations
or
solution paths.

 Is embedded in a
contrived context.


Teacher:

 Expects students to


Teacher:

(teacher does thinking)
Does not expect
students to interpret
representations.
Expects students to
memorize procedures
withno connection to
meaning.
model and interpret

tasks using a single

representation.
Explains connections
between procedures and
meaning.











Construct viable
arguments and
critique the
reasoning of
others.


Task:

Task:
Is either ambiguously
stated.
Teacher:


Does not ask students to
present arguments or
solutions.
Expects students to
follow a given solution
path without
opportunities to
make conjectures.

Is not at the appropriate
level.
Teacher:



Does not help students
differentiate between
assumptions and logical
conjectures.
Asks students to present
arguments but not to
evaluate them.
Allows students to make
conjectures without
justification.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions
PROFICIENT
EXEMPLARY
(teacher mostly models)
(students take ownership)


Task:
Task:
 Has realistic context.
 Has relevant realistic
context.
 Requires students to

frame solutions in a


Teacher:
context.


 Expects students to
 Has solutions that can be


interpret, model, and
expressed with multiple


connect multiple


representations.
representations.


 Prompts students to


Teacher:
articulate connections
 Expects students to

between algebraic


interpret and model
procedures and contextual
using multiple



meaning.


representations.


 Provides structure for

students to connect


algebraic procedures to


contextual meaning.


 Links mathematical
solution with a


question’s answer.


Task:
Teacher:
 Avoids single steps or
 Helps students
routine algorithms.
differentiate between
assumptions and logical
Teacher:
conjectures.
 Identifies students’
 Prompts students to
evaluate peer arguments.
assumptions.
 Expects students to
 Models evaluation of
formally justify the validity
student arguments.
of their conjectures.
 Asks students to explain
their conjectures.
Summer 2011
Page 12 of 162
PRACTICE
Task:
Model with
mathematics.
EMERGING
NEEDS IMPROVEMENT

Task:
Requires students to
identify variables and to
perform necessary
computations.
Teacher:


(teacher does thinking)
Identifies appropriate
variables and procedures
for students.
Does not discuss

Requires students to
PROFICIENT
(teacher mostly models)
Task:

Requires students to
identify variables and to
compute and interpret
results.
Teacher:


Verifies that students have
identified appropriate
variables and procedures.
Explains the
appropriateness of model.
appropriateness of model.


identify variables, compute
and interpret results, and
report findings using a
mixture of
representations.
Illustrates the relevance of
the mathematics involved.
Requires students to
identify extraneous or
missing information.
Teacher:


EXEMPLARY
(students take ownership)
Task:

Requires students to
identify variables, compute
and interpret results,
report findings, and justify
the reasonableness of their
results and procedures
within context of the task.
Teacher:


Expects students to justify
their choice of variables
and procedures.
Gives students opportunity
to evaluate the
appropriateness of model.
Asks questions to help
students identify
appropriate variables and
procedures.
Facilitates discussions in
evaluating the
appropriateness of model.
Use appropriate
tools strategically.
Task:

Task:
Does not incorporate
additional learning tools.
Teacher:

Does not incorporate
additional learning tools.


Task:
Lends itself to one learning
tool.
Does not involve mental
computations or
estimation.
Teacher:

Demonstrates use of


Task:
Lends itself to multiple
learning tools.
Gives students opportunity
to develop fluency in
mental computations.
Teacher:

appropriate learning tool.


Chooses appropriate
learning tools for student
use.
Models error checking by
estimation.

Requires multiple learning
tools (i.e., graph paper,
calculator, manipulative).
Requires students to
demonstrate fluency in
mental computations.
Teacher:


Allows students to choose
appropriate learning tools.
Creatively finds
appropriate alternatives
where tools are not
available.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions
Summer 2011
Page 13 of 162
PRACTICE
Attend to
precision.
EMERGING
NEEDS IMPROVEMENT

Task:
Task:

(teacher does thinking)
Gives imprecise
instructions.


Has overly detailed or
wordy instructions.

Teacher:


Look for and make
use of structure.
Teacher:

Does not intervene
when students are being
imprecise.
Does not point out
instances when students
fail to address the
question completely or
directly.
Requires students to
automatically apply an
algorithm to a task
without evaluating its
appropriateness.








Teacher:


Teacher:



Inconsistently intervenes
when students are
imprecise.
Identifies incomplete
responses but does not
require student to
formulate further
response.

Task:
Task:





Does not recognize
students for developing
efficient approaches to
the task.
Requires students to
apply the same
algorithm to a task
although there may be
other approaches.












Requires students to
analyze a task before
automatically applying
an algorithm.
Identifies individual
students’ efficient
approaches, but does
not expand
understanding to
the rest of the class.
Demonstrates the same
algorithm to all related
tasks although there
may be other more
effective
approaches.
Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions
PROFICIENT
EXEMPLARY
(teacher mostly models)
(students take ownership)


Task:
Task:
 Has precise instructions.
 Includes assessment
Teacher:
criteria for communication


of ideas.

Consistently
demands


Teacher:
precision
in


communication and in

 Demands and models
precision in
mathematical
solutions.


communication and in

Identifies
incomplete


mathematical solutions.
responses
and
asks



Encourages students to
student
to
revise
their


identify when others are
response.


not addressing the


question completely.





Task:

Task:





Requires students to
analyze a task and
identify more than one
approach
to the problem.












Facilitates all students in
developing reasonable
and
efficient ways to
accurately perform basic
operations.
Continuously questions
students about the
reasonableness of their
intermediate results.
Requires students to
identify the most efficient
solution to the task.

Teacher:



Teacher:














Prompts students to
identify mathematical
structure of the task in
order to identify the most
effective solution path.
Encourages students to
justify their choice of
algorithm or solution path.

Summer 2011
Page 14 of 162
PRACTICE
Look for and
express regularity
in repeated
reasoning.
EMERGING
NEEDS IMPROVEMENT


Is disconnected from
prior and future
concepts.
Has no logical
progression that leads to
pattern recognition.
Teacher:


(teacher does thinking)
Does not show evidence
of understanding the
hierarchy within
concepts.
Presents or examines
task in isolation.


Is overly repetitive or
has gaps that do not
allow for development
of a pattern.
future concepts.
Teacher:


Teacher:

 Hides or does not draw

connections to prior or


EXEMPLARY
(teacher mostly models)
(students take ownership)


Task:
Task:
 Reviews prior knowledge
 Addresses and connects to
prior knowledge in a nonand requires cumulative


routine way.
understanding.



Requires recognition of

Lends
itself
to


pattern or structure to be
developing
a


completed.
pattern
or
structure.



Task:
Task:
PROFICIENT













Teacher:
Connects concept to
prior and future
concepts to help
students develop an
understanding of
procedural shortcuts.
Demonstrates
connections between
tasks.





























Institute for Advanced Study/Park City Mathematics Institute
Secondary School Teachers Program/Visualizing Functions



Encourages students to
connect task to prior
concepts and tasks.
Prompts students to
generate exploratory
questions based on the
current task.
Encourages students to
monitor each other’s
intermediate results.
Summer 2011
Page 15 of 162
Mathematical Practices: A Walk-Through Protocol
*Note: This document should also be used by the teacher for planning and self-evaluation.
Mathematical Practices
Observations
MP.1. Make sense of problems
and persevere in solving them
Students are expected to______________:
 Engage in solving problems.
 Explain the meaning of a problem and restate in it their own words.
 Analyze given information to develop possible strategies for solving the problem.
 Identify and execute appropriate strategies to solve the problem.
 Check their answers using a different method, and continually ask “Does this make sense?”
Teachers are expected to______________:
 Provide time for students to discuss problem solving.
MP.2. Reason abstractly and
quantitatively.
Students are expected to______________:
 Connect quantity to numbers and symbols (decontextualize the problem) and
create a logical representation of the problem at hand.
 Recognize that a number represents a specific quantity (contextualize the problem).
 Contextualize and decontextualize within the process of solving a problem.
Teachers are expected to______________:
 Provide appropriate representations of problems.
MP.3. Construct viable arguments
and critique the reasoning of
others.
Students are expected to____________________________:
 Explain their thinking to others and respond to others’ thinking.
 Participate in mathematical discussions involving questions like “How did you get that?” and
“Why is that true?”
 Construct arguments that utilize prior learning.
 Question and problem pose.
 Practice questioning strategies used to generate information.
 Analyze alternative approaches suggested by others and select better approaches.
 Justify conclusions, communicate them to others, and respond to the arguments of others.
 Compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is.
CCSSM
National Professional Development
Page 16 of 162
Mathematical Practices
Observations
Teachers are expected to______________:
 Provide opportunities for students to listen to or read the conclusions and
arguments of others.
MP.4. Model with mathematics.
MP 5. Use appropriate
tools strategically
Students are expected to______________:
 Apply the mathematics they know to solve problems arising in everyday life,
society, and the workplace.
 Make assumptions and approximations to simplify a complicated situation,
realizing that these may need revision later.
 Experiment with representing problem situations in multiple ways, including numbers,
words (mathematical language), drawing pictures, using objects, acting out, making a
chart or list, creating equations, etc.
 Identify important quantities in a practical situation and map their relationships
using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.
 Evaluate their results in the context of the situation and reflect on whether their results
make sense.
 Analyze mathematical relationships to draw conclusions.
Teachers are expected to______________:
 Provide contexts for students to apply the mathematics learned.
Students are expected to______________:
 Use tools when solving a mathematical problem and to deepen their understanding of
concepts (e.g., pencil and paper, physical models, geometric construction and measurement
devices, graph paper, calculators, computer-based algebra or geometry systems.)
 Consider available tools when solving a mathematical problem and decide when
certain tools might be helpful, recognizing both the insight to be gained and their
limitations.
 Detect possible errors by strategically using estimation and other mathematical knowledge.
Teachers are expected to______________:
CCSSM
National Professional Development
Page 17 of 162
Mathematical Practices
MP.6. Attend to precision.
Observations
Students are expected to______________:
 Use clear and precise language in their discussions with others and in their own reasoning.

Use clear definitions and state the meaning of the symbols they choose, including using the
equal sign consistently and appropriately.

Specify units of measure and label parts of graphs and charts.
Calculate with accuracy and efficiency based on a problem’s expectation.
Teachers are expected to______________:
 Emphasize the importance of precise communication.
Students are expected to______________:
 Describe a pattern or structure.
 Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in
written form.

MP.7. Look for and make use of
structure.

Relate numerical patterns to a rule or graphical representation
 Apply and discuss properties.
Teachers are expected to______________:
 Provide time for applying and discussing properties.
Students are expected to______________:
 Describe repetitive actions in computation
MP.8. Look for and express
regularity in repeated
reasoning.

Look for mathematically sound shortcuts.

Use repeated applications to generalize properties.

Use models to explain calculations and describe how algorithms work.

Use models to examine patterns and generate their own algorithms.
 Check the reasonableness of their results.
Teachers are expected to______________:
CCSSM
National Professional Development
Page 18 of 162
High School Common Core Math II
Curriculum Timeline
Topic
Intro Unit
Similarity
Trigonometric
Ratios
Other Types of
Functions
Comparing
Functions and
Different
Representations
of Quadratic
Functions
Modeling Unit
and Project
Quadratic
Functions:
Solving by
Factoring
Quadratic
Functions:
Completing the
Square and the
Quadratic
Formula
Probability
Geometric
Measurement
Geometric
Modeling Unit
and Project
Standards Covered
G – SRT 1
G – SRT 1a
G – SRT 1b
G – SRT 6
G – SRT 2
G – SRT 3
G – SRT 4
G – SRT 7
G – SRT 5
Grading
Period
1
1
No. of
Days
5
20
G – SRT 8
1
20
A – CED 1
A – CED 4
A – REI 1
N – RN 1
N – RN 2
N – RN 3
F – IF 4
F – IF 5
F – IF 6
F – IF 7
F – IF 7a
F– IF 9
F – IF 4
F – IF 7b
F – IF 7e
F – IF 8
F – IF 8b
F– BF1
A– CED 1
A– CED 2
F– BF 1
F– BF 1a
F – BF 1b
F– BF 3
F – BF 1a
F – BF 1b
F – BF 3
A – SSE 1b
N–Q2
2
15
F – LE 3
N– Q 2
S – ID 6a
S – ID 6b
A – REI 7
2
20
2
10
A – APR 1
A – REI 1
A – REI 4b
F – IF 8a
A – CED 1
A – SSE 1b
A – SSE 3a
3
20
A – REI 1
A – REI 4
A – REI 4a
A – REI 4b
A – SSE 3b
F – IF 8
F – IF 8a
A – CED 1
N – CN 1
N – CN 2
N – CN 7
3
20
S – CP 1
S – CP 2
S – CP 3
G – GMD 1
S – CP 4
S – CP 5
S – CP 6
G – GMD 3
S – CP 7
4
20
4
10
G – MG 1
G – MG 2
G – MG 3
4
15
Page 19 of 162
High School Common Core Math II
1st Nine Weeks
Scope and Sequence
Intro Unit – IO (5 days)
Topic 1 – Similarity (20 days)
Geometry (G – SRT):
1) Similarity, Right Triangles, and Trigonometry:
Understand similarity in terms of similarity transformations.
 G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale
factor.

G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel
line, and leaves a line passing through the center unchanged.

G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale
factor.

G – SRT 2: Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and the
proportionality of all corresponding pairs of sides.

G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for
two triangles to be similar.
Geometry (G – SRT):
2) Similarity, Right Triangles, and Trigonometry:
Prove theorems involving similarity.
 G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem
proved using triangle similarity.

G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Topic 2 – Trigonometric Ratios (20 days)
Geometry (G – SRT):
3) Similarity, Right Triangles, and Trigonometry:
Define trigonometric ratios and solve problems involving .right triangles
 G – SRT 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 7: Explain and use the relationship between the sine and cosine of complementary
angles.

G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
Page 20 of 162
High School Common Core Math II
2nd Nine Weeks
Scope and Sequence
Topic 3 – Other Types of Functions (15 days)
Creating Equations (A – CED):
4) Create equations that describe numbers or relationships
 A – CED 1: Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.
 A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning
as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Reasoning with Equations and Inequalities (A – REI):
5) Understand solving equations as a process of reasoning and explain the reasoning.
 A – REI 1: Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original equation
has a solution. Construct a viable argument to justify a solution method.
The Real Number System (N – RN):
6) Extend the properties of exponents to rational exponents.
 N – RN 1: Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a notation for
radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
 N – RN 2: Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
The Real Number System (N – RN):
7) Use properties of rational and irrational numbers.
 N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum
of a rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
Interpreting Functions (F – IF):
8) Interpret functions that arise in applications in terms of the context.
 F – IF 4*: For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.*
Interpreting Functions (F – IF):
9) Analyze functions using different representations.
 F – IF 7b: Graph square root, cube root, and absolute value functions.

F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.
Page 21 of 162

F – IF 8: Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.

F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions.
For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Building Functions (F – BF):
10) Build a function that models a relationship between two quantities.
 F – BF 1: Write a function that describes a relationship between two quantities.

F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation
from a context.

F – BF 1b: Combine standard function types using arithmetic operations. For example, build
a function that models the temperature of a cooling body by adding a constant function to a
decaying exponential, and relate these functions to the model.
Building Functions (F – BF):
11) Build new functions from existing functions.
 F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x +
k) for specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
Seeing Structure in Expressions (A – SSE):
12) Interpret the structure of expressions.
 A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1 + r)n as the product of P and a factor not
depending on P.
Quantities (NQ):
13) Reason quantitatively and use units to solve problems.
 N – Q 2: Define appropriate quantities for the purpose of descriptive modeling.
Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20
days)
Interpreting Functions (F – IF):
14) Interpret functions that arise in applications in terms of the context.
 F – IF 4*: For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.*

F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would be
an appropriate domain for the function.*
Page 22 of 162

F – IF 6: Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph.
Interpreting Functions (F – IF):
15) Analyze functions using different representations.
 F – IF 7: Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.

F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

F – IF 9: Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a graph of one quadratic function and an algebraic expression for another, say which
has the larger maximum.
Creating Equations (A – CED):
16) Create equations that describe numbers of relationships.
 A – CED 1: Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.

A – CED 2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Building Functions (F – BF):
17) Build a function that models a relationship between two quantities.
 F – BF 1: Write a function that describes a relationship between two quantities.

F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation
from a context.

F – BF 1b: Combine standard function types using arithmetic operations. For example, build
a function that models the temperature of a cooling body by adding a constant function to a
decaying exponential, and relate these functions to the model.
Building Functions (F – BF):
18) Build new functions from existing functions.
 F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x +
k) for specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
Linear and Exponential Models (F – LE):
19) Construct and compare linear and exponential models and solve problems.
 F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial
function.
Quantities (N-Q):
20) Reason quantitatively and use units to solve problems.
 N – Q 2: Define appropriate quantities for the purpose of descriptive modeling.
Page 23 of 162
Interpreting Categorical and Quantitative Data (S – ID):
21) Summarize, represent, and interpret data on two categorical and quantitative variables.
 S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the
context of the data. Use given functions or choose a function suggested by the context.
Emphasize linear and exponential models.

