Trigonometry Unit:
Honors Geometry Grade 9
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Instructor:
Mike Bensley
Table of Contents
Introduction…………………………………………………….. Page 3
Curriculum Matrix………………………………………........... Page 4
Day 1 Lesson Plan………………………………………........... Page 6
Day 2 Lesson Plan……………………………………………... Page 11
Day 3 Lesson Plan……………………………………………... Page 15
Day 4 Lesson Plan……………………………………………... Page 18
Day 5 Lesson Plan……………………………………………... Page 21
Unit Exam……………………………………………………... Page 24
Grading Rationale……………………………………………... Page 27
Grade Tracker…………………………………………………. Page 28
References…………………………………………………….. Page 29
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Introduction
This unit is an introduction to trigonometry that takes place in a
ninth grade geometry class. The duration of this unit is five ninety
minute class periods.
In this unit students will use their prior knowledge of the
Pythagorean theorem to derive the distance formula. Then they will
use right triangle trigonometry to learn basic trigonometric functions
and their relationships to the sides and angles in a right triangle. The
heart of this unit focuses on using both the Pythagorean theorem and
trigonometry to solve right triangles and the final segment introduces
special right triangles and the unit circle.
Students will form the foundation of trigonometry to be later
used in Pre Calculus and Calculus. They will gain an understanding
of how to apply trigonometry to the real world through the use of
maps and by collecting data within their own homes and classrooms.
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Curriculum Matrix
Standard
15.0-Students use
the Pythagorean
theorem to
determine distance
and find missing side
lenghts of sides of
right triangles
17.0- Students prove
theorems by using
coordinate geometry,
including the
midpoint of a line
segment, the
distance formula, and
various forms of
equations of lines
and circles.
18.0- Students know
the definitions of the
basic trigonometric
functions defined by
the angles of a right
triangle. They also
know and are able to
use elementary
relationships
beteween them.
Lesson 1
Lesson 2


Lesson 3

Lesson 4
Lesson 5



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19.0- Students use
trigonometric
functions to solve for
an unknown length of
a side of a righ
triangle, given an
angle and a length of
a side.
20.0- Students know
and are able to use
angle and side
relationships in
problems with special
right triangles, such
as 30, 60 and 90
degree triangles and
45, 45 and 90 degree
triangles





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Day 1 Lesson Plan
Teacher: Mike Bensley
Grade Level: 9th (Geometry)
Lesson Title: Distance and Midpoint Formula
Overview:
Students will learn and apply the Pythagorean theorem in order to derive the
distance formula. They will then use the distance formula and the midpoint
formula to find distances and exact middles between cities on a map.
Connection to the Curriculum:
This lesson is taught in Geometry and prepares students for their
introduction into Trigonometry. Students will also be exposed to Geography
through the use of maps.
Connection to Standards:
California High School Geometry Standards#15 and #17
Multiple Intelligences/Modalities:
Students will develop their spatial intelligence through the use of maps.
They will also engage each other with their linguistic intelligence through
discussion. Logical and mathematical intelligence will be emphasized when
deriving and using the Pythagorean theorem, the distance formula and the
midpoint formula.
Technology:
This lesson will incorporate the use of a smartboard projector in order to
make accurate diagrams. Students will be using graphing calculators.
Time:
This lesson is set for a ninety minute block period.
Objectives:
 Students will comprehend the derivation of the midpoint and distance
formulas through the use of the Pythagorean theorem.
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 Students will apply the midpoint and distance formulas in the domain
of mapping.
 Students will evaluate the effectiveness of the distance and midpoint
formula through discussion.
 Students will synthesize new ideas on where the distance and
midpoint formula can be applied in their own lives.
Suggested Procedure:
Opening: 10 minutes
Students will be given maps and asked to come up with ideas on how they
might find the distance between cities if they could only measure horizontal
and vertical distances. They will pick two cities to come back to later in the
lesson.
Development:
Direct Instruction- 20 minutes
 Have students construct a right triangle using compass and ruler.
Students will then identify and label hypotenuse and legs of their
particular right triangle.
 Show derivation of Pythagorean theorem
Group Participation- 20 minutes
 Students will be put into groups of four and work on a five question
problem set applying the Pythagorean theorem.
 While in groups, students will divide the section in their textbook on
the distance formula into four parts. Students will outline their
sections by making specific note of vocabulary and examples.
