Soh Cah Toa
3 Day Lesson Plan
Jen Thomas
21 October 2010
Jen Thomas
CI 403
10/21/10
3 DAY LESSON PLAN
Discovering and Proving Soh Cah Toa
Day 1 Review of sine and cosine and introduction of soh cah toa
Day 2 Continuation of soh cah toa
Day 3 Soh Cah Toa Extension: Measuring the Earth’s Radius
I.
Unit Introduction/Rationale
This unit is for the end of the year of a high school geometry class or the beginning of the
year lesson for a high school trigonometry class. In my 3 day lesson unit I am covering a review
of cosine and sine, right triangles, soh cah toa, and an extension to apply what they learned.
After everything, I plan on giving a short quiz on the fourth day. I peer taught my last lesson, the
extension. I chose that because I thought it was a very interesting problem that can be solved in
class that relates to the real world. I liked how what at first seems like a very advanced problem,
can be solved using high school math. I used many different types of instructional strategies in
my lesson plans. Every day I have worksheets for the students to follow along with or do
practice problems with for immediate assessment. I also include warm ups everyday so the
students can have a quick review and will be able to more easily connect previous lessons to the
current topic, creating a larger picture. The warm up is also a way of training my students to
come in and immediately start working quietly. This way the class can be productive the whole
period. The last day, I have an activity to motivate and involve the students more than usual.
This way they are more enticed to work on the problem and use their critical thinking skills to
solve it. I intentionally asked very vague questions in the hopes that the students will be trained
throughout the year to find what they know, what they want, and how to figure out a problem. I
think this is essential because I want my students to learn how to logically think. This is one of
the most important parts that everyone single person in the classroom can learn and bring into the
real world. I also want them to work in groups so they can verbalize math and discuss important
ideas. Learning and understanding math is important, but I think if a student can explain it out
loud shows they truly have a deep understanding of the topic.
Throughout my lesson I may not constantly be using hands on activities, but I do
continually seek feedback via questions and worksheets and made an effort to include many
graphics. Including more visuals helps the students be able to see what they are learning about.
This way I incorporate different kinds of learners into my classroom. I also want to focus on
writing in an organized fashion on the board, so then I can erase less regularly and keep all of the
important information up front.
I am not planning on using technology in the classroom an abundant amount. I do not want
my students to use calculators as a crutch, but they will need them for the extension day. I will
let calculators be available, but they will have to come get the calculators themselves. This way,
the calculator is not just sitting on their desks when they are solving a problem, but instead they
are hopefully more likely to sit there and figure it out rather than get up to retrieve a calculator.
Depending on what access the classroom has, I would like to use a projector or smart board or
sorts to present this information, but it is not necessary.
II.
Objective/Purpose
After these 3 days I expect my students will know and be able to demonstrate their
knowledge of sine, cosine, finding missing angle or side in a right triangle by using soh cah toa,
and finding the radius of the earth under certain circumstances. These are the mathematical
concepts that I want them to be able to understand. One of my main goals of the extension is
having my students applying their critical thinking skills. This day is less guided so the students
can try to analyze the problem and use their previous knowledge to discover the answer. I want
my students to be able to know what they want and how to get it because these are skills they can
use in everyday life. This will be very useful for students who intend on furthering their math
career, but also for other students who plan on using critical and logical thinking skills in their
future, which hopefully is all of my students. If students are frustrated and do not understand
why they are learning a certain topic or how it relates to them or real life, I will explain that a
main goal is to teach them how to think logically. This is best learned from going through math
problems, step by step.
III.
Meeting the Needs of your Students
It depends on what languages my ELL students speak for me to decide what I would do with
them. If they all speak the same language, I would group them together. I would encourage
them to work together and interpret words or definitions that are hard, especially for the student
who speaks minimal English. Together, I believe they would be able to keep one another on
track. If the students spoke different languages, I still may keep them together. It all depends on
the specific students. If the students are more comfortable with other students who do not speak
the dominant language, then I would keep them together with some of my more patient students
who speak English fluently. This way, my ELL students do not feel like they are the only one
going through this experience, but also have students they can trust that are willing to help.
I am assuming if I am given a Spanish speaking aid, that my ELL students speak Spanish.
This way I can put them in a group in the back or to the side of the classroom with the Spanish
speaking aid always nearby. I do not necessarily want to put them in back, but keeping them
together and not the center of attention may help them. This way they can see what everyone
else is doing and copy their actions if they do not understand what is happening. This way, they
can still be involved in the class without being humiliated. When I make lessons, I can also
make different ones or ask the Spanish aid if there are any words that I could put the translation
next to that would help the students. If so, I would work with the aid to make the worksheets
more understandable for the ELL students. I would also make sure the aid was aware that I am
always open to ideas or suggestions that would improve my lessons and my ELL students’
understanding. I am not an expert on teaching ELL, unlike my aid. I should want to use all of
the resources, which means that I should listen or ask for their advice from the aid.