S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals.
Reasoning with Equations and Inequalities (A – REI):
22) Solve systems of equations.
 A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in
two variables algebraically and graphically. For example, find the points of intersection
between the line y = -3x and the circle x2 + y2 = 3.
Modeling Unit and Project –(10 days)
Page 24 of 162
High School Common Core Math II
3rd Nine Weeks
Scope and Sequence
Topic 5–Quadratic Functions – Solving by factoring (20 days)
Arithmetic with Polynomials and Rational Expressions (A – APR):
23) Perform arithmetic operations on polynomials.
 A – APR 1: Understand that polynomials form a system analogous to the integers, namely,
they are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
Reasoning with Equations and Inequalities (A – REI):
24) Understand solving equations as a process of reasoning and explain the reasoning.
 A – REI 1: Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original equation
has a solution. Construct a viable argument to justify a solution method.
Reasoning with Equations and Inequalities (A – REI):
25) Solve equations and inequalities in one variable.
 A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial form
of the equation. Recognize when the quadratic formula gives complex solutions and write
them as a ± bi for real numbers a and b.
Interpreting Functions (F – IF):
26) Analyze functions using different representations.
 F – IF 8a: Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
Creating Equations (A – CED):
27) Create equations that describe numbers of relationships.
 A – CED 1: Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.
Seeing Structure in Expressions (A – SSE):
28) Interpret the structure of expressions.
 A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1 + r)n as the product of P and factor not depending
on P.
Seeing Structure in Expressions (A – SSE):
29) Write expressions in equivalent forms to solve problems.
 A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines.
Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days)
Reasoning with Equations and Inequalities (A – REI):
30) Understand solving equations as a process of reasoning and explain the reasoning.
Page 25 of 162

A – REI 1: Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original equation
has a solution. Construct a viable argument to justify a solution method.
Reasoning with Equations and Inequalities (A – REI):
31) Solve equations and inequalities in one variable.
 A – REI 4: Solve quadratic equations in one variable.

A – REI 4a: Use the method of completing the square to transform any quadratic equation in
x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic
formula from this form.

A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial form
of the equation. Recognize when the quadratic formula gives complex solutions and write
them as a ± bi for real numbers a and b.
Seeing Structure in Expressions (A – SSE):
32) Write expressions in equivalent forms to solve problems.
 A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Interpreting Functions (F – IF):
33) Analyze functions using different representations.
 F – IF 8: Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.

F – IF 8a: Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
Creating Equations (A – CED):
34) Create equations that describe numbers of relationships.
 A – CED 1: Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.
The Complex Number System (N – CN):
35) Perform arithmetic operations with complex numbers.
 N – CN 1: Know there is a complex number i such that i 2  1 , and every complex number
has the form a+bi with a and b real.

N – CN 2: Use the relation i 2  1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
The Complex Number System (N – CN):
36) Use complex numbers in polynomial identities and equations.
 N – CN 7: Solve quadratic equations with real coefficients that have complex solutions.
Page 26 of 162
High School Common Core Math II
4th Nine Weeks
Scope and Sequence
Topic 7 –Probability (20 days)
Conditional Probability and the Rules of Probability (S – CP):
37) Understand independence and conditional probability and use them to interpret data.
 S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
other events (“or,” “and,” “not”).

S – CP 2: Understand that two events A and B are independent if the probability of A and B
occurring together is the product of their probabilities, and use this characterization to
determine if they are independent.

S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability of A given B is
the same as the probability of A, and the conditional probability of B given A is the same as
the probability of B.

S – CP 4: Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to
decide if events are independent and to approximate conditional probabilities. For example,
collect data from a random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly selected student from
you school will favor science given that the student is in the tenth grade. Do the same for
other subjects and compare the results.

S – CP 5: Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations. For example, compare the chance of having
lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Conditional Probability and the Rules of Probability (S – CP):
38) Use the rules of probability to compute probabilities of compound events in a uniform
probability model.
 S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that
also belong to A, and interpret the answer in terms of the model.

S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer
in terms of the model.
Topic 8 – Geometric Measurement (10 days)
Geometric Measurement and Dimension (G – GMD):
39) Explain volume formulas and use them to solve problems.
 G – GMD 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 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Page 27 of 162
Geometric and Modeling Project-(15 days)
*Modeling with Geometry (G – MG):
40) Apply geometric concepts in modeling situations.
 G – MG 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 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 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).*
Page 28 of 162
COLUMBUS CITY SCHOOLS
HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE
TOPIC 1
CONCEPTUAL CATEGORY
TIME RANGE
20 days
Similarity 1, 1a, 1b, 2, 3, 4, Geometry
5
Domain: Geometry: Similarity, Right Triangles, and Trigonometry (G – SRT):
Cluster
1) Understand similarity in terms of similarity transformations.
2) Prove theorems involving similarity.
GRADING
PERIOD
1
Standards
1) Understand similarity in terms of similarity transformations.
 G – SRT 1: Verify experimentally the properties of dilations given by a center and a
scale factor.

G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.

G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.

G – SRT 2: Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.

G – SRT 3: Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
2) Prove theorems involving similarity.
 G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity.

G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 29 of 162
TEACHING TOOLS
Vocabulary: AA, center of dilation, corresponding parts, cross product, dilation, extremes, figure, image,
irregular polygon, means, midsegment, proportion, proportional, ratio, regular polygon, rotational
symmetry, scale factor, similar polygons, tessellation, transformations, transversal segments, similarity
Teacher Notes
Dilations
A dilation is a transformation that produces an image that is the same shape as the original, but is a
different size. A dilation used to create an image larger than the original is called an enlargement. A
dilation used to create an image smaller than the original is called a reduction.
The website for the Topic index for dilations at Regents Prep is sited below. It includes lessons, practice
and teacher support.
http://www.regentsprep.org/Regents/math/geometry/GT3/indexGT3.htm
The website below provides a lesson with a warm-up, vocabulary, and examples with solutions for
dilations.
http://www.chs.riverview.wednet.edu/math/aitken/Integrated-Old/int1-notes/Unit6/Int1_6-6Dilations-notes.pdf
At the website below, teachers can look at a tutorial for dilations. There are four different ones: dilating a
triangle; invariants in dilation; dilations in the coordinate plane; and problem solving with dilations.
http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course
Geometry with Cabri Jr. and the TI-84 Plus
 Module 9 DILATIONS
 Lesson 1 - dilating a triangle
 Lesson 2 - invariants in a dilation
 Lesson 3 - dilations in the coordinate plane
 Lesson 4 - problem solving with dilations
Similarity
Similar polygons are two polygons with congruent corresponding angles and proportional corresponding
sides. If the cross product is equal, then the corresponding sides are proportional. Similarity of polygons
can be proven in three different ways: Angle-Angle Similarity, Side-Side-Side Similarity, and Side-AngleSide Similarity. A-A Similarity is used when two pairs of corresponding angles are congruent. S-S-S
Similarity is used when all three pairs of corresponding sides are proportional. S-A-S Similarity is used
when two pairs of corresponding sides are proportional and their included angles are congruent.
Below the website listed contains lessons, practice and teacher resources on similarity.
http://www.regentsprep.org/Regents/math/geometry/GP11/indexGP11.htm
A tutorial on the Pythagorean Theorem and trigonometry can be found at the website below.
https://activate.illuminateed.com/playlist/resourcesview/rid/50c56098efea65b540000000/id/50c4c151
efea65fd18000003/bc0/user/bc1/playlist/bc0_id/4fff3767efea650023000698
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 30 of 162
The website below has cliff notes on AA triangle similarity.
http://www.cliffsnotes.com/study_guide/Similar-Triangles.topicArticleId-18851,articleId-18812.html
The website below gives examples of SAS, AA, and SSS triangle similarity.
http://www.analyzemath.com/Geometry/similar_triangles.html
This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice,
and hands on activities for similar polygons.
http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_2.html
This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice,
and hands on activities for similar triangles.
http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_3.html
The scale factor is the ratio of lengths of two corresponding sides of similar polygons. The phrase “scale
factor” is used in different ways.
Example1:
If the length of a side of Square A is 4 and the length of a side of Square B is 7, then the scale factor of
Square A to Square B is 4/7.
Example2:
If the length of a side of Square A is 4 and Square A is enlarged by a scale factor of 2, then the length of a
side of the new square is 8.
Scale factor is used to produce dilations, which can be smaller or larger than the original figure.
Real life applications include reading maps, blueprints, and varying recipe sizes.
This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice,
and hands on activities for using proportions.
http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_1.html
This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice,
and hands on activities for proportional parts.
http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_4.html
The TI-84 and Cabri Jr. can be used for special triangles. An on-line tutorial can be found at the website
below.
Module 11 SPECIAL TRIANGLES - Lesson 3 - constructing a right triangle
http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course'
The TI-84 and Cabri Jr. can be used for special triangles. A tutorial can be found at the website below.
Module 14 PROPORTIONS - Lesson 1 - similar triangles
http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course
This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice,
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 31 of 162
and hands on activities for Pythagorean Theorem.
http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_2.html
Below a website is listed for a video tutorial for solving a triangle using SAS.
http://patrickjmt.com/solving-a-triangle-sas-example-1/
Below a website is listed for a video tutorial for another example of solving a triangle for SAS.
http://patrickjmt.com/solving-a-triangle-sas-example-2/
The website below has a video tutorial to find the missing side and angles of triangle using SAS.
http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-1/
Another example of finding the missing side and angles of a triangle using SAS can be found at the
website below.
http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-2/
Misconceptions/Challenges:
 Students do not match up the corresponding sides of figures, and therefore incorrectly set up
proportions between similar polygons, which cause them to get the incorrect side lengths or
transversal segments.
 Students believe that adding a particular value to all sides of a polygon will create a similar
polygon.
 Students mix up the possible values of the scale factors for enlargements and reductions.
 Students do not multiply the scale factor by all sides in the polygon.
 Students think that all polygons of a particular shape (for example all right traingles, or all
rectangles) are similar; they do not recognize that they can have different corresponding angles.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 32 of 162
Instructional Strategies:
SRT 1

Analyzing Congruence Proofs.
http://map.mathshell.org.uk/materials/lessons.php?taskid=452
This lesson focuses on the concepts of congruency and similarity, including identifying
corresponding sides and corresponding angles within and between triangles. Students will identify
and understand the significance of a counter-example, and prove and evaluate proofs in a
geometric context

Key Visualizations, Geometry:
http://ccsstoolbox.agilemind.com/animations/standards_content_visualizations_geometry.html
This website has an animation where students can explore dilations of lines by selecting points along the line
and thinking about point-by-point dilations. Students make a connection between dilations and ratios.

Photocopy Faux Pas
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
The website below provides a lesson on the essential features of dilation. It is Classroom Task: 6.1 found on
pages 4 – 10.
SRT 1a
 Dialations
http://psdsm2ccss.pbworks.com/w/page/56495542/GSRT1%20Verify%20experimentally%20properti
es%20of%20dilations
Create a dilation of a line segment AB through point C with a scale factor of 2:1 to create segment EF. Find
lengths of all segments, EF, AB, BC, CE and CF.

Dilate and Reflect
http://education.ti.com/xchange/US/Math/AlgebraII/16008/Transformations_Dilating_Functions_Teac
her.pdf
Students will use the Nspire Handheld to dilate and reflect different types of functions by grabbing
points. Students will understand the effect of the coefficient on the vertical stretch or shrink of
the function

Properties of Dilations
http://education.ti.com/en/us/activity/detail?id=0C732215F7EC479AB1A6350A64B161B2
Students explore the properties of dilations and the relationships between the original and image figures.

Playing with Dilations
http://www.cpalms.org/RESOURCES/URLresourcebar.aspx?ResourceID=SSGunMzEork=D
Students explore dilations and rotations using Virtual Manipulates

Dilation and Scale Factor
http://www.illustrativemathematics.org/illustrations/602
Give student a copy of the picture so they can draw the points A’, B’ and C’. Provide extra space below the
picture. This task enables the students to verify that a dilation takes a line that does not pass through the
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 33 of 162
center of a line parallel to the original line and the dilation of the line segment is longer or shorter by a scale
factor. Rulers may be useful for duplication of lengths without formal constructions

Properties of Dilations
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Lesson 5-1 page 5. Activity to investigate properties of dilations using geometry software. Properties
include: dilations preserve angle measure, betweenness, collinearity, maps a lines not passing through the
center of the dilation to a parallel line and leaves a line passing through the center unchanged, and a dilation
of a line segment is longer or shorter in the ratio given by the scale factor.

Analogy of Dilation to zoom
http://www.geogebra.org/cms/
Draws an analogy of dilation to zoom-in and zoom-out of a camera, a document camera, an iPad, or using
geometry software programs such as Geogebra.
SRT 1b
 It is beneficial to use real-life data to discuss ratios with students. You can ask students to compare the
number of male students to female students, the number of students in tennis shoes to students not in tennis
shoes, and the number of students with homework to students without homework.

Have students complete the activity “Are You Golden?” (Included in this Curriculum Guide). Divide
students into groups of 3-4. This activity will allow students to discover the golden ratio by finding the
ratios of various body parts.
Take two triangles that are congruent. The sides can be 3, 4, and 5 units long. Set up ratios comparing
1 . Introduce similar triangles. All congruent triangles are
corresponding sides. The ratios all reduce to1 =
1
similar triangles with a scale factor of 1:1. The corresponding sides of similar triangles are proportional and
corresponding angles are congruent. Take two similar triangles. One has sides 6, 8, and 12. The other has
sides 9, 12, and 18. Each ratio of the corresponding sides reduces to 2:3. Next, we can present situations
with similar triangles where the length of one side is missing. We can demonstrate how we can set up ratios
comparing corresponding sides and use properties of proportions to calculate the missing side.

Similarity and Triangles.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid
motions to map the image of triangle ABC to triangle DEF ( This lesson can also be found in SRT 3)

Discuss what a blueprint is and the purpose it serves. Have students do the following activity in small
1
cooperative learning groups. Ask them to make a blueprint of the classroom. Use the scale: inch = 1 foot.
4
Use quarter-inch graph paper for this activity. Have students measure the length and width of the room.
Point out those decisions that will need to be made, such as where doors and windows should be located on
the scale drawing. As an extension, a scale drawing of the building or the cafeteria could be done. Ask
students if the same scale should be used. Ask them to explain why or why not. Discuss options.

The link below contains an explanation about dilations.
http://www.frapanthers.com/teachers/zab/Geometry(H)/GeometryinaNutshell/GeometryNutshell2005/
Text/Dilations.pdf
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 34 of 162

The website below has a practice sheet for dilations.
http://mathematicsburns.cmswiki.wikispaces.net/file/view/DilationsTranslations+activity+worksheet+
for+2-20.pdf

Have students complete the activities “The Gumps” and “The Gumps and Similar Figures” (included in
this Curriculum Guide) to lead students into discovering that mathematically similar figures have congruent
angles and proportional The Gumps sides. Divide students into groups of 3-5. Each group should create one
set of figures based on the coordinates given in the chart. Graph paper is required and some figures may
require more than one sheet. The sample figures drawn in this Curriculum Guide use a scale factor of 2 in
order for each figure to fit on one sheet of paper. Transparencies can be made of the figures to overlay them
in order to show that the angles of Giggles, Higgles, and Ziggles are congruent.

Are They Similar?
http://www.illustrativemathematics.org/illustrations/603
The activity includes a picture of two triangles that appear to be similar but to prove similarity
they need further information. Ask students to provide a sequence of similarity transformations
that map one triangle to the other one. Remind students that all parts of one triangle get mapped
to the corresponding parts of the other one. An additional task includes asking the students to
prove or disprove that the triangles are similar in each problem using properties of parallel lines
and the definition of similarity.

Transformations and Similarity
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Similarity practice. Lesson 5-3. Students find that two plane figures are similar if and only if one
can be obtained from the other by transformations.

Geometry Problems: Circles and Triangles,
http://map.mathshell.org/materials/lessons.php?taskid=222
Students solve problems by determining the lengths of the sides in right triangles. They also determine the
measurements of shapes by decomposing complex shapes into simpler ones.

Scale (or Grid) Drawings and Dilations.
http://www.regentsprep.org/Regents/math/geometry/GT3/DActiv.htm
Students work with scale (or grid) drawing to reinforce the concept of scalar factor.


Angles and Similarity
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13153
Students use technology (TI-Nspire or Nspire CAS) to experiment with the measures of the angles of similar
triangles to determine conditions necessary for two triangles to be similar.

Corresponding Parts of Similar Triangles
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13150
Students use technology(TI-Nspire or Nspire CAS) to change the scale factor (r) between similar triangles,
identify the corresponding parts, and establish relationships between them.

Nested Similar Triangles
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13152
Students use technology (TI-Nspire or Nspire CAS) to discover the conditions that make triangles similar by
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 35 of 162
moving the sides opposite the common angles.

Demonstrate to students the properties of similarity. Draw a triangle and ask students “How would you
draw a triangle similar to the triangle shown?” Include in your discussion that angle measures are the same
and sides are proportional. Have students draw two triangles one that is similar to and larger than the
original and one that is similar to and smaller than the original. Follow up this introduction to similarity
with the “Draw Similar Triangles” activity (included in this Curriculum Guide). Students will need a
protractor, straightedge, calculator, and a copy of the worksheet. Students can do this activity in partners or
individually.

Quilts are a beautiful, practical, and historically significant use of geometric shapes. Students will work
with triangles in historic quilt patterns by creating triangles similar to those in a quilt block and then creating
their own pattern with the new triangles as described in the activity “Similar Quilt Blocks” (included in this
Curriculum Guide). Students may work individually or in groups. Students will need a copy of the “Similar
Quilt Blocks” sheet, the “Quilt Calculations” sheet (included in this Curriculum Guide), the “Quilt Design
#1” sheet (included in this Curriculum Guide), the “Quilt Design #2” sheet (included in this Curriculum
Guide), a ruler, a protractor, and materials to make their quilt design (e.g., construction paper, scissors, etc.).
After students have completed this activity, have students share their creations and any challenges they may
have had in creating their new pattern. Students who express an interest in this art form may find additional
information by searching the web using the keyword “quilt”.

Scale Factor Area Perimeter
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13154
Students use technology (TI-Nspire or Nspire CAS) to explore the relationship of perimeter and area in
similar triangles when the scale factor is changed.

Transformations with Lists,
http://education.ti.com/calculators/downloads/US/Activities/Detail?id=10278
Students use list operations to perform reflections, rotations, translations, and dilations on a figure and graph
the resulting image using a scatter plot..

Dilations.
http://www.frapanthers.com/teachers/zab/Geometry(H)/ClassNotes/14.6Dilations.pdf
The website below has practice for dilations.
SRT 2
 Triangle Similarity.
https://ccgps.org/G-SRT_9DRF.html
This website offers internet resources for triangle similarity.

