Assessment Mathematics 20-1 Trigonometry - Rubric Level Criteria Excellent 4 Proficient 3 Adequate 2 Limited* 1 Insufficient / Blank* Performs Algebraic Operations and Verification using Sine Law (Parts B and C) Student is able to determine and verify all angles and sides in both oblique triangles Student is able to determine and verify four or five out of six angles and sides in both oblique triangles Student is able to determine and verify three out of six angles and sides in both or either oblique triangles Student is able to determine and/or verify one or two angles or sides in either or both oblique triangles Student is unable to determine or verify any angles or sides in either oblique triangle Performs Algebraic Operations and Verification using Cosine Law (Parts B and C) Student is able to determine and verify all angles and sides in both oblique triangles Student is able to determine and verify four or five out of six angles and sides in both oblique triangles Student is able to determine and verify three out of six angles and sides in both or either oblique triangles Student is able to determine and/or verify one or two angles or sides in either or both oblique triangles Student is unable to determine or verify any angles or sides in either oblique triangle Solving for Reference, Principle, Negative Angles and Coordinates (Part D) Student is able to solve all angles and coordinates for each location Student is able to solve twelve out of sixteen angles and coordinates for each location Student is able to solve eight out of sixteen angles and coordinates for each location Student is able to solve four out of sixteen angles and coordinates for each location Student is unable to solve any angles or coordinates for each location Presentation (Parts A – D) Student has presented all clear and accurate diagrams solutions, and Provides relevant reasons for locations chosen Student has presented most clear and accurate diagrams, solutions, and provides relevant reasons for locations chosen. Student has presented some clear and accurate diagrams, solutions, and provides relevant reasons for locations chosen. Student has presented no clear and accurate diagrams, solutions, and fails to provide relevant reasons for locations chosen Student has not presented.
As a demonstration, draw a Cartesian plane, choose a point in quadrant I, and draw a terminal arm through the point. Determine: distance from the origin to the point the exact value of the sin θ, cos θ, and tan θ the reference angle Student Pairs Activity: Students pick any point in quadrant I and draw a terminal arm. Students should then swap papers and determine: distance from the origin to the point the exact value of the sin θ, cos θ, and tan θ the reference angle Going Beyond Continue the above lesson using negative angles and angles beyond 360 Resources Math 20-1 (McGraw-Hill Ryerson: sec 2.2, pages 88-99) source: Supporting Assessment Glossary angle in standard position - The location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate initial arm - For an angle in standard position, the arm along the positive x-axis quadrantal angle – an angle in standard position where the terminal arm is on the x- or Examples are 0°, 90°, 180°, 270° and 360°. reference angle – The acute angle formed by the terminal arm of an angle in standard position and the x-axis rotation angles: positive angle - An angle in standard position swept out by a counterclockwise rotation of its terminal arm negative angle - An angle in standard position swept out by a clockwise rotation of its terminal arm reference triangle – A right triangle with a reference angle as one of its vertices terminal arm – For an angle in standard position, the arm that is free to rotate
- Common endpoint of two rays that form the angle. Other Lesson 3 Applying the CAST Rule STAGE 1 BIG IDEA: The sine law and cosine law allows us to work with non right triangles. Applications of trigonometry include surveying, navigation, construction, and calculus. ENDURING UNDERSTANDINGS: Students will understand … Angles in a circle can be expressed in a variety of ways. Each primary trigonometric ratio is positive in two quadrants and negative in two quadrants between 0o and 360o (CAST). ESSENTIAL QUESTIONS: What is triangulation? How many different ways can you estimate the height of a mountain? Why is it easier to find the exact value of some trigonometric ratios? KNOWLEDGE: Students will know … the sign of a given trigonometric ratio for a given angle, without the use of technology, and explain the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30º, 45º or 60º the patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360° some contextual problems can be solved using trigonometric ratios SKILLS: Students will be able to … determine the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ determine, without the use of technology, the value of sin θ, cos θ or tan θ, given any point P (x, y) on the terminal arm of angle θ, where θ = 0º, 90º, 180º, 270º or 360º determine the sign of a given trigonometric ratio for a given angle, without the use of technology, and explain solve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real number determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30º, 45º or 60º describe patterns in and among the values of the sine, cosine and tangent ratios for angles from 0° to 360° solve a contextual problem, using trigonometric ratios Lesson Summary
Hook Consider allowing students to explore a resource that restricts examples to right angle triangles. At the end of the exploration illicit that there are other kinds of triangles (oblique) that we need to be able to calculate angle and side values for. Lesson Goal Students can see the relationship between angles and the length of opposite sides and use the sine law to find unknown sides and angle measures. Students will realize that sine law can be used when a triangle does not have a 90º angle, but an angle, its opposite side, and at least one other angle or side is known. Activate Prior Knowledge Students will be using a protractor to determine angle size(s) and a ruler to determine side length(s). They will need to set up a ratio using appropriate sides and angles. Lesson Discuss the relationship between the sine of the angle and the opposite side. Teach the sine law to show how this relationship can be used to find unknown sides and angles. Stress to students that the sine law can only be used if you are given (or can find) the angle, its opposite side, and at least one other angle or side. Go through examples finding: unknown sides unknown angles. Provide examples to be given where sine law must be used to solve triangles. Problems should include examples where students must sketch a diagram and solve a problem using the sine law. Going Beyond Resources Math 20-1 (McGraw-Hill Ryerson: sec 2.3, pages 100-113)