Since I have 2 student’s that have IEP’s for extended testing time due to reading
proficiency, there are a couple things I would do. I want to make sure when I have an
explanation on a worksheet, that the statements are short and concise. Also if I have notes, I
would make them so they are fill in the blank so they can still pay attention without being
affected.
Having students with various levels of academic motivation in class also effects how I
will teach. I want to make the class more interactive. Lecturing the majorty of time will not
keep less motivated students involved. I have to do a little lecture with practice and worksheets
constantly throughout. This way I can keep the most students engaged as possible. I also want
to vary the activities, so my students do not become bored. By the last day’s activity, there is
practically no lecture and almost entirely group work. This way they can get involved and try to
creatively come up with ways to measure the radius of the earth.
I have written throughout my lesson plans when I do certain things just for certain
students.
Connection to NCTM Standards
Mathematics Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
This is done throughout all 3 days of my lessons. During the first day, they have
to find the missing side of the triangle. After realizing they need to find a side, they need
to determine what given information is given to decide if they should use sine, cosine, or
tangent. Once they figure that out, they can solve for the variable and discover the
missing side. The second day, the students find out how to solve if an angle is missing.
First they need to decide if they want to solve for an angle or side, then with the
information given if they should solve it using sine, cosine, or tangent. Once all of that is
determined, the students need to solve for the variable. On the third day they are given a
problem to find the earth’s radius, which is much more challenging. This is teaching the
students how to truly persevere through a problem, even though they are given little
guidance. Eventually, the students will solve it in groups and as a class.
2. Reason abstractly and quantitatively.
Each day the students use abstract thoughts when they solely use variables and
use quantitative reasoning when they use numbers to solve the problem. On the first day,
after having a discussion about trigonometric functions, they discover how they can solve
for a missing side using only variables. This is more abstract since they are not using
any specific measurements. Once they understand the concept, then they can apply it
with numbers and real life situations. The second day is a very similar situation. Except
at first I use numbers and compare to the day before to see if the students can use
quantitative reasoning to discover the angle. Later on in the lesson I show them that the
inverse of sin is sin^(-1), which is more abstract thinking. The last day uses abstract
reasoning because the students need to discover how to use the information given to draw
the correct diagram and solve the equation they create. Once they are able to set up the
equation, they use quantitative reasoning to solve the problem.
3. Construct viable arguments and critique the reasoning of others.
The first day I ask the students where chart of sine and cosine numbers come
from. They have to come up with arguments why these numbers exist, then we discuss as
a class how they come from triangles. In the discussion, students can agree and disagree
with one another. The second day the students have to come up with ways to discover the
missing angle, instead of the missing side. The last day the students have to construct a
diagram and equation from a situation. Working in groups will force the students to
discuss the situation and thereby reasoning and critiquing one another’s ideas.
Content Standards
Geometry Standards for Grades 9–12

use geometric ideas to solve problems in, and gain insights into, other disciplines
and other areas of interest such as art and architecture.
The first day, when the students learn about missing side, there are word problems they
have to do that takes right triangles into real world situations. This also applies on the
second day. On the last day, the whole period is devoted to using geometry to gain
insights to solving other problems. They are given a situation and asked to find the
radius of the earth. It takes a deep understanding of right triangles and soh, cah, toa to
understand how to even create the diagram.
Process Standards
Reasoning and Proof Standard for Grades 9–12
 make and investigate mathematical conjectures;
Throughout the 3 days of lessons, the students use new and old information to create new
conjectures.
Communication Standard for Grades 9–12
 organize and consolidate their mathematical thinking through communication;
The students do this when they have mathematical discussions when trying to make a new
conjecture as well as when they have to work together on the last day to figure out the
radius of the earth.
 communicate their mathematical thinking coherently and clearly to peers, teachers,
and others;
Communication is essential in math. I think this is most emphasized on the last day when
they have to communicate to their peers in their group what they are thinking and to the
teacher if they have questions or are confused. I also have students explain their
answers, which is another way to vocalize what they knew to the rest of the class.
Connections Standard for Grades 9–12
 recognize and apply mathematics in contexts outside of mathematics.
This is most evident in the third day’s lesson when they have to use right triangles and
soh, cah, toa from the past few days to find the radius of the earth.
DAY 1
Title: Sine and Cosine Review and Soh Cah Toa Introduction
Grade Level: High school Geometry
Length of Class: 45-50 minutes
Number of Students: 18 students
Prior Knowledge: Know what a hypotenuse is, know how to cross multiply, are able to find the
missing side of a right triangle using Pythagorean Theorem, and know a little bit about sine,
cosine, and tangent.
Objectives:
1. Students will be able to understand what input and output of trigonometric functions are.
2. Students will identify whether to solve the missing side of a right triangle using sine,
cosine, or tangent.