Investigating Triangles with Two Pairs of Congruent Angles (AA similarity): Have students complete the
activity “Investigating Triangles with Two Pairs of Congruent Angles” (included in this Curriculum
Guide). Students should discover the AA Similarity Theorem from this activity. Students will need
protractors and straightedges to complete this activity.

Draw a triangle on the chalkboard. Label the vertices of the triangle A, B, and C. Double the length of AB
from point A. Label the resulting endpoint B'. Double the length of AC from point A. Label the resulting
endpoint C'. Connect B' and C'. Compare ABC and AB'C'. Discuss with students whether or not the
triangles are similar. (They are similar because of SAS for ~ ’s.)
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 36 of 162

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are
similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Have students complete the activity “Similar Triangles Application” (included in this Curriculum Guide)
to use their skills with similar triangles in a real life situation. Meter sticks and mirrors are required
materials. Students should stand several feet away from an object, placing the mirror on the ground between
themselves and the object. The student should place themselves or the mirror in such a way that s/he can
spot the top of the object in the mirror. A partner should take three measurements: the distance the student is
standing from the mirror, the distance from the mirror to the base of the object,, and the distance from the
students line of sight to the ground. Using proportions and similar triangles, the students should be able to
indirectly calculate the height of the object.

Have students do the activity “Find the Scale Factor” (included in this Curriculum Guide) for more practice
in using scale factor to solve similarity problems. Before doing this activity, discuss scale factors with the
students. For example, discuss with the students how the scale factor is 5:1 not 4:1 in the figure below.
24
6

Falling Down a Rabbit Hole Can Lead to a King Sized Experience - Exploring Similar Figures Using
Proportions,”
http://alex.state.al.us/lesson_view.php?id=30067
Students explore similarity. They simplify ratios, solve proportions using cross products, and use properties
of proportions to solve real-world problems.

Similarity Transformation
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf:
Students find that two plane figures are similar if and only if one can be obtained from the other by
transformations (reflections, translations, rotations, and/or dilations Lesson 5-3, page 13 (This lesson is also
found in SRT 1b)

Triangle Dilations
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students examine relationships of proportions in triangles that are known to be similar to each other based
on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 5.)

Similar Triangles and Other Figures
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students compare the definitions of similarity based on dilations and relationships between corresponding
sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.)
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 37 of 162
SRT3
 The website below contains lessons for SRT3.
https://ccgps.org/G-SRT_AVKU.html

Similarity and Triangles.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid
motions to map the image of triangle ABC to triangle DEF. (This lesson was also provided in SRT 1b)

Practice with Similarity Proofs,
http://www.regentsprep.org/Regents/math/geometry/GP11/PracSimPfs.htm
Eight formative assessment questions are provided.

Similar Triangles and Other Figures
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students compare the definitions of similarity based on dilations and relationships between corresponding
sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.)
SRT4
 Pythagorean Theorem
https://ccgps.org/G-SRT_G6QQ.html
A power point presentation on the Pythagorean Theorem.

A Proportionality Theorem.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides.
The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in
SRT 5.)

Proving the Pythagorean Theorem
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Students will use their knowledge about similar triangles to prove the Pythagorean Theorem.

Applying Angle Theorems
http://map.mathshell.org/materials/lessons.php?taskid=214
Students use geometric mean properties to solve problems using the measures of interior and exterior angles
of polygons

Have students complete the activity “Let’s Prove the Pythagorean Theorem” (included in this Curriculum
Guide) to construct a proof of the Pythagorean Theorem.

Have students complete the activity “Proving the Pythagorean Theorem, Again!”(included in this
Curriculum Guide) to reinforce the proof of the Pythagorean Theorem.

Proofs of the Pythagorean Theorem
http://map.mathshell.org/materials/lessons.php?taskid=419&subpage=concept Below link: Students
interpret diagrams, link visual and algebraic representations, and produce a mathematical argument
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 38 of 162

Cut by a Transversal
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students examine proportional relationships of segments when two transversals intersect sets of parallel
lines. Classroom Task: 6.4 pages 30-37

Measured Reasoning
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students apply theorems about lines, angles, and proportional relationships when parallel lines are crossed
by multiple transversals. Classroom Task6.5 pages 38-45. (This lesson can also been found at SRT 5.)

Pythagoras by Proportions
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in
right triangles. Classroom Task 6.6 pages 36-52(This lesson can also been found at SRT 5.)

Finding the Value of a Relationship
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages
67-74 (This lesson can also been found at SRT 5.)
SRT5
 Proving the Pythagorean Theorem
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Students will use their knowledge about similar triangles to prove the Pythagorean Theorem. (Lesson 5-7,
page 27)

Measured Reasoning,
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf Students apply theorems about lines, angles, and proportional relationships when parallel lines are
crossed by multiple transversals. Classroom Task 6.5 pages 38-45 (This lesson can also been found at
SRT 4.)

Finding the Value of a Relationship
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages
67-74 (This lesson can also been found at SRT 4.)

Pythagoras by Proportions
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf
Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 39 of 162
right triangles. Classroom Task 6.6 pages 36-52 (This lesson can also been found at SRT 4.)

How Tall is the School’s Flagpole
https://ccgps.org/G-SRT_AVKU.html
Students will apply math concepts concerning similar triangles and trigonometric functions to real life
situations. The students will find measurements of objects when they are unable to use conventional
measurement. (This lesson can also be found at SRT 3)

A Proportionality Theorem.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides.
The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in
SRT 4)

Solving Problems Using Similarity.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf
Students use proportionality of corresponding sides to find side lengths of two similar polygons. Lesson 5-5

Solving Geometry Problems: Floodlights
http://map.mathshell.org/materials/lessons.php?taskid=429&subpage=problem
Students make models, draw diagrams, and identify similar triangles to solve problems.

.https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/MathII_Unit2_%20Student_E
dition_revised_8-10-09.pdf
This website contains a set of lessons on right triangle trigonometry. These lessons include discovering
special right triangles, discovering trigonometric ratio relationships, and determining side or angle measures
using trigonometry

How Far Can You Go in a New York Minute?
http://illuminations.nctm.org/LessonDetail.aspx?id=L848
Students use proportions and similar figures to adjust the size of the New York City Subway Map so that it
is drawn to scale.

http://education.ti.com/en/us/activity/detail?id=A760474813204FBB944031327521B742&ref=/en/us/ac
tivity/search/subject?d=6B854F0B5CB6499F8207E81D1F3A25E6&s=B843CE852FC5447C8DD8
8F6D1020EC61&sa=71A40A9FD9E84937B8C6A8A4B4195B58&t=3CC394B76E4347CF8C
EFCADAACAE9754
Students will explore the ratio of perimeter, area, surface area, and volume of similar figures in twodimensional figures using graphing technology.

Triangle Dilations
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091
3.pdf Students examine relationships of proportions in triangles that are known to be similar to each other
based on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 2.)
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 40 of 162
Reteach:
 Construct  ABC with sides 5, 8, and 10 units. Tell students that the scale factor of  ABC to  DEF is
1
2 . Ask the students to construct  DEF.

Construct ABC with sides 5, 8, and 10 units. Tell students that the scale factor of  ABC to  RST is 2.
Ask the students to construct  RST.

Construct two equilateral triangles of different sizes on the chalkboard. Ask students to determine if they
are congruent or similar. Ask the students to justify their answers. (The equilateral triangles that were
drawn are not congruent because the sides do not have the same length. They are similar because the
angles all have a measure of 60° and the ratios of the lengths of the corresponding sides are the same.)
Extensions:

Use coordinate geometry and graph paper to draw the dilation of a figure.

Use construction tools to construct the dilation of a figure.

Take a map of Ohio or the United States. Make a transparency of the map. Place it over a coordinate
plane. Write the coordinates of many of the border points. Have groups multiply each coordinate by a
1
scale factor. Have some groups use a scale factor of 3. Have others use a scale factor of 3 . Tape pages
of graph paper together. Have students graph the new image. Discussion: Are the maps proportional?
Textbook References:
Textbook: Geometry, Glencoe (2005): pp. 282-287, 288
Supplemental: Geometry, Glencoe (2005):
Chapter 6 Resource Masters
Study Guide and Intervention, pp. 295-296
Skills Practice, p. 297
Practice, p. 298
Reading to Learn Mathematics, pp. vii-viii, 299
Enrichment, p. 300
Textbook: Geometry, Glencoe (2005): pp. 289-297
Supplemental: Geometry, Glencoe (2005):
Chapter 6 Resource Masters
Study Guide and Intervention, pp. 301-302
Skills Practice, p. 303
Practice, p. 304
Reading to Learn Mathematics, pp. vii-viii, 305
Textbook: Geometry, Glencoe (2005): pp. 298-306, 307-315, 316-323
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 41 of 162
Supplemental: Geometry, Glencoe (2005):
Chapter 6 Resource Masters
Learning to Read Mathematics, pp. ix-x
Study Guide and Intervention, pp. 307-308, 313-314, 319-320
Skills Practice, pp. 309, 315, 321
Practice, pp. 310, 316, 322
Reading to Learn Mathematics, pp. 311, 317, 323
Enrichment, pp. 312, 318, 324
Textbook: Geometry, Glencoe (2005): pp. 490 – 493
Textbook: Algebra 1, Algebra 1 (2005): pp. 197 – 203
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 42 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
Are You Golden?
Materials: meter stick or tape measure, calculator
For each group members, measure the length from the shoulder to the tip of the fingers and the length
from the elbow to the tip of the fingers. Record the data below.
Column A
Column B
TABLE 1
Column C
Column D
Group Members’
Names
Length from
shoulder to tip of
fingers (cm)
Length from
Elbow to tip of
fingers (cm)
Find the Ratio of
Column B to
Column C
Column E
Decimal form of
Column D
Round to 2
decimal places.
1. Examine Column E. What do you notice about all of the decimals?
2. Find the average of all the decimals in Column E. Round to two decimal places.
Now, measure each group member’s height and the height of the navel from the ground (make
sure to take off your shoes). Record the data below.
Column A
Column B
Group Members’
Names
Height (cm)
TABLE 2
Column C
Column D
Height of Navel
from the Ground
(cm)
Find the Ratio of
Column B to
Column C
Column E
Decimal form of
Column D
Round to 2
decimals places.
3. Examine Column E. What do you notice about all of the decimals?
4. Find the average of all the decimals in Column E. Round to two decimal places.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 43 of 162
5. How do the decimal averages in #2 and #4 compare?
The ratios that you found are very close to what is known as the
1 5
“Golden Ratio”, which is
. The decimal approximation
2
of the “Golden Ratio” is 1.618033989… Set up proportions to
answer the following questions based on the “Golden Ratio”.
6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this
person’s elbow to the tip of the fingers?
7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person?
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 44 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
Are You Golden?
Answer Key
Materials: meter stick or tape measure, calculator
For each group members, measure the length from the shoulder to the tip of the fingers and the length
from the elbow to the tip of the fingers. Record the data below.
Column A
Column B
TABLE 1
Column C
Column D
Group Members’
Names
Length from
shoulder to tip of
fingers (cm)
Length from
elbow to tip of
fingers (cm)
Find the ratio of
Column B to
Column C
Column E
Decimal form of
Column D
Round to 2
decimal places.
1. Examine Column E. What do you notice about all of the decimals?
Answers Will Vary.
2. Find the average of all the decimals in Column E. Round to two decimal places.
Answers Will Vary.
Now, measure each group member’s height and the height of the navel from the ground (make
sure to take off your shoes). Record the data below.
TABLE 2
Column A
Column B
Column C
Column D
Column E
Decimal form of
Height of navel
Find the ratio of
Group Members’
Column D
Height (cm)
from the ground
Column B to
Names
Round to 2
(cm)
Column C
decimal places.
3. Examine Column E. What do you notice about all of the decimals?
Answers Will Vary.
4. Find the average of all the decimals in Column E. Round to two decimal places.
Answers Will Vary.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 45 of 162
5. How do the decimal averages in #2 and #4 compare?
Answers Will Vary
Students could conclude that the decimal average is very close to 1.6.
The ratios that you found are very close to what is known as the
1 5
“Golden Ratio”, which is
. The decimal approximation
2
of the “Golden Ratio” is 1.618033989… Set up proportions to
answer the following questions based on the “Golden Ratio”.
6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this
person’s elbow to the tip of the fingers?
68
 1.618
x
x  42.03 cm
7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person?
x
= 1.618
105
x  169.89 cm
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 46 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
The Gumps
There are imposters lurking among the family of Gumps. Using the following criteria, you will
create a set of characters. They will all look somewhat alike but only some of them are considered to
be mathematically similar.
Each group should create a set of characters in order to answer the questions that follow. Every
graph within the group should be drawn using the same scale in order to see the changes between the
Gumps. More than one piece of graph paper may be needed for a particular character.
Plot each point on graph paper. For the points in SET 1 and SET 3, connect them in order and
connect the last point to the first point. For SET 2, connect the points in order but do not connect the
last point to the first point. For SET 4, make a dot at each point.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 47 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
The Gumps
Giggles
(x,y)
SET 1
(4,0)
(4,6)
(2,4)
(0,4)
(4,8)
(2,10)
(2,14)
(4,16)
(5,18)
(6,16)
(8,16)
(9,18)
(10,16)
(12,14)
(12,10)
(10,8)
(14,4)
(12,4)
(10,6)
(10,0)
(8,0)
(8,4)
(6,4)
(6,0)
SET 2
(4,11)
(6,10)
(8,10)
(10,11)
SET 3
(6,11)
(6,12)
(8,12)
(8,11)
SET 4
(5,14)
(9,14)
Higgles
(2x,2y)
SET 1
Wiggles
(3x,y)
SET 1
Ziggles
(3x,3y)
SET 1
Miggles
(x,3y)
SET 1
SET 2
SET 2
SET 2
SET 2
SET 3
SET 3
SET 3
SET 3
SET 4
SET 4
SET 4
SET 4
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 48 of 162
The Gumps
Answer Key
Giggles
(x,y)
SET 1
(4,0)
(4,6)
(2,4)
(0,4)
(4,8)
(2,10)
(2,14)
(4,16)
(5,18)
(6,16)
(8,16)
(9,18)
(10,16)
(12,14)
(12,10)
(10,8)
(14,4)
(12,4)
(10,6)
(10,0)
(8,0)
(8,4)
(6,4)
(6,0)
SET 2
(4,11)
(6,10)
(8,10)
(10,11)
SET 3
(6,11)
(6,12)
(8,12)
(8,11)
SET 4
(5,14)
(9,14)
Higgles
(2x,2y)
SET 1
(8,0)
(8,12)
(4,8)
(0,8)
(8,16)
(4,20)
(4,28)
(8,32)
(10,36)
(12,32)
(16,32)
(18,36)
(20,32)
(24,28)
(24,20)
(20,16)
(28,8)
(24,8)
(20,12)
(20,0)
(16,0)
(16,8)
(12,8)
(12,0)
SET 2
(8,22)
(12,20)
(16,20)
(20,22)
SET 3
(12,22)
(12,24)
(16,24)
(16,22)
SET 4
(10,28)
(18,28)
Wiggles
(3x,y)
SET 1
(12,0)
(12,6)
(6,4)
(0,4)
(12,8)
(6,10)
(6,14)
(12,16)
(15,18)
(18,16)
(24,16)
(27,18)
(30,16)
(36,14)
(36,10)
(30,8)
(42,4)
(36,4)
(30,6)
(30,0)
(24,0)
(24,4)
(18,4)
(18,0)
SET 2
(12,11)
(18,10)
(24,10)
(30,11)
SET 3
(18,11)
(18,12)
(24,12)
(24,11)
SET 4
(15,14)
(27,14)
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Ziggles
(3x,3y)
SET 1
(12,0)
(12,18)
(6,12)
(0,12)
(12,24)
(6,30)
(6,42)
(12,48)
(15,54)
(18,48)
(24,48)
(27,54)
(30,48)
(36,42)
(36,30)
(30,24)
(42,12)
(36,12)
(30,18)
(30,0)
(24,0)
(24,12)
(18,12)
(18,0)
SET 2
(12,33)
(18,30)
(24,30)
(30,33)
SET 3
(18,33)
(18,36)
(24,36)
(24,33)
SET 4
(15,42)
(27,42)
Miggles
(x,3y)
SET 1
(4,0)
(4,18)
(2,12)
(0,12)
(4,24)
(2,30)
(2,42)
(4,48)
(5,54)
(6,48)
(8,48)
(9,54)
(10,48)
(12,42)
(12,30)
(10,24)
(14,12)
(12,12)
(10,18)
(10,0)
(8,0)
(8,12)
(6,12)
(6,0)
SET 2
(4,33)
(6,30)
(8,30)
(10,33)
SET 3
(6,33)
(6,36)
(8,36)
(8,33)
SET 4
(5,42)
(9,42)
Columbus City Schools
6/28/13
Page 49 of 162
Giggles
Higgles
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 50 of 162
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 51 of 162
Wiggles
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 52 of 162
Ziggles
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 53 of 162
Miggles
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 54 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
The Gumps and Similar Figures
1. Use a protractor to measure the following angles of the Gumps’ bodies.
Giggles
Higgles
Wiggles
Ziggles
Miggles
Top of Ear
Under Arm
Neck
Smile
Do you notice anything about the above measurements? If so, explain.
Count the length of the following sides of the Gumps’ bodies.
Giggles
Higgles
Wiggles
Ziggles
Miggles
Width of Head
Length of Leg
Width of Hand
Width of Waist
Total Height
Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you
notice.
Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to
be mathematically similar.
What other Gump(s) fit this description. Why?
Complete the following table.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 55 of 162
Nose
Width
Nose
Length
Width
Length
Nose
Perimeter
Nose
Area
Giggles
(Gump 1)
Higgles
(Gump 2)
Ziggles
(Gump 3)
Prediction
for
Gump 4
Prediction
for
Gump 5
.
.
.
Prediction
for
Gump 10
Prediction
for
Gump 20
Prediction
for
Gump 100
Wiggles
Miggles
Make ratios using the nose perimeter for the following figures:
Gump 2:Gump 1
Gump 3:Gump 1
Gump 4:Gump 1
Gump 5:Gump 1
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 56 of 162
Make a comparison between the scale factor of objects and the ratio of their perimeters.
Make ratios using the nose area for the following figures:
Gump 2:Gump 1
Gump 3:Gump 1
Gump 4:Gump 1
Gump 5:Gump 1
Make a comparison between the scale factor of objects and the ratio of their areas.
Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the relationship
between scale factor of objects and the ratio of their perimeters and areas.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 57 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
The Gumps and Similar Figures
Answer Key
1. Use a protractor to measure the following angles of the Gumps’ bodies.
Giggles
Higgles
Wiggles
Ziggles
o
o
o
Top of Ear
53
53
112
53o
Under Arm
45o
45o
72o
45o
o
o
o
Neck
90
90
37
90o
Smile
153o
153o
171o
153o
Miggles
19o
18o
143o
124o
Do you notice anything about the above measurements? If so, explain.
Giggles, Higgles and Ziggles have the same angle measurements. They are the same shape
just different sizes which preserves their angle measurements. The other two figures are stretched
because only one of their dimensions was changed.
Count the length of the following sides of the Gumps’ bodies.
(Remember to count by 2 on the sample drawings since the scale is 2!)
Giggles
Higgles
Wiggles
Ziggles
Width of Head
10
20
30
30
Length of Leg
4
8
4
12
Width of Hand
2
4
6
6
Width of Waist
6
12
18
18
Total Height
18
36
18
54
Miggles
10
12
2
6
54
Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you
notice.
All of Higgles’ measurements are two times that of Giggles’. All of Ziggles’ measurements are three
times that of Giggles’. Wiggles’ widths only are three times larger than Giggles’ widths because
only the x-values were multiplied by 3. Miggles’ lengths only are three times larger than Giggles’
lengths because only the y-values were multiplied by 3.
Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to
be mathematically similar.
Two figures are mathematically similar if their angle measures are the same and all of
their dimensions are proportional.
What other Gump(s) fit this description. Why?
Ziggles is also mathematically similar to Giggles and Higgles because they have the same
angle measurements and their sides are all proportional.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 58 of 162
Complete the following table.
Nose
Width
Nose
Length
Width
Length
Nose
Perimeter
Nose
Area
Giggles
(Gump 1)
1 cm
2 cm
1
2
6 cm
2 cm2
Higgles
(Gump 2)
2 cm
4 cm
2 1
=
4 2
12 cm
8 cm2
Ziggles
(Gump 3)
3 cm
6 cm
3 1
=
6 2
18 cm
18 cm2
4 cm
8 cm
4 1
=
8 2
24 cm
32 cm2
5 cm
10 cm
5 1
=
10 2
30 cm
50 cm2
10 cm
20 cm
10 1
=
20 2
60 cm
200 cm2
20 cm
40 cm
20 1
=
40 2
120 cm
800 cm2
100 cm
200 cm
100 1
=
200 2
600 cm
20,000 cm2
1 cm
6 cm
1 1
=
6 2
14 cm
6 cm2
2 cm
3 cm
2 1
=
3 2
10 cm
6 cm2
Prediction
for
Gump 4
Prediction
for
Gump 5
.
.
.
Prediction
for
Gump 10
Prediction
for
Gump 20
Prediction
for
Gump 100
Wiggles
Miggles
Make ratios using the nose perimeter for the following figures:
Gump 2:Gump 1
Gump 3:Gump 1
12 2

