What About Triangles? - PYTHAGORASREADS - home About Triangles? We see them everywhere, they are one of the most common shapes in our world. This unit will focus on what we need to

What About Triangles?

We see them everywhere, they are one of the most common shapes in our world. This unit will focus on what we need to know about triangles to begin to understand their place in art, structure, and history.

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Introduction Differentiation to a Mathematics Triangles Unit9th grade TrigonometryEducating Students with Disabilities in Secondary SettingsRachel KopelsThe State University of New York at Albany

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Message to the Reader / Teacher

My name is Rachel Kopels and I am in my first year of a Masters in Secondary Education Program at the University at Albany. I look forward to teaching Math in either public, private, or charter high schools. Prior to the MSSE program, I was a consultant at IBM Global Business Services. My teaching experience is

limited to teaching adults at companies computer programs and tutoring my friends upon request. I have always loved math and I felt that I could share my interest and views about math with others and teach them how applicable it will really be in their lifetime.

I chose to teach a unit on Introducing Trigonometry. So many students hate trigonometry as they do not understand the base concepts, which halts their progress in following units. I would like to present the unit so that the students can feel comfortable moving forward into more complex trigonometric units. Trigonometry is everywhere really, and the ability to do basic trig tasks will

allow students to become more versatile adults. Some of the many fields that use trigonometry or trigonometric functions include astronomy (hence navigation on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging, pharmacy, chemistry, number theory (hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying

and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development. To bring this home, students will be required to talk about math and triangles with their family and adult family friends to discover how useful it is in everyday life and work.

In this lesson, I attempt to focus on differentiation of lessons such that students who experience more difficulties in mathematics can still succeed while the students who excel in math can still be challenged. I would like for peers to work together and teach each other how they came to understand the concepts. Additionally I would like the students to be more proactive in math practice, review, reflection, and correction of their mistakes, as that is how people learn.

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Standards and BenchmarksThe following THE UNIVERSITY OF THE STATE OF NEW YORK: THE STATE EDUCATION DEPARTMENT (http://www.emsc.nysed.gov) Mathematics Standards for grades K-12 will be covered:

Through think-aloud modeling: PK.CM.1 Understand how to organize their thought processes with teacher guidanceThrough peer interaction: PK.CM.3 Listen to solutions shared by other students

Through Interviews and Journal Entries:PK.CM.4 Formulate mathematically relevant questions with teacher Guidance

Through vocabulary review and repetition:PK.CM.5 Use appropriate mathematical terms, vocabulary, and language

Within Assignments, Assessments, and Pre-Instructional Sets:PK.CN.1 Recognize the presence of mathematics in their daily lives PK.CN.3 Recognize and apply mathematics to objects and pictures

During class activities:PK.R.1 Use multiple representations, including verbal language, acting out or modeling a situation, and drawing pictures as Representations

Through Word Problem Usage versus Pictures versus Conceptual Practice:A2.PS.1 Use a variety of problem solving strategies to understand new mathematical content A2.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept

Content:A2.A.55 Express and apply [three of ]the six trigonometric functions as ratios Functions of the sides of a right triangle A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º... angles A2.A.57 Sketch and use the reference angle for angles in standard position

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Planning Pyramid

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• How to solve complex

“Soh Cah Toa” Word Problems

• How to discover the need for the

pythagorean theorem and use it for a word problem

• Significance of the two important right triangles (30:60:90, 45:45:90)

• How acute and obtuse angles relate to the definitions of acute and obtuse

•How the right triangle fits on a graph•How to use Soh Cah Toa for lines on a graph

•How to solve simple Soh Cah Toa word problems•How to solve simple Pythagorean Theorem word

problems

• Triangle Side Classification: Equilateral, Isosceles, Scalene•Triangle Angle Classification: Acute, Obtuse, Right

•How to Draw each type of Triangle•How to Label Triangle Sides and Angles

•Hypotenuse of a Right Triangle•How to Calculate a Side of a Right Triangle using Pythagorean Theorem: a2 + b2 = c2

•How to locate the Opposite and Adjacent Sides of a Triangle using a reference Angle•The mnemonic device “Soh Cah Toa”

•How to use Sine, Cosine, and Tangent to discover unknown angles and lengths of sides of a right triangle.