3. Students will understand why to apply soh, cah, toa to right triangles.
4. Students will utilize the mnemonic device, soh cah toa.
Materials:
1. Chalkboard or a white board
2. Chalk or markers
3. Worksheet for every students (warm up, exit slip, follow along)
4. Pencils
5. Projector or white board would be useful but not necessary
Motivation/Hook
1. Start the lesson telling the students how Chief Soh-cah-toa has helped students and then
they have to find the missing side of the triangle.
Lesson Procedure (45-50 minutes):
1. Have students come in and sit in their assigned seats. They are accustomed to taking out
their homework and silently working on the warm up questions.
2. (3 minutes) Have them work on the warm up worksheet by themselves.
3. While the students are doing the warm up, they will also have their homework out. I will
go around checking to see if a few answers are correct and that it looks like they put forth
effort on the homework. I will also keep an eye on the warm up for assessment to make
sure that the students remember and understand previous concepts. This way I am
assessing what they know from yesterday via the homework and the warm up. To make
sure the students truly do their homework and ask questions I will have random
homework quizzes.
4. (3 minutes) After I have checked everyone’s homework and everyone has finished the
warm up I will have different students go to the board to answer the questions and discuss
it with the class. I will use mathematically significant terms and revoice what the
students are saying. I will also tell the students to give me a thumbs up on their desk
when they are done with the warm up. This way I can tell who is done and it is more
subtle, so people who are still working will not be distracted.
5. (5 minutes) After going over the warm up, I will then ask if there are any questions about
the homework. If so, then I will have students answer the questions on the board and
have a discussion.
Transition: I will tell the class something along the lines of, “before we start learning
today’s lesson, I need to tell everyone about the single most important figure in the
history of Trigonometry, who was not some mathematician like Blaise Pascal or René
DeCartes, but a mythical American Indian chief named Chief Soh-cah-toa. This man’s
history and accomplishments are clouded by time, but over the years he has singlehandedly helped countless Geometry students memorize the formulas for angle
functions in a right triangle.”
6. (2 minutes) Then I will review trigonometric functions with my students by saying
something such as, “Before we meet Chief Soh-cah-toa, we need to review our
trig functions.”
A. I will begin by asking my students what the three primary trig functions are.
B. Hopefully they will answer sine, cosine, and tangent.
I.
If not, I will remind them of one and then they
should be able to come up with at least one if not
the other two.
II.
I assume that they already know a little bit about
the functions.
5. (10 minutes) When the students say the functions, I will write their whole names
out and then write the equal sign and then their abbreviation.
I.
I will also explain that sine is our f(x), or our function, we need to
include (x) so it looks like sin(x).
II.
Then I will go into more detail explaining how even though this is what
we mean, it is common for the (x) part to be dropped, but the students
cannot forget that it is supposed to be there.
1. If the students are confused I will use a simple equation and compare
the two’s graph or table to show that it is the same idea.
III.
Ex: sine = sin(x) = sin
IV.
I will discuss with the class like in f(x), how x is the input, x is the input
for sin(x) and represents an angle (for this specific example in a right
triangle)
V.
The table down below will be shown on the board, I will explain this will
come in handy but will not need to be memorized because after using
these common ones so many times it will happen automatically.
0.0000 = sin(0 °)
1.0000 = cos(0 °)
0.0000 = tan(0 °)
0.5000 = sin(30 °) 0.8660 = cos(30 °) 0.5773 = tan(30 °)
0.7071 = sin(45 °) 0.7071 = cos(45 °) 1.000 = tan(45 °)
0.8660 = sin(60 °) 0.5000 = cos(60 °) 1.7320 = tan(60 °)
1.0000 = sin(90 °) 0.0000 = cos(90 °) +infinity = tan(90 °)
VI.
VII.
I will then ask the students where do the numbers in the chart come
from?
How did we get those numbers?
Transition: After a short discussion, I will tell the students that we will discover this
using trigonometric functions and right triangles.
7. (15 minutes) I will hand out a worksheet that the students can follow along with.
This way students who are better audio learners can pay attention and worry less
about having to write everything down.
8. I will project on the board, a right triangle with sides a, b, and c (c being the
hypotenuse) and wanting to find angle A, which is between sides b and c.
c
a
A
b
9. After drawing, labeling, and explaining what we want to find, I will ask the
students if they have any ideas how to find the angle. I talk through each step to
mimic what I want my students to do on the homework and all of their problems.
A. I will hint that they should think about the trig functions that we just discussed
if they are having difficulty thinking of any ideas.
B. After discussing it, I will explain there are multiple trig functions we can use to
discover angle A depending on what information we know.
C. Let’s first discuss what another name for side c, the side that is always the
longest and directly across from the 90 degree angle. The word hypotenuse
was on their warm up, but I want to emphasize this mathematical term.
I.
To emphasize that I think this term is important,
and for students who may not understand English
well, I will write next to the c = hypotenuse.
D. I will then write down how sine, cosine, or tangent can be used to find the
angle.
I.
I will have a worksheet that I also project in the classroom to show
when you use each function.
II.
I will make sure to write out, for example, sin(x) = opposite/hypotenuse
and highlight the first letter of each word then write SOH next to it.