Students will compile their notes as a group to form one group
outline. P. 19-20 and 34-35 in Geometry textbook
Direct Instruction- 10 minutes
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 Show two examples of the use of the distance formula and midpoint
formula.
Back to the Maps- 10 minutes
 Students will use the maps from opening activity, along with the
distance and midpoint formula to find the distance between the two
cities that they identified on the map.
Closing: 10 minutes
In groups of two students will come up with three examples of where they
can use the distance and/or midpoint formula in real life.
Student Assessment: 10 minutes
Assign problem set from the book and select specific problems for student to
work on as the class comes to an end. Roam the room and observe students
work as they work on the problem set.
P. 22 # 34, 35 and P. 38 # 17, 18 from Geometry Textbook
Extending the Lesson:
Students will apply distance formula to objects in their houses and confirm
its validity by measuring the actual distance and then comparing calculations
with actual distance.
Additional Resources:
www.purplemath.com/modules/distform.htm
Adaptations for Diverse Learners:
Provide students that operate visually with graphic organizers from
textbook. There is a lot of information and formulas that go into this lesson
and for this reason some students can benefit from a graphic organizer.
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Additional Material for Day 1 Lesson Plan
Name:__________________
Date:__________________
Pythagorean Theorem is Everywhere: Even in Your House
You will need a tape measure and a calculator
1. Find two objects in your house that you can measure the distance between using a tape
measure (They should both be sitting on the floor). Label the drawing and measure the
distances x and y.
Object A:__________________
X:____
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Object B:__________________
Y:____
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2. Use X and Y along with the Pythagorean theorem to find the distance between the two
objects. Remember your units (ft. , in. , cm….)
3. Now measure the actual distance between object A and object B using your tape
measure and compare your results. Think of potential places that could have resulted in
error.
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Day 2 Lesson Plan
Teacher: Mike Bensley
Grade Level: 9th (Geometry)
Lesson Title: Equations of Circles and Basic Trigonometric Functions
Overview:
Students learn to define the equation of circles and use basic trigonometric
functions to solve trigonometric identities.
Connection to the Curriculum:
This continues to develop student’s comprehension of trigonometry and will
also start to form a foundation for conic sections in Algebra II. Circles are
also an inherent part of an artists toolkit.
Connection to Standards:
California Geometry Standards#17 and #18
Multiple Intelligences/Modalities:
Students are engaged on a bodily kinesthetic level through the construction
of the equation of a circle. Students will also be stimulated on a linguistic
level through discussion. The discussion will also give students a means of
using their interpersonal intelligence.
Technology:
I will use the smartboard and projector to demonstrate direct instruction and
the internet to show the derivation of the equation of a circle.
Time:
This lesson will take ninety minutes.
Objectives:
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 Students are able to comprehend the derivation of the equation of a
circle
 Students will be able to construct a circle using compass and ruler
 Students will engage each other in a discussion of how to apply
trigonometric functions.
Suggested Procedure:
Opening: 5 minutes
In pairs students will discuss how they can find the missing side of a right
triangle given two other sides. They will also discuss the relationship
between side lengths and angles in a right triangle.
Development:
Construction- 10 minutes
 Students will construct a right triangle using compass and ruler.
 They will label one of the acute angles as angle A
 They identify the side opposite angle A, the side adjacent to angle A
and the hypotenuse (label each side appropriately)
Direct Instruction- 20 minutes
 Students will learn the sine, cosine and tangent ratios on a right
triangle.
Practice Problems- 10 minutes
 In groups of two students will work on a four question problem set in
which they will create the three trigonometric ratios discussed above.
P. 562 # 10-13 from Geometry Textbook
Discussion- 5 minutes
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 In the same groups of two, students will discuss the anatomy of a
circle. They will come up with two parts of a circle that they feel are
most critical
Direct Instruction- 10 minutes
 I will derive the equation of a circle on the smartboard.
Guided exploration- 15 minutes
 Students will follow a guided exploration out of the textbook to derive
simple trigonometric functions.
P. 567 Activity at bottom of page from Geometry textbook
Closing: 10 minutes
On a fresh sheet of paper, students will draw a right triangle, label one of the
acute angles angle A, measure angle A using a protractor and finally define
all three trigonometric functions of the angle A.
