Soh Cah Toa

3 Day Lesson Plan

Jen Thomas

21 October 2010

Jen ThomasCI 40310/21/10

3 DAY LESSON PLANDiscovering and Proving Soh Cah Toa

Day 1 Review of sine and cosine and introduction of soh cah toaDay 2 Continuation of soh cah toaDay 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 StandardsReasoning 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 1Title: Sine and Cosine Review and Soh Cah Toa IntroductionGrade Level: High school GeometryLength of Class: 45-50 minutesNumber of Students: 18 studentsPrior 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 board2. Chalk or markers3. Worksheet for every students (warm up, exit slip, follow along)4. Pencils5. Projector or white board would be useful but not necessary

Motivation/Hook1. 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 single-handedly 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) = sinIV. 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. I will then ask the students where do the numbers in the chart come

from?VII. 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.

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

b

c

A

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  ABC =AC/ CB

VI. 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).

VII. 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.

VIII. 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 worksheetA. 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.

Assessment1. 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 worksheet2. 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 1Solve for x.

1. 34 = 12

x Cross multiply so 3x = 48 x = 16

2. x20

=65 Cross multiply so 5x = 120 x = 24

3. Since it is opposite/hypotenuse it is sin(x) = 7/14 = ½sin−1( 1

2 )=30

X = _______ 30What length is the hypotenuse? __________ 14

Day 1 WorksheetName: ____________________Date: ____________________

14

x

7

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 O H C A H T O AI p y o d y a p dN p p s j p n p jE o o I a o g o a

Identifying Opposite, Adjacent and Hypotenuse

1. Identify the side that is opposite of YZX: Line YX 2. Identify the side that is adjacent to YZX: Line XZ

3. Identify the sides that are opposite and adjacent to IHU. Opposite Side: Line IUAdjacent Side: Line HI

Part II

1. How long is the side opposite of ACB? 122. How long is the hypotenuse? 134. How long is the side adjacent to ACB? 9

6. How long is the side opposite of 1? 247. How long is the hypotenuse? 25 9. How long is the side adjacent to 1? 7

1. What side is adjacent to MLN? Line ML2. What is the hypotenuse? Line LN

3. Calculate cos(MLN): adjacent/hypotenuse = 8/10 cos−1( 45 )=36.8699

4. Calculate cos(LMN) adjacent/hypotenuse = 6/10 = 3/5 cos−1( 35 )=53.1301

More challenging Problems:

Find the sine, cosine and tangent of a.

5. sin(a): opposite/hypotenuse = 12/13 sin−1( 1213 )=67.38

6. cos(a) : adjacent/hypotenuse = 9/13 cos−1( 913 )=46.1869

7. tan(a) : opposite/adjacent = 12/9 = 4/3 tan−1( 43 )=53.1301

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 1Name: ____________________Date: ____________________

Exit Slip1. 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, HypotenuseCosine, Adjacent, HypotenuseTangent, Opposite, Adjacent

3. How are the sine, cosine, and tangent ratios related?

All come from a right triangle.

DAY 2Title: Finding the angle of right triangles using trigonometric functionsGrade Level: High school Geometry Length of Class: 45-50 minutesNumber of Students: 18 studentsPrior 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 board2. Chalk or markers3. Worksheet for every students4. Graphing Calculators5. Pencils

Motivation/Hook1. 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 problem2. (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.

E. How do we find x?F. I will then guide the students to discover the answer with open

questions.A. What do we want?B. 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”

C. I then assume that a student will realize that we are given the opposite and hypotenuse sides of the triangle.

D. 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.

E. Eventually, I assume the students will realize that we need to use the sine function to discover the angle.

F. After making that realization, I will tell my students to plug in what we learned from the day before.

27

x

G. I will ask a student to give me their equation and either they or I will write it up on the board.III. 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.

H. Sin(x) = opposite/hypotenuse = 2/7II. 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. 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?”