This way, it will be easier for students to connect and understand
where the mnemonic is coming from, especially students that have
reading disabilities or do not speak much English.
III.
Sine Function
A. The sine of an angle is always the ratio of the (opposite
side)/(hypotenuse).
Sine of
ABC=AC /AB
(opposite side)/(hypotenuse)
IV.
Cosine Function
A. The cosine of an angle is always the ratio of the (adjacent side/
hypotenuse)
The cosine of
ABC =CB/AB
(adjacent side)/(hypotenuse)
V.
Tangent Function
A. The tangent of an angle is always the ratio of the (opposite
side)/(adjacent side)
The tangent of
VI.
VII.
VIII.
ABC =AC/ CB
I will remind my students that opposite means the side directly across
from the angle that we want. The adjacent side is the shorter side of
the triangle touching the angle (but is NOT the hypotenuse).
After writing these down, showing on a triangle, and writing (and
underlining the first letter in each word to show the correlation), I will
ask students what the three mean.
By having the words along with the pictures, this encompasses
students who do not speak English well to use it as a visual aid,
students who have reading disabilities, along with student who prefer
learning methods of visual or audio.
IX.
I will then tell them that they just met the helpful Chief Soh-cah-toa and
hope for a prosperous friendship.
10. (5 minutes) After making sure the students understand the concept, I will pass
out a worksheet
A. I will make sure that I leave the notes on the board, so there is an easy
reference if the students do not remember which function to use.
B. As a class, we will do the first few problems together
I.
This will help me gauge the class to see how much of the lesson they
understand with immediate feedback.
C. When I feel the class understands the concept, I will let them get in groups of
2 or 3 and work on the worksheet together.
I.
This is another way to assess the class, especially as the problems
become more difficult.
11. I will have the class come back together and do a quick overview/summary of the
lesson today as a class to make sure they understand the concept.
I.
Having the students summarize is another assessment and encourages
them to vocalize math while using math terms.
12. Depending on time and how much I like the problems given in the book, I will
give either the worksheet or some problems out of the book as homework.
13. The last few minutes of class I plan on handing out an exit slip so I can assess
the students to see if they understood the overall concept.
I.
I want to make sure there are not a lot of words or it uses wording taught
in class for my students with reading disabilities and the ones that do not
speak a lot of English.
Assessment
1. We walk around the classroom to make sure that the students are doing the worksheet
correctly and stay on track.
a. We do this while the students are working on the warm up and the worksheet
2. We ask questions aloud to the students and ask for explanations why (and have wait time so
all of the students have time to actually think about the answer in their head first)
a. We discuss with the class the warm up questions
b. When introducing the main topic
3. We make sure the class remembers the main concepts from the previous lesson by the warm
up. We then have them write on the same sheet the summary of today’s lesson for the exit slip.
We will have them hand these in so we can get a general idea if the students understand the
major concepts.
Name: ____________________
Date: ____________________
Warm-Up Day 1
Solve for x.
1.
3
2.
𝑥
4
=
20
12
𝑥
=
6
5
Cross multiply so 3x = 48  x = 16
Cross multiply so 5x = 120  x = 24
3.
14
Since it is opposite/hypotenuse it is
sin(x) = 7/14 = ½
x
𝑠𝑖𝑛−1 (2) = 30
7
1
X = _______ 30
What length is the hypotenuse? __________
14
Day 1 Worksheet
Name: ____________________
Date: ____________________
SOHCAHTOA WORKSHEET
(Sine, cosine and tangent)
Sine of
ABC=AC /AB
(opposite side)/(hypotenuse)
The cosine of
ABC =CB/AB
(adjacent side)/(hypotenuse)
The tangent of
ABC =AC/ CB
(only soh cah toa on ws, not enough room to write it all out)
S
I
N
E
O
p
p
o
H
y
p
o
C
o
s
I
A
d
j
a
H
y
p
o
T
a
n
g
O
p
p
o
A
d
j
a
Identifying Opposite, Adjacent and Hypotenuse
1. Identify the side that is opposite of
2. Identify the side that is adjacent to
Line YX
Line XZ
YZX:
YZX:
3. Identify the sides that are opposite and adjacent to
Opposite Side: Line IU
Adjacent Side: Line HI
Part II
1. How long is the side opposite of
ACB? 12
2. How long is the hypotenuse?
13
4. How long is the side adjacent to
ACB? 9
6. How long is the side opposite of
1?
7. How long is the hypotenuse? 25
9. How long is the side adjacent to
1?
24
7
IHU.
1. What side is adjacent to
MLN? Line ML
2. What is the hypotenuse? Line LN
3. Calculate cos(MLN):
4
adjacent/hypotenuse = 8/10  𝑐𝑜𝑠 −1 (5) = 36.8699
3
4. Calculate cos(LMN) adjacent/hypotenuse = 6/10 = 3/5  𝑐𝑜𝑠 −1 ( ) = 53.1301
5
More challenging Problems:
Find the sine, cosine and tangent of a.
