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Grade: Senior
Subject: Mathematics

#3043. Trig Fcns to Find Sides of Triangle

Mathematics, level: Senior
Posted Thu Jan 29 18:38:53 PST 2004 by L. Fisanick (
University of Pittsburgh at Johnstown, Johnstown, PA, USA
Materials Required: Transparencies/Markers/Projector/Paper/Rulers/Protractors/Calculators/Handouts

Class: Eleventh Grade Trigonometry Grade Level: 11th Grade
Unit: Trigonometric Functions Lesson: Trig Fcns to Find Sides of Triangle

PA Academic Standards: 2.10.11.B Identify, create and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem.
2.3.11.C Demonstrate the ability to produce measures with specified levels of precision (if time permits activity).

Goal of this lesson: *To help students to review their previous knowledge of the Pythagorean Theorem.
*To help students understand that triangle sides could also be found through the use of trigonometric functions.
*To help students understand that when you have a right triangle with one known angle and one known side, the other angle and the other two sides can be solved for easily.
*To help students understand that real life problems could involve trigonometry.
*To help students to understand that the Pythagorean Theorem does not involve the angles of a right triangle.

Materials: Transparencies/Markers/Projector
Chalk Board or Dry Erase Board; Chalk/Erasers
Pre-Tests, Princess Problem, Review Sheets, Note Takers, Homework Sheets, Answer Keys
Princess Problem Slide, Triangle Problem Slide
Paper (enough for each student to have two sheets)
Rulers (enough for each student to have one)
Protractors (enough for each student to have one)
Scientific Calculators (on hand in case if any student needs one)

Clerical/Administrative Tasks: *Take roll.
*Collect homework.
*Make copies of pre-tests.
*Make copies of princess problem (only for learning disabled student).
*Make question list (only for learning disabled student).
*Make copies of review sheets.
*Make copies of note takers.
*Make copies of homework problems.
*Make answer key for the pre-test, princess problem, & homework problem.
*Make triangle slide and princess problem slide.
*Gather materials for the "if time permits activity."

Instructional Objectives (Student-centered, observable, and precise statements of what students will be able to do):
1) When given the pre-test, the students will be able to use yesterday's lesson on "soh cah toa" to figure out the trigonometric functions.
2) When given the lesson opener problem (princess), the students will be able to think in their heads about what the problem asks.
3) When the unsolved side of a right triangle problem is posted on the projector, the students will be able to explain how to solve using the Pythagorean Theorem.
4) When asked how to solve the same problem (when given an angle) without using the Pythagorean Theorem, the students will be able to verbally express their thoughts and ideas to me in a logical manner.
5) When given the review sheet on yesterday's lesson, the students will be able to recall the old facts but use the princess problem to think about what's ahead.
6) When given the review sheet, the students will be able to solve for the missing side using the previously learned trigonometric functions.
7) When asked to open their books, the students will be able to retrieve theirs quietly and open to the correct page.
8) When we come back to solve the lesson opener problem (princess problem), the students will be able to explain what they would do to solve.
9) When asked to solve the princess problem in their notebooks, the students will be able to show all their work by working through the problem on their own.
10) When given their homework problem, the students will be able to apply today's lesson to solve.

Introduction (attention getter, anticipatory set, discrepant event, open-ended problem scenario, engagement)
*Take roll.
*Hand out the pre-test for the students to complete. Inform the class that the pre-test will be collected, but will not count as a certain amount of points. The point of the pre-test is to see if everyone understands what is going on in class.
*While the students are taking their pre-test, collect homework from the previous day and pass back old homework.
*Collect the pre-tests and again, calm the students' nerves down by stating that there will not be a numerical grade attached to the test. However, like always, class participation will be awarded.
*Place the princess/castle slide on the projector. State that we know how high the castle is and that we know how far the prince is from the castle. (Anticipatory Set of Hunter Model)

ASK: How long does the prince's ladder need to be to reach the princess (rhetorical question)? Leave time for students to think. Tell the students that we will return to this problem later on in class.

*Post the slide up containing a right triangle with an unsolved side. Cover up the second part of the slide.

ASK: How would we figure this problem out normally? Respond to all questions until you here "Pythagorean Theorem."

*Pass out the note takers.
*Solve the problem in front of the class--through student interaction--using the Pythagorean Theorem.


Developmental Activities (Instructional components that provide opportunities for students to make progress toward intended instructional objectives):
*Remind the students about what we learned yesterday about sine, cosine, and tangent. Explain to the students that trigonometric functions can be used to solve for missing sides of a right triangle.
*When given an angle and one side in a right triangle, the other angle and the other two sides can be solved for. (Teaching to an Objective of Hunter Model)
*Move the slide up until the same right triangle pictured before is displayed--however, with a solved angle added also.

