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Grade: Senior
Subject: Mathematics
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Brett Tyrrell
Tyler Rational Unit Plan
Dr. Janak
March 26, 2012
The general topic of this unit plan is going to be about trigonometric ratios and problem solving with trigonometry. The definitions of sine, cosine, and tangent for acute angles are found on right triangles and similarity; in conjunction with the Pythagorean Theorem, these are fundamental in many real-world and theoretical problem situations. Since the Pythagorean Theorem can only be used with right triangles, and every geometrical problem we deal with is not so kind to give us a ninety degree angle with every triangle the Law of Cosines can be used to generalize to non-right triangles. Partnered with the Law of Sines, the Law of Cosines embodies the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. By completely solving triangles I mean find every angle and side length. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion.
Students can use trigonometry to enrich their mind and expand there thinking, but more practically trigonometry can be used in surveying, engineering problems, heights of building or trees, stock market trends, or maybe business cycles. These students are eighth and ninth grade students in an advance geometry course at Laramie Junior High School which is part of Albany County School District #1. This unit plan will be implemented in three separate class all of 19 to 23 students. The community is in Laramie, Wyoming, one of the few college towns in Wyoming. There is a high amount of academic rigor stressed in the community, in particular, the parents of many of these "advanced" students. There are three other junior high level schools in Laramie, but the Laramie Junior High School has more junior high students than the other three schools combined.
Prior to teaching the trigonometry unit the students have utilized Pythagorean Theorem, similarity and congruency conjectures, and ratios and proportions. This is the last unit in the geometry course prior to taking a final exam. All three of the previously mentioned topics will be necessary to have success with trigonometry. Only a couple of students have seen trigonometry outside/prior to this unit, this includes the students who found out from self-interest or parental involvement.
To create this unit I start by making a two-dimensional graph with behavior and content objectives. I used Tyler's example (p.50) as a reference for this unit plan. While planning the lessons, I looked at the Common Core Standards to make sure I was aligning with them. I also focused on learning experiences within each lesson plan, rather than just giving myself a script or notes I included what the students should be doing during the lesson. Learning experiences are the interactions between the learner and external conditions that can be interacted with (Tyler, p. 62). Tyler also stresses that learning takes place through the active behavior of the student; it is what the student does that is learned...not the teacher.
Tyler talks about the importance of more than one "appraisal" so that you are able to identify progress or changes; to implement this I have created two quizzes to take after some of the material has been covered. To create the quiz and test I referred to the two dimensional chart (included later in this write-up). Tyler states that ". . .the two-dimensional analysis which served as a basis for planning the learning experiences also serves as the basis for planning the evaluation procedures." (p. 110). He goes on to say that the evaluation should "observe the degree to which the objectives are actually being realized." (p. 112). I would then cross reference the two-dimensional chart, the lesson objectives, and the assessments (both formative and summative). These assessments were made towards the end of the planning stage; I finished the unit by working on the unit test. Note: some questions on the test came from the textbook, Discovering Geometry while others were utilized to meet specific objectives stated within the lesson plans.
The broad goals of my unit had influence from the Common Core Standards. Students will be provided learning experiences to understand similarity and side ratios in right triangles leading to definitions of trigonometric ratios for acute angles. Goal two is for the students to be able to explain and use the relationship between the sine and cosine of complementary angles. Perhaps the most important goal that will be included in this unit is for students to be able to apply trigonometric ratios and the Pythagorean Theorem to solve problems both theoretical and applicable. As a final goal for this trigonometric unit students will be able to apply the Law of Sines and the Law of Cosines in real and theoretical problems.
This unit builds on three previously done units: Pythagorean Theorem, Triangle Congruency, and Similarity. These units have proved a background and a base to launch off of into trigonometry. Many of the students have seen topics such as rations, proportions, similarity, and even Pythagorean Theorem in fifth, sixth, and/or seventh grade. I know my students have seen this before because we have a vertical meeting for mathematics once a year as well as heavy collaboration at the 7-9 levels in the junior high school. Though this is the last unit in my geometry course, this will prepare them for the Advance Placement Prep course all of these students will be taking next year or the year following. This AP Prep course will delve deeper into trigonometry while my unit is giving an overview and some particulars that apply directly to previous units.