12
5. sin(a): opposite/hypotenuse = 12/13  𝑠𝑖𝑛−1 ( ) = 67.38
6. cos(a) : adjacent/hypotenuse = 9/13  𝑐𝑜𝑠
−1
13
9
(13) = 46.1869
4
7. tan(a) : opposite/adjacent = 12/9 = 4/3  𝑡𝑎𝑛−1 ( ) = 53.1301
3
Are all of angle a’s the same? Why? No, I think this is because they are all slightly
different due to how you measure it.
Day 1
Name: ____________________
Date: ____________________
Exit Slip
1. How do you know which side is the hypotenuse?
_________________________________________________________
_________________________________________________________
It is always the longest side and across from the right angle.
2. What do these stand for?
S O H
C A H
T O A
Sine, Opposite, Hypotenuse
Cosine, Adjacent, Hypotenuse
Tangent, Opposite, Adjacent
3. How are the sine, cosine, and tangent ratios related?
All come from a right triangle.
DAY 2
Title: Finding the angle of right triangles using trigonometric functions
Grade Level: High school Geometry
Length of Class: 45-50 minutes
Number of Students: 18 students
Prior Knowledge: Students should know how to solve for a missing leg given the length of the
other leg using Soh Cah Toa, and can see a right triangle and set up the equation to find the
missing side or angle.
Objectives:
1. Students will understand how to apply soh, cah, toa to real life situations.
2. Students will apply their knowledge of soh, cah, toa to discover the missing angle.
3. Students will be able to decide what way of solving to find the missing side or angle.
Materials:
1. Chalkboard or a white board
2. Chalk or markers
3. Worksheet for every students
4. Graphing Calculators
5. Pencils
Motivation/Hook
1. We will challenge the students how to find a missing angle if they already know all three
sides. It is very similar to yesterday’s problem, but they have to go about a different way
to solve for the angle.
Lesson Procedure (45-50 minutes):
1. When students come in make sure they take out their homework and quietly work on
the warm up problem
2. (3 minutes) I will walk around the classroom and make sure that a few homework
problems are correct and that the students are on the right track for the warm up
problems.
A. This way I can assess their knowledge from yesterday by looking at their work on
the warm up problem and the homework. This is very important since today’s
lesson will be off of that lesson and showing another way to solve for angles
using soh, cah, toa.
3. (5 minutes) I will then allow an open discussion and students come up to the board to
show their work for the warm up. As before, I will make sure to emphasize important
terminology and revoice what the students are saying.
A. Throughout discussions, I will also get a better idea of what they know and how
my students know how to verbalize math concepts and if they know how to use
the terms in sentences.
4. (3 minutes) After the warm up, I will see if anyone has any questions about the
homework.
A. These can be answered by other students; unless everyone is
confused then I will personally review the topic.
B. In case the concept of soh cah toa is confusing, I might have to
do a quick review and a few practice problems with
explanations; this will depend on my assessments earlier in the
class.
Transition: Now that you know how to solve for the missing side of a right triangle with
trigonometric functions, how do we solve for a missing angle if we know all of the side
measurements?
C. (25 minutes) I will first ask for any ideas on how this can be
done.
D. Then I will draw out what we are talking about.
I.
This way my students that barely speak English can see
what is going on and better able understand.
7
2
x
A.
B.
C.
D.
E.
F.
G.
E. How do we find x?
F. I will then guide the students to discover the answer with open
questions.
What do we want?
What sides are given?
I.
If the students just say the numbers, then I will ask using what we learned
yesterday with opposite, adjacent, and hypotenuse sides to tell me what
sides we have according to the angle that we want.
II.
I can prompt them with a statement such as “lets use the words we used in
soh cah toa, from class yesterday”
I then assume that a student will realize that we are given the opposite and
hypotenuse sides of the triangle.
After we make this realization, I will ask what we do next.
I.
If no students answer or are not sure about the answer I can rephrase my
question.
II.
What function uses opposite and hypotenuse?
III.
If still confused, I will encourage the students to look at their notes from
yesterday and depending on how much of a struggle it is possibly asking
their new friend Chief Soh-cah-toa. I will not just tell them the answer
because I do not want them to expect that and what to teach them how to
think if they are really confused.
Eventually, I assume the students will realize that we need to use the sine function
to discover the angle.
After making that realization, I will tell my students to plug in what we learned
from the day before.
I will ask a student to give me their equation and either they or I will write it up
on the board.
III.
H.
I.
J.
K.
L.
M.
N.
O.
P.
By writing it on the board while saying it will help make connections for
the students who do not speak English well. For them, I want to make
sure to write and draw out as much as possible so they are not entirely lost.
I can also label where they come from so no one gets confused.
Sin(x) = opposite/hypotenuse = 2/7
II.
After I write this down I will have one of my students
explain why we used that equations from looking at the
figure we drew earlier.
I will then ask the class, “Are we done?”