ASK: How can we easily find the missing angle?

ASK: How would we solve for the missing side without using the Pythagorean Theorem? The hint is to use what we learned yesterday.

*Hand out the review sheets on yesterday's lesson as a refresher.
*Tell the students to use the old information to apply to the triangle.

ASK (again): How would we solve for the missing side without using the Pythagorean Theorem?

*Have the students actively solve the problem using the trigonometric functions (on their note takers). Record the answer on the board. (Presentation of New Material/Modeling of Hunter Model)
*Ask many questions--including another explanation of how to use the trigonometric functions. Ask questions and make sure half of the class has their hands raised. (Checking for Understanding of Hunter Model)
*Ask the students to get out their books.
*While the students are doing their example problem, monitor by walking around the room, doing a spot check to make sure the students are on the right track. (Guided Practice of Hunter Model)
*When the students are done practicing, have them start a new page in their notebooks for the next activity. *Ask for their attention, as you post the princess/castle problem back up on the projector.

ASK: Now that we have discussed a few example problems, can anyone now answer how we would solve this problem? Accept student's explanations.

*Have the students show their work in their notebooks. Inform the students that you will be checking over their work at the beginning of next class. (Independent Practice of Hunter Model)
*While the students are working independently, write the homework assignment up on the board. Hand out the homework sheets. While you pass out the sheets, do an eye scan on the students' work--responding to any problems.


Assessment/Evaluation (How you and the students will know that they learned. May be formative or summative):
*I will know that the students learned by grading the homework problems due today.
*I will know that the students learned by asking student-specific questions during the beginning review.
*I will know that the students learned by looking over their pre-tests (diagnostic). The pre-test is basically in the constructed response format.
*I will know that the students learned by observing their efforts put towards the lecture problems on the note taker (formative).
*I will know that the students learned by grading their "if time permits activity."
*I will know that the students learned by grading the homework problems the following day.
*I will know that the students learned by doing a notebook check--looking at the princess/castle problem work.
*I will know that the students learned by grading their weekly quiz on sine, cosine, and tangent (summative). *I will know that the students learned by reading their journal entries on the lesson.
*I will know that the students learned by grading their actual unit test (summative).

Conclusion (Closure; a planned wrap-up for the lesson):
*All students should be quietly seated.
*Remind the students that their homework assignment is due on the next day of class.
*Remind the students that when given an angle and one side in a right triangle, the other angle and the other two sides can be solved for.
*Tell the students that the Pythagorean Theorem is only helpful when there are 2 sides given.
* Preview tomorrow's lesson by stating that the Pythagorean Theorem will be no use for finding the two missing angles in a right triangle. State that tomorrow we will use trigonometric functions to use two side's lengths to solve for an angle.


Accommodations/Adaptations for Students with Special Needs:
*When accommodating a student with a learning disability, I will be sure to have note takers for the student. It is important for this student to be able to see what we are doing. However, the note taker will not be completely filled in. During this lesson, I present the princess problem to the students. The students don't get a copy of the actual castle; it will be posted on the projector. However, I printed off a copy of the castle for the student with a learning disability. I will also provide a list of questions that can be answered by the learning disabled student. The question list will not contain the answers. However, it will contain hints.

If Time Permits Activity:
* Give the students 2 pieces of blank paper, a ruler, a protractor, and a scientific calculator (if needed). Have the students construct their own right triangle problem that requires the use of trigonometric functions. Have the students make up a solution sheet for their problem. Suggest that one of these problems might be randomly chosen for the next quiz. This activity will be handed in and graded. Students who receive full credit will receive 10 points. Points will be deducted based on error.


Reflective Notes:
*How was timing in general?
*Overall, how was the lesson?
Technology Integration
If I was teaching in a classroom with only one computer with a large screen projector that was hooked up to the Internet, I would begin class with a virtual reality tour of a medieval castle. This would help to get students interested in the first application problem (a princess trapped in a castle) presented in the lesson.
Having only one computer should not be seen as a disadvantage. By me doing the actual procedure, the students can watch and learn. Virtual tours are exciting and can be seen as a learning experience. If there was extra time at the end of the class, I would show the students a preview of the next day's lesson either through a short PowerPoint or brain teaser related to the topic.

If I was teaching in a classroom with six computers hooked up to the Internet, I would have groups of four students each come up with their own trigonometric triangle problem. I would have instruction sheets explaining what each student was required to do. I would also give a brief overview of how to use the Word drawing tools. At the next class day, the groups would exchange their print-outs and try to solve the other group's problem. This would be a great review of the previous day's lesson!

If I was teaching in a lab-style classroom where all students had a computer to use, I would have the students explore the following website:
On this website, there is a review of the basic terminology. In addition, there are graphics and diagrams that can be moved. The formulas change whenever the diagrams are changed.