UNIT PLAN th Grade Pre-Calculus

Transcription

1 UNIT PLAN Subject/Grade Level: th Grade Pre-Calculus Unit #: 1 Unit Name: Trigonometric Functions Big Idea/Theme: Using trigonometric functions to model real-life situations will prepare students for future careers in engineering, medicine, music, liberal arts, and business. Culminating Assessment: Develop and construct a biorhythm chart from their personal data. (See attached project description.) Unit Understanding(s) Students will understand The relationship of degree and radian measure of any triangle can be graphically displayed using the six trigonometric functions. Graphing trigonometric functions and their inverses is accomplished by analyzing intercepts, periodic behavior, domain, range, asymptotes, and graphs of reciprocal functions. Trigonometric equations are solved graphically. Trigonometric identities and laws of sines and cosines must be used to evaluate expressions, solve trigonometric equations and solve applied problems. Calculating the area of a triangle is accomplished when the lengths of two sides and the measure of the included angle is given so the slope of a line given the angle of inclination can be computed. Points can be plotted in the polar coordinate system using a specific procedure. Unit Essential Question(s): What is the relationship of degree and radian measure of triangles and how would you use the information to graphically display the six trigonometric functions? How do you graph trigonometric and their inverse functions by analyzing intercepts, periodic behavior, domain, range, asymptotes, and graphs of reciprocal functions? How do you solve trigonometric equations graphically and algebraically? How can you use trigonometric identities and laws of sines and cosines to evaluate expressions, solve trigonometric equations and solve applied problems? How would you calculate the area of a triangle when given the lengths of two sides and the measure of the included angle, and compute the slope of a line given the angle of inclination? How procedure is used to plot points in the polar coordinate system? 1

2 Students will know / Students will be able to Demonstrate through the processes an the behaviors of trigonometric functions. Understand and utilize trigonometric identities to solve trigonometric equations algebraically and graphically. South Carolina Academic Standards: PC-1.1 Communicate knowledge of algebraic and trigonometric relationships by using terminology appropriately. PC-1.2 Connect algebra and trigonometry with other branches of mathematics. PC-1.5 Demonstrate an algebraic and trigonometric relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic). PC-1.6 Understand how algebraic and trigonometric relationships can be represented in concrete models, pictorial models, and diagrams. PC-1.7 Understand how to represent algebraic and trigonometric relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs). PC-5.1 Understand how angles are measured in either degrees or radians. PC-5.2 Carry out a procedure to convert between degree and radian measures. PC-5.3 Carry out a procedure to plot points in the polar coordinate system. PC-5.4 Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions. PC-5.5 Carry out procedures to determine the characteristics of trigonometric functions (including domain, range, intercepts, and asymptotes). PC-5.6 Apply a procedure to evaluate trigonometric expressions. PC-5.7 Analyze given information to write a trigonometric function that models a given problem situation involving periodic phenomena. PC-5.8 Analyze given information to write a trigonometric equation that models a given problem situation involving right triangles. PC-5.9 Carry out a procedure to calculate the area of a triangle when given the lengths of two sides and the measure of the included angle. PC-5.10 Carry out a procedure to solve trigonometric equations algebraically. PC-5.11 Carry out a procedure to solve trigonometric equations graphically. PC-5.12 Apply the laws of sines and cosines to solve problems. PC-5.13 Apply a procedure to graph the inverse functions of sine, cosine, and tangent. PC-5.14 Apply trigonometric relationships (including reciprocal identities; Pythagorean identities; even and odd identities; addition and subtraction formulas of sine, cosine, and tangent; and double angle formulas) to verify other trigonometric identities. PC-5.15 Carry out a procedure to compute the slope of a line when given the angle of inclination of the line. 2

3 Interim Assessment (formative) Board work Class discussion Graphing Calculator/Computer Activities Group Activities Journals Observations Question and answer Quizzes Tests Key Criteria (to meet the standard/rubric) See rubric following project description sheet. 3

4 Biorhythm Project Example 1 On the day of your birth all three cycles began at the same time. To find out where you are in your three cycles consider the following example. Suppose you were born on August 13, 1950 and today is (insert today s date). Calculate your 3 graphs. Do each one on a separate graph 1. First, calculate the total number of days you have lived. a) Multiply 365 by your age. (43 in this example) 365 X 43 = b) You must add an extra day for each leap year. Remember, leap years are divisible by 4 ( 52, 56, 60, 64, 68, 72, 76) There are 11 in this example = c) Since it is (insert today s date), we must add the number of days since your last birthday. In this case it will be 329 days = d) To compute where you are in your 23 day physical cycle, divide the total number of days lived by 23. You are only interested in your remainder /23 = 697 remainder 4 you would be on the 4 th day of your physical cycle on July 7, 1994 e) Since this is your 4 th day then July 4 was your first day of your physical cycle. So start your graph on July 4 and count forward to the 23 rd day. This is the last day of your cycle. f) Now, draw one (or more) sine graph(s) starting on July 4 and ending on July 27. (To draw more periods, add 23 more days.) g) Do the same thing with your emotional and intelligence cycles. Emotional: 16035/28 = 572 remainder 19 You would be on the 19 th day of your emotional cycle July 7, Since this is closer to the middle of your cycle, count back to day 14, July 2. So, July 2 would be where you would begin your downward part of your sine curve for your emotional cycle. Now, draw one (or more) sine curve(s) starting on July 2 and ending on July 30. Intelligence: 16035/33 = 485 remainder 30 You would be on the 30 th day of your intelligence cycle, July 7. Since this is closer to the end of your cycle, count forward to the first day of your next cycle, July 11. Now, draw one (or more) sine curve(s) starting on July 11 and ending on August 13. h) If your remainder is 0, then you are on your last day of your cycle. 4

5 Math - Problem Solving CATEGORY Mathematical Concepts Mathematical Reasoning shows complete the concepts used to solve the Uses complex and refined reasoning. Mathematical Errors % of the steps and solutions have no Strategy/Procedures Typically, uses an efficient and effective strategy shows substantial the concepts used to solve the Uses effective reasoning Almost all (85-89%) of the steps and solutions have no Typically, uses an effective strategy shows some the concepts needed Some evidence of reasoning. Most (75-84%) of the steps and solutions have no Sometimes uses an effective strategy to solve problems, but does not do it consistently. shows very limited the underlying concepts needed problem(s) OR is not written. Little evidence of reasoning. More than 75% of the steps and solutions have Rarely uses an effective strategy to solve problems. 5