High School MATHEMATICS Trigonometry

Transcription

1 High School MATHEMATICS Trigonometry Curriculum Curriculum Map USD 457 Math Framework Performance/Open-Ended Response Assessments o First Semester (Tasks 1-3) o Second Semester (Tasks 1-4) Available on Intranet High School Assessment Guide (flip chart) State Standards Scope and Sequence (Math) USD 457 Garden City, KS

2 Trigonometry - Curriculum Map MATH Bold Indicator tested in current grade Underline Indicator tested in Next Grade Regular Indicator appears in the State Standards and recommended for inclusion Sans Comic Indicator is Current District Wording 1 st Quarter Indicators Trig Functions K1 changes from radian to degree measures and vice versa K2 finds reference and coterminal angles K7 finds six circular/trig functions K1 solves right triangles General Triangle Trigonometry K2 uses law of sines K3 uses laws of cosines K3 finds area of triangles using trigonometry 2 nd Quarter Indicators Local uses length of arc and area of sector formulas Graphs of Trig Functions K8 finds and graphs amplitude, period, and phase shift of trig functions K9 writes equations of trig functions given amplitude, period and phase shift and graphs of trig functions K5 evaluates inverse of trig functions K6 finds principal values of inverse trig functions 3 rd Quarter Indicators Trig Identities K7 uses basic trig identities (reciprocal, quotient, and Pythagorean theorem identities) K8 Uses sum and difference identities K9 Uses double-angle and half-angle identities Trig Equations K4 solves trig equations 4 th Quarter Indicators Vectors K10 adds and subtracts vectors geometrically K11 Adds, subtracts, multiplies and finds magnitude and direction of vectors algebraically Polar Coordinates K12 Graphs polar coordinates and equation K13 Converts from polar to rectangular coordinates and vice versa K1 Change complex numbers from rectangular to polar form and vice versa 1

3 High School Performance/Open-Ended Response Assessments Trigonometry First Semester QUARTER ITEM # TASK BENCHMARK 1 1 Trigonometric Functions 1 2 General Triangle Trigonometry 2 3 Graphs of Trig Functions Changes from Radian to degree measures & vice versa Finds reference and coterminal angles Finds six circular/trig functions Solves right triangles Uses Law of Sines Uses Law of Cosines Finds area of triangles using trigonometry Finds & graphs amplitude, period, and phase shift of trig functions Writes equations of trig functions given amplitude, period, and phase shift and graphs of trig functions Evaluates inverse trig functions Finds principal values of inverse trig functions All tests will be filed in student portfolios for documentation For scoring: rubrics (5, 4, 3, 2, 1) are for district assessment purposes, and the percents are to be used as the grade 2

4 Item #1 LOCAL INDICATORS Task: Trigonometric Functions Trigonometry X K Understands radian and degree measures Understands reference and coterminal angles Understands circular and trigonometric functions Applies sine, cosine, and tangent formulas (e.g., right triangle) PERFORMANCE TASK The teacher will design a test that measures trig functions. At least 5 test items will cover conversion between radians and degrees. At least 5 test items will cover both reference and coterminal angles. At least 5 items will require students to apply trig functions to circle problems, finding both ordered pairs and central angles. At least 5 test items will assess the ability of students to apply sine, cosine, and tangent formulas to right triangle situations. On all items points will be awarded for both process and answer. Sample items are attached. *The assessment must be filed in the student s portfolio for documentation. **The second attempt must be a different exam or project. KNOWLEDGE/SKILLS Conversion between radian and degree measures Angles Reference Coterminal Application of trig functions to circles Finding ordered pairs on the unit circle Finding central angle given the radius Trig Functions Sine Cosine Tangent SCORING GUIDE 3

5 Radian/Degree Measures Item #1 Include at least 5 items of each type of problem. SAMPLE ITEMS TRIGONOMETRY Trig Functions 1. Convert the angle θ = ' 18" to (degree) decimal form. 2. Convert to radians : 25 4

6 Reference/Coterminal Angles Item #1 SAMPLE ITEMS TRIGONOMETRY Trig Functions Include at least 5 items. 1. Find the reference angle for θ = Find two coterminal angles (one positive and one negative) for the angle θ = 57. 5