J. 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.

K. Anyone have any idea what the inverse is of sin?L. 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).M. 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)) = 1III. I will ask the students what this meansIV. 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.

N. What do we plug into the sin^(-1)? X, the angle, or the ratio, the sides?O. 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).

P. (10 minutes) Draw a diagram to show when to use sine or its inverse and what to plug in

sinKnown Angle

Known sides Sin^(-1)

II. Ex: sin(30) = .5 and sin^(-1)(.5) = 30

III. 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.

G. Hand out the worksheet to everyone in class.H. Go over a few problems as a class, during this time I can assess

to see if the students understand the new concept.I. 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.

J. 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

K. 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.

L. By using a comparison I am testing their understanding and assessing to see if they can compare and contrast 2 similar concepts

M. 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 worksheet2. 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 21. Label on the right triangle:

hypotenuse Hypotenuse Short Leg long leg Long Leg Right Angle right angle Vertices (all 3 points/intersection of lines) short leg

State the reciprocals of the following:

2. 87 7

8 3.ba b

a 4. -34 −4

3 5. HypotenuseOpposite Opposite

Hypotenuse

Evaluate and leave the answer in exact form:

6. 3√2

3√2 * √2

√2 = 3√22

Worksheet Day 2:

Name: ____________________Date: ____________________

Sine, Cosine, Tangent

8. What is X? tan(52) = x

18 1.2799 = x18 x = 1.2799*18 = 23.0389

9. How long is the hypotenuse of this triangle?

Cos(52) = 18H .61566 =

18H .65155H = 18 H =

18.65155 = 29.2368

10. What is X? cos(16) = 24X .96126 =

24X .96126x = 24 x =

24.96126 =

24.9672

11. How long is the side IJ? Sin(16) = IJ24 .2756 =

IJ24 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

2) What is the length of x? Cos(63) = 3X .45399 =

3X .45399X = 3 X =

3.45399 = 6.608

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

3. What is Y? (In words!) It appears to be the long leg, but we would have to calculate to be sure

4. What is the length of y? tan(55) = Y22 1.428 =

Y22 Y = 1.428 * 22 = 31.4193

5. Find X. sin(16) = 14x .2756 =

14x .2756x = 14 x =

14.2756 = 50.7914

6. What is another way to find x? cos(16) = 48x .9612 =

48x .9612x = 48 x =

48.9612 =

49.9347. Are your answers exactly the same? Why? No because I had to round a little bit, but the two

answers are very close.

Properly use soh cah toa to find XDAY 3: ExtensionTitle: Using the sunset, timer, and cosine to find the radius of the earthGrade Level: High school Geometry Length of Class: 45-50 minutesNumber of Students: 18 studentsPrior 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 board2. Chalk or markers3. Worksheet for every students4. Graphing Calculators5. Pencils6. Globe7. Figure8. Flashlight

Motivation/Hook2. 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 ΘC. 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 roundA. 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 storyA. 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) DemonstrationA. 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 WorkA. 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 figurei. “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 Equationsa. Have trouble with the side that is height + radiusb. 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-Up2. Have groups come together3. Will go through solutions to the problem by having the

students vocalize what they gota. 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 SlipA. 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?”

Assessment1. 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 worksheet2. 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.

6

r t

r

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.

cos(t) = r

r+6

cos(1

8640 ) = r

r+6

.999999 = r

r+6.99999(r + 6) = r.99999r + 5.9994 = r5.9994 = r - .99999r.59994 = .00001r.59994/.00001 = rR = 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?

In minutes In hours In daysT = 10 secs 10/60 10/(60*60) 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 C A H T O A

Sine Cosine TangentOpposite Adjacent OppositeHypotenuse Hypotenuse Adjacent

2. Find x.

tan(x) = 312 =

14

x = tan−1(14 ) = 14.0364

3. Find x.

Cos(60) = 9x

12 =

9x

x = 9*2 = 18

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.

3

x

12

60

x

9