SOME students will

know

MOST students will

know

ALL students will know

Lesson One: Triangle Geometry Review and Identification,

Introduction to Right Triangles

Objectives and Standards:‣ The student will be able to state the differences between equilateral, isosceles, scalene, acute, obtuse, and right triangles. ‣The student will be able to appropriately label triangle sides and angles.‣The student will be able to identify the hypotenuse of a right triangle.

Pre-Instructional Set: Prior to class, student homework would have been to research and brainstorm all the different ways triangles are used. They must find three ways. They can choose whether to bring in pictures, write it down on paper, or even bring in examples. The first activity we do will be to pair up then square to discuss what they discovered. We discuss as a class after

this activity.Input: * Review triangle side classification: Equilateral, Isosceles, & Scalene Triangles. Students provided visual aid during instruction. * Briefly review similar triangles* Using tick marks to label the sides that are congruent is explained.* Triangle angle classification is also reviewed: Acute, Obtuse, Right. * Using tick marks to label the sides that are congruent is explained.* The angle label and the right angle label is reviewed. * The hypotenuse of the right triangle is reviewed (longest side, across from the right angle).

Modeling: Materials: Many cut out sets of triangles of the different classifications in different colors and sizes. Sticky or magnetic strips to attach the triangles to the board with. Using a set of triangles, as a think aloud, model how I would organize triangles in angles and sides. Then how I would think about what was similar, so as to organize in sequence: 3 sides the same, 2 sides the same, no sides the same, small (acute), medium (right), and large (obtuse) angles.

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Checking for Understanding: Have the students guide me with the next set of cut out triangles on where they should go. Draw different versions of the triangles and have the students tell me where they would put them.

Guided Practice: Materials: Cut out colorful triangles for the class to be split into groups of 3 and have the students work together to organize them on their own onto a large sheet of paper. Activity: Students will label the triangle sides and angles (and hypotenuses), write what kind of triangle they are, and then attach the paper to the wall when they are finished and stand by it. (I walk around to provide helpful hints) Colorful markers will be provided. Students will go to each paper and do a peer review and tell the class one thing that was great and one thing that could be improved for each group’s assignment.

Closure: I close the lesson for the day by reviewing the organization of the classification of triangles, hypotenuse, and labeling. Students will have ten minutes to complete a required “Ticket out the Door” before leaving for the day. They will be instructed that it is important that they complete this seriously so I can try to make lessons more fun for them and easier to follow.

TICKET OUT THE DOOR

1. What did you like about today’s lesson?

2. What would have made today’s lesson more fun?

3. What you understood about today’s lesson?

4. Something you did not understand about today’s lesson?

Independent Practice: Students get together in groups of four and combine all of their triangles they gathered for homework and draw them on a piece of poster board. They will be required to label the sides and angles by congruency and name each by what side and angle definition that fits. Example below.

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Lesson Two: Right Triangle Sides - Pythagorean Theorem

Objectives and Standards:‣The student will be able to label the sides of a Right Triangle a, b, and c, such that c is the Hypotenuse.‣The student will be able to use the Pythagorean Theorem (a2 + b2 = c2) to find the length of one unknown side of a Right Triangle.

Pre-Instructional Set: (To set focus on right triangles and sides)Have five students go up to the board and draw a right triangle. Have a different five students go up and label the right angle of the right triangle. Have another five students go up and circle the longest side of the right triangle. Have the last few go up and label the hypotenuse of the right

triangle with the letter “c”.

Input: What we want to do for this lesson is show the students how to find the length of a side of a right triangle when we already know the lengths of the other two sides.