6 1
Gump 4:Gump 1
24 4

6 1
18 3

6 1
Gump 5:Gump 1
30 5

6 1
Make a comparison between the scale factor of objects and the ratio of their perimeters.
The ratio of the perimeters of two objects is the same as the scale factor.
CCSSM II
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Page 59 of 162
Make ratios using the nose area for the following figures:
Gump 2:Gump 1
Gump 3:Gump 1
18 9

2 1
8 4

2 1
Gump 4:Gump 1
Gump 5:Gump 1
32 16

2
1
50 25

2
1
Make a comparison between the scale factor of objects and the ratio of their areas.
The ratios of the areas of two object is equal to the square of the scale factor.
12.
Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the
relationship between scale factor of objects and the ratio of their perimeters and areas.
Perimeter of Gump 10 10

Perimeter of Gump 1
1
Area of Gump 10 102
 2
Area of Gump 1
1
Perimeter of Gump 20 20

Perimeter of Gump 1
1
Area of Gump 20 202
 2
Area of Gump 1
1
Perimeter of Gump 100 100

Perimeter of Gump 1
1
Area of Gump 100 1002
 2
Area of Gump 1
1
x 10

6 1
x 100

2
1
x 20

6 1
x 400

2
1
x 100

6
1
x 10,000

2
1
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Perimeter of Gump 10 = 60 cm
Area of Gump 10 = 200 cm2
Perimeter of Gump 20 = 120 cm
Area of Gump 20 = 800 cm2
Perimeter of Gump 100 = 600 cm
Area of Gump 100 = 20,000 cm2
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G –SRT 1b
Name ______________________________ Date ____________ Period ________
Draw Similar Triangles
Instructions: In each problem, draw a triangle similar to the one shown. Remember, corresponding
angles of similar triangles have the same measure. Sides of similar triangles are proportional. Show
all calculations that verify the triangles are similar.
1.
2.
3.
4.
5.
_________________________________________________________________________
CCSSM II
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6.
7.
8.
9.
10.
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Page 62 of 162
G –SRT 1b
Name ______________________________ Date ____________ Period ________
Similar Quilt Blocks
Quilts are a beautiful, practical, and historically significant use of geometric shapes.
Create a quilt block using triangles that are similar to the triangles in the quilt block you selected.
Select one of the quilt blocks shown on the “Quilt Design” pages. Your quilt block may be a replica
of the given quilt block or it may be of your own design. Verify that your triangles are similar and
show calculations on the “Quilt Calculations” page. Draw your quilt design in the space below.
CCSSM II
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G –SRT 1b
Name ______________________________ Date ____________ Period ________
Quilt Calculations
Measure the sides and angles of each triangle in the quilt block. Record these values in the “Original
Triangle” section of the chart. Draw a triangle similar to the one you just measured. Measure the
sides and angles and record these values in the “New Triangle” section of the chart. Verify that the
sides of the similar triangles are proportional and place those calculations in the “Calculations” area.
Original Triangle
angle A
angle B
angle C
side a
side b
side c
New Triangle
angle A
angle B
angle C
side a
side b
side c
Calculations
Original Triangle
angle A
angle B
angle C
side a
side b
side c
New Triangle
angle A
angle B
angle C
side a
side b
side c
Calculations
Original Triangle
angle A
angle B
angle C
side a
side b
side c
New Triangle
angle A
angle B
angle C
side a
side b
side c
Calculations
Original Triangle
angle A
angle B
angle C
side a
side b
side c
New Triangle
angle A
angle B
angle C
side a
side b
side c
Calculations
CCSSM II
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Page 64 of 162
Quilt Design #1
Hopscotch Grandma’s
is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State University.
CCSSM II
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Columbus City Schools
6/28/13
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Quilt Design #2
Laced Star is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State
University.
G –SRT 2
CCSSM II
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Quarter 1
Columbus City Schools
6/28/13
Page 66 of 162
Name ______________________________ Date ____________ Period ________
Investigating Triangles with Two Pairs of Congruent
Angles
Given: a triangle with one angle measure of 40o and another angle measure of 50o:
1.
Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B,
and the third angle C.
2. Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a centimeter.
AB=
BC=
AC=
3.
Draw a second triangle that has the same angle measurements but is not congruent to triangle
ABC. Label this triangle A'B'C'.
4.
Use a ruler to find the length of each side of triangle A’B’C’ to the nearest tenth of a
centimeter.
A'B'=
B'C'=
A'C'=
5.
How do the sides of triangle A'B'C' compare to the sides of triangle ABC?
6.
How does the measurement of angle C compare to the measurement of angle C'?
CCSSM II
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Page 67 of 162
7.
What conclusion can be drawn about triangle ABC compared to triangle A'B'C'?
Given: a triangle with one angle measure of 80o and another angle measure of 60o:
8.
Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N,
and the third angle O.
9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a
centimeter.
MN=
NO=
MO=
10. Draw a second triangle that has the same angle measurements but is not congruent to
triangle MNO. Label this triangle M'N'O'.
11. Use a ruler find the length of each side of triangle M'N'O' to the nearest tenth of a centimeter.
M'N'=
N'O'=
12.
M'O'=
How do the sides of triangle M'N'O' compare to the sides of triangle MNO?
13.
How does the measurement of angle O compare to the measurement of angle O'?
14.
What conclusion can be drawn about triangle MNO compared to triangle M'N'O'?
15.
What can you conclude about two triangles given two pair of congruent angles?
CCSSM II
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Quarter 1
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6/28/13
Page 68 of 162
Name ___________________________________ Date __________________ Period ________
Investigating Triangles with Two Pairs of Congruent
Angles
Answer Key
Given a triangle with one angle measure of 40o and another angle measure of 50o.
1.
Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B,
and the third angle C.
Answers may vary.
2.
Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a
centimeter.
AB= Answers may vary.
BC= Answers may vary.
AC= Answers may vary.
3.
Draw a second triangle that has the same angle measurements but is not congruent to
ABC . Label this triangle A'B'C'.
Answers may vary.
4.
Use a ruler to find the length of each side of triangle A'B'C' to the nearest tenth of a
centimeter.
A'B'= Answers may vary.
B'C'= Answers may vary.
A'C'= Answers may vary.
5.
How do the sides of triangle A'B'C' compare to the sides of triangle ABC?
They are proportional.
6.
How does the measurement of angle C compare to the measurement of angle C'?
They are congruent.
7.
What conclusion can be drawn about triangle ABC compared to triangle A'B'C'?
They are similar
Given a triangle with one angle measure of 80o and another angle measure of 60o.
8.
Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N,
and the third angle O.
Answers may vary.
9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a
centimeter.
MN= Answers may vary.
NO= Answers may vary.
MO= Answers may vary.
10. Draw a second triangle that has the same angle measurements but is not congruent to triangle
MNO. Label this triangle M'N'O'.
Answers may vary.
11.
Use a ruler to find the length of each side of triangle M'N'O' to the nearest tenth of a
centimeter.
CCSSM II
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Columbus City Schools
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Page 69 of 162
M'N'= Answers may vary.
N'O'= Answers may vary.
M'O'= Answers may vary.
12.
How do the sides of triangle M'N'O' compare to the sides of triangle MNO?
They are proportional.
13.
How does the measurement of angle O compare to the measurement of angle O'?
They are congruent.
14.
What conclusion can be drawn about triangle MNO compared to triangle M'N'O'?
They are similar.
15.
What can you conclude about two triangles given two pair of congruent angles?
They are similar.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
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6/28/13
Page 70 of 162
G –SRT 2
Name ______________________________ Date ____________ Period ________
Similar Triangles Application
Use a mirror, a meter stick, and similar triangles to calculate the height of three objects in the room.
Think about what information will be needed and how to accurately collect it. Describe the object,
its location, and the measurements taken.
Description of Object
Distance From Student
To Mirror
Distance From The
Mirror To The Base Of
The Object
Distance From Line Of
Sight To Ground
Draw a sketch of each situation and explain why this scenario involves similar s.
Label your picture with your measurements and use proportions or scale factor to calculate the height
of each object. Record your calculated heights below.
Now, measure the actual height of each object. Record the actual (measured) heights below.
Describe how well the calculated height matches the actual height. If there is a significant
discrepancy, explain where any error may have occurred and if it can be corrected.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 71 of 162
G –SRT 2
Name ______________________________ Date ____________ Period ________
Find the Scale Factor
For each exercise, find the scale factor of figure A to figure B and solve for x.
1.
24
A
A
B
6
3
x
2.
5
6
A
x
x+4
B
3.
A
CCSSM II
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Page 72 of 162
20
B
15
x
4
4.
A
8
10
B
x
x+1
CCSSM II
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Quarter 1
Columbus City Schools
6/28/13
Page 73 of 162
Name ___________________________________ Date __________________ Period ________
Find the Scale Factor
Answer Key
For each exercise, find the scale factor of figure A to figure B and solve for x.
1.
Scale Factor = 5; x = 12
24
A
6
B
3
x
2.
Scale Factor =
5
x
1 ; x = 20
5
6
x+4
A
B
3.
Scale Factor = 6; x = 3
CCSSM II
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Page 74 of 162
A
20
B
15
4
x
4.
Scale Factor = 3; x = 4
A
8
10
B
x
x+1
CCSSM II
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Quarter 1
Columbus City Schools
6/28/13
Page 75 of 162
G –SRT 4
Name ______________________________ Date ____________ Period ________
Let’s Prove the Pythagorean Theorem
Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA
such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on
BC such that BH = AE. Once you have marked all the new points, connect them to create another
square that is inscribed in square ABCD. Label each side of the new smaller square x.
A
B
D
C
Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label
each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and
CH .
How many right triangles do you see? Name all of them.
In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)?
Represent the area of square ABCD in terms of s and l. Simplify the expression.
Represent the combined area of all the triangles in terms of s and l. Simplify the expression.
Represent the area of the smaller square in terms of x.
CCSSM II
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Page 76 of 162
Write an expression for the total area of all the right triangles and the smaller square. What should
this total area be equal to and why?
Write an equation that relates part E to Part C. Identify and eliminate any common terms on each
side of the equation. Explain what each variable in the new equation represents.
You have just proven the Pythagorean Theorem!
CCSSM II
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Quarter 1
Columbus City Schools
6/28/13
Page 77 of 162
Name ___________________________________ Date __________________ Period ________
Let’s Prove the Pythagorean Theorem
Answer Key
Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA
such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on
BC such that BH = AE. Once you have marked all the new points, connect them to create another
square that is inscribed in square ABCD. Label each side of the new smaller square x.
Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label
each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and
CH .
How many right triangles do you see? Name all of them.
Four triangles -
AEF,
BEH,
CGH,
DFG (students could label these differently)
In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)?
s is a leg, l is a leg and x is the hypotenuse
Represent the area of square ABCD in terms of s and l. Simplify the expression.
Area = (s + l)2 = s2 + 2sl + l2
Represent the combined area of all the triangles in terms of s and l. Simplify the expression.
Area = 4(½)sl = 2sl
Represent the area of the smaller square in terms of x.
Area = x2
Write an expression for the total area of all the right triangles and the smaller square. What should
this total area be equal to and why?
Total area = 2sl + x2
CCSSM II
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Quarter 1
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Page 78 of 162
This total area should equal to the area of the original square ABCD because the original square
consists of the 4 right triangle and the inscribed square.
Write an equation that relates part E to Part C. Identify and eliminate any common terms on each
side of the equation. Explain what each variable in the new equation represents.
s2 + 2sl + l2 = 2sl + x2
s2 + l2 = x2
s and l are the legs of the right triangle and x is the hypotenuse.
You have just proven the Pythagorean Theorem!
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 79 of 162
G –SRT 4
Name ______________________________ Date ____________ Period ________
Proving the Pythagorean Theorem, Again!
a
2
b
c
3
c
a
1
b
(base1  base2 )height
If the formula for finding the area of a trapezoid is
, find the area of the above
2
trapezoid. Simplify the expression.
If the formula for finding the area of a triangle is
base height
, find the areas of each triangle in the
2
picture above.
Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so
that only the Pythagorean Theorem remains.
CCSSM II
Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5
Quarter 1
Columbus City Schools
6/28/13
Page 80 of 162
Name ___________________________________ Date __________________ Period ________
Proving the Pythagorean Theorem, Again!
Answer Key
a
2
b
c
3
c
a
1
b
(base1  base2 )height
If the formula for finding the area of a trapezoid is
, find the area of the above
2
trapezoid. Simplify the expression.
(a  b)(a  b) a 2  ab  ab  b 2 a 2  2ab  b 2


base1 = a
Area of trapezoid =
2
2
2
base2 = b
height = a + b
If the formula for finding the area of a triangle is
picture above.
Area of triangle1 = ½ ab
base height
, find the areas of each triangle in the
2
Area of triangle2 = ½ ab
Area of triangle3 = ½ c2
Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so
that only the Pythagorean Theorem remains.
Area of triangle1 + Area of triangle2 + Area of triangle3 = Area of trapezoid
2
2
1
1
1 2 a +2ab+b
ab + ab +
c =
2
2
2
2
2
2
ab + ab + c = a + 2ab + b2
2ab + c2 = a2 + 2ab + b2
c2 = a2 + b2
CCSSM II
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Quarter 1
Columbus City Schools
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Page 81 of 162
COLUMBUS CITY SCHOOLS
HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE
TOPIC 2
CONCEPTUAL CATEGORY
TIME RANGE
GRADING
20 days
Trigonometric Ratios Geometry
PERIOD
G-SRT 6, 7, 8
1
Domain: Similarity, Right Triangles, and Trigonometry (G – SRT):
Cluster
3) Define Trigonometric ratios and solve problems involving right triangles.
Standards
3) Define Trigonometric ratios and solve problems involving similarity.
 G – SRT 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 7: Explain and use the relationship between the sine and cosine of
complementary angles.

