Doing Mathematics



|Learning Goals |Main Topic |Outline of the Lecture |

|To sketch the graphs of secant and cosecant |What is the main procedure or topic that students will|What are the relevant components of the main topic? In what order will they be presented?|

|functions |be working on in this lesson? |What examples will be used as illustrations? |

| |Graphs of secant and cosecant functions |Answer questions on 4.6 Day 1 homework assignment. If students don’t ask a question. Will |

| | |lead a short discussion on how students can sketch the graph of #29 from the homework |

| | |assignment. Where are the endpoints of the first period? Make a general sketch with |

| | |students input. |

| | |Introduction: Today we will be discussing how to sketch secant and cosecant functions. The|

| | |best way to understand how to graph these two functions is by connecting those functions |

| | |to the graphs of sine and cosine functions. Then if there is time we are going to make a |

| | |chart that summarizes the graphs of all six trigonometric functions. |

| | |Notes will be done on the whiteboard. Let’s start with graphing the sine function to |

| | |understand the cosecant function graph. Ask a student to come up and graph the function |

| | |y=sinx. The sine functions will be in green and the cosecant function will be in black. |

| | |From there examine what the reciprocal would look like starting with x-intercepts then |

| | |from a maximum point then a minimum point. What do you notice happens with the y-values |

| | |between two reciprocals? What do you think the graph will look like at the minimum value? |

| | |We can do the same for graphing a secant function by using cosine. What do you think the |

| | |Domain is? Range? Period? Where are the vertical asymptotes? |

| | |Take out your calculators, Right now I what you to graph y=2sin(x+ pi/4) and y=2csc(x+ |

| | |pi/4). How can we graph the cosecant function on the calculator? Make sure the mode is in |

| | |radians and not degrees. How should we set the window to? What do you notice between the |

| | |two graphs? What do you notice about the maximum and minimum values of the sine function |

| | |compare to the maximum and minimum values of the cosecant function? The a tells us the |

| | |vertex of the “mini-parabolas” that happen at the maximum and minimum values of the sine |

| | |function. If you can graph the sine or cosine function of the corresponding secant or |

| | |cosecant function, you can graph the secant or cosecant function. Let’s use the next |

| | |example to show what I mean. |

| | |We want to sketch the graph y=sec2x. What cosine function do you think corresponds to |

| | |that? What do we know about the graph of y=cos2x? Where are the end points of one period |

| | |of the cosine function? The amplitude? Where do you think the vertical asymptotes are? |

| | |Remember just graph two periods. |

| | |Summarize |

| | |If time have students get out two pieces of paper. What you are going to do in the next |

| | |few minutes is create either a secant or cosecant function that someone else will have to |

| | |sketch the graph of. For instance, I created y=1/2secant(2x-pi/4). After you create the |

| | |function I want you to sketch the graph yourself to make sure it works and you have a |

| | |“key” for it. After you are done, I want you to exchange the function you created with |

| | |someone around you and I want you to sketch the graph of their function. After you have |

| | |completed your sketch, give it back to the person. The person will then grade your work by|

| | |saying two things that you did well in sketching the graph and one thing that you should |

| | |work on with your graphing. (I will put up bulleted directions on Doc-Cam so that |

| | |students know what they will be doing for about the next 10-15 minutes). |

| | |Have students work on homework assignment |

|Conceptual Connections |Prior Knowledge |Lesson Concepts |Example(s) |

|What will students understand conceptually about the|How will prior knowledge connect to and influence the |How do you sketch the graphs of |Examples are in the notes outline for 4.6 section |

|procedures they are learning? |development of the main topic? |secant and cosecant functions? |notes |

|Sine and cosecant functions are reciprocals as well |I will build on students’ knowledge of sketching | | |

|as cosine and secant functions. Understanding what |graphs of sine and cosine functions in order for | | |

|the sine and cosine functions look like that |students to understand sketching graphs of secant and | | |

|corresponds to the cosecant or secant functions will|cosecant functions. If students can sketch sine and | | |

|help you graph secant and cosecant functions. |cosine graphs, students will be able to sketch secant | | |

| |and cosecant functions. | | |

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|Evidence |Representations |Key Errors |

|What will students say, do, produce, etc. that will |What are the various ways in which the concept will be|What errors do you expect students to make when working independently? How will you |

|provide evidence of their understandings? |represented throughout the lecture? |address these potential errors in lecture? |

|Students can sketch the graphs of secant and |Fill-in the blank notes |Students have a tendency to interchange the reciprocal of trig functions with the inverse |

|cosecant functions |Equations |of trig functions- Make it explicit and clear that the reciprocal of sine is cosecant and |

|Students can do their homework and if they get the |Verbal |the inverse of sine is theta. We know that inverses undo the operations of each other, |

|problem wrong can find the mistake and redo the |Words |whereas the reciprocals are flips of each other. The graphs look completely different from|

|problem |graphs |each other. Today we are just focusing on analyzing the graph of the reciprocal of sine |

| | |(secant function) and the reciprocal of cosine (cosecant function). Showing the connection|

| | |between sine and cosecant functions and cosine and secant functions by graphing sine and |

| | |cosine functions then taking the reciprocal. This will make the connection of what a |

| | |reciprocal trigonometric function is even stronger. |

| | |Graphing secant and cosecant functions- Where sine and cosine functions have x-intercept |

| | |is where secant and cosecant will have vertical asymptotes. Secant and cosecant functions |

|Standard(s) Addressed: | |will have the same period as sine and cosine which is generally 2pi. How do you think we |

|F-TF.5- Choose trigonometric functions to model | |can graph cosecant and secant functions? During the lesson, going to have students graph |

|periodic phenomena with specified amplitude, | |sine and cosecant on the calculator and have students as pairs talk about the similarities|

|frequency, and midline.★ | |of the two graphs. Graphing secant and cosecant functions is much easier when you graph |

| | |the sine and cosine function that corresponds to them. During the lesson, especially in |

| | |the beginning will start with sine and cosine functions to create the graphs of secant and|

| | |cosecant functions. |

| | |Students will be confused why the vertical asymptotes do not make up the period for secant|

| | |or cosecant. For tangent and cotangent between two consecutive vertical asymptotes which |

| | |was pi, but with secant and cosecant it is a little different. The reason for that is that|

| | |since secant and cosecant are reciprocals of sine and cosine they have the same period |

| | |which for the parent functions (secant and cosecant) are 2pi. In fact, you can use your |

| | |knowledge of sketching graphs of sine and cosine to sketch graphs of secant and cosecant. |

| | |The period for cosecant has three vertical asymptotes in a period- it starts at an |

| | |asymptote, ends at an asymptote and half between there is an asymptote (where the |

| | |x-intercepts of the sine function are). This consists of two “u”s (one up and one down). |

| | |For secant a period starts with a “u” and ends with a “u” half way between is the opposite|

| | |“u” and have way between the “u”s are vertical asymptotes (relates to starts at a max ends|

| | |at a max, half way between is a min, half way between max and min are the x-intercepts of |

| | |a cosine function). |

|Summary: What are the big “take-away” ideas of the lesson? How do these connect to what you will do in the next lesson? |

|Today we talked about how to sketch the graphs of secant and cosecant functions. We also talked about how those graphs relate to sine and cosine graphs. As I said yesterday, we will not be doing |

|part 2 notes of 4.6, because you will not be tested on it and it is not part of Algebra 3 curriculum. After today’s lesson, we can sketch the graphs of all six trigonometric functions. Is there any |

|trig function that you want more review in graphing? |

Homework assignment: p. 341 #13-19, 23, 27

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