Introduce the Pythagorean Theorem (a2 + b2 = c2). Understanding what the hypotenuse of the right triangle is the most important piece as the hypotenuse is “c”. It is the longest side, across from the largest angle which is why it is what equals the other two

sides squared and added. The other two sides are “a” and “b” (order is unimportant).

Modeling: Show the students that what we did at the beginning of class is what we want to do first when we practice using the Pythagorean Theorem: make sure it is a right triangle, locate the right angle, locate the hypotenuse (notice it is the longest side!), label the hypotenuse “c” and the other two sides “a” and “b”. Write down the formula a2 + b2 = c2. Plug in the values and solve the equation for the missing value.

Checking for Understanding: Do a blind call and ask students questions about how to locate the hypotenuse and what the Pythagorean Theorem is and what it is used for. Ask what kind of triangles it is specific to.

Guided Practice: Students get into five groups and measure the walls and ceiling of the classroom and compute the length of the diagonal of each. Require they write down their work and the steps they followed in their journals. (Fun Computer Game - http://www.themathlab.com/Algebra/pythagorean%20theorem%20intro%20to%20trig/pythgpbs.htm)

Closure: Review the theorem, remind of Right Triangles, Hypotenuse, the largest side corresponding to the largest angle (the right angle). Write down the steps they should go through to complete the problems.Independent Practice: (SEE Learner Activities: Lesson 2 Independent Practice)

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Lesson Three: Right Triangle Angles - “Soh Cah Toa”

Objectives and Standards:‣The student will be able to label the sides of a right triangle hypotenuse, opposite, and adjacent using a reference angle.‣The student will be able to use the mnemonic device “Soh Cah Toa” to remember Sine(θ) = Opp/Adj , Cosine(θ) = Adj/Hyp , and Tangent(θ) = Opp/Adj, where “θ” is the reference angle.‣The student will be able to find an unknown (reference) angle in a Right Triangle using Sine, Cosine, and Tangent.

Pre-Instructional Set: For homework, students were instructed to go out into their environment and find 8 right triangles and draw and measure them. In class, 10 students are chosen at random to put one example on the board.

Input: Students first taught how to choose a reference angle and name the opposite and adjacent sides to the reference angle. Next they are introduced to Sine, Cosine, and Tangent. They learn the mnemonic device “Soh Cah Toa”. They are also introduced to special triangles (30:60:90) and (45:45:90) and the standard values associated with sine, cosine, tangent.

Modeling: I model a think aloud problem for them with a picture and numbers. Next I model a word problem with a picture and no numbers. Last I model a word problem where I must draw the picture and apply the numbers.

Checking for Understanding: Students instructed to go up to the board and choose a reference angle on each of the problems on the board that they put up there at the beginning of the class. They label opposite and adjacent, then write the formulas for sine, cosine, and tangent. The remainder of the students are instructed to choose a triangle that they wrote down for homework and do the same. I guide them through the process where needed. Calculations are not required yet.Guided Practice: Students get to go to the computer lab and play Soh Cah Toa games. They have to write down their work and questions as they work in their class Journals. I walk around and view problems and address them as a class to assist all.Closure: Pair up and complete the following “Ticket out the door”. Show your work. If you cannot finish the problem, show work for doing as much of the problem as you can, then ask questions for where you got stuck. Additional questions about today’s lesson are welcome.

Ticket Out the Door!

A 4 foot tall mailbox creates a 3 foot long shadow. (Hint: Draw and Label this!!)What is the length of the ray of the sun from the top of the mailbox to the top of the shadow of the mailbox?What is the angle of the shadow to the ray of the sun?QUESTIONS:

Independent Practice: (SEE Learner Activities: Lesson 3 Independent Practice)

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Special Angles Sin Cos Tan

30° ½ √³⁄₂ √³⁄₃

45° √²⁄₂ √²⁄₂ 1

60° √³⁄₂ ½ √3

Teacher LibraryMath Lab Site

<http://www.themathlab.com/Algebra/pythagorean%20theorem%20intro%20to%20trig/pythgpbs.htm>

Students can go to this site

and put their friends’ and

classmates names in to make

the problems a bit more silly

and fun. This is great online

practice for the Pythagorean

Theorem.