Student Assessment: 5 minutes
Students will start their problem set from the textbook. I will pick specific
problems from the book for them to work on as the class comes to a close. I
will roam the room looking for comprehension.
P. 563 # 31, 35, 38 from the Geometry Textbook
Additional Resources:
Equation of a Circle youtube:
http://www.youtube.com/watch?v=HjN9TTRrQiA
Equation of a Circle Webquest:
http://www.zunal.com/process.php?w=17526
Adaptations for Diverse Learners:
There is a lot of difficult language in this lesson. For students that have
difficulty with pronunciation I will suggest different websites that focus on
pronunciation and demonstrate people using the language in context. The
above mentioned websites are examples of these resources.
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Additional Material for Day 2 Lesson Plan
Name:_________________
Date:_________________
What my angle of elevation?
1. Find one flight of stairs in the school. With a ruler, measure the length of the rise and
the length of the run.
Rise=
Run=
2. Find the angle of elevation of the stairs using inverse trigonometric functions. Be sure
to show all work.
Rise=
Angle of elevation=
Run=
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Day 3 Lesson Plan
Teacher: Mike Bensley
Grade Level: 9th (Geometry)
Lesson Title: Solving Right Triangles
Overview:
Students will learn to solve for the missing sides and angles of a right
triangle in two cases:
Case 1- Given one side and one angle
Case 2- Given two sides
Connection to the Curriculum:
Solving right triangles is a mathematical application that appears in Algebra
II and Calculus preparation courses. This lesson will come back in the later
scope of Geometry and the above mentioned courses.
Connection to Standards:
California Geometry Standard#15 and #19
Multiple Intelligences/Modalities:
This lesson is both spatial and kinesthetic. Students will see various solving
techniques presented during lecture that require a logical mathematical
intelligence and will construct triangles with compass and ruler.
Technology:
I will use the smartboard and projector to present direct instruction and
students will be using both trigonometric and inverse trigonometric
functions on their calculators to solve for missing sides and angles in right
triangles.
Time:
This will be a ninety minute lesson.
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Objectives:
 Students apply both trigonometric and inverse trigonometric functions
to solve for the missing sides and angles in right triangles.
 Students analyze the effectiveness of trigonometry in the real world.
Suggested Procedure:
Opening: 10 minutes
Students will be put in pairs. Each pair will cut a right triangle from a sheet
of paper. They will then measure all the sides of the triangle using a ruler
and the angles using a protractor. We will come back to these groups later in
the development.
Development:
Direct Instruction- 20 minutes
 Teach students what it means to solve a right triangle.
 Show student how to use trigonometric functions to solve a right
triangle given one side and one angle.
Group Section Outline- 15 minutes
 In groups of fours students will break the section on solving right
triangles into four parts. Each student will outline the section that they
are responsible for. They will come together as a group and
collaborate to make a chapter outline, by noting key definitions and
examples.
P. 567- 569 in Geometry Textbook
 After the outline is done students will be given two problems from the
book and will work the problem out as a team. I will roam the
classroom and facilitate.
P. 570-571 # 23, 32 in Geometry Textbook
Direct Instruction- 15 minutes
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 Show students how to use inverse trigonometric functions to solve
right triangles when they are given two sides and no angles.
Come back to Opening Exercise- 15 minutes
 Students will look at the original triangle that they created out of
paper and confirm the accuracy of their measurements using
trigonometric functions and inverse trigonometric functions.
Closing: 10 minutes
Students will identify one triangle that they can see on the walls of the
classroom and then will discuss in a five sentence paragraph how they would
solve their particular triangle.
Student Assessment: 5 minutes
Students will start their problem set from the textbook. I will pick specific
problems from the book for them to work on as the class comes to a close. I
will roam the room looking for comprehension.
P. 570-571 # 22, 27, 33, 35, 39 from Geometry Textbook
Extending the Lesson:
Students will do a worksheet in which they will identify different flights of
stairs within the school and find their angles of elevation using a measuring
stick and their new knowledge of inverse trigonometric functions. Using
clinometers from class students can find the heights of tall objects such as
trees and buildings.
Adaptations for Diverse Learners:
Students that have a hard time using a compass or a protractor will be given
the opportunity to come into class after school for a thirty minute workshop.
It is difficult to visit every student during a lesson that might need help with
the tools used in class. This workshop will really aid students that are not
mechanically driven.