III.
I assume that someone will say no because we never
found the angle and if no one realizes that I will point
out what we decided we wanted to find earlier.
IV.
I will then ask, “Why not?”
V.
“What should be our next step?”
VI.
If the students do not make the connection that they
need x by itself I will ask, “How do we get x by itself?”
or “We now have an equation, how would we solve this
algebraically?”
We usually divide by the number there, or in other words, multiply by its inverse.
Using the word inverse, after rephrasing what the students say emphasizes the
vocab word.
Anyone have any idea what the inverse is of sin?
After a short discussion saying when a number is multiplied by its inverse it is
equal to one we will eventually come to the conclusion that the inverse is sin^(-1).
Another way of saying sin^(-1) is 1/sin(x).
I.
I will write it out on the board sin^(-1) = 1/sin(x)
II.
I will then multiply sin(x)*(1/sin(x)) = 1
III.
I will ask the students what this means
IV.
I will assume that they will recall from our previous discussion that it is its
inverse if it equals 1 when multiplied together. If there are no answers, I
will make sure to suggest to see if it agrees with the definition we just
discussed.
What do we plug into the sin^(-1)? X, the angle, or the ratio, the sides?
By giving the student multiple options, I want to make them think and reconfirm
that x stands for the angle, so we need to plug in the sides of the triangle because
we are going backwards (want the angle, x).
(10 minutes) Draw a diagram to show when to use sine or its inverse and what to
plug in
sin
Known Angle
Known sides
Sin^(-1)
II.
Ex: sin(30) = .5 and sin^(-1)(.5) = 30
III.
G.
H.
I.
J.
K.
L.
M.
This confirms that sin^(-1) is the inverse of sin, this will
confirm for any students that are hesitant to believe that
sin^(-1) is the inverse of sin.
IV.
By drawing out the diagram and including a few terms
will help my students that don’t speak much English
understand the lesson better.
Hand out the worksheet to everyone in class.
Go over a few problems as a class, during this time I can assess
to see if the students understand the new concept.
Have students participate by coming up to the board and
answering your questions, trying to get all students involved in
at least solving part of a problem if they are confused.
Allow the students to finish the worksheet by themselves or
with a partner, while they are finishing walk around the
classroom to keep assessing their understanding
Bring the class back together to do a quick review of the two
different ways using trig functions to find angles or sides in a
right triangle using a Venn diagram.
By using a comparison I am testing their understanding and
assessing to see if they can compare and contrast 2 similar
concepts
After the class discussion, I will hand out the exit slip for the
students to do before they leave, which is another way to
assess. I will also not want there to be lots of writing on it
because then it won’t do a good job of assessing my students
with reading disabilities or the ones that do not speak a lot of
English.
Assessment
1. We walk around the classroom to make sure that the students are doing the worksheet
correctly and stay on track.
a. We do this while the students are working on the warm up and the worksheet
2. We ask questions aloud to the students and ask for explanations why (and have wait time so
all of the students have time to actually think about the answer in their head first)
a. We discuss with the class the warm up questions
b. We discuss how to find the angle and the difference between finding a side and an
angle.
c. During the wrap up we have the students discuss their solutions and compare and
contrast.
3. We make sure the class remembers the main concepts from the previous lesson by the warm
up. We then have them write on the same sheet the summary of today’s lesson for the exit slip.
We will have them hand these in so we can get a general idea if the students understand the
major concepts.
Name: ____________________
Date: ____________________
Warm-Up Day 2
1. Label on the right triangle:
 Hypotenuse
 Short Leg
long leg
 Long Leg
 Right Angle
right angle
 Vertices (all 3 points/intersection of lines)
hypotenuse
short leg
State the reciprocals of the following:
2.
8
7
7
8
3.
𝑏
𝑏
𝑎
𝑎
4. -
3
4
−
4
3
5.
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Evaluate and leave the answer in exact form:
6.
3
√2
3
√2
*
√2
√2
=
3√2
2
Worksheet Day 2:
Name: ____________________
Date: ____________________
Sine, Cosine, Tangent
𝑥
𝑥
8. What is X? tan(52) = 18 1.2799 = 18  x = 1.2799*18 = 23.0389
9. How long is the hypotenuse of this triangle?
18
18
18
Cos(52) =  .61566 =  .65155H = 18  H =
= 29.2368
𝐻
10. What is X? cos(16) =
𝐻
24
𝑋
.65155
 .96126 =
𝐼𝐽
24
𝑋
24
 .96126x = 24  x = .96126 = 24.9672
𝐼𝐽
11. How long is the side IJ? Sin(16) = 24  .2756 = 24  IJ = .2756*24 = 6.615
Using SOH CAH TOA to find a side of a triangle.