Site: <http://zonalandeducation.com/mmts/trigonometryRealms/introduction/rightTriangle/trigRightTriangle.html>

Students can go to this site to get another explanation of Trigonometry and Right Triangles.

Regents Prep Site<http://www.regentsprep.org/regents/math/algebra/AT1/PracPyth.htm>

Students can go to this site to receive extra

instruction on and practice using the Pythagorean

Theorem for the Math Regents. If they do not

understand, it provides and explanation of how to

perform the problems.

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Zone Land Education Site <http://zonalandeducation.com/mmts/trigonometryRealms/solvers/

rtTriSolvers/rtTriSolversHome.htm>

Students can go to this site to practice using the Pythagorean Theorem.

They click on a side of the triangle they would like to practice with then it takes them to a problem page.

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Learner Activities

Lesson 2 Independent Practice (Basic, Challenging)

Basic - Directions: Draw and Label the triangles in your Journal. Then Solve the following in your Journal. Show your work! Every step!

(1-3) Locate the hypotenuse on the following triangles. Label it “c”. Label the other two sides “a” and “b”.

1. 2. 3.

4. What is the Pythagorean theorem? ______________________.

(5-7) Find the value of the missing side using the Pythagorean Theorem.

5. 6. 7.

8. Ms. Spires tells you that a right triangle has a hypotenuse of 13 and a leg of 5. She asks you to find the other leg of the triangle. What is your answer?

9. Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet

long and the bottom of the house is 6 feet across. What is the height of his dog house, in feet, at its tallest point?

10. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase to the

nearest tenth of a foot?

11

3

4

x

2 1y

12

13z

Challenging Directions: Solve the following in your Journal. Draw and label and show your work!

1. Ms. Spires tells you that a right triangle has a hypotenuse of 13 and a leg of 5. She asks you to find the other leg of the triangle. What is your answer?

2. Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet

long and the bottom of the house is 6 feet across. What is the height of his dog house, in feet, at its tallest point?

3. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase to the

nearest tenth of a foot?

4. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the

nearest meter, how many meters would be saved if it were possible to walk through the pond?

5. A baseball diamond is a square with sides of 90 feet. What is the shortest

distance, to the nearest tenth of a foot, between first base and third base?

6. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?

7. In a computer catalog, a computer monitor is listed as being 19 inches. This distance is the diagonal distance

across the screen. If the screen measures 10 inches in height, what is the actual width of the screen to the nearest inch?

8. In construction, floor space must be given for staircases. If the second floor is 3.6 meters above the first floor

and a contractor is using the standard step pattern of 28 cm of tread for 18 cm of rise then how many steps are needed to get from the first to the second floor and how much linear distance will need to be used for the

staircase?

9. When the ancient Egyptians were building the great pyramids at Giza, they checked each course or level of stones to make sure they were being laid square by measuring the diagonals. If each course of stones has the

length of the square reduced by 2 meters what is the reduction in the length of each diagonal?

10. Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore he started from but ends up 100 meters down river from where he started because of the current. How far did he

actually swim from his starting point?

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Lesson 3 Independent Practice (Basic, Standard, Challenging)

Basic - Directions: Copy the problem and write down the answers to the following in your Journal. Show your work! Every step!

1. Label the opposite and adjacent sides to the Reference Angles.

2. What does “Soh Cah Toa” stand for? _________________________________________

3. We know that the reference angle θ = 30°. We also know that the side opposite of θ (b) = 3. What is the length of the hypotenuse (c)?

4.We know that the reference angle θ = 30°. We also know that the side adjacent of θ (a) = 5. What is the length of the opposite side (b)?

5. We know that the reference angle θ = 45°. We also know that the side adjacent of θ (a) = 1. What is the length of the hypotenuse (c)?

