Project Based Instruction

Transcription

1 Project Based Instruction -Find the Missing Side Find the Missing Side Author(s): Description of Class: Name of Class: Geometry Grade Level: 10th Honors or Regular: Regular Source of Lesson: Nelson, B., & Barnett, J. C. (2009). Analytic Trigonometry and Trigonometric Applications. SpringBoard Mathematics With Meaning Precalculus (pp ). New York: The College Board. Roberts, D. (n.d.). Area of Triangle Using Trigonometry. Oswego City School District Regents Exam Prep Center. Retrieved February 25, 2010, from TeacherTube Videos - Using Trig in Real Life. (2009, April 27). TeacherTube - Teach the World Teacher Videos Lesson Plan Videos Student Video Lessons Online Teacher Made Videos TeacherTube.com. Retrieved February 25, 2010, from Concepts: Develop and apply formulas based upon trigonometry relationships to find the area of a triangle; apply trigonometry to solve real world problems. Objectives: Students will be able to develop formulas based upon trigonometry relationships for the area of a triangle. Students will be able to apply formulas to find the area of a triangle using trigonometry. Students will be able to apply trigonometry to real world problems. State Standards: TEKS: Geometry (One Credit). (b) Knowledge and skills. (11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: (C) Develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and 2007 The University of Texas at Austin

2 CCRS: III.Geometric Reasoning A. Figures and their properties 3. Recognize and apply right triangle relationships including basic trigonometry. Materials List and Advanced Preparations: Math Journals TI 84+ Calculators Rulers (measuring feet/centimeters) Protractors Safety: Only use rulers for measuring. No throwing them. Should only be two at each table. Only use protractors for measuring angles, nothing else. Should only be two at each table. Use calculators appropriately. Each student should only have one calculator. ENGAGEMENT What the Teacher Will Do Hello, we are going to start today s lesson with a video. com/viewvideo.php?vi deo_id= Provide video for students to watch. Probing/Eliciting Questions What do you think geometry is used for in the real world? How do you think finding the area of right triangles is using geometry is used outside of math class? Time: 5 Minutes It s not [Architecture, other real world applications students might say.] Page 2 of 8

3 EXPLORATION What the Teacher Will Do On the smartboard, draw triangle ABC: Probing/Eliciting Questions What is the formula for the area of a triangle? What are the 3 basic trigonometric functions? Time: 15 Minutes A=base x height [A=1/2(base)(height)] [Sin(Θ)=opposite/hypotenuse] [Cos(Θ)=adjacent/hypotenuse] [Tan(Θ)=opposite/adjacent] After drawing the triangle, add measurements for side a, side b and angle B. We will pass out triangles (one right triangle, and one not right triangle) and a worksheet to each group. I am passing out triangles and worksheets to each of your groups. You and your partners will need to find the base, height, and area for your triangles, and come up with a way to find the area using geometry and trigonometry. Measuring the lengths and the angles of your triangles is a great place to start. What does A, B, C and a, b, c stand for? Numbers [A, B, C represent angles and a, b, c represent sides.] Follow the directions on your worksheet for the activity. Remember, you are measuring the big triangles, not the ones on the worksheet. As soon as I place the materials on your table, your group may get started. You have 15 minutes to complete this activity. When you are finished sit quietly with your group and go over your solutions. We are going to select three groups to come to the front of the class and explain their answers at the end of the 15 minutes, so be Page 3 of 8

4 prepared. As students are working I will be walking from group to group giving them clues and helping them to see a formula. How can you use the formula for the area of a triangle to find the area of your triangles? Are there any other formulas that can help you? Page 4 of 8

5 EXPLANATION What the Teacher Will Do Call on 3 groups to present their work. Review each groups work and then develop a formula together as a class. From triangle ABC, how can you develop a new formula for the area? Remember: A, B, and C are angles and a, b, and c are sides. We know that the area of a triangle is ½(base) (height). We can draw an altitude from angle C to side c to find the height. An altitude is a straight line drawn from a vertex and perpendicular to the opposite side. Probing/Eliciting Questions How do you find the height of a triangle? Does anybody know what an altitude is? Time: 15 Minutes Add all the sides added together [Area= ½(base)(height)] A line cutting the triangle in half [A straight line from angle C to side c from a vertex and perpendicular to the opposite side.] Now, we need to solve for h. h=b (sina). Since we solved for h, we can now plug h=b (sina) into A=1/2bh. Our new equation looks like: A= 1/2c x bsina. Notice that c is the base. The formula depends on what angle is given: 1/2acsinB 1/2absinC These formulas are used for side angle side triangles. Have a student come to the board and use their triangle as an example. How can we find the height of this triangle? [A= 1/2c x bsina] Page 5 of 8

6 Have students find the area of one of their triangles from the exploration. ELABORATION What the Teacher Will Do The formula depends on what angle is given: 1/2acsinB 1/2absinC These formulas are used for side angle side triangles. Probing/Eliciting Questions What kind of triangle is the area formula we ve been working with used for? Time: 5 Minutes All triangles [Triangles that have 2 sides given and 1 angle.] EVALUATION What the Teacher Will Do Probing/Eliciting Questions Time: 10 Minutes I am passing out an activity. This is to do individually, which means there should be no talking. When you are finish place the worksheet face down on your desk and one of us will come pick it up. Page 6 of 8

7 EXPLORING TRIANGLES NAME: For each triangle, answer the question and find out: What is the base? What is the height? What is the area? (You may write on the cut out triangle and on the worksheet. Show all work and label units!) 1. A dirt bike is heading towards a ramp. How long is the ramp? 2. Two soccer players are deciding who should take the shot. How far from the goal is each player? 3. GPS satellites use angles to determine your position. If you are on the ground with the two satellites shown above, find all the angles involved. Page 7 of 8

8 NAME: End of Lesson Activity Draw each triangle and find the area of each triangle using trigonometry. Show your work and round to the nearest hundredth! 1) A = 57, b = 10, c = 22 2) a = 5, b = 7, c = 9 3) C = 145, a = 78, b = 32 Page 8 of 8