7 Right Triangle Trigonometry Item #1 SAMPLE ITEMS TRIGONOMETRY Trig Functions Test will include at least 5 items. 1. What is the ordered pair on the unit circle that is associated with π t =? 6 2. A central angle θ of a circle with a radius 16 inches subtends an arc of inches. Find θ. 6

8 Right Triangle Trigonometry Item #1 SAMPLE ITEMS TRIGONOMETRY Trig Functions Include at least 5 items. 1. A ladder is leaning against the side of a house. The base of the ladder is 5 feet from the wall and makes an angle of 39 with the ground. Find the length of the ladder. 2. Find x for the triangle shown at the right x 7

9 Item #2 LOCAL INDICATORS Task: General Triangle Trigonometry Trigonometry Applies the law of sines to problem situations involving general triangles Applies the law of cosines to problem situations involving general triangles Figures area of triangles using trigonometric functions X K PERFORMANCE TASK The teacher will design a test, which assesses application of general trigonometry to triangles. At least 5 items will require students to apply the law of sines to problem situations, including both triangles and circles. At least 5 items will cover the application of the law of cosines where students are given a variety of side and/or angle measures. At least 5 items will require students to find the area of a triangle given a variety of side and/or angle measures, including the use of Heron s formula. Points will be given for both steps and answers. Sample items are attached. *The assessment must be filed in the student s portfolio for documentation. **The second attempt must be a different exam or project. KNOWLEDGE/SKILLS Law of Sines - Triangles Circles (arcs, chords, radius, central angles) Finding radius Finding length of arc Law of Cosines - Application given both angle and/or side measures Area of Triangle Cosine Tangent Money value SCORING GUIDE P , , and % points earned % points earned % points earned % points earned 1 Less than 65% points earned Application given both angle and/or side measures Heron s formula 8

10 Law of Sines Item #2 SAMPLE ITEMS TRIGONOMETRY General Triangle Trigonometry Include at least 5 items. 1. Use the given information to find the remaining sides and angles of the triangle. Use Law of Sines. C 15 A 105 b 35 a c B 2. The circular arc of a railroad curve has a chord of length 3000 feet and a central angle of 80. Find a) the radius r of the circular arc b) the length s of the arc 9

11 Law of Cosines Item #2 SAMPLE ITEMS TRIGONOMETRY General Triangle Trigonometry Include at least 5 items. 1. Find the three angles of the triangle whose sides have length a=8 ft., b=19 ft., and c=14 ft. 2. A ship travels 60 miles due east, then adjusts its course 15 northward. After traveling 80 miles in that direction, how far is the ship from its point of departure. 10

12 Area of Triangles with Trigonometry SAMPLE ITEMS Item #2 TRIGONOMETRY General Triangle Trigonometry Include at least 5 items. 1. Given a triangle with c=24.19, a=91.6, and B= 37, find the area. 2. Given a triangle with a=80, b=90, and c=110, use Heron s formula to find the area. 11

13 Item #3 LOCAL INDICATORS Task: Graphs of Trig Functions X Trigonometry K Uses the effects of parameter changes on the graphs of trig functions Derives the equation relating to trigonometric functions Find principal values of inverse trigonometric functions PERFORMANCE TASK The teacher will design a test that measures a student s understanding of trig function graphs. At least 5 items will cover the effects of parameter changes on trig function graphs. These items will include both graphing functions and determining functions given a graph. At least 5 items will items will cover basic concepts related to amplitude, period, and phase shift. At least 5 items will cover principal values. Points will be awarded for both steps and answers. Sample items are attached. *The assessment must be filed in the student s portfolio for documentation. **The second attempt must be a different exam or project. KNOWLEDGE/SKILLS Parameter Changes Amplitude Period Phase Shift Sketching the Graph of a Function (given the function) Derive the Equation of a Function (given the graph or given information about the function) SCORING GUIDE , and % points earned % points earned % points earned % points earned 1 Less than 65% points earned 12

14 Parameter Changes Item #3 SAMPLE ITEMS TRIGONOMETRY Graphs of Trig Functions Include at least 5 items. 1. Determine the period of the function f(x) = -7cos3x 2. Determine the amplitude of the function f(x) = -7cos3x 13