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Day 4 Lesson Plan
Teacher: Mike Bensley
Grade Level: 9th(Geometry)
Lesson Title: Special Right Triangles
Overview:
In this lesson students will explore their previously acquired trigonometric
skills and use them to explore two types of special right triangles: 30, 60, 90
degree triangle and 45, 45, 90 degree triangles.
Connection to the Curriculum:
This is a building block for topics to come in Trigonometry.
Connection to Standards:
California Geometry Standards
#15, #19, and #20
Multiple Intelligences/Modalities:
Students will use linguistic intelligence both in writing and discussion. They
will operate their logical sense in the relationship between angle measure
and side lengths in right triangles. Finally they will use their spatial
intelligence to recognize patterns inherent in solving right triangles.
Technology:
I will use smartboard technology to present direct instruction. Students will
watch a short film on the formation of special right triangles.
Time:
This is a ninety minute lesson.
Objectives:
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 Students will comprehend the relationship between solving right
triangles and special right triangles.
 Students will analyze the types of angles can be used to find exact
values of trigonometric functions.
 Students will apply special right triangles to the unit circle.
Suggested Procedure:
Opening: 5 minutes
Show a short video clip on special right triangles.
http://www.youtube.com/watch?v=NeIm_aSFd3I
Development:
Discussion- 5 minutes
 Students will discuss the types of triangle that they saw in the video
and any other numerical information that they saw in groups of two
Direct Instruction- 15 minutes
 Teach students about the relationships of the side lengths in a 45, 45,
90 degree triangle.
Group practice- 20 minutes
 In their same groups of two students will show why the relationship is
true for a 45, 45, 90 degree triangle with use of the Pythagorean
theorem.
P. 552 example 2 from Geometry Textbook
 Students will then construct a 45, 45, 90 degree triangle using a
protractor and a ruler and show that their initial conjectures are true.
Direct Instruction- 15 minutes
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 Teach students the relationships between sides in a 30, 60, 90 degree
triangle.
Group practice- 20 minutes
 Students are put into new pairs of two students and will show why the
relationship is true for a 30, 60, 90 degree triangle with use of the
Pythagorean theorem.
P. 552 example 3 from Geometry Textbook
 Students will then construct a 30, 60, 90 degree triangle using a
protractor and a ruler and show that their initial conjectures are true.
Closing: 5 minutes
Have students write down the physical triangles and the relationships that
exist amongst side lengths and angles. Students will turn this sheet as a
formative assessment.
Study Guide: 5 minutes
Students will start their study guide for the unit exam to follow.
P. 582-584 #1-21 from Geometry Textbook
Student Assessment:
Students will go to
http://www.regentsprep.org/Regents/math/algtrig/ATT2/PracSpecial.htm
and complete the online worksheet and then print their results to bring to
class.
Additional Resources:
Extremely informative website on special right triangleshttp://mathworld.wolfram.com/RightTriangle.html
Adaptations for Diverse Learners:
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There are many memorization techniques that go along with the topics
taught in this lesson. Student that are having a hard time remembering all the
information taught will be given a guided activity that takes them through
the construction of these memorization techniques.
Day 5 Lesson Plan
Teacher: Mike Bensley
Grade Level: 9th(Geometry)
Lesson Title: Introduction to Unit Circle
Overview:
Students will apply their knowledge of special right triangles to find the
exact trigonometric values of angles on the unit circle.
Connection to the Curriculum:
This lesson lies at the heart of trigonometry and will aid students in their
exploration of Algebra II and Calculus.
Connection to Standards:
California Geometry Standards#19 and #20
Multiple Intelligences/Modalities:
 Students use their logical intuition to relate the unit circle to the
previous lesson on special right triangles.
 Students use their spatial skills to find patterns that can be generalized
from one quadrant of the unit circle to the remaining three quadrants.
 Students will engage their linguistic skills in discussion.
Technology:
Direct instruction will be presented through use of the smartboard and
projector. Students will use their calculators during the direct instruction
portion of the lesson and during the assessment.
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Time:
 The direct instruction portion of this lesson will last only twenty
minutes.
 The assessment will take one hour.
 The total lesson will take place in a ninety minute session.
Objectives:
Suggested Procedure:
Opening: 10 minutes
Students will formulate two questions each from any of the material covered
in the last four lessons. In groups of four students will explore each other’s
questions. The class will disperse from groups and I will take five questions
from the students that they were not able to answer in groups.