1) What is x? (Use words!) X is the hypotenuse
3
3
3
2) What is the length of x? Cos(63) = 𝑋  .45399 = 𝑋  .45399X = 3  X = .45399 = 6.608
3. What is Y? (In words!) It appears to be the long leg, but we would have to calculate to be sure
𝑌
𝑌
4. What is the length of y? tan(55) = 22  1.428 = 22  Y = 1.428 * 22 = 31.4193
5. Find X. sin(16) =
14
𝑥
 .2756 =
14
𝑥
 .2756x = 14  x =
48
14
.2756
48
= 50.7914
48
6. What is another way to find x? cos(16) = 𝑥  .9612 = 𝑥  .9612x = 48  x = .9612 = 49.934
7. Are your answers exactly the same? Why? No because I had to round a little bit, but the two
answers are very close.
Summary:
What did you learn?
How to find the missing length of a triangle using soh cah toa and that there is more than one
way to find x.
I can…
 Find the missing side length of a triangle

Find x two different ways

Properly use soh cah toa to find X
DAY 3: Extension
Title: Using the sunset, timer, and cosine to find the radius of the earth
Grade Level: High school Geometry
Length of Class: 45-50 minutes
Number of Students: 18 students
Prior Knowledge: Just learned soh, cah, toa. Can see a right triangle and set up the equation to
find the missing side or angle.
Objectives:
1. Students will be able to create a diagram from a story.
2. Students will identify useful information from a story.
3. Students will understand how to apply soh, cah, toa to real life situations.
4. Students will utilize cosine to discover the length of a right triangle.
5. Students will apply their knowledge of soh, cah, toa to discover the length of the
Earth’s radius.
Materials:
1. Chalkboard or a white board
2. Chalk or markers
3. Worksheet for every students
4. Graphing Calculators
5. Pencils
6. Globe
7. Figure
8. Flashlight
Motivation/Hook
2. We are introducing the students to a story and having them finish and solve it. It applies
to real life and is posed more as a challenge, which will hopefully motivate them to want
to discover the answer.
Lesson Procedure (40 minutes):
A. When students come into class make sure they are only sitting 3 to 4 people at a table.
B. (3 minutes) Have the warm up written on the board when the students come in so they
can sit down and start working on it immediately. We will draw out a right triangle with
sides x, y, and z and angle Θ. Next to it we will write: What is sin Θ, cos Θ, and tan Θ
in terms of x, y, and z?
x
y
z
Θ
C. When class starts tell the students that they need to get out a piece of paper and can cut it
in half and share with a neighbor if they want to.
D. We will give the students a few minutes to work on it and during this time we will walk
around the room to make sure students recall this from the lesson before.
E. After the majority of the class appears to be done we will ask for students to tell us their
answer and we will write them on the board.
Transition: We are going to use these to solve a very interesting problem…how big is
the earth?
F. (5 minutes) Discussion on why Earth is round
A. Before we discover how to calculate the Earth’s size, let’s discuss how we know the
Earth is round.
B. Scientists used to believe that the Earth is flat. Falling off of waterfalls definitely
makes it seem like there is an edge to the world. Other than being told, how do you
know that the earth is round?
I.
We are expecting some crazy answers such as “we were told,” pictures of
Earth from space, etc.
II.
If no one brings it up we can suggest watching a ship disappear when it sail
out to sea (it looks like it gradually sinks, with the hull disappearing first and
then the sail)
Transition: Now that we are convinced that the Earth truly is round, Aaron is going to
tell you a story that needs to be solved to find the size of the Earth.
G. (3 minutes) Aaron’s story
A. I want to tell you about one indication that we have that the earth is round. My oldest
son Spencer sometimes gets up VERY early in the morning. We spent some time
in Chicago this last summer and one such morning Spencer and I went down to the
lake (Michigan) to see the sunrise. When I first say the rays of sunlight peaking over
the water, I pointed it out to Spencer. He claimed not to see the sun and when I bent
down to show him: Sure enough, the sun could no longer be seen. The earth is
round, and if I am higher up than Spencer, I will be able to see further along the
curvature of the earth (and for this reason, the sun rise comes a few seconds earlier to
me than to Spencer).
Transition: We'll now describe an experiment that can be done to quantify this
phenomenon; this can then be used to determine the approximation the radius of the
earth.
H. (3 minutes) Demonstration
A. We will hand out the worksheets and tell the students to write down what information
they find out from the demonstration (figurine is 6 feet and the difference between the
2 sunrises is 10 seconds)
B. Using a globe, a flashlight, and a figurine, we will act out the following experiment:
A person watches the sunrise over a large body of water (say Lake Michigan). The
figure will stand near the edge of the lake with its’ eyes exactly 6 ft above water level.
The instant the sun is visible, the figure starts its stopwatch and lies down and waits
again for the sun. Its’ eyes are very close to the water level and the instant the sun is
again visible she stops her stop watch. The stopwatch reads 10 seconds.
B. We will then tell the class to stay in their groups of 3 to 4 people at their table and
work on the worksheet. If they have any problems please raise their hands to ask for
help.
C. The challenge we have for you is this: Use the information from this experiment to
give an approximate value for the radius of the earth.