15 Concepts Item #3 SAMPLE ITEMS TRIGONOMETRY Graphs of Trig Functions Include at least 5 items. 1. Sketch the graph of the function: 1 π f ( x) = sec x Sketch the graph of the function: f ( x) = x sec x 3 3. Using the internal, 2 2, write the equation for this function. 14

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17 Performance/Open-Ended Response Assessments Trigonometry Second Semester QUARTER ITEM # TASK BENCHMARK 3 1 Trig Identities/Verifying Uses basic trig identities (Reciprocal, quotient, and Pythagorean Theorem identities) Uses sum and difference identities Uses double-angle & half-angle identities 3 2 Trig Equations Solves trig equations 4 3 Vectors Add and subtract vectors geometrically Add, subtract, multiply, and find magnitude and direction of vectors algebraically 4 4 Polar Coordinates Graphs polar coordinates and equations Converts from polar to rectangular coordinates and vice versa Changes complex numbers from rectangular to polar form and vice versa All tests will be filed in student portfolios for documentation For scoring: rubrics (5, 4, 3, 2, 1) are for district assessment purposes, and the percents are to be used as the grade 16

18 2 nd Semester Item #1 LOCAL INDICATORS Task: Trigonometric Identities Trigonometry X K Verifies basic trigonometric identities Use the sum and difference identities for sine, cosine and target functions Use double and half angles identities for the sine, cosine, and target functions PERFORMANCE TASK The teacher will design a test that measures student mastery of ability to identify trig identities. At least 10 items will be included in the assessment. Items will include reciprocal, quotient, and Pythagorean identities. At least 5 items will cover sum and difference identities. At least 5 items will assess double and half angle identities. Points will be awarded for both process and answers. Sample items are attached. *The assessment must be filed in the student s portfolio for documentation. **The second attempt must be a different exam or project. KNOWLEDGE/SKILLS Identify Verification Reciprocal Quotient Pythagorean Identities Sum Difference Double Half Angle Money value Penny Nickel SCORING GUIDE % points earned % points earned % points earned 1 Less than 65% points earned % points earned 17

19 Verifying Trig Identities 2 nd Semester Item #1 SAMPLE ITEMS TRIGONOMETRY Trigonometric Identities Include at least 5 items. 1. Verify the identity: sec x csc 2 x - csc 2 x = sec x 1+ cos x x 2 2 tan Verify the identity: = csc x 2 tan x 3. Use trigonometric identities to simplify: 2 csc xcos x 1+ csc x 18

20 2 nd Semester Item #2 LOCAL INDICATORS Solves trig functions Task: Trig Equations Trigonometry X K PERFORMANCE TASK The teacher will design a test, which measures student mastery of ability to identify trig equations. At least 10 items will be included in the assessment. Items will assess solving trigonometric functions. Points will be awarded for both process and answers. KNOWLEDGE/SKILLS Trigonometric Equations Money value Penny Nickel Dime Quarter Half Dollar $1.00 $5.00 SCORING GUIDE % points earned % points earned % points earned % points earned 1 Less than 65% points earned 19

21 2 nd Semester Item #3 Task: Vectors LOCAL INDICATORS Solves trig functions Trigonometry X K PERFORMANCE TASK The teacher will design a test, which measures student mastery of vectors. At least 10 items will be included. Items will cover solving vectors geometrically and algebraically. KNOWLEDGE/SKILLS Geometric add subtract Algebraic add, subtract, multiply Magnitude SCORING GUIDE % points earned % points earned % points earned % points earned 1 Less than 65% points earned 20

22 2 nd Semester Item #4 LOCAL INDICATORS Task: Polar Coordinates Trigonometry X K Graph polar coordinates and equations Convert from polar coordinates to rectangular coordinates and vice versa. PERFORMANCE TASK The teacher will design a test, which measures student mastery of polar coordinates. At least 10 items will be included. Items will cover graphing coordinates/equations and polar/rectangular coordinate conversion. KNOWLEDGE/SKILLS Graph Coordinates equations Conversions between polar and rectangular SCORING GUIDE % points earned % points earned % points earned % points earned 1 Less than 65% points earned 21

23 Reference/Coterminal Angles Item #1 SAMPLE ITEMS TRIGONOMETRY Trig Functions Include at least 5 items. 22