Development:
Direct Instruction- 20 minutes
 Students learn how to recognize different components of the unit
circle.
 Students learn to relate special right triangles to the unit circle.
 Students learn to find reference angles.
Study Session- 5 minutes
 Students will have five minutes to relax and prepare for the exam. I
will take any questions for the students during this period (before the
test I like to play a little inspirational music. The Beatles “We can
work it out”.)
Closing: 55 minutes
Students will have the whole remainder of the period to finish their exams.
Unit assessment. The exam should take about one hour.
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Extending the Lesson:
Students will read and take notes on the section from their textbook
regarding the unit circle.
P. 585-590 from Geometry Textbook
Additional Resources:
Ways to remember the whole unit circlehttp://www.youtube.com/watch?v=cIVpemcoAlY
Adaptations for Diverse Learners:
For students that need more time on their exams I will allot time during
lunch in the following days for them to finish.
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Unit Exam
Name:__________________
Date:__________________
Trigonometry Unit Exam
1) Find the midpoint between the two points (2,-7) and (-4, 12).
2) Find the distance between the two points (-5,8) and (4, 7) (Write you solutions as an
exact answer using radicals).
3) Pick two cities on the map below and use the Pythagorean theorem to find the distance
between them (Be sure to convert your answer into miles using the scale on the map).
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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4) What is the equation of a circle with center (7,-3) and a radius of 12?
5) If cos()=3/5 find the other five trigonometric ratios of .
6) Solve the right triangle below:
5 ft
23
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7) Solve the right triangle below:
7 in.
20 in.
8) Give exact answers from the unit circle based on the questions below
sin(30)=?
cos(45)=?
sin(120)=?
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Grading Rationale
Grading Rationale for Honors Geometry: Trigonometry Unit
Homework(20%), Unit Exam(40%), constructions(15%), attendance(10%), in class
assignments(15%)
A math course requires many components to learning. Geometry is a hands on topic and
for this reason my students will be actively getting their hands dirty. Throughout this
course students will be engaged in projects, constructions and in class assignments. This
will be conducted in class and at home.
Each of these components will comprise fifteen percent of a student’s grade. It is critical
that students actively work with the material in geometry. Each of these components is
worth a letter grade so that students do not feel as though they were minor. Together they
comprise a total of forty five percent.
Another large component to any math class is practice. Students will be doing anywhere
from one to two hours of work from the book each night. Homework from the book will
focus on applying formulas in order to solve equations and defining key terms.
Homework is worth fifteen percent of a student’s total grade. In high school math classes
students need to start getting used to homework comprising less of their grade. They
need to see homework as a way to prepare for their exams and not a way to earn a grade.
For this reason I made homework worth twenty percent and put much of the rest of the
weight into assessments.
There will be two types of summative assessments in my class: unit tests and quarterly
exams. Chapter tests will take place at the end of each chapter and quarterly exams will
take place at the end of each quarter. High school students in a college preparatory class
such as geometry need to get used to taking grade-making assessments. Each quarter
there will be three chapter tests worth a total of twenty percent of the total grade and one
quarterly exam worth another twenty percent of the total grade. In this way tests will
comprise forty percent of a student’s total grade. There is no way to pass with bad grades
on exams.
Students cannot learn math without the help of guided instruction. For this reason
attendance will be worth ten percent of their total grade.
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Grade Tracker:
Trigonometry Unit
Categrory:
Homwork:(20%)
Trigonometry Unit
Assignment 1
Assignment 2
Assignment 3
Assignment 4
/100
/100
/100
/100
Assessments:
Chapter tests:(40%)
Unit Exam
total/400*.2=
A
/100
total/100*.4=
B
Constructions:(15%)
Perpendicular Bisector
Parallel Lines
Equilateral Triangle
Medains of Triangle
Angle bisector
/100
/100
/100
/100
/100
total/500*.15=
C
In Class Assignments:(15%)
In Class 1
In Class 2
In Class 3
In Class 4
/100
/100
/100
/100
total/400*.15=
D
Attendance:(10%)
/100
total/100*.1=
E
Total grade= 100*(A+B+C+D+E)
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References
Larson, R., Boswell, L., & Stiff, L. (2004). Geometry . Evanston, Ill.: McDougal Littell.
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