G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 82 of 162
TEACHING TOOL
Vocabulary: acute angle, adjacent, angle of depression, angle of elevation, complementary angles,
corresponding sides, cosine, geometric mean, hypotenuse, opposite, proportion, Pythagorean
Theorem, Pythagorean triple, ratio, right triangle, similar triangles, sine, solving a triangle, special
right triangles, tangent, trigonometric ratios, trigonometry
Teacher Notes
Properties of Radicals
n
An expression that contains a radical sign is called a radical expression, a . The expression under
the radical sign is the radicand and the numeric value, n is the index. We read this as “the nth root of
5
a. Looking at the radical expression 3x , 3x is the radicand and 5 is the index.
1. c is a square root of a, if c2 = a, e.g., 2 is a square root of 4 because 22 = 4 and
-2 is a square root of 4 because (-2)2 = 4.
Because there are two values that satisfy the equation x2 = 4, we take the term square root to mean the
principal square root which has a non-negative value. In this case 4  2 is the principal square
root. Mathematically, we express this as:
a2  a
2. c is a cube root of a if c3 = a, e.g., 3 is a cube root of 27 because 33 = 27 and -3 is a cube root of 27 because (-3)3 = -27.
The cube root of a negative number is negative.
The cube root of a positive number is positive.
3. c is an nth root of a if cn = a. Note that if the index is odd and the radicand is negative then the
principal root is negative. For example, 32  2 because (-2)5 = -32. The following are general
rules for taking the roots of positive and negative numbers.
5
The answer is the principal root.

The answer is the opposite of the principal root.

The answer is both roots, the positive and the negative root.
odd number
even number
odd number
even number
negative number
negative number
positive number
The answer is a negative number.
There is no real solution.
The answer is the principal root.
positive number
The answer is the principal root.
For any value of x and any even number n,
= -5, then
8
n
xn  x . For example, if x = 4, then
6
46  4  4 . If x
(5)8  5  5 . For any value of x and any odd number n greater than 1,
example, if x = 4, then . If x = -5, then
4. product rule:
n
9
n
x n  x . For
(5)9  5 .
a  n b  n ab , e.g.
3
7  3 5  3 7  5  3 35 , and
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 83 of 162
( x  6)  ( x  6)  ( x  6)( x  6)  x 2  36
The product rule can be used for factoring to simplify radical expressions as shown
below.
50  25  2  25  2  5 2
and
72  2 2 2 3 3
2  2 2   3 3  2 3 2
6 2
19  19
72 can be written as the product of prime factors, and then simplified, but 19 is a prime
number so it is already in its simplest form.
5. quotient rule: given
n
a and
n
b,b  0 ,
n
a na

, e.g.
b nb
x2
x2 x


16
16 4
6. principle of powers: if a = b then an = bn
This website offers a teacher resource that includes a power point presentation for operations with
radical expressions.
http://teachers.henrico.k12.va.us/math/hcpsalgebra1/module11-3.html
This website has an on-line explanation of radicals.
http://www.regentsprep.org/Regents/math/algtrig/ATO3/simpradlesson.htm
The website below is a teacher resource that has lessons, practice and a tutorial.
http://www.regentsprep.org/Regents/math/ALGEBRA/AO1/indexAO1.htm
The two following websites have practice with operations with radicals.
http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-2.xml
http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-3.xml
Right Triangles
Remind students that it is better to remember the Pythagorean Theorem as leg2 + leg2 = hypotenuse2
rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse.
There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1:1: 2 .
The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 .
Solving special right triangles
http://www.youtube.com/watch?v=nVTtSE5nv7c
http://www.youtube.com/watch?v=NsNaYwHtowA
Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the opposite
side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
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6/28/13
Page 84 of 162
The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent (tan) of an
angle is the ratio of the opposite side to the adjacent side.
There are many different ways to help your students remember the sine, cosine, and tangent functions.
 Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief was also a great
mathematician. And he developed sine, cosine, and tangent to match his name.
SOH (sin = opp / hyp)

CAH (cos = adj / hyp)
TOA (tan = opp / adj)
The following phrase could also be used.
Some
Caught
Taking
Old
Horse
Another Horse
Oats
Away
The geometric mean is the square root of the product of two numbers. In right triangles, an altitude
drawn to the hypotenuse is the geometric mean of the measures of the two segments of the hypotenuse.
Each leg of a right triangle is the geometric mean of the measure of the adjacent segment of the
hypotenuse and the total measure of the hypotenuse.
angle of depression
The angle of elevation is the angle between the line of sight and the horizontal when looking up. The
angle of depression is the angle between the line of sight and the horizontal when looking down. It is
helpful to remember that the angle of elevation and the angle of depression are alternate interior angles
to each other.
angle of elevation
Real life applications are architecture and engineering.
Right triangle trigonometry is one of the more practical day-to-day applications of mathematics. Used
to find lengths and angles, it is a necessity in construction and home improvement. For example, if
you wish to build a deck that is a regular polygon, you only need the length of one side to find the area
using trigonometry and simple geometry.
The three trigonometric functions, sine, cosine, and tangent are simply ratios of the sides of right
triangles. These values can be found in a table, in a calculator, or in a textbook. By the Angle-Angle
Similarity Theorem, if the measures of two of the angles of a pair of triangles are equal, then the
triangles are similar. Since we are working with right triangles only, all triangles with a second angle
of the same measure are similar and their sides are proportional.
The given angle is called “theta” and is represented by the symbol . The side of the triangle across
from  is the “opposite side”. The side of the triangle next to  is the “adjacent side”.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 85 of 162
hypotenuse
opposite side

adjacent side
The trigonometric ratios are:
=
sin
opposite
=o
hypotenuse h
θ=
cos
Words
Symbol
Trigonometric
sine θ
sin θ
Ratios
cosine θ
cos θ
tangent θ
tan θ
adjacent
=a
hypotenuse h
θ=
tan
opposite o
=
adjacent a
Definition
opposite
sin  
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
If the angle measure is 30°, 45° or 60° in a right triangle, special trigonometric relationships exist.
θ
sin θ
1
2
cos θ
tan θ
csc θ
sec θ
cot θ
3
3
2 3
2
3
2
3
3
2
2
45˚
1
1
2
2
2
2
1
3
2 3
3
60˚
2
3
2
2
3
3
2
2
Remind students that it is better to remember the Pythagorean Theorem as leg + leg = hypotenuse2
rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse.
30˚
There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1 :1 : 2 .
The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 .
45o
30o
x 2
x
2x
x 3
45
o
x
x
x
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
60o
Columbus City Schools
6/28/13
Page 86 of 162
m∠A + m∠B = 90o
B
sin A= opposite side of A  a
sin B= opposite side of B  b
cos A= adjacent side of A  b
cos B= opposite side of B  a
hypotenuse
c
a
c
hypotenuse
C


b
a2 + b2 = c2
c
A tan A= opposite side of A a

adjacent side of A
b
hypotenuse
hypotenuse
c
c
tan B= opposite side of B  b
adjacent side of B
a
Students must understand that triangles with congruent angles are similar triangles.
Students must understand that the ratio of two sides in one triangle is equal to the ratio of the
corresponding two sides of all other similar triangles.
Right Triangles
Right Triangle Trigonometry
http://patrickjmt.com/right-triangles-and-trigonometry/
A website video tutorial on right triangle trigonometry.
Evaluating Trigonometric functions
http://patrickjmt.com/evaluating-trigonometric-functions-for-an-unknown-angle-given-a-pointon-the-angle-ex-1/
Evaluating trigonometric functions for an unknown angle given a point on the angle.
Right Triangle Trigonometry
http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_4.html
Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on
activities for right triangle trigonometry.
Special Right Triangles
http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_3.html
Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on
activities for special right triangles.
Basic Trigonometry
http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593
Students define basic terms relating to trigonometry and use trigonometric ratios using their TI-84
calculator.
Module 16 Trigonometric Ratios
http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outlineand-take-this-course
Using TI – 84 and Cabri Jr for special triangles
Sine and Cosine of Complementary Angles http://learni.st/users/60/boards/3370-sine-and-cosineCCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 87 of 162
of-complementary-angles-common-core-standard-9-12-g-srt-7#/users/60/boards/3370-sine-andcosine-of-complementary-angles-common-core-standard-9-12-g-srt-7
Tutorials to explain the relationship between the sine and cosine of complementary angles.
Co-Functions
http://www.regentsprep.org/Regents/math/algtrig/ATT6/cofunctions.htm
Practice and warm ups to explain co-functions
Finding Height Using Trigonometry
http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-1/
Tutorial on finding the height of an object using trigonometry example 1
Finding Height Using Trigonometry
http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-2/
Example 2
Finding Height Using Trigonometry
http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-3/
Example 3
Finding Height Using Trigonometry
http://patrickjmt.com/trigonometry-word-problem-finding-the-height-of-a-building-example-1/
Word Problem 1
Finding Height Using Trigonometry
http://patrickjmt.com/trigonometry-word-problem-example-2/
Word Problem 2
Misconceptions/Challenges:
SRT 6
 Students struggle labeling the opposite, adjacent and hypotenuse. Sometimes they use the
shortest leg as the opposite leg or confuse adjacent and hypotenuse.
 Students get confused of where the angle of depression is located.
 Students confuse the difference on how to use the calculator when finding values of a missing side
or missing angle.
 Students may apply the ratios of the special right triangles to all right triangles.
 Once trigonometry is taught, students like to use that instead of the ratios of special triangles. But
to get exact values, they must use the ratios.
SRT 8
 Students may not substitute the hypotenuse in for ‘c’ in the Pythagorean Theorem.
 Angle of depression is often mislabeled as the angle between the vertical and hypotenuse
 Students incorrectly identify corresponding legs when using hypotenuse-leg congruence for right
triangles.
 Students do not understand that equilateral triangles are also equiangular and vice versa.
 Students do not realize that congruent angles in an isosceles triangle are opposite the congruent
sides.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 88 of 162
Instructional Strategies:
 This is an entire unit that covers all three standards. There are many references to everyday objects
in the lessons.
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_912_AccelCoorAlgebraAnalyticGeom_Unit8SE.pdf

This link has instructional strategies and sample formative assessment tasks as well as key
concepts and vocabulary.
http://www.schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-5-G-SRT-6.aspx

project ideas
http://ccss.performanceassessment.org/taxonomy/term/1045

The following website has practice on simplifying radical expressions.
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Simplifying%20Radicals.pd
f
SRT 6
 Stay in Shape
http://www.teachengineering.org/view_activity.php?url=http://www.teachengineering.org/col
lection/cub_/activities/cub_navigation/cub_navigation_lesson03_activity1.xml
Lesson on how triangles and The Pythagorean Theorem are used in measuring distance.

Fit by Design
https://access.bridges.com/usa/en_US/choices/pro/content/applied/topic/aom14CX.html
lesson relates actual and calculated measures of right triangles to objects created by mechanical
drafters or designers

Calculating Volumes of Compound Objects
http://map.mathshell.org/materials/lessons.php?taskid=216
Decomposing shapes into simpler ones and using right triangles to solve real-world problems.

Geometry Problems: Circles and Triangles
http://map.mathshell.org/materials/lessons.php?taskid=222
Students determine the lengths of sides in right triangles to solve problems.

Hopewell Geometry.
http://map.mathshell.org/materials/tasks.php?taskid=127&subpage=apprentice
How the Hopewell people constructed earthworks using right triangles.

Have students complete the activity “Exploring Special Right Triangles 45-45-90” (included in
this Curriculum Guide) to reinforce the properties of 45-45-90 triangles.

Have students complete the activity “Exploring Special Right Triangles 30-60-90” (included in
this Curriculum Guide) to reinforce the properties of 30-60-90 triangles.

Have students complete the activity “Discovering Trigonometric Ratios” (included in this
curriculum guide) to develop their understanding of trigonometry. Students will need centimeter
rulers and protractors to measure the parts of the given triangles.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 89 of 162

Have students complete the activity “Make a Model: Trigonometric Ratios” (included in
Curriculum Guide) to discover that the trigonometric ratios of any right triangle with specific acute
angles are the same regardless of the lengths of the sides. Calculators may be helpful for this
activity.

Eratosthenes Finds the Circumference of the Earth
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students examine a diagram and verify the two triangles are similar. Page 12

Discovering Special Triangles
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use real-world situations to discover special right triangles. Page 16

Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students find right triangles such as a ramp. Page 20

Create Your Own Triangles
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use paper, compass, straight edge and protractor to create right triangles and verify the
measurements. Page 22

The Tangent Ratio
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB.
pdf
Students solve real-world problems using the tangent ratio

Are Relationships Predictable
.http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students develop and use right triangle relationships based on similar triangles. Classroom Task:
6. pages 53-59. (This strategy can also be found in SRT8.)

Relationships with Meaning
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students find relationships between the sine and cosine ratios for right triangles, including the
Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in
SRT7.)

Solving Right Triangles Using Trigonometric Relationships
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found
on pages 75-81 (This strategy can also be found in SRT7.)
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 90 of 162
SRT 7
 Relationships with Meaning
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students find relationships between the sine and cosine ratios for right triangles, including the
Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in SRT
6.)

Solving Right Triangles Using Trigonometric Relationships
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found
on pages 75-81 (This strategy can also be found in SRT 6.)

Create Your Own Right Triangles
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use paper, compass, straight edge and protractor to create right triangles and verify the
measurements. Page 22.

Discovering Trigonometric Ratio Relationships
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use degree measurements of acute angles from right triangles to determine trigonometric
ratios. Page 27

Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students find right triangles such as a ramp. Page 20

The Sine and Cosine Ratios
https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf
Students use sine and cosine ratios of acute angles of right triangles to solve real-world problems.
Page 7

Special Right Triangles
https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf
Students investigate special right triangles. Page 11
SRT 8
 Have students complete the activity “Application of Trigonometry” (included in this Curriculum
Guide) to practice using trigonometry to solve indirect measurement questions.

Horizons
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use trigonometric ratios to determine distance to the horizon from different locations.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 91 of 162
Page 11

Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students find right triangles such as a ramp. Page 20

Create Your Own Triangles
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGP
S_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use paper, compass, straight edge and protractor to create right triangles and verify the
measurements. Page 22.

Find that Side or Angle
https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf
Students use graphing technology to find the values of sine and cosine in real-world situations.

Access Ramp
http://www.achieve.org/files/CCSS-CTE-Task-AccessRamp-FINAL.pdf
Students design an access ramp which complies with the Americans with Disabilities Act (ADA)
requirements and includes pricing based on local costs.

Land Surveying Project
http://alex.state.al.us/lesson_view.php?id=25108
Students learn the basics of civil engineering in land surveying.

The Clock Tower
http://alex.state.al.us/lesson_view.php?id=25107
Students use the Pythagorean Theorem, and Sine, Cosine, and Tangent to find unknown heights of
objects.

Solving Right Triangles.
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB.
pdf
Students determine the angle measures in right triangles. Lesson 6-4 Page 15

Determine the Missing Sides of Special Right Triangles
http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8Special%20Right%20Triangles.pdf
Students practice finding missing sides of special right triangles

Applied Trigonometry
http://learni.st/users/60/boards/3453-trig-ratios-and-the-pythagorean-theorem-common-corestandard-9-12-g-srt-8
Several tutorials on trigonometry

Solving Right Triangles
https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB.
pdf
Students determine how to find unknown angle measures of a right triangle. Page 15
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 92 of 162

Have students complete the activity “Let’s Measure the Height of a Flagpole” (included in this
Curriculum Guide) to practice solving problems involving indirect measurement. A clinometer
can be built from a piece of cardboard, a drinking straw, a piece of string, a washer, and a
protractor as shown on the first page of the activity record sheet. Templates for protractors are
included “Grids and Graphics” (included in this Curriculum Guide). Note: Answers will vary.

Have students practice finding the missing side or angle of a right triangle with the “Find the
Missing Side or Angle” activity (included in this Curriculum Guide). Students will need a copy of
the activity and a scientific or graphing calculator.

Remind students of the legend of Soh Cah Toa, the “great trigonometry leader”. Embellish the
story yourself or ask students to spin their own tall tale, write a poem or create a rap that includes a
description of the trigonometric ratios: sine = opposite/hypotenuse, cosine =adjacent/hypotenuse,
and tangent = opposite/adjacent. Review how the trigonometric ratios can be used to find missing
angle measures or side lengths in right triangles.

Arrange students in groups of three to play “paper football”. After each group has made its
football by folding a sheet of paper, the group will assign duties and measure its field (the distance
along the ground from where the ball is kicked to the uprights). One student will be the kicker, one
will hold up their hands as the uprights, and one will measure the height of the “football” from the
ground as it crosses the uprights. Each group will draw its field, record the measurements, and
calculate the angle of elevation the ball makes with the ground for each kick, for at least 5 kicks.
Follow up with the “Between the Uprights” activity (included in this Curriculum Guide). Remind
students that the angle the goal post makes with the ground is 90. Students will need a copy of the
activity, a sheet of paper to use to make a football, and a calculator. Discuss with students the
effect of a five or ten yard penalty on the results for each situation. How significantly would the
angle or distances be changed?

Arrange students into groups of two. Student will practice their right triangle solving skills with
the exercise “Solve the Triangle” (included in this Curriculum Guide). Students will need a copy
of the activity, a calculator, a ruler, and a protractor. Have students take turns explaining to their
partner how they solved one of the problems on the sheet.

Students design their ideal city park in the activity “Right Triangle Park” (included in this
Curriculum Guide). The catch is that their “ideal” park can be made up of only right triangles.
Students will then measure two parts of each triangle and calculate the remaining parts using
trigonometry. Students will need a calculator, ruler, protractor, and a copy of the worksheet.
Allow students time to share with the class their design. Ask students to point out several of the
right triangles in their design and to explain how they calculated the lengths and/or angles for a few
of the triangles.

Have students complete the activity “Applications of the Pythagorean Theorem” (included in
this Curriculum Guide) to explore real-life applications of the Pythagorean Theorem.

Students can reinforce their similar triangle skills by using the properties of similar triangles to
measure objects around school. Have the students select three objects they want to measure and
use a mirror to create a pair of similar triangles as instructed in the “Find the Height” activity
(included in this Curriculum Guide). Each group of two or three students will need a tape measure,
mirror, and a copy of the activity instructions. Each student will need a copy of the activity data
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 93 of 162
sheet. After students have completed the activity, discuss as a class their strategies for finding the
height of each object.