The cross-curricular integration opportunities are limited within the junior high setting. However music, business, and science enthusiasts are able to include trigonometry within a unit; however, since most of the junior high students will not see this material until 10th grade, possibly as late as 12th grade non-math teachers shy away from getting into many units that could include trigonometry.
This unit reflects my teaching philosophy in the sense that I have the students exploring, working, talking, and discovery. In essence the students are doing in my classroom rather than just listening. This unit, as most of my other units, includes a heavy portion on applied problems. Though I find theoretical math exciting and captivating, I realize not all fourteen year olds feel the same way. However, I do include theoretical problems in addition to real-life problem situations. My students in the classroom should be engaged with some type of math problem for the duration of class.
Common Core Standards for Mathematics
Similarity, Right Triangles, and Trigonometry G-STR
Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply trigonometry to general triangles
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Behavior Aspect of the Objectives
Content Aspect of the Objectives 1. Understand the Important facts and principles 2. Ability to interpret data 3. Ability to apply principles Develop cooperative behavior/attitudes
1.Trigonometric Ratios X X X X
a. Sine X X X X
b. Cosine X X X X
c. Tangent X X X X
2. Angles of depression and elevation X X X X
3. Law of Sines X X X X
4. Law of Cosines X X X X
ReferencesCommon Core State Standards for Mathematics. (2011). Common Core State Standards. Retrieved March 26, 2012, from http://www.corestandards.org/standards
Randall, C. I., Kennedy, D., Hall, B. (2012). Geometry Common Core. Boston, MA: Pearson
Serra. M. (2003). Discovering Geometry: An Investigative Approach. Emeryville, CA: Key Curriculum Press.
Tyler, R. W. (1969). Basic Principles of Curriculum and Instruction (Paperback ed.). Chicago, IL: The University of Chicago Press. (Original work published 1949)
Organization of content/Instructional plan
This section will give a detailed description of each lesson with in the trigonometry unit. In total there are five lesson plans, after each lesson plan there is the included pages from Discovery Geometry (Serra, 2003) as well as some supplemented pages from Geometry Common Core (Randall et al., 2012). These are included to give a better understanding to the types of problems and homework the students will be expected to understand.
The objectives within the unit are broken down within each lesson plan; each objective is written as illustrated in Tyler's (1969) Basic Principles of Curriculum and Instruction. Each homework assignment is included below. This serves two purposes: 1) It allows the students to see what type of work is expected of them from the homework problems 2) It will be used as the answer key to go over the homework the following class period. The students will not receive any handouts or worksheets during this unit unless they are absent, in which case they will be given a copy of the notes.
The order of my lessons are important because Tyler (1949, p. 85) states that "Sequence as a criterion emphasizes the importance of having each successive experience build upon the preceding one but to go more broadly and deeply into the matters involved." The Laws of Sines and Cosines must build upon an understanding of the trigonometric ratios, namely, sine and cosine. A focal point of the unit is problem-solving thus application problems are included in the introduction, each lesson, and an entire section devoted to problem solving skills.
Assessment/Evaluation PlanWithin all my unit plans certain types of formative assessment will be included. The cooperative learning environment will require and encourage students to communicate their mathematic learning knowledge. As the teacher I will be monitoring these discussion, listening for misconceptions and understanding of the material. One quiz and one unit test will be given as formal assessments, each of these are included at the end of the document. Another formal assessment for this unit is the Geometry of Baseball project, also included below.
"The process of evaluation begins with the objectives of the educational program," (Tyler, 1949, p. 110). During each lesson, investigation, and homework I will be monitoring each student's progress towards successful completion. At any time I deem necessary I will bring individual students or groups of two to four in for extra help. The times will include before school, after school, lunch, and planning periods (if it coincides with their study hall). One class prior to taking the unit test will be devoted to a class review. The review will include teacher and students discussing questions and practicing questions by means of individual white board practice. The questioning techniques and discussions utilized within the everyday classroom provide the basis for my formative assessments, while work on homework problems and problems done on white boards will add to the formative assessment during the unit.