Standard (Combination of Basic and the following)

A

B

C

5. On the above Triangle, let AB = 1 and ∠BCA = 45°. Find the length of Triangle side BC

6. On the above Triangle, let AC = 27.22 and ∠ABC = 32°. Find the length of Triangle sides BC and AB.

7. A right triangle has a sides of z and y, and a hypotenuse of 9. The angle opposite z is 27°. Find the lengths of

sides z and y and the measure of the angle opposite y.8. A right triangle has legs x and 3. A 16º angle is opposite the side whose length is 3. Find the length of side x

and the measure of its opposing angle, X.

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Triangle ABC

Reference Angle

b

aθ

c

Challenging

Directions: Draw the triangle in your Journal and add given values. Label your reference angle, then the opposite and adjacent sides to that angle. Solve the equation using the necessary “Soh Cah Toa” function.

1. On the above Triangle, let BC = 4 and ∠ABC = 60°. Find the length of Triangle side AC.

2. On the above Triangle, let AB = 1 and ∠BCA = 45°. Find the length of Triangle side BC

3. On the above Triangle, let AC = 27.22 and ∠ABC = 32°. Find the length of Triangle sides BC and AB.

4. A right triangle has a sides of z and y, and a hypotenuse of 9. The angle opposite z is 27°. Find the lengths of

sides z and y and the measure of the angle opposite y.5. A right triangle has legs x and 3. A 16º angle is opposite the side whose length is 3. Find the length of side x

and the measure of its opposing angle, X.6. A right triangle has sides 11 and x with a hypotenuse of 17.112. Find the length of side x and the measure of the

angle opposite the side whose length is 11.7. A ladder rests against a wall. the top of the ladder reaches 15 feet up the wall, and makes an angle of 53 with the

wall. how long is the ladder?8. A man on a 135-ft vertical cliff looks down at an angle of 16 degrees and sees his friend. How far away is the

man from his friend? How far is the friend from the base of the cliff?9. A ski slope at a mountain has an angle of elevation of 25.2 degrees. The vertical height of the slope is 1808 feet.

How long is the ski slope?10. Amelia sees a jet heading south away from her at 42 degree angle of elevation. Twenty seconds later the jet is

still moving away from her, heading south at a 15 degree angle of elevation. If the jets elevation is constantly 6.3 km, how fast is it flying in kilometers per hour?

11. From the top of a lighthouse 160 feet above sea level, the angle of depression to a boat at sea is 25 degree. To the nearest foot, what is the horizontal distance from the boat to the base of the lighthouse?

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A

B

CTriangle ABC

AssessmentThere is a wealth of evidence that suggests that we move away from solely using paper and pencil tests towards more alternative methods of assessment such as reflection, peer assess- ment, rubrics, self-assessment, among others (Paris & Eyres, 1994; O'Donnell & Topping, 1998; Andrade & Boulay, 2002). Specifically, multiple-choice tests take away any assessment of stu- dent constructs, or reasoning processes. In math especially, it is important to measure reasoning and process in order to fully assess understanding (Moskal & Leydens, 2000).

Students will keep journals for their homework, in which they will be responsible to answer their own questions through peer and teacher interaction, research, reflection and re-work. Through this work, re-work, reflection and research, I will be able to see each student’s process evolve, and more importantly, the student will be able to see it and review it.

Journal Example

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Every two weeks students will be required to do an interview of an adult in their lives and ask them how they use mathematics in their everyday life. It will be a short one page write up in their Journals in order to get the students to think about math in a real world concept and to bring writing into the classroom.

Informal student- and teacher- formative assessment can be very helpful for students to understand where they are in learning the material and how they can improve. Informal formative assessment does not produce a grade, but it assesses student progress during the process and allows the teacher and student see where strengths are and what needs more attention. I will use this type of assessment by using a Survey and using Tickets Out the Door. This can assess behavioral characteristics or understanding of concepts and topics. The survey example is used to assess behavior.

