|TITLE: Graphs of Tan, Cot, Sec, and Csc
TASK DEVELOPER: Pete Anderson
GRADE AND CONTENT AREA: Grade 12, Precalculus
SCHOOL: Mergenthaler Vocational Technical Senior High School
Maryland Core Learning Goals, Expectations, and Indicators
Goal 1 Functions and Algebra: The student will demonstrate the ability to investigate, interpret, and communicate solutions to mathematical and real-world problems using patterns, functions, and algebra.
Expectation 1.1: The student will analyze a wide variety of patterns and functional relationships using the language of mathematics and appropriate technology.
Indicator 1.1.4: The student will describe the graph of a non-linear function and discuss its appearance in terms of the basic concepts of maxima and minima, zeros (roots), rate of change, domain and range, and continuity.
- A coordinate graph will be given with easily read coordinates.
- Zeros refers to the x-intercepts of a graph, roots refers to the solution of an equation in the form p(x) = 0.
- Problems will not involve a real-world context.
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The students will explain how transformations of tan, cot, sec, and csc graphs are related to a basic trigonometric graph, determine the period, domain, range, zeros, and asymptotes (if any) of that graph, and sketch that graph. The students will solve trigonmetric equations involving tan, cot, sec, and csc in given intervals graphically and analytically.
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Real World Setting: You are an engineer for Lockheed Martin. You are faced with finding the coverage of a lighthouse on a small island 350 feet from the shore of the mainland. You must express the distance d as a function of the angle x. Once you have completed this job, you will find d if the angle x is 1.55 radians. Then, you report your findings to your company project manager to verify that there is enough light coverage for this danagerous commute by boat from the island to the mainland.
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Level I: Acquiring Data - Data students will acquire in this standards-based task:
Vocabulary: tangent function, cotangent function, secant function, cosecant function
Images: graphs of the above functions
Skills: Explain how transformations of these above graphs are related to a basic trigometric graph. Determine the period, domain, range, zeros, and asymptotes (if any). Then, sketch a graph of the function with and without the use of a graphing calculator. Solve trigonometric equation involving these functions in all four quadrants graphically and analytically by using the inverse function.
Concepts: Knowing important features of all trigonometric graphs, and learning how to use them to solve problems.
Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task: Graphs of trigonometric functions
Organizing: The similarities and differences of the trigometric graphs and their transformations
Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task: Graph the basic trigonometric graphs
Solving problems: Graph transformations of tangent, cotangent, cosecant and secant graphs. Solve trigonometric problems involving these functions without and with real world situations
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Task Approach - The students will complete the task individually and working in cooperative groups.
Task Delivery - The task will be organized and delivered in the student classwork during class time.
Assistive Technology Accommodations - Students with special needs will require the following electronic devices: Calculator.
Use of Resources - The school will provide classroom time to complete the task.
Use of Resources - The students will provide: classroom materials such as pencils, paper, notebooks, and homework time.
Customer for Student Work - The students will present their work as evidence of task completion to the teacher.
Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: the student's teacher.
Reporting Results - The assessment results will be reported as a letter grade. Proficient will be 80% or better on the 6 classwork problems.
Timeline - The estimated time needed to plan, teach, and score this task is one 90 minute class period.
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The Directed Teaching Activity (DTA)
Opening - Estimated Time: 18 minutes
Focusing Student Attention:
Statement of objective: The student will know how to generate graphs for the tangent, cotangent, secant, and cosecant functions, to explain transformations of these graphs, and to solve problems involving these trigonometric functions either graphically or analytically using the inverse functions.
- Determine the measure of the angle coterminal to an angle of -70degrees if 360 < the angle < 720
- Evaluate all six trigonometric functions in a right triangle if the hypotenuse is 13 and one leg is 5.
- The angle is in standard position with P(-3,-4) on the terminal side. Find sin and cot.
- What transformations are there for y = -3sin(4x+pie/4) compared to y = sin x ?
Step 1: How will you engage the students in learning? Plot key points of tan using tan = sin / cos using -pie/2, -pie/3, -pie/4, -pie/6, 0, pie/6, pie/4, pie/3, pie/2 and connect them with a smooth curve. Now, graph y = tan x on your graphing calculator using zoom trig. What are the period, domain, range, vertical asymptotes, and zeros ?