For those students that do not meet the objectives and standards on the unit test I will offer a retake. However, prior to being able to have a second chance they will have to come in for remediation type instruction on their own time. Tyler (1949) reasons that we must give students an opportunity to show they have acquired a behavior. By bringing each student in that did not meet the requirements the first time it allows a one-on-one correspondence, making it easier to target areas of struggle and gives the students a verbal way to show they have meet the objectives. I also will only give up to an 82 percent on a retake.
Laramie Junior High School Weekly Instruction Plan
Teacher's Name: Brett Tyrrell
Nature of Lesson: Trigonometric RatiosStandards Being Addressed:
Define trigonometric ratios and solve problems involving right trianglesObjective: 1. To discover important facts and principles with trigonometric ratios, including sine, cosine, and tangent.
2. To interpret data using sine, cosine, and tangent.
3. To understand the usefulness of trigonometry and develop cooperative behavior.Assessments to be used:
Allow students to work in groups or pairs on problems in class with teacher guidance
Listen to student questions
Homework, Quiz, TestCorrectives to be used:
Help students' correct mistakes in class while working on homeworkContent to do in class:
Teacher:
Introduction to Trigonometry. . .pose the Leaning Tower of Pisa problem on document camera and ask, "If you were to drop an object from the top of the Leaning Tower of Pisa how far away from the base would it land?"
PowerPoint (slides attached)
Monitor InvestigationStudents (experiences the teacher is expecting the students to be having): Discuss Leaning Tower of Pisa problem including what information they need to begin and what mathematical properties are capable of solving such a problem.
During the PowerPoint presentation the students will be taking notes and asking questions. Students will then do the Investigation: Trigonometric Tables with the objective of understanding sine, cosine, and tangent.
Homework p. 624 #1-13 odds and 14-20
Laramie Junior High School Weekly Instruction Plan
Teacher's Name: Brett Tyrrell
Nature of Lesson: Trigonometric RatiosStandards Being Addressed:
Define trigonometric ratios and solve problems involving right trianglesObjective: 1. To be able to apply principles to problems involving trigonometric ratios
2. To interpret data involving angles of depression and elevation
3. To develop cooperative behavior and problem-solving skillsAssessments to be used:
Allow students to work in groups on problems in class with teacher guidance
Listen to student questions
Homework, Quiz, TestCorrectives to be used:
Help students' correct mistakes in class while working on homework.
"Lunch Bunch"Content to do in class:
Teacher: Introduce angles of elevation and depression with the theatre problem. Include a drawing on the board when differing between the angle of elevation and depression; stress that these two angles are alternate interior angles and will therefore be congruent.
Instruct the students to read page 627 in their textbook and discuss their understanding with a partner.
Have the students stand-up, find a partner and demonstrate each angle with multiple partners by making eye contact with other students. Questions will include: "Who will have to have an angle of elevation between a short student and tall student?" "Which students will have the most extreme angles of elevation and depression in the class?" "How does the distance between the two people affect the angles?"Students: The learning opportunities for students to have in this lesson will include audio, visual, and kinesthetic. The students will be seeing theatre lights as well as other visual models of the two types of angles being discussed. In the second part of the lesson the students will be asked to read the first page of section 12.2 from the text book (included below). The students will then be engage in communicating their understanding in groups of two-three. Prior to beginning to work on the application problems involving trigonometric ratios the students will be able to demonstrate an angle of depression and angle of elevation while moving about the classroom.
Homework p. 628 #2-4, 8
Laramie Junior High School Weekly Instruction Plan
Teacher's Name: Brett Tyrrell
Nature of Lesson: Law of SinesStandards Being Addressed:
Geometry, Communication, Measurement, Algebra, Representation, Connections, Reasoning, Problem SolvingObjective: 1. To understand the important facts and principles involving the Law of Sines
2. To interpret data using the Law of Sines
3. To use the Law of Sines to solve practical problems.Assessments to be used:
Allow students to work in groups on problems in class with teacher guidance
Listen to student questions
Homework, Quiz, TestCorrectives to be used:
Help the students through the investigations. Help students' correct mistakes in class while working on homework.Content to do in class:
Teacher: Instruct the student to do the Investigation 1: Area of a Triangle step 1 (a and b); step 2. Monitor the students so they are on the correct track of coming up with the SAS Triangle Conjecture. Question to ask: "How do you find the area of a triangle?" "Are there any patterns or relationships that are appearing?"