I. (15 minutes) Group Work
A. We will walk around during this time assisting the groups with any questions they
might have while working on the worksheet. Since there will be approximately 16
students, we should have about 4 groups.
B. The objective is for the students to do the heavy work of coming up with a plan to
solve the problem. The computations are relatively easy in comparison with this.
C. Problems we expect them to encounter:
I.
Students having trouble knowing where to start
a. We want to help guide them with questions, not statements. So we can ask
them “Can you draw our demonstration?” or “What information do you know
from our demonstration?”
b. If they are confused about the two times the sun rises we can ask “Would it be
easier to understand if you drew out the two times with two different
pictures?”
II.
Students having difficulties labeling their figure
i. “What shapes did we use in yesterday’s lesson or today’s warm up?”
“How can you create that shape in your diagram?”
ii. “Can you label the diagram with the information you already know?”
III.
Setting Up the Equations
a. Have trouble with the side that is height + radius
b. sin Θ = R/(R + Θ)
c. Students will have trouble if they round too much (make sure Θ is not
equal to 1!)
IV.
If the students still seem very confused after 5 minutes we will have the class
come back together. We will have a student literally lay on the ground and
stand up for another presentation and draw the diagram together.
a. Hopefully this will help the students and will let them
struggle, but not enough for them to give up.
J. (10 minutes) Wrap-Up
2. Have groups come together
3. Will go through solutions to the problem by having the
students vocalize what they got
a. We will make sure to revocalize, have wait time, and
write the main points on the board
b. We will know what solutions the students have from
walking around, so if there are multiple solutions we
will encourage the groups to tell the rest of the class
about their alternative way.
c. This will also depend on how the groups do during the
class and how far they get on the worksheet
d. If the students go through the worksheet quickly, there
is a challenge problem on the back
e. And if they get through all of that we plan on asking
them to think of ways they can measure the distance of
the moon from the earth using similar ideas.
K. (1 minute) Exit Slip
A. Depending on time we will have an exit slip question that can be written on the back
of their warm up (which we will collect once they leave the classroom.)
B. “Describe, without computations, how the initial information (6 feet and 10
seconds) can be used to determine the radius of the Earth.
I.
We want to see if the students got the general idea of the day’s lesson
by having them summarize it without using numbers.
L. Homework
A. We are not going to give them actual homework, just something to think about that
previews the next day’s lesson
B. “Is there any reason to believe that the moon is closer to the Earth than the
sun?”
C. “Is there any reason to believe that the moon is closer to the Earth than the
sun?”
Assessment
1. We walk around the classroom to make sure that the students are doing the worksheet
correctly and stay on track.
a. We do this while the students are working on the warm up and the worksheet
2. We ask questions aloud to the students and ask for explanations why (and have wait time so
all of the students have time to actually think about the answer in their head first)
a. We discuss with the class the warm up questions
b. We discuss how we know the earth is round
c. During the wrap up we have the students discuss their solutions
3. We make sure the class remembers the main concepts from the previous lesson by the warm
up. We then have them write on the same sheet the summary of today’s lesson for the exit slip.
We will have them hand these in so we can get a general idea if the students understand the
major concepts.
Name:_____________________
Period:__________
The Earth is Round?!
1. What do you know?
Batman is 6 feet tall
Takes him 10 seconds to see it when lying down
2. What do you want to know?
Radius of the Earth
3. Find what you want to know.
6
𝑟
cos(t) = 𝑟+6
r
1
𝑟
cos(8640) = 𝑟+6
t
r
𝑟
.999999 = 𝑟+6
.99999(r + 6) = r
.99999r + 5.9994 = r
5.9994 = r - .99999r
.59994 = .00001r
.59994/.00001 = r
R = 599994 feet
Helpful Questions



Can you draw a helpful figure?
How is what you know connected to what you want to know?
Is there an equation that models the demonstration?
T = 10 secs
In minutes
10/60
In hours
10/(60*60)
In days
10/(60*60*24) = 1/8640
Day 4:
There will be a summative quiz over the last 3 days to make sure the students have mastered soh
cah toa. It will be a short quiz to assess the students and make sure that they understood the last
few days. Here are some possible questions that I may ask to make sure that the students have a
good understanding of this small unit.
Day 4: Quiz
Name:__________________
Date: __________________
1. What do these stand for?
S
O
H
Sine
Opposite
Hypotenuse
C
A
Cosine
Adjacent
Hypotenuse
2. Find x.
H
T
O
A
Tangent
Opposite
Adjacent
3
3
1
x
tan(x) = 12 = 4
1
x = 𝑡𝑎𝑛−1 (4) = 14.0364
12
3. Find x.
x
9
Cos(60) = 𝑥
1
9
=𝑥
x = 9*2 = 18
2
60
9
4. Summarize what you learned from finding the radius of the earth.
5.
I learned that I could find the radius of earth by using how long it takes to see the sun
from when I am standing and lying down and soh cah toa that we learned in class.