Close Enough
http://www.teachengineering.org/view_activity.php?url=http%3A%2F%2Fwww.teachengin
eering.org%2Fcollection%2Fcub_%2Factivities%2Fcub_navigation%2Fcub_navigation_les
son04_activity1.xml
Hands-on activity shows how accurate measurement is important as students use right triangle
trigonometry and angle measurements to calculate distances

Six Trigonometric Ratio Values of Special Acute
http://illuminations.nctm.org/LessonDetail.aspx?id=L383
A puzzle for practicing knowledge of all six trigonometric ratios. Two activities involve angle of
elevation and angle of declination.

Solving Problems Using Trigonometry
http://education.ti.com/en/us/activity/detail?id=EB3E2581FFEC4FDA8FC94C3AA51F3D31
Students use TI-84 calculator to find the angle of elevation or the angle of depression.

Basic Trigonometry
http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593
Students define basic terms relating to trigonometry and use trigonometric ratios.

Are Relationships Predictable
http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig
_te_040913.pdf
Students develop the right triangle trigonometric relationships based on similar triangles. (This
strategy can also be found in SRT8.)
Reteach:
 Have students complete the activity “Memory Match” (included in this Curriculum Guide) to
reinforce right triangle terminology. Students can work in groups of three or four. First place all
cards facedown and then have each student take turns drawing two cards. If the two cards drawn
go together as a pair the student will keep it as a match. Students take turns drawing. The student
with the most pairs or matches wins. Students will need a scientific calculator.

Additional practice in solving proportions, using the properties of similar triangles and right
triangle trigonometry is available in the “Similar Right Triangles and Trigonometric Ratios”
activity (included in this Curriculum Guide).

Have students complete the activity “Hey, All These Formulas Look Alike” (included in this
Curriculum Guide) to investigate the tangent relationship. Note: For the visual learner, the use of
highlighted notes may lead to greater understanding. Highlighting (x2 – x1 ) x and ( y2 – y1) y
in yellow and pink, respectively, may aide the visual learner in the formula comparisons.
Extensions:
 Have students complete the activity “Similar Right Triangles and Trigonometric Ratios”
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 94 of 162
(included in Curriculum Guide) to make connections between similar triangles and trigonometric
ratios

Have students complete the activity “Problem Solving: Trigonometric Ratios” (included in
Curriculum Guide) to apply their knowledge of trigonometric ratios.

Given the sides of a right triangle inscribed in a circle and circumscribed about another circle,
students find the radii of each circle. Also, students analyze sample solutions and compare
their own solutions to those given.
http://map.mathshell.org/materials/lessons.php?taskid=403#task403
Textbook References:
Textbook: Geometry, Glencoe (2005): pp. 349, 350-356
Supplemental: Geometry, Glencoe (2005):
Chapter 7 Resource Masters
Study Guide and Intervention, pp. 357-358
Skills Practice, p. 359
Practice, p. 360
Enrichment, p. 362
Textbook: Geometry, Glencoe (2005): pp. 357-363
Supplemental: Geometry, Glencoe (2005):
Chapter 7 Resource Masters
Study Guide and Intervention, pp. 363-364
Skills Practice, p. 365
Practice, p. 366
Enrichment, p. 368
Textbook: Geometry, Glencoe (2005): pp. 364-370
Supplemental: Geometry, Glencoe (2005):
Chapter 7 Resource Masters
Study Guide and Intervention, pp. 369-370
Skills Practice, p. 371
Practice, p. 372
Enrichment, p. 374
Textbook: Geometry, Glencoe (2005): pp. 342-348 371-376
Supplemental: Geometry, Glencoe (2005):
Chapter 7 Resource Masters
Study Guide and Intervention, pp. 351-352, 375-376
Skills Practice, pp. 353, 377
Practice, pp. 354, 378
Enrichment, pp. 356, 380
Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002):
Teacher’s Resources for Transfer Students, pp. 39-40
Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002):
Skills Bank, p. 104
Overhead Visuals, folders A, 10
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 95 of 162
Textbook: Algebra 1, Glencoe (2005): pp. 622 – 630
Textbook: Algebra 1, Glencoe (2005): pp. 698 – 708
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 96 of 162
G-SRT 6 SRT- 6
Name ___________________________________ Date __________________ Period ________
Exploring Special Right Triangles (45-45-90)
Given the isosceles right triangle below.
l
h
l
1. What is the measure of each acute angle? Explain.
2. a) If the length of each leg is 1 unit, find the length of the hypotenuse.
Leave answer exact and simplified.
b) What are the side-length ratios of leg: leg: hypotenuse?
3. a) If the length of each leg is 2 units, find the length of the hypotenuse.
Leave answer exact and simplified.
b) How many times longer is the hypotenuse than the leg?
c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio.
4. a) If the length of each leg is 5 units, find the length of the hypotenuse.
Leave answer exact and simplified.
b) How many times longer is the hypotenuse than the leg?
c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio.
5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles right
triangle?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 97 of 162
Name ___________________________________ Date __________________ Period ________
Exploring Special Right Triangles (45-45-90)
Answer Key
Given the isosceles right triangle below.
l
h
1. What is the measurel of each acute angle? Explain how you know.
Each acute angle is 45o. Since this is an isosceles right triangle, each angle opposite the legs
are congruent. Since there’s a total of 90o for both acute angles and they are congruent,
they must be 45o each.
2. a) If the length of each leg is 1 unit, find the length of the hypotenuse.
Leave answer exact and simplified.
Hypotenuse = 2
b) What are the side-length ratios of leg: leg: hypotenuse?
1: 1: 2
3. a) If the length of each leg is 2 units, find the length of the hypotenuse.
Leave answer exact and simplified.
Hypotenuse = 8 = 2 2
b) How many times longer is the hypotenuse than the leg?
The hypotenuse is 2 times longer than the leg.
c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio.
2: 2: 2 2 which is really 1: 1: 2
4. a) If the length of each leg is 5 units, find the length of the hypotenuse.
Leave answer exact and simplified.
Hypotenuse = 50 = 5 2
b) How many times longer is the hypotenuse than the leg?
The hypotenuse is 2 times longer than the leg.
c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio.
5: 5: 5 2 which is really 1: 1: 2
5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles
right triangle?
They will always be 1: 1: 2 .
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 98 of 162
G-SRT 6
Name ___________________________________ Date __________________ Period ________
Exploring Special Right Triangles (30-60-90)
1. Given the equilateral triangle below whose sides are 2 units long.
A
2
2
B
2
a) What is the angle measure of each acute angle?
C
b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along
the crease. Label point D where the new segment intersects segment BC . What is special
about AD ? What is the length of BD (label it in the diagram above)? Explain your
reasoning.
c) What does AD do to A ?
d) Examine ABD . What is the measure of BAD and BDA (label it in the diagram
above)?
e) Find the length of AD . Leave exact and simplified.
f) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times
longer is the longer leg than the shorter leg?
g) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg:
hypotenuse?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 99 of 162
2. Given the equilateral triangle below whose sides are 4 units long.
A
4
4
B
C
4
a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along
the crease. Label point D where the new segment intersects BC . What is the length of
BD (label it in the diagram above)? Explain your reasoning.
b) Examine ABD . What is the measure of BAD and BDA (label it in the diagram
above)?
c) Find the length of AD . Leave exact and simplified.
d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times
longer is the longer leg than the shorter leg?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 100 of 162
e) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg: hypotenuse?
3. Repeat the steps above using a different number for the length of the side of the equilateral
triangle. What can you conclude about the side-length ratios of short leg: long leg:
hypotenuse for any 30o-60o-90o triangle?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 101 of 162
Name ___________________________________ Date __________________ Period ________
Exploring Special Right Triangles (30-60-90)
Answer Key
1. Given the equilateral triangle below whose sides are 2 units long.
A
2
B
2
D
C
a) What is the angle measure of each acute angle?
2
60o
b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along
the crease. Label point D where the new segment intersects BC . What is special about
AD ? What is the length of BD (label it in the diagram above)? Explain your reasoning.
AD is the altitude,  bisector, bisector, and median of ABC . BD = 1 because
AD bisects BC
c) What does AD do to A ?
It bisects A
d) Examine ABD . What is the measure of BAD and BDA (label it in the diagram
above)?
m BAD = 30o and m BDA = 90o
e) Find the length of AD . Leave exact and simplified.
AD = 3
f) In ABD, how much longer is the hypotenuse than the shorter leg? How much longer is the
longer leg than the shorter leg?
The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as
long as the shorter leg.
g) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg:
hypotenuse?
1:
3:2
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 102 of 162
2. Given the equilateral triangle below whose sides are 4 units long.
A
4
4
B
C
D
4
a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along
the crease. Label point D where the new segment intersects BC . What is the length of
BD (label it in the diagram above)? Explain your reasoning.
BD = 2 because AD bisects BC
b) Examine ABD . What is the measure of BAD and BDA (label it in the diagram
above)?
m BAD = 30o and m BDA = 90o
c) Find the length of AD . Leave exact and simplified.
AD = 12 = 2 3
d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times
longer is the longer leg than the shorter leg?
The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as
long as the shorter leg.
e) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg: hypotenuse?
1: 3 : 2
3. Repeat the steps above using a different number for the length of the side of the equilateral
triangle. What can you conclude about the side-length ratios of short leg: long leg:
hypotenuse for any 30o-60o-90o triangle?
They will always be 1: 3 : 2 .
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 103 of 162
G-SRT 6
Name ___________________________________ Date __________________ Period ________
Discovering Trigonometric Ratios
For the following right triangles, find the indicated ratios.
C
C'
A'
B'
B
A
Find each length to the nearest quarter of an inch and after division round the quotient to three
decimal places.
1.
Length of AB
=
Length of AC
Length of AB
=
Length of AC 
2.
Length of BC
=
Length of AC
Length of B / C /
=
Length of A/ C /
3.
Length of AB
=
Length of BC
Length of A/ B /
=
Length of B / C /
Triangles ABC and A/B/C/ are similar triangles.
4. From the above experiment, what can you conclude about these ratios?
Find the measures of  C and  C/ to the nearest tenth of a degree.
5. m  C =
6. m  C/ =
Using your calculator, find the following using the value of  C from above. (#5)
7. sin  C =
8. cos  C =
9. tan  C =
10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3?
11. Match sine, cosine, and tangent to the three ratios in #1 - #3.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 104 of 162
Find the measures of  A and  A/.
12.  A =
13.  A/ =
Using your calculator, find the following using the value of  A from above. (#12)
14. sin  A =
15. cos  A =
16. tan  A =
17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and
why?
18. Write a general equation for sine, cosine, and tangent that could be used with any right
triangle.
19. Make your own triangles to test the above equations.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 105 of 162
G-SRT 6
Name ___________________________________ Date __________________ Period ________
Discovering Trigonometric Ratios
Answer Key
For the following right triangles, find the indicated ratios.
C
Find each length to the nearest quarter of an inch and after division round the quotient to three
decimal places.
2.5
1.
Length of AB
2 = 0.8
1.5 =
Length of AC 2.5
2.
Length of BC
1.5
B = 2.5 = 0.6
Length of AC
2
Length of A/ B /
1 = 0.8 C'
=
1.25
Length of A/ C /
1.25
.75
/ /
Length of B C
.75 = 0.6
A' =
B'
A
1.25
Length of A/ C /
1
3.
Length of AB
2 = 1.333
=
Length of BC 1.5
Length of A/ B /
1 = 1.333
=
.75
Length of B / C /
DUE TO HUMAN AND ROUNDING ERRORS THESE MAY BE CLOSE BUT NOT BE
EXACT.
Triangles ABC and A/B/C/ are similar triangles.
4. From the above experiment, what can you conclude about these ratios?
Both ratios in #1 are close to being the same.
Both ratios in #2 are close to being the same.
Both ratios in #3 are close to being the same.
Find the measures of  C and  C/ to the nearest tenth of a degree.
5. m  C = 53.1o
6. m  C/ = 53.1o
Using your calculator, find the following using the value of  C from above. (#5)
7. sin  C = 0.7997
8. cos  C = 0.6004
9. tan  C = 1.3319
10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3?
The value in #7 is close to the value in #1
The value in #8 is close to the value in #2
The value in #9 is close to the value in #3
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 106 of 162
11. Match sine, cosine, and tangent to the three ratios in #1 - #3.
Sine matches with the ratio in #1
Cosine matches the ratio in #2
Tangent matches the ratio in #3
Find the measures of  A and  A/.
12.  A = 36.9o
13.  A/ = 36.9o
Using your calculator, find the following using the value of  A from above. (#12)
14. sin  A = 0.6004
15. cos  A = 0.7997
16. tan  A = 0.7508
17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and
why?
Yes this changed. When looking at #14 - #16, sine matches with #2, cosine matches with
#1, and tangent matches with the reciprocal of #3.
18. Write a general equation for sine, cosine, and tangent that could be used with any right
triangle.
sin  =
opposite
hypotenuse
cos  =
adjacent
hypotenuse
tan  =
opposite
adjacent
19. Make your own triangles to test the above equations.
Answers may vary.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 107 of 162
G-SRT 6
Name ___________________________________ Date __________________ Period ________
Make a Model: Trigonometric Ratios
Materials:
protractor, metric ruler, compass, plain or graph paper, and scientific
calculator for each person in the group.
Directions:
Everyone in the group should do Steps 1 - 5 individually. Steps 6 – 8
should be done collectively.
Step 1: On a sheet of graph or plain paper, use a protractor to make as large a right
triangle ABC as possible with mB = 90°, mA = 20°, and mC = 70°.
Label the vertices appropriately.
Step 2: Use your ruler to measure sides AB, AC, and BC to the nearest millimeter.
AB = ______ mm
AC = ______ mm
BC = ______ mm
Step 3: Recall by definition:
r
hypotenuse
y
leg opposite 