Instruct the students to do Investigation 2: The Law of Sines Steps 1-7
"When can you use the Law of Sines?" (ASA or AAS)Students: Participate in the investigation with one other person; concluding with the
SAS Triangle Area Conjecture- The area of a triangle is given by the formula A = ½ ab sinC, where a and b are the lengths of two sides and C is the angle between them.
The will then actively take part in the second investigation with the intent of finding:
Law of Sines- For a triangle with angles A, B, and C and side lengths of a, b, and c (a is opposite A, b is opposite B, and c is opposite C), sinA/a=sinB/b=sinC/c
The homework problems will include application and theorectical problems involving the Law of Sines.Homework p. 637 #1, 2, 5-11
Laramie Junior High School Weekly Instruction Plan
Teacher's Name: Brett Tyrrell
Nature of Lesson: Law of Cosines
Standards Being Addressed:
Geometry, Communication, Measurement, Algebra, Representation, Connections, Reasoning, Problem SolvingObjective: 1. To understand the important facts and principles involving the Law of Cosines
2. To interpret data using the Law of Cosines
3. To use the Law of Cosines to solve practical problems.Assessments to be used:
Allow students to work in groups on problems in class with teacher guidance
Listen to student questions
Homework, Quiz, TestCorrectives to be used:
Help students' correct mistakes in class while working on homeworkContent for the teacher to do in class:
Teacher: Draw a right triangle ABC, where C is the 90 degree angle. Pick any measure for angle A and find
(sin A)2 + (cos A)2 = . (It will be 1 every time). "Compare your answer to those around you."
"What are the ratios for sin A and cos A? a/c and b/c?"
"What does the Pythagorean Theorem tell us?"
Then (sin A)2 + (cos A)2 = (a/c)2 + (b/c)2
(a2 +b2)/c2 is equal to 1. Then (a2 +b2) = c2
Law of Cosines- For any triangle with sides of lengths a, b, and c, and with C the angle opposite the side with length c,
c2 = a2 + b2 -- 2ab cos(C)
"When can use the Law of Cosines?" (SSS and SAS).Students: The students will be writing in their notes while the teacher is giving them the problem situation and they will be expected to participate in the discussion. The students will be asked to give their thoughts while the teacher is doing the problem at the board and the students are doing the problem at the desks.
Homework p. 643 #1-9
Laramie Junior High School Weekly Instruction Plan
Teacher's Name: Brett Tyrrell
Nature of Lesson: Application Problems and Baseball Project
Standards Being Addressed:
Geometry, Communication, Measurement, Algebra, Representation, Connections, Reasoning, Problem SolvingObjective: 1. To interpret data using trigonometric ratios, Law of Sines, and Law of Cosines
2. To apply prinicples of trigonometric ratios, Law of Sines, and Law of Cosines to solve application problems
3. To develop cooperative behavior and problem-solving skillsAssessments to be used:
Allow students to work in groups on problems in class with teacher guidance
Listen to student questions
Homework, Quiz, TestCorrectives to be used:
Help students' correct mistakes in class while working on homeworkContent to do in class:
The students will begin class with The Geometry of Baseball Project: in groups of three or four, they will be discussing, analyzing, and interpreting data. The students will be working on application problems for the duration of the class. They will be paired up and expected to explain their thinking to others (including the teacher).Homework: p. 648 #1-8
Chapter 12 Quiz
Name: Date: Period:
P
Angle P is 90 degrees.
sin Q = 24
cos Q =
tan Q =
Q 30 RIn problems 2-5, round your answers to the nearest tenth.
12 m
n40̊ B
Area of triangle ABC =
Angel A = 70̊ ; AB is congruent to BC
A C
17 cm 10 mmx
15 cmx =
As Juliet stands 15 feet off the ground on her balcony, she spots Romeo in the distance, standing on the ground. She uses her angle measuring device and finds his position is 3̊ from the horizontal. To the nearest foot, how far away is Romeo from the base of her balcony?
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