Student Reflection & Teacher Informal Formative Assessment Example

Student/Teacher Progress Assessment

Name ______________________________ Date ________________________Assign a number between 1-4, where 1 - Rarely, 2 - Sometimes, 3 - Often, 4 - Always

Student/Teacher Progress Assessment

Name ______________________________ Date ________________________Assign a number between 1-4, where 1 - Rarely, 2 - Sometimes, 3 - Often, 4 - Always

Student/Teacher Progress Assessment

Name ______________________________ Date ________________________Assign a number between 1-4, where 1 - Rarely, 2 - Sometimes, 3 - Often, 4 - Always

Description Student Teacher

I pay attention in class

I show respect for my classmates and my teacher

I show my work on my assignments

I ask questions where I have them

I perform my assignments to the best of my ability

I complete assignments on time

I am comfortable with the material we are learning

I feel confident in my math abilities

Teacherʼs Comments on Student Strengths

___________________________________________________________________________________________

___________________________________________________________________________________________

Teacherʼs Comments on Student Strengths

___________________________________________________________________________________________

___________________________________________________________________________________________

Teacherʼs Comments on Student Strengths

___________________________________________________________________________________________

___________________________________________________________________________________________

Teacherʼs Comments on Possible Improvements

___________________________________________________________________________________________

___________________________________________________________________________________________

Teacherʼs Comments on Possible Improvements

___________________________________________________________________________________________

___________________________________________________________________________________________

Teacherʼs Comments on Possible Improvements

___________________________________________________________________________________________

___________________________________________________________________________________________

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Peer Assessment can be a fantastic mode of student learning on both sides of the assessment if conducted in an effective manner. This type of assessment is not intuitive to most; students must learn to approach it in a constructive, positive way. When organized and taught effectively, peer feedback can confirm what a student thought the task was asking, it can help students add any knowledge they had missing on the topic, and can correct any false theories (O’Donnell & Topping, 1998). For this class, we will use Peer Journal Reviews as informal formative assessment. Students can learn how to positively provide feedback and share their knowledge.

Peer Journal Assessment Checklist Example

Journal ReviewReviewer Name: ______________________ Journal Owner Name:_______________________

Review:

1. Something my partner did well was ___________________________________________

________________________________________________________________________

2. Something my partner may have missed was ___________________________________

________________________________________________________________________

3. Something interesting about what my partner wrote was ___________________________

________________________________________________________________________

4. How I can use my partner’s work to improve my own _____________________________

_______________________________________________________________________________

_______________________________________________________________________________

CITATIONS

✦ Andrade, H., & Boulay, B. (2002). The role of rubric-referenced self-assessment in learning to write. Manuscript under consideration.

✦ Moskal, B., & Leydens, J. (2000). Scoring rubric development: Validity and reliability. Practical Assessment, Research and Evaluation, 7(10). Available online: http://pareonline.net/getvn.asp?v=7&n=10.

✦ O'Donnell, A., & Topping, K. (1998). Peers assessing peers: Possibilities and problems. In K. Topping & S. Ehly (Eds.), Peer-assisted learning. Mahwah, NJ: Lawrence Erlbaum Associates.

✦ Paris, S., & Ayres, L. (1994). Becoming reflective students and teachers with portfolios and authentic assessment. Washington, DC: American Psychological Association.

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Modifications: Planning for Academic Diversity

LEARNING BARRIER POSSIBLE SOLUTIONS RESOURCES

Student cannot read at grade level

Provide reading materials at lower grade levels. Simplify directions. Use pictures often

See outlines below

Student has difficulty comprehending the material

Outline the material, provide more basic instructions.

See outlines below

Student has difficulty mastering the vocabulary of the unit

Provide Flash Cards, repeat, review, clarify.

Student has difficulty with handwriting (speed or accuracy)

Provide extra time to perform work. Make homework shorter.

Student has difficulty with calculating activities

Allow student to use calculator. Focus student on concepts. Homework practice for calculating separately from concepts.