sin  =
length of leg opposite θ
y
=
length of hypotenuse
r
cos  =
length of leg adjacent θ
x
=
length of hypotenuse
r
tan  =
length of leg opposite θ
y
=
length of leg adjacent θ
x
x
leg adjacent to 
Step 4: Use the information obtained in Step 2 to complete the following statements.
Write the following ratios in fraction form and decimal form to the nearest
thousandth.
Fraction
Decimal
sin 20° =
length of leg opposite A
length of hypotenuse
cos 20° =
length of leg adjacent to A
= __________ = __________
length of hypotenuse
tan 20° =
length of leg opposite A
= __________ = __________
length of leg adjacent to A
= __________ = __________
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 108 of 162
sin 70° =
length of leg opposite C
= __________ = __________
length of hypotenuse
cos 70° =
length of leg adjacent C
= __________ = __________
length of hypotenuse
tan 70° =
length of leg opposite C
= __________ = __________
length of leg adjacent C
Step 5: In the tables below, record your ratios in decimal form to the nearest
thousandth in the appropriate boxes (Individual) under sin A, cos A, tan A,
sin C, cos C, and tan C.
Step 6: Compare the ratios obtained by the members of your group. Calculate the
average of each of the ratios found by the members of your group. In the
tables below, record the ratios in the appropriate boxes (Group Averages)
under sin A, cos A, tan A, sin C, cos C, and tan C.
Step 7: Use a calculator to check your group’s results. Calculate sin 20, cos 20, tan
20, sin 70, cos 70, and tan 70. Record the results in your tables. How do
the trigonometric ratios that were found by measuring the sides compare with
the trigonometric ratios that were found by using a calculator?
______________________________________________________________
m  A = 20 
Ratios (Individual)
sin A
cos A
tan A
sin C
cos C
tan C
Ratios (Group Averages)
Ratios (Calculator)
mC = 70
Ratios (Individual)
Ratios (Group Averages)
Ratios (Calculator)
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 109 of 162
Step 8:
Questions for Discussion
A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain.
__________________________________________________________________
B. For any two right triangles with acute angles measuring 20° and 70°:
The sin 20°, cos 20°, and tan 20° are ____________________________ the same.
sometimes, always, or never
The sin 70°, cos 70°, and tan 70° are ____________________________ the same.
sometimes, always, or never
C. Why are the trigonometric ratios of any right triangle with acute angles measuring
20° and 70° the same regardless of the lengths of the sides?
__________________________________________________________________
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 110 of 162
Name ___________________________________ Date __________________ Period ________
Make a Model: Trigonometric Ratios
Answer Key
The responses given in Steps 1-6 are based on the lengths of the sides of the different triangles
that are drawn by individual students. The responses in Steps 1-6 may vary.
Step 7:
Use a calculator to check your group’s results. Calculate sin 20, cos 20, tan
20, sin 70, cos 70, and tan 70. Record the results in your tables. How do
the trigonometric ratios that were found by measuring the sides compare with
the trigonometric ratios that were found by using a calculator?
They are the equal or approximately equal to each other.
sin A
cos A
tan A
mA = 20
Ratios (Individual)
May Vary
May Vary
May Vary
Ratios (Group Averages)
May Vary
May Vary
May Vary
Ratios (Calculator)
.342
.940
.364
mC = 70
Ratios (Individual)
Ratios (Group Averages)
Ratios (Calculator)
sin C
May Vary
May Vary
.940
cos C
May Vary
May Vary
.342
tan C
May Vary
May Vary
2.747
Step 8: Questions for Discussion
A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain.
Yes. Two triangles are similar if their corresponding angles are congruent.
B. For any two right triangles with acute angles measuring 20° and 70°:
The sin 20°, cos 20°, and tan 20° are always the same.
The sin 70°, cos 70°, and tan 70° are always the same.
C. Why are the trigonometric ratios of any right triangle with acute angles measuring
20° and 70° the same regardless of the lengths of the sides?
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
All right triangles with acute angles measuring 20° and 70° are similar;
therefore, the ratio of any two sides of one triangle will equal the ratio of the
corresponding two sides of another.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 111 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Let’s Measure the Height of the Flagpole
50
60
40
0
13
0
12
0
14
30
0
15
20
10
0
16
0
17
The clinometer is used to measure the heights of objects. It is a simplified version of the quadrant, an
important instrument in the Middle Ages, and the sextant, an instrument for locating the positions of
ships. Each of these devices has arcs which are graduated in degrees for measuring angles of
elevation. The arc of the clinometer is marked from 0 to 90 degrees. When an object is sighted
through the straw, the number of degrees in angle BXY can be read from the arc. Angle BAC is the
angle of elevation of the clinometer. Angle BXY on the clinometer is equal to the angle of elevation,
angle BAC.
70
0
11
X
20
30
40
50
10
0
17
0
16
0
15
0
13
0
12
60
70
0
14
0
11
0
10
80
90
0
10
80
Drinking Straw
B
A
Y
C
Objective:
You will use your skills of right triangle trigonometry to measure the height of the school’s
flagpole.
Materials:
- clinometer, meter stick, calculator
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 112 of 162
Procedures & Questions:
- Pick a certain distance (in meters) that you want to stand from the flagpole. Record it
below.
_____________
meters
-
Use the clinometer and look through the straw to locate the top of the flagpole. Record
the angle measure that is created from the string below.
______________
degrees
-
Draw a picture of this situation and label all parts clearly.
-
Use your knowledge of right triangle trigonometry to find the height of the flagpole.
Show algebraic work. Round answer to two decimal places.
-
Can you think of another method to find the height of the flagpole? Explain clearly and
be very specific.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 113 of 162
G-SRT 8
Name _____________________________ Date ______________ Period ________
Applications of Trigonometry Using Indirect
Measurement
1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure
distances by recording the time required for a signal to reflect off the object. They use the
equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of
the Hocking Hills than the right part?
T
M
950 ft
880 ft
60o
50o
B
C
A
2. You are designing a jet plane as shown. In preparing the documentation for your design, you are
required to find the measures of  RPQ and  PQR in the wing (triangle PQR). What are the
measures?
P
30 ft
12 ft
R
Q
3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers
in 1903.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 114 of 162
A
B
C
E
D
G
F
Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is a
rectangle,  BFC is equilateral,  AEF   DGF, EF = 15, and BC = 9. Round your answers to
two decimal places.
a) BF
b) AE
c) AF
d) AB
e)  AFE
f)
g)  ABF
h)  FBC
 FAB
In #3h, you can find m  FBC in two ways. Describe the two ways. Do they yield the same
value?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 115 of 162
4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an
electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and
the distance to the top of the dam to be 922 feet.
922 ft
55º
500 ft
x ft
a) Use the diagram to find the height of the dam.
b) If you are standing 500 feet from the base of the dam, find x.
5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation
is 70o. Find the height of the Eiffel Tower.
70º
382.5 ft
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 116 of 162
6. A yacht is sailing toward the lighthouse and a airplane is flying toward the lighthouse as well.
The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300
feet from the lighthouse and has the same height as the top of the lighthouse.
300 ft
y
250 ft
x
400 ft
Find the angle of elevation of the yacht and the angle of depression of the airplane.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 117 of 162
Name ___________________________________ Date __________________ Period ________
Application of Trigonometry Using Indirect
Measurement
Answer Key
1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure
distances by recording the time required for a signal to reflect off the object. They use the
equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of
the Hocking Hills than the right part?
T
M
950 ft
880 ft
60o
50o
B
C
A
In the right triangle ∆ATB, you can use the sine ratio to find the length of TB .
TB
TB
sin  TAB =
sin 60o =
950(sin 60o) = TB
822.72  TB
950
TA
Use the same procedure to find the length of MC in  AMC.
MC
MC
sin  MAC =
sin 50o =
880(sin 50o) = MC
674.12  MC
MA
880
From these two approximations, you can conclude that the difference in the heights is:
822.72 – 674.12 = 148.6 feet.
2. You are designing a jet plane as shown. In preparing the documentation for your design, you are
required to find the measures of  RPQ and  PQR in the wing (triangle  PQR). What are the
measures?
P
30 ft
12 ft
R
Q
To find the measure of  RPQ, you can use the tangent ratio.
RQ
30 = 2.5
tan P =
tan P =
m  P  68.2o
RP
12
Because  P and  Q are complementary, you can determine the measure of  Q to be
m  Q = 90o – 68.2o = 21.8o
3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers
in 1903.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 118 of 162
A
B
C
E
D
G
F
Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is
a rectangle,  BFC is equilateral,  AEF   DGF, EF = 15, and BC = 9. Round your
answers to two decimal places.
a) BF
b) AE
9
7.79 or 4.5 3
c)
AF
16.90
d) AB
10.5
e)  AFE
27.46o
f)
 FAB
27.46o
g)  ABF
120o
h)  FBC
60o
In #3h, you can find m  FBC in two ways. Describe the two ways. Do they yield the same
value?
Method 1: Each angle of equilateral  FBC is 60o.
4.5 = 1
Method 2: cos  FBC =
9
2
So m  FBC = 60o; yes.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 119 of 162
4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an
electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and
the distance to the top of the dam to be 922 feet.
922 ft
55º
500 ft
x ft
a) Use the diagram to find the height of the dam.
opp
hyp
opp
sin 55o =
922
sin 55o =
922(sin 55o) = opp
922(.819)  opp
755.12  opp = the height of the dam
b) If you are standing 500 feet from the base of the dam, find x.
adj
hyp
500 + x
cos 55o =
922
cos 55o =
922(cos 55o) = 500 + x
922(.573)  500 + x
528.31  500 + x
28.31  x
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 120 of 162
5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation
is 70o. Find the height of the Eiffel Tower.
opp
adj
opp
tan 70o =
382.5
tan 70o =
382.5(tan 70o) = opp
382.5(2.747)  opp
1050.7 ft  opp = height of Eiffel Tower
70º
382.5 ft
6. A yacht is sailing toward the lighthouse and an airplane is flying toward the lighthouse as well.
The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300
feet from the lighthouse and has the same height as the top of the lighthouse.
300 ft
y
250 ft
x
400 ft
Find the angle of elevation of the yacht and the angle of depression of the airplane.
Angle of Elevation:
tan x =
Angle of Depression:
opp
adj
tan y =
opp
adj
tan x = 250
400
tan y = 250
300
tan x = .625
tan y  .833
x  32o
y  39.81o
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 121 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Find the Missing Side or Angle
Instructions: Find the missing side or angle as indicated in each of the right triangles below.
1.
2.
 = ___________
x = ___________
10
23
28
x
18

3.
4.
45
9
x
30
x = ___________
c
c = ___________
70

5.
x
55
6.
 = ___________
x = __________
25
11
2
38
17
7.
8.
8
a
65
a = ___________
b = __________
b
9. Describe a situation when you would use sine. Use illustrations to support your answer.
10. Describe a situation when you would use cos-1. Use illustrations to support your answer.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 122 of 162
Name ___________________________________ Date __________________ Period ________
Find the Missing Side or Angle
Answer Key
Instructions:
Find the missing side or angle as indicated in each of the right triangles below.
1.
2.
10
x=
18.81
=
23
28
18
x
51.5o

3.
4.
9
x
x=
45
8.46
30
70
c=
42.42
c

5.
6.
11
=
x
55
10.30
x=
14.34
25
2
7.
8.
17
a
65
a=
38
7.93
b=
6.25
8
b
9. Describe a situation when you would use sine. Use illustrations to support your answer. When
you know the measure of an angle and the measure of either the opposite side or the hypotenuse.
x
15
25
10. Describe a situation when you would use cos-1. Use illustrations to support your answer.
When you know the measure of the adjacent side and the hypotenuse and want to find the measure of
the angle.
10
x
5
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 123 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Between the Uprights
Using the picture below:
Find the angle of elevation the ball makes with the ground when it is kicked.
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights
(hypotenuse).
24 feet
60
50
40
30
20
10
yards
Using the picture below:
Find the angle of elevation the ball makes with the ground when it is kicked.
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights
(hypotenuse).
20 feet
40
30
20
10
yards
Using the picture below:
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 124 of 162
Find the angle of elevation the ball makes with the ground when it is kicked.
Find the length of the most direct path from where the ball is kicked to where it hits the uprights
(hypotenuse).
10 feet
50
40
30
yards
10
20
Using the picture below:
If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance
from the ground will it cross the uprights?
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights
(hypotenuse).
?
30
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
20
yards
10
Columbus City Schools
6/28/13
Page 125 of 162
Write your own problem for the picture below. Label all parts. Solve the problem showing all
calculations.
___ feet
30
20
yards
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
10
Columbus City Schools
6/28/13
Page 126 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Between the Uprights
Answer Key
Using the picture below:
Find the angle of elevation the ball makes with the ground when it is kicked.
7.59
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights
(hypotenuse).
60.53 yds
24 feet
60
50
40
20
30
10
yards
Using the picture below:
Find the angle of elevation the ball makes with the ground when it is kicked.
9.46
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights
(hypotenuse).
40.55 yds
20 feet
40
30
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
20
10
yards
Columbus City Schools
6/28/13
Page 127 of 162
Using the picture below:
Find the angle of elevation the ball makes with the ground when it is kicked.
3.81
Find the length of the most direct path from where the ball is kicked to where it hits the uprights
(hypotenuse).
50.11 yds
10 feet
50
20
30
yards
40
10
Using the picture below:
If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance
from the ground will it cross the uprights?
45.86 ft = 15.29 yds
Find the length of the most direct path from where the ball is kicked to where it crosses the uprights.
33.67 yds
?
30
20
yards
10
Write your own problem for the picture below. Label all parts. Solve the problem showing all
calculations.
Answers will vary.
___ feet
30
20
yards
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
10
Columbus City Schools
6/28/13
Page 128 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Solve the Triangle
Instructions:
Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right
triangles shown below.
Example:
A
b
C
c
a
B
Given
Side or
Angle
A
a
Measure
Calculated
Side or
Angle
B
b
c
Measure
Once you have completed your measurements, solve each triangle (find all missing sides and angles),
placing values in the table.
Trade papers with your partner and check each other's completed triangles using the following
checklist:
____ all calculations are correct
____ the sum of all angles of each triangle is 180o, accuracy within 1o
____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse,
i.e., a2 + b2 = c2
1.
A
c
b
C
a
B
Given
Side or
Angle
B
c
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Measure
Calculated
Side or
Angle
Measure
Columbus City Schools
6/28/13
Page 129 of 162
2.
A
c
b
a
C
3.
B
A
Given
Side or
Angle
b
c
c
b
C
Measure
Measure
Calculated
Side or
Angle
Calculated
Side or
Angle
Measure
Measure
B
a
4.
A
Given
Side or
Angle
A
b
c
b
B
Given
Side or
Angle
A
c
C
a
Measure
Calculated
Side or
Angle
Measure
5.
C
a
B
b
c
A
Given
Side or
Angle
B
b
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Measure
Calculated
Side or
Angle
Measure
Columbus City Schools
6/28/13
Page 130 of 162
6.
b
A
C
a
c
Given
Side or
Angle
A
a
Measure
Calculated
Side or
Angle
Measure
B
7.
A
c
b
C
8.
B
a
C
Given
Side or
Angle
B
a
Measure
Calculated
Side or
Angle
Measure
B
a
b
c
Given
Side or
Angle
a
b
Measure
Calculated
Side or
Angle
Measure
A
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 131 of 162
Name ___________________________________ Date __________________ Period ________
Solve the Triangle
Answer Key
Instructions:
Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right
triangles shown below.
Example:
A
c
b
C
B
a
Given
Side or
Angle
A
a
Measure
o
30
1.5 cm
Calculated
Side or
Angle
B
b
c
Measure
60o
2.6 cm
3 cm
Once you have completed your measurements, solve each triangle (find all missing sides and angles),
placing values in the table.
Trade papers with your partner and check each other’s completed triangles using the following
checklist:
____ all calculations are correct
____ the sum of all angles of each triangle is 180o, accuracy within 1o
____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse,
i.e., a2 + b2 = c2
1.
A
c
b
C
a
B
Given
Side or
Angle
B
c
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Measure
27o
2.8 cm
Calculated
Side or
Angle
A
a
b
Measure
63o
2.50 cm
1.27 cm
Columbus City Schools
6/28/13
Page 132 of 162
A
2.
a
C
3.
Given
Side or
Angle
A
c
c
b
B
Measure
o
50
3.3 cm
Calculated
Side or
Angle
B
a
b
Measure
40o
2.53 cm
2.12 cm
A
Given
Side or
Angle
b
c
c
b
C
B
a
Measure
2.5 cm
2.7 cm
Calculated
Side or
Angle
A
B
a
Measure
22.19o
67.81o
1.02 cm
A
4.
c
b
B
C
a
Given
Side or
Angle
A
b
Measure
o
65
1.9 cm
Calculated
Side or
Angle
B
a
c
Measure
25o
4.07 cm
1.72 cm
5.
a
C
B
b
c
Given
Side or
Angle
B
b
Measure
21o
1.6 cm
A
Calculated
Side or
Angle
A
a
c
Measure
69o
4.17 cm
4.46 cm
6.
b
A
c
C
Given
Side or
Angle
A
a
a
Measure
42o
2.5 cm
Calculated
Side or
Angle
B
b
c
Measure
48o
2.78 cm
3.74 cm
B
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 133 of 162
7.
A
b
8.
c
C
a
B
C
a
B
b
c
Given
Side or
Angle
B
a
Given
Side or
Angle
a
b
Measure
29o
2.6 cm
Measure
2.5 cm
4 cm
Calculated
Side or
Angle
A
b
c
Calculated
Side or
Angle
A
B
c
Measure
61o
1.44 cm
2.97 cm
Measure
32o
58o
4.72 cm
A
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 134 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Right Triangle Park
Because of your reputation for drawing and your keen mathematical ability, you have been selected
to design a very special park for your neighborhood! This park will be designed using only right
triangles!
Instructions:
Design a city park using only right triangles. Your park must include at least 4 different components
such as picnic tables, swing sets, slides, gardens, skating ramps, etc.
Draw your design in the area provided below.
Measure one side and one acute angle of each triangle in your design. Solve and label each triangle,
using trigonometry to find the missing sides and angles.
Right Triangle Park
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 135 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Find the Height
When you see an image in a mirror, the angle your line of sight makes with the ground is the same as
the angle the top of the object being reflected makes with the ground as shown below.
You and your work group will use this fact and your knowledge of similar triangles to find the
heights of structures in your school yard.
Instructions:
Select 3 tall objects you wish to measure (tree, flagpole, smokestack, school, goalpost, etc).
Place a mirror on the ground between yourself and the object whose height you are calculating.
Stand so you can see the top of the object in the mirror.
While you stand, your partner will measure the ground distance from you to the mirror and from the
mirror to the object.
Record the measurements on the “Find the Height” data sheet. Record the mirror watcher's height on
the data sheet.
Set up your proportion and find the height of the object.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 136 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Find the Height Data Sheet
Object measured
Sketch your reflection experiment in the box below. Label all measurements.
Proportion _____________________________ Height of Object _________________________
Object measured
Sketch your reflection experiment in the box below. Label all measurements.
Proportion _____________________________ Height of Object _________________________
Object measured
Sketch your reflection experiment in the box below. Label all measurements.
Proportion _____________________________ Height of Object _________________________
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 137 of 162
G-SRT 8
Name ___________________________________ Date __________________ Period ________
Applications of the Pythagorean Theorem
For each of the following word problems, draw a picture to represent the situation, write an equation
and solve for the missing parts.
A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the
house, how high up will the ladder reach?
A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft
from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support
wire?
Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s
ladder?
The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to
second base, how far does the catcher throw?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 138 of 162
Name ___________________________________ Date __________________ Period ________
Applications of the Pythagorean Theorem
Answer Key
For each of the following word problems, draw a picture to represent the situation, write an equation
and solve for the missing parts.
A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the
house, how high up will the ladder reach?
x2 + 152 = 252
x = 20 ft
25 ft
x ft
15 ft
A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft
from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support
wire?
2
2
125 ft
2
x ft
125 + 58 = x
x = 137.8
58 ft
ft
Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s
ladder?
4.32 + x 2 = 72
x = 5.5
xm
7m
4.3 m
The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to
second base, how far does the catcher throw?
902 + 902 = x 2
x = 127.3 ft
90 ft
x ft
90 ft
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 139 of 162
Reteach
Name ___________________________________ Date __________________ Period ________
Memory Match – Up
Students can be put into groups of 3 – 4.
First place all cards face down and have each student take turns drawing two cards. If the two cards
drawn go together as a pair, then the student will keep it as a match. The student with the most
matches wins.
Note: There are 3 cards that say “1”. There are 2 cards that say “ 3 ”. There are 2 cards that say
3 ”. There are 2 cards that say “ 1 ”. Make sure that students know that two cards with the exact
“
2
2
same expression on them are not considered a match. For example: A card with a “1” on it does not
match a card with a “1” on it. A card with a “1” on it is a match with a card that has “tan 45º” on it.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 140 of 162
Memory Match – Up Cards
Pythagorean
Theorem
2
45o
1
?
45o
30o
45o
?
?
60o
45o
?
1
3
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 141 of 162
3
1
2
2
2
1
tan 45º
sin 45º
sin 30º
cos 30º
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 142 of 162
30o
30o
?
60o
60o
?
It can be used to
solve for an acute
angle in a right
triangle.
3
2
sin
1
2
3
2
3
3
-1
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 143 of 162
sin 
cos 
tan 
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
3
1
2
leg2 + leg2 =
hypotenuse2
o
tan 45
o
sin 30
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Page 144 of 162
Columbus City Schools
6/28/13
oo
45
sin 30
cos
tan 30º
sin 60º
cos 60º
tan 60º
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 145 of 162
Memory Match-Up
Answer Key
leg2 + leg2 = hypotenuse2
Pythagorean Theorem
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
sin 
cos 
tan 
1
2
3
2
3
3
2
2
sin 30º
cos 30º
tan 30º
sin 45º
tan 45º
1
3
2
1
2
sin 60º
cos 60º
3
tan 60º
It can be used to solve for an acute angle in
a right triangle.
sin -1
?
45
º
1
45º
?
?
45º
2
45º
?
30º
2
60º
1
30º
60º
?
?
3
30º
60º
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 146 of 162
Reteach
Name ___________________________________ Date __________________ Period ________
Similar Right Triangles and Trigonometric Ratios
Draw a right triangle. Label it ABC, with C being the right angle. Measure the sides in centimeters
and the angles in degrees. Complete this chart by filling in the measurements for each angle and
each side.
Side or
Angle
Measure
A
B
A
B
C
Remember that the trigonometric ratios are defined as shown below.
sin  =
length of opposite leg
length of hypotenuse
cos  =
length of adjacent leg
length of hypotenuse
tan  =
length of opposite leg
length of adjacent leg
Complete the chart from your measurements. Use the trigonometric functions on your calculator to
find the values. If the two sets of values are not about the same, measure and compute again.
Trigonometric Value
From Measurement
From Calculator
sin A
cos A
tan A
sin B
cos B
tan B
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 147 of 162
Draw a right triangle, DEF, whose angles are the same as those in triangle ABC, but whose sides are
twice as long. Complete the chart as you did for triangle ABC.
Side or
Measure
Angle
D
E
d
e
f
Trigonometric Value
From Measurement
From Calculator
sin D
cos D
tan D
sin E
cos E
tan E
Make a triangle GHI, that is similar to the other two triangles, with side GH measuring 20 cm long.
Show how you find the length of the other two sides. What do you know about the sine, cosine, and
tangents of angles G and H?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 148 of 162
Extension
Name ___________________________________ Date __________________ Period ________
Similar Right Triangles and Trigonometric Ratios
Given: rt. ABC ~ rt. DEF
A
D
b
e
c
f
d
E
a
F
B the fact that the triangles are similar
C to find the missing term (?). Write the missing term
Part A: Use
in the space provided.
__________
1.
a d

b ?
__________
2.
? f

b e
__________
3.
e ?