Step 2: How will you connect the lesson to their prior knowledge? a,b,h,k influence the graph of y = a tan(b(x-h))+k in much the same way that they do for the graph of y = a sin (bx + c) + d = a sin (b(x-h)) + k when you factor it out.
Heart of the Lesson - Estimated Time:54 minutes
Introductory and/or Developmental Activities
Teacher Directed Activities - Estimated Time: As needed
Step 1: How will you aid students in constructing meaning of new concepts? With the graphing calculator and refering to the four examples in the textbook.
Step 2: How will you introduce/model new skills or procedures? With the overhead graphing calculator.
Teacher Monitored Activities - Estimated Time: 15 minutes
Step 1: What will students do together to use new concepts or skills? p442-446:Examples 1 - 4
Step 2: How will you assist students in this process? Circulate, give individual help, let them work in groups of four, put up the examples on the board, and get individual students to explain the examples outloud. Add additional examples if ncessary.
Independent Activities and/or Meaningful use Tasks
Step 1: What will students do together to use new concepts or skills? p447-449:4,12,20,28,44,45
Step 2: How will you assist students in this process? Circulate, give individual help, let them work in groups of four, and have students put up the classwork work and answers on the front board.
Extension, Refinement, and Practice Activities- Estimated Time: As needed
Extension p449:48,50,52 This also serves as differentiated instruction.
Step 1: What opportunities will students have to use the new skills and concepts in a meaningful way? Homework - p447-449:8,16,24,32,40,46
Step 2: How will students demonstrate their mastery of the essential learning outcomes? I will fully grade the drill, classwork, and homework the next day.
The role of the engineer that is stated in the real world setting is introduced when going over problem 45 on the classwork. This real world setting is continued in a similar engineering problem in problem 46 which is assigned for homework.
Closing - Estimated Time: 3 minutes
Step 1: Ongoing Assessment - How will you monitor student progress throughout the lesson? By calling on almost every student in the class to explain a problem or put it on the board, by circulating and giving individual help, and by having students put the classwork on the board.
Step 2: Culminating Assessment - How will you ensure that all students have mastered the identified learning indicators? How will you assess their learning? I will grade the drill, classwork, and homework the next day. I will also use two BCR's that come from this section on the practice test and test to be given in a few days.
Step 3: Closure Activities: Through this teacher-guided activity, how will you assist students in reflecting upon what they learned today and preparing for tomorrow's lesson? What homework will be assigned to help students practice, prepare, or elaborate on a concept or skill? I will ask, with hints if necessary, a student to give a summary, and the homework will be p447-449:8-40 every 8th one and p449 46 and to read and copy the examples in the next section. Also, to differentiate instruction, students may do any of the even problems in the Extending the Ideas section (problems 48,50,52,54,56) in place of any regular homework problem.
Source: Standards for Excellence: A Framework for Teaching in Prince George'sCounty Public Schools.
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Student Performance One:
Assessment Benchmarking Example: Students will correctly show all work and solve the warmup problems. Students will correctly show all work and solve the classwork p446-449:4,12,20,28,46. Students will correctly show all work and solve the homework p446-449:8,16,24,32,40,46. In a few days, students will have two BCR's related to this section on the practice test and the test.
Classwork: Student responses from another semester that meets the top 3 score
4. y = cot(x-.5)+3 The transformations to y = cotx are 3up, .5 units right so this is the dotted graph and the solid one is y = cot x.
12. y = 3sec4x possible window (-pie/2,pie/2) by (-15,15)
correct graph not shown on this lesson here, but the student did show the correct graph
20. y = sec(-x) The transformation to y = sec(x) is reflect across the y-axis. The period is 2pie, the domain is x not= npie + pie/2 n any integer, the range is (-infinity,-1) union (1,infinity), there are no zeros, and the vertical asymptotes are at x = npie + pie/2.
28. y = sec-1/2x The transformations starting with y = sec x are horizontally strectch by a factor of 2, and reflect across the y-axis. The period is 4pie, the domain is x not= (2n-1)pie where n is any integer, the range is (-infinity, -1) union (1,infinity), there are no zeros, and the vertical asymptotes are x = (2n+1)pie , n any integer.