Student needs help with conducting research

Develop a buddy system. Have peers help student do research. Provide Instructions

Student needs the instructional material in a language other than English

This can be accommodated. Provide in English and other language. Have student work with a translator in class.

Student needs additional challenge

Provide more complex problems. Allow student to teach themselves. Allow student to teach material to others.

See Teacher Library

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Lesson 1: Triangle Review - Outline

Triangles have 3 sides and 3 angles.

The angles within a triangle add up to 180°.

Organization of Triangles: Sides/Angles

Sides/Angles - Equilateral, Isosceles, Scalene (Vocabulary) Equilateral Isosceles Scalene

All 3 sides and angles are same 2 sides and angles are same No sides and angles are same

Angles - Acute, Right, Obtuse (Vocabulary)

Acute Right Obtuse

All angles are less than 90° There is one 90° angle There is an angle greater than 90°

Some definitions of Acute - 1. characterized by sharpness or severity; 2. sudden onset, short course; 3. Being, providing, requiring short-term medical care; 4. Responsive to slight impressions of stimuli [Note: all has to do with sharp, intense, small, short-term, slight]Some definitions of Obtuse - 1. Lacking sharpness or quickness of sensibility or intellect; 2. Difficult to comprehend, not clear or precise in thought or expression; 3. Blunt, not pointed [Note: all has to do with NOT sharp, clear, or pointed]

Right Triangles (Note: Angle looks like an “L”, Looks like half a rectangle or square)(30°: 60°: 90°) (45°: 45°: 90°)

Hypotenuse - Longest side of a right triangle, opposite the right angle

19

A B

θ1

θ2

θ3

θ1 + θ2 + θ3 = 180°

C

hypotenusehypotenuselegleg

legleg

hypotenuse

Lesson 2: Pythagorean Theorem - Outline

EXAMPLE: Say we know that side AB = 3, side AC = 4, and we want to find side BC, can we use the Pythagorean Theorem to do this? Yes!

Ask ourselves... 1. Is this a right triangle? Yes!2. Label the right angle... Done!3. Where’s the hypotenuse? Side BC! (That’s the side we want!)4. What’s the formula? a2 + b2 = c2

5. Plug in the side... AB = a, AC = b, BC = c6. Therefore... AB2 + AC2 = BC2

7. Solve! 32 + 42 = BC2 9 + 16 = BC2

25 = BC2

√25 = √BC2

5 = BC

A

B

C

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PYTHAGOREAN THEOREMLegs: a, b Hypotenuse: c

Formula a2 + b2 = c2

hypotenuse

Lesson 3: “Soh Cah Toa” - Outline

Reference Angle: Angle being used or Known angle Opposite: across from ref angleDef’n-facing; contrastingAdjacent: next to ref. angleDef’n - neighboring; next door

Sine, Cosine, Tangent

Sin (θ) = opposite⁄hypotenuse Cos (θ) = adjacent⁄hypotenuse Tan (θ) = opposite⁄adjacent

“Soh Cah Toa” - S = O⁄H ; C = A⁄H ; T = O⁄A

Special Triangles MODEL

30°:60°:90° 45°:45°:90°

Table Special Angles Sin Cos Tan

30° ½ √³⁄₂ √³⁄₃

45° √²⁄₂ √²⁄₂ 1

60° √³⁄₂ ½ √3

θ

21

X

Y

θ

opposite

adjacent opp

adjθ

hyphypotenuse

1

2

√360°

30°1

1

√245°

45° 4

X

30°STEPS:1. Label Reference Angle2. Label opp, adj, hyp3. What do we know?4. What do we want to know?5. Find Formula6. Plug and Solve

3. We know θ (ref angle) = 30° and opposite = 44. We want to know X (hypotenuse)5. Sin (θ) = opposite⁄hypotenuse 6. Sin (30°) = 4⁄X (x⁄1) 1⁄2 = 4⁄X (x⁄1)

(2) X⁄2 = 4 (2) X = 8

adj

opp

hyp

Ref ∠