f c
__________
4.
d f

? c
__________
5.
a ?

c f
__________
6.
b e

? f
__________
7.
? b

d a
__________
8.
f c

d ?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 149 of 162
Part B: Describe two ways that similarity proportions can be formed.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Part C: Each of the proportions below are true for the two similar triangles given. The ratios that
form the proportions can be written as trigonometric ratios. Complete the statements below that
correspond to the given proportions to make them true.
A
D
b
e
c
f
d
E
a
1.
3.
5.
Bd
a
=
b
e
sin A = sin _____
C c
f
2.
=
b
e
sin C = sin _____
c
f
=
b
e
cos A = cos _____
a
d
=
c
f
tan A = tan _____
F
4.
6.
a
d
=
b
e
cos C = cos _____
c
f
=
a
d
tan C = tan _____
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 150 of 162
Part D: Use the information in Part C to complete the following statements.
1.
sin C = cos _____
2.
cos A = sin _____
3.
sin A = cos _____
4.
cos C = sin _____
5.
Describe the relationship that exists between angles A and C?
________________________________________________________________________
6.
sin D = cos _____
7.
cos F = sin _____
8.
sin F = cos _____
9.
cos D = sin _____
10.
Describe the relationship that exists between angles D and F?
________________________________________________________________________
Conclusion:
________________________________________________________________________
________________________________________________________________________
Part E: Complete the following statements.
1.
sin 20° = cos _____
2.
cos 35° = sin _____
3.
sin 10° = cos _____
4.
cos 45° = sin _____
5.
sin 30° = cos _____
6.
cos 60° = sin _____
7.
sin x° = cos _____
8.
cos y° = sin _____
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 151 of 162
Name ___________________________________ Date __________________ Period ________
Similar Right Triangles and Trigonometric Ratios
Given: rt. ABC ~ rt. DEF
Answer Key
A
D
b
e
c
f
d
E
a
B
F
C
Part A: Use the fact that the triangles are similar to find the missing term (?). Write the missing term
in the space provided.
e
1.
a d

b ?
c
2.
? f

b e
b
3.
e ?

f c
a
4.
d f

? c
d
5.
a ?

c f
c
6.
b e

? f
e
7.
? b

d a
a
8.
f c

d ?
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 152 of 162
Part B: Describe two ways that similarity proportions can be formed.
1.
One way: Each ratio of the proportion compares a side of one triangle with the corresponding
side of the other triangle.
2.
Another way: Each ratio of the proportion compares two sides from the same triangle with
two corresponding sides of the other triangle.
Part C: Each of the proportions below are true for the two similar triangles given. The ratios that
form the proportions can be written as trigonometric ratios. Complete the statements below that
correspond to the given proportions to make them true.
A
D
b
e
c
f
d
E
a
B
a
d
1.
=
b
e
sin A = sin D
C
2.
c
f
=
b
e
cos A = cos D
3.
5.
F
4.
a
d
=
c
f
tan A = tan D
6.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
c
f
=
b
e
sin C = sin F
a
d
=
b
e
cos C = cos F
c
f
=
a
d
tan C = tan F
Columbus City Schools
6/28/13
Page 153 of 162
Part D: Use the information in Part C to complete the following statements.
1.
sin C = cos A 2.
cos A = sin C
3.
sin A = cos C 4.
cos C = sin A
5.
Describe the relationship that exists between angles A and C?
Angles A and C are complementary angles. The sum of their measures equals 90°.
6.
sin D = cos F 7.
cos F = sin D
8.
sin F = cos D 9.
cos D = sin F
10.
Describe the relationship that exists between angles D and F?
Angles D and F are complementary angles. The sum of their measures equals 90°.
Conclusion:
In a right triangle, the sine of one of the acute angles equals the cosine of the other acute
angle (complement of the angle).
Part E: Complete the following statements.
1.
sin 20° = cos 70°
2.
cos 35° = sin 55°
3.
sin 10° = cos 80°
4.
cos 45° = sin 45°
5.
sin 30° = cos 60°
6.
cos 60° = sin 30°
7.
sin x° = cos (90 – x)° 8.
cos y° = sin (90 – y)°
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 154 of 162
Reteach
Name ___________________________________ Date __________________ Period ________
Hey, All These Formulas Look Alike!
Show your work. Include formulas in your explanations.
Consider the  ABC.
Show thatCBA is a right angle.
C (3, 4)
A (0, 0)
B (3, 0)
Complete the chart.
Slope, AC
Distance, AC
Pythagorean
Theorem, ABC
Tan CAB
Write Formulas
Substitute Values
and Simplify
Compare the expressions and the values for the slope of AC and tan CAB. Are the formulas the
same? Are the values equal? Support your answer by showing your work.
This exercise utilized the first quadrant only. Predict if your conclusions will hold if the triangle is
rotated 90, about the origin, counterclockwise. Test your prediction.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 155 of 162
Name ___________________________________ Date __________________ Period ________
Hey, All These Formulas Look Alike!
Answer Key
Show your work. Include formulas in your explanations.
Consider the  ABC.
Show thatCBA is a right angle.
Solution: If CBA is a right angle, then CB  BA .
If CB  BA , then m1 * m2 = -1. This is a
special case in which the slope of one of the
perpendicular lines is undefined and the slope
of the other line is zero.
40 4
00 0
 

Slope BC =
Slope BA =
33 0
30 3
C (3, 4)
A (0, 0)
B (3, 0)
2. Complete the chart.
Slope, AC
Write
Formulas
m
y2  y1
x2  x1
Pythagorean
Theorem, ABC
(x2-x1)2+(y2-y1)2=AC2
Distance, AC
3  4  AC
2
Substitute
Values and
Simplif
40 4

30 3
length of opp. side
length of adj. side
( x2  x1 )2  ( y2  y1 )2
(3  0)2  (4  0)2  AC
2
9  16  AC
25  AC
5 = AC
Tan CAB
(3-0)2+(4-0)2=(AC)2
32 + 42 = (AC)2
9 + 16 = (AC)2
25 = (AC)2
40 4

30 3
25  ( AC )2
5 = AC
3.
Compare the expressions and the values for the slope of AC and tan CAB. Are the
formulas the same? Are the values equal? Support your answer by showing your work.
The formula for the slope of AC and the equation for tan CAB are the same.
Slope of AC =
y2  y1
x2  x1
Tan CAB =
length of opposite side y2  y1

x2  x1
length of adjacent side
The values are the same. The slope AC of and tan CAB both equal
40 4

30 3
4.
This exercise utilized the first quadrant only. Predict if your conclusions will hold if the
triangle is rotated 90, about the origin, counterclockwise. Test your prediction.
Answers will vary.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 156 of 162
Extension
Name ___________________________________ Date __________________ Period ________
Problem Solving: Trigonometric Ratios
Materials: scientific calculator
Use the information given in the figure below to determine the sine, cosine, and tangent of .
Explain your answer.
Sin  = _______ Cos  = _______ Tan  = _______
(0,5)
B
(3,4)

A
C
(5,0)
Use the information given in the figure below to determine the perimeter of rectangle ABCD.
Support your answer by showing your work.
B
C
100 cm
35
A
D
Perimeter = ____________
3.
Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a
value greater than 1? Why is it that the values of the other two trigonometric ratios can never be
greater than 1? Explain.
4.
John, an employee of the U.S. Forestry Service has been asked to determine the height of a
tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 157 of 162
elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33.
He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the
nearest meter? Support your answer by showing your work and including a diagram.
5.
Determine the perimeter to the nearest centimeter and the area to the nearest square
centimeter of the triangle shown below. Support your answer by showing your work and giving an
explanation.
B
10 cm
A
C
6.
Use what you know about the side lengths of special right triangles to complete the following
table. Express your answers in simplified radical form.
30
45
45
30
45
60
60
Sin
Cos
Tan
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 158 of 162
Name ___________________________________ Date __________________ Period ________
Problem Solving: Trigonometric Ratios
Answer Key
Use the information given in the figure below to determine the sine, cosine, and tangent of .
4
3
4
Explain your answer. Sin  =
Cos  =
Tan  =
(0,5)
5
5
3
B
(3,4)
Solution: The lengths of AC and BC can be determined by
using the coordinates of point B(3,4). The length of AB can be
determined by using the fact that it is a radius of a circle. AB =

5, BC = 4, and AC = 3. By definition:
A
C (5,0)
BC 4
AC 3
BC 4
sin θ =
= ; cos θ =
= ; and tan θ =
=
AB 5
AB 5
AC 3
Use the information given in the figure below to determine the perimeter of rectangle ABCD.
Support your answer by showing your work.
Solution: By definition:
57.36 cm
B
C
AB
AB
sin 35° =
; .5736 
; AB  57.36
100
100
BD
BD
cos 35° =
; .8192 
; BD  81.92
100
100
81.92 cm
100 cm
35
A
D
The perimeter of the rectangle = 2(57.36) +2(81.92) = 278.56 cm.; Perimeter = 278.56 cm
Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value
greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater
than 1? Explain.
Solution: The tangent of an angle can be greater than 1. The sine of an acute angle of a right triangle
length of leg opposite the angle
is defined as
and the cosine of an acute angle of a right triangle is
length of hypotenuse
length of leg adjacent to the angle
defined as
. The length of a leg of a right triangle will always
length of hypotenuse
be less than the length of the hypotenuse. If the numerator of a fraction is less than the denominator,
the fraction is always less than 1.
Therefore, the sine and cosine of an angle will never be greater than 1 by definition of the sine and
cosine ratios.
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 159 of 162
4.
John, an employee of the U.S. Forestry Service has been asked to determine the height of a
tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of
elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33.
He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the
nearest meter? Support your answer by showing your work and including a diagram.
Solution:
h
tan 33° =
24
h
h
.6494 
24
h  16 m
33
24 m
5. Determine the perimeter to the nearest centimeter and the area to the nearest square centimeter of
the triangle shown below. Support your answer by showing your work and giving an
explanation.
Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore
the acute angles each have a measure of 45. The ratio of the sides of a 45- 45- 90 triangle is
10 10 2
B
1:1: 2 . The length of each leg is
=
= 5 2.
2
2
The perimeter of the triangle is
45
10 + 5 2 + 5 2 = 10 + 10 2  24 cm.
10 cm
The area of the triangle is
1
1
• 5 2 • 5 2 = • 25 • 2  25 cm 2 .
2
2
45
A
C
Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore
the acute angles each have a measure of 45. The lengths of the legs can be found by using the sine
and cosine ratios.
B
AC
10
AC = sin 45° •10  .707 •10  7.07 cm
AB
cosB = cos 45° =
10
AB = cos 45° •10  .707 •10  7.07 cm
sinB = sin 45° =
45
10 cm
45
A
C
The perimeter of the triangle is 7.07 + 7.07 + 10 = 24.14  24 cm.
The area of the triangle is (.5)(7.07)(7.07) = 24.99  25 cm2
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
Columbus City Schools
6/28/13
Page 160 of 162
6. Use what you know about the side lengths of special right triangles to complete the following
table. Express your answers in simplified radical form.
30
45
45
60
30
Sin
1
2
Cos
Tan
3
2
1
3
=
3
3
45
1
2
=
2
2
1
2
=
2
2
1
=1
1
CCSSM II
Trigonometric Ratios G-SRT 6, 7, 8
Quarter 1
60
3
2
1
2
3
= 3
1
Columbus City Schools
6/28/13
Page 161 of 162
Grids and Graphics
Addendum
CCSSM II
Comparing Functions and Different Representations of Quadratic Functions FIF 3, 4, 5, 6, 7, 7a, 9, F-BF 1, 1a, 1b, A-CED 1, 2, , F-LE 3, , N-NQ 2, , S-ID
6a, 6b, A-REI 7
Quarter 2
Page 162 of 162
Columbus City Schools
6/28/13
Algebra Tiles Template
Grids and Graphics
Page 1
of 18
Columbus Public Schools 6/27/13
10 by 10 Grids
Grids and Graphics
Page 2
of 18
Columbus Public Schools 6/27/13
20 by 20 Grids
Grids and Graphics
Page 3
of 18
Columbus Public Schools 6/27/13
Small Coordinate Grids
Grids and Graphics
Page 4
of 18
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Dot Paper
Grids and Graphics
Page 5
of 18
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Isometric Dot Paper
Grids and Graphics
Page 6
of 18
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Quarter-Inch Grid
Grids and Graphics
Page 7
of 18
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Half-Inch Graph Paper
Grids and Graphics
Page 8
of 18
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One-Inch Grid Paper
Grids and Graphics
Page 9
of 18
Columbus Public Schools 6/27/13
Centimeter Grid
Grids and Graphics
Page 10
of 18
Columbus Public Schools 6/27/13
Pascal’s Triangle Template
Grids and Graphics
Page 11
of 18
Columbus Public Schools 6/27/13
Probability Spinners
Grids and Graphics
Page 12
of 18
Columbus Public Schools 6/27/13
Protractor
80
70
60
50
120
130
140
30
150
20
160
10 170
110
100
90
100
110
80 70
120
130
140
30
150
20
160
10 170
40
110
80
100
90
100
80
110
70
60
Grids and Graphics
50
120
130
140
30
150
20
160
10 170
140
150
20
160
10 170
110
90
100
100
80
110
70
130
50
120
40
130
140
30
150
20
160
10 170
140
40
150
30
20
160
10 170
110
100
90
100
80
110
70
130
50
120
130
140
30
150
20
160
10 170
140
40
40
150
30
20
100
80
110
70
120
160
10 170
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of 18
130
60
140
150
30
20
160
10 170
110
80
90
100
100
80
110
70
120
130
60
140
50
40
150
30
20
160
10 170
70
50
100
90
40
60
120
60
80
50
70
50
80
110
60
120
60
110
120
80 70
130
60
140
50
40
150
30
160
20
40
30
70
120
130
140
30
150
20
160
10 170
130
100
10 170
70
40
80
110
60
120
50
60
40
160
10 170
70
50
150
20
60
120
40
130
140
30
150
20
160
10 170
120
130
140
30
150
20
160
10 170
100
90
40
30
40
50
50
140
50
70
50
130
60
40
60
80
70
60
120
110
80
100
90
100
80
110
70
120
130
60
140
50
40
150
30
20
160
10 170
Columbus Public Schools 6/27/13
Tangram Template
Grids and Graphics
Page 14
of 18
Columbus Public Schools 6/27/13
Blank 11
Grids and Graphics
11 Geoboards
Page 15
of 18
Columbus Public Schools 6/27/13
Blank Number Lines
Grids and Graphics
Page 16
of 18
Columbus Public Schools 6/27/13
Rulers
mm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
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1
2
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7
8
9
10
11
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13
14
15
16
mm
mm
mm
mm
mm
Grids and Graphics
Page 17
of 18
Columbus Public Schools 6/27/13
Websites for Graph Paper
and More!
Below you will find great web sites to visit for graph paper and other things to use in your math
activities.
http://www.mathematicshelpcentral.com/graph_paper.htm (requires Adobe Acrobat Reader
version 5.0 or higher to view or print graphs)
This is a wonderful collection of all different kinds of graphs from full-page format to several per
page for multiple problems. You will also find a page set up specifically for proofs and graph
paper for 3-space, polar coordinates, and logarithms.
http://mathpc04.plymouth.edu/gpaper.html
At this site you will find several versions of coordinate, semi-logarithmic, full logarithmic, polar,
and triangular graph paper.
http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm
This is site to go to if you need to make math manipulatives. Cutouts are available for pattern
blocks, geometric shapes, base-ten and base-five blocks, xy blocks, attribute blocks, rods, and
color tiles. Graph paper can be printed as well.
http://www.handygraph.com/free_graphs.htm
Several forms of coordinate graphs and number lines sized just right for homework and tests.
http://donnayoung.org/frm/spepaper.htm
Not only does this site have graph paper it contains notebook paper, Lego design paper, music
paper, and award certificates.
http://www.lib.utexas.edu/maps/map_sites/outline_sites.html#W
Outline maps for states, countries, regions, and the world.
Grids and Graphics
Page 18
of 18
Columbus Public Schools 6/27/13