45. d = 350secx = 350/cosx x = 1.55 radians d = 350sec1.55 = 16831.1083 feet.
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Maryland High School Mathematics Rubric: Brief Constructed Response Items
The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or
justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."
Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.
Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.
Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.
Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
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Cognitive Information: Items to keep in mind for students:
- I will collect the following information: the transformations, the period, domain, range, zeros, asymptotes (if any), the sketch of the function and the equation's solution in the interval graphically and analytically for the classwork problems in the 15 minutes allotted. I will do likewise for the homework referring back to my classwork and the examples we reviewed in class.
- I will explain how I solved the goal, problem, or issue in this task in words and/or symbols.
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Organize: I will develop a chart to show the following data for each time that I assess and score the same learning standard(s):
- number of students who performed at or above the proficient level on my scoring tool - see the reflection below.
Analyze: I will examine the data in the chart to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.
- Trends: see the reflection below
Reflect: I will consider two or more of the following stems to reflect on the instructional practices I used and others I might benchmark and apply in the future. Then, I'll write a summary paragraph about my findings, contributing factors, and implications for improvement.
I noticed that I was unable to get to problem 45 as a result of it being a short period (about 65 minutes) because of a two hour late snow day. Nevertheless, I included this in the lesson because I think that in 90 minutes there would be enough time. However, 20 out of 23 students were proficient on the drill ( 3 out of 4 completely correct as in a 3 for a BCR) and classwork (3 out of 4 completely correct as in 3 for a BCR) without that problem, and 17 out of 23 students were proficient on the homework (4 out of 5 completely correct as in a 3 for BCR)without assigning problem 46.
However, when I gave the summative test, which was a total of 15 BCR's and two BCR's on this particular section, only two students were proficient on the first BCR and ten students were proficient on the second BCR for this section out of the 23 students. The first of these two BCR's is explain how the graph of the function y = -sec2x is related to a basic trigonometric graph, determine the period, domain, range, zeros, and asymptotes (if any), and then sketch the graph. I found when grading this that even after the practice test, which had two very similar BCR's to these that I gave and which I reviewed the day before the test, most students did not do the easiest part which was to list the transformations. In addition, most of them got the domain and asymptotes wrong. Obviously, I need to spend more time on that in my lesson.
The second of these BCR's is solve the equation cot x = -3.2 for x in the interval pie < or equal to x < or equal to 3pie graphically, and estimate answers to the nearest hundredth. Since I stressed doing this problem analytically with inverse trigonometric functions and reference angle rules, most student got the calculator answer. However, almost all students mixed up the final answers using the reference angles. Since this problem was like homework problem 40 in the textbook, I was surprised at this result. However, we did only have 2 minutes to spend on this problem before the end of the period even though I followed up with a similar problem on a future drill. As a result of this analysis, I would in the future when I teach this lesson again assign the additional classwork problem p449:42. I try to leave the odd problems, for which the answers are in the textbook, to students to get extra practice/credit if they wish. In addition, maybe the graphing method which I just briefly alluded to as in y = cot x and y = -3.2 with a window of pie to 3pie would have been easier for students. I will spend more time with this method next time I teach this lesson.
Overall, the median for this test on chapter 6 in my Precalculus class was 76%. Each BCR counted for five points, and there was a 25 point minimum grade. None of the BCR's in this lesson's section dealt with a real world situation; so, I doubt that missing that in the lesson was the cause of the poor results on these two BCR's. Instead, I think the real cause for the poor performance on this lesson's BCR's on the test was the short period and the poor attendance that day because of the snow. I marked proficiency on these BCR's as a two or three out of a possible three.
Finally, how do these benchmark scores compare to the increase of 10 percentage points that are Mergenthaler's goal on the Algebra HSA/MSA? Well, a 10 percentage point increase for this 2005-2006 school year would be 27%. For the drill, my drill was about 87% proficient (which was BCR's on topics from from previous lessons) and my classwork was about 73% proficient. Both of which were way above the 27% target. However, on the summative test about one week later, the students' proficiency on the first BCR on this objective was about 9% and on the second BCR on this objective it was about 43%. Therefore, there was some long term cumulative progress compared to a very different Algebra 1 HSA/MSA 10 percentage point gain goal, but this remains a work in progress.