5/6 Lesson: Angles, measurement, right triangle trig, and Pythagorean theorem


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1 5/6 Lesson: Angles, measurement, right triangle trig, and Pythagorean theorem I. Lesson Objectives: Students will be able to recall definitions of angles, how to measure angles, and measurement systems and conversion. Students will be able to recall right triangle trigonometry (covered previously in class) and solve real world examples with right triangle trig. Students will be able to recall the Pythagorean Theorem and prove it using areas of triangles and squares. II. Assessment: Formative: Review activities. Success indicators: correctly recalling theorems and definitions, effectively apply theorems and definitions to solve problems. Summative: Blog post #1. Success indicators: Written well and in their own words, no grammatical or spelling errors, at least one image. III. Standards: OCCSS GSRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. OCCSS 4.MD 5: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement. OCCSS 4.MD 6: Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. OCCSS 4.MD 1: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. OCCSS 8.G 6: Explain a proof of the Pythagorean Theorem and its converse. OCCSS 8.G 7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. IV. Time: One 90 minute class period V. Materials: Notes on angles, measurement, right triangle trig, and the Pythagorean Theorem. Protractors and rulers. Measurement conversion charts In class right triangle trig example worksheet Square papers (x2 for each student) Right triangle trig and Pythagorean Theorem worksheet VI. Beginning the Lesson: [25 min] Warm up: Review of angles. Have students define what an angle is and how to measure an angle. [5 min] Go over warm up and make a precise definition of angle. [5 min] Brainstorm length measurement units. Create a web of conversions between the units students generate. Pass out conversion charts. [15 min] VII. Middle of the Lesson: [45 min] Review right triangle trig. Notes for definitions and examples of applied examples [25 min]
2 Prove the Pythagorean Theorem using two squares, one with a cxc square skewed in the middle and four right triangles around it (an a leg, a b leg, and a c hypotenuse); the other with an axa square in one corner, and a bxb square in the opposite corner, and two bxa rectangles, each with a c length diagonal. (See Notes for more details) [20 min] VIII. End of the Lesson: [20 min] Allow groups to start collaborating and writing their blog post. Blog post due digitally through GoogleDocs by tomorrow. [20 min]
3 Angle, measurement, right triangle trig, and Pythagorean Theorem notes Defn: Angle: The space between two rays that share the same endpoint. It is measured by the degree amount of rotation needed to rotate one of the rays exactly on top of the other. We use a protractor to measure angles. (Demonstrate measuring an angle) Students draw three different angles and measure them. Brainstorm: What tools do we use to measure lengths? What units do we use to measure lengths? Pass out measurement conversion chart. Examples: 23m = cm 17in = ft 4 in = cm 23 km = mi Right triangle trig: Trig ratios: sina = a/c cosa = b/c tana = a/b a is the opposite side from angle A. b is the adjacent side to angle A. c is the hypotenuse in the right triangle. So we have the acronym SOHCAHTOA to help us remember. Vocabulary used in word problems: Angle of elevation: The angle from the ground to the top of a high place. Angle of depression: The angle from the top of a high place to the ground. congruent. These angles are actually the same measurement, they are just measured differntly. Notice that the two horizontal lines are parallel, so the angle of elevation and the angle of depression are alternate interior angles, so they are Example 1:
4 A squirrel sees the top of a tree at an angle of elevation of 55. He is 30 ft away from the tree. Draw a picture of the situation and find the height of the tree using SOHCAHTOA. Example 2: A hang glider is looking at the edge of a lake at an angle of depression of 30. He is 300ft above the ground. How far away from the edge of the lake is he? Draw a picture. Pass out right triangle trig example worksheet. Pythagorean Theorem Review: Pythagorean Theorem: a2 + b2 = c2 for any right triangle. In other words: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. In a right triangle, if: 1. a = 2, b = 5, what is c? 2. a = 3, c = 7, what is b? Examples: Do squares activity.
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6 Solve the following word problems. For each question, draw a diagram to help you. 1) An airplane is flying at an altitude of 6000 m over the ocean directly toward a coastline. At a certain time, the angle of depression to the coastline from the airplane is 14. How much farther (to the nearest kilometer) does the airplane have to fly before it is directly above the coastline? 2) From a horizontal distance of 80.0 m, the angle of elevation to the top of a flagpole is 18. Calculate the height of the flagpole to the nearest tenth of a meter. 3) A 9.0 m ladder rests against the side of a wall. The bottom of the ladder is 1.5 m from the base of the wall. Determine the measure of the angle between the ladder and the ground, to the nearest degree. 4) The angle of elevation of the sun is 68 when a tree casts a shadow 14.3 m long. How tall is the tree, to the nearest tenth of a meter? 5) A wheelchair ramp is 4.2 m long. It rises 0.7 m. What is its angle of elevation to the nearest degree? 6) A person flying a kite has released 176 m of string. The string makes an angle of 27 with the ground. How high is the kite? How far away is the kite horizontally? Answer to the nearest meter.
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