Developing Meaning in Trigonometry

Transcription

1 Developing Meaning in Trigonometry Valerie May, Scott Courtney Abstract Trigonometric concepts and ideas continue to be an important component of the high school mathematics curriculum. In spite of its importance to both high school and advanced mathematics and science, research has shown that trigonometry remains a difficult topic for both students and teachers. In this article, we describe a sequence of activities designed for students to identify relationships between the sine, cosine, and tangent functions; we derive trigonometric identities related to the sum of angles; and we connect trigonometry to other areas of mathematics. The sequence of activities is designed to support students development of coherent trigonometric meanings by building on students prior knowledge in an active learning environment. Keywords: trigonometry, trigonometric functions, trigonometric identities Published online by Illinois Mathematics Teacher on February 29, In alignment with the Common Core State Standards for Mathematics (Common Core State Standards Initiative and others, 2010), trigonometric functions and other trigonometric ideas and concepts (e.g., trigonometric ratios, the laws of sines and cosines) continue to be important components of the high school mathematics curriculum. Further, the capacity to work with trigonometric functions is requisite for study in both advanced mathematics (e.g., Fourier series and the Fourier transform) and the sciences (e.g., modeling the behavior of light and sound waves). Yet despite its importance to both high school and advanced mathematics and science, research has shown that trigonometry remains a difficult topic for both students and teachers (Brown, 2006; Thompson et al., 2007; Weber, 2005). Thompson describes the difficulties that middle and secondary school students in the United States have with trigonometry, resulting from an incoherence of foundational meanings developed in grades 5 through 10 (Thompson, 2008, p. 8). Rather than developing a meaning of angle measure that supports a single trigonometry, encompassing both triangle similarity and periodic be- Corresponding author havior, typical curricula develop them separately and unrelatedly. Specifically, middle school and secondary mathematics textbooks develop two unrelated approaches to trigonometry: the trigonometry of triangles and the trigonometry of periodic functions. In this article, we describe a sequence of activities (a multiple-day lesson) designed for students to: conjecture relationships between the sine, cosine, and tangent functions; derive trigonometric identities related to the sum of angles; and connect trigonometry to other areas of mathematics. In an active learning environment, and by building on students prior and developing understandings and ways of thinking, the sequence of activities is designed to support the development of coherent meanings for several trigonometric concepts and ideas, including trigonometric functions. The activities are designed with several important goals. Firstly, the activities help students connect concepts in geometry and algebra. Secondly, the activities help students develop an understanding of how mathematics ideas develop. Thirdly, students come to see that mathematics principles and properties have their foundations in logic and are not just arbitrary rules made by mathematicians contrary to what many stu- Illinois Mathematics Teacher 25

2 Valerie May, Scott Courtney Conceptual Category Domain Cluster Standard Functions (F) Functions (F) Functions (F) Geometry (G) Trigonometric Functions (TF) A. Extend the domain of trigonometric functions using the unit circle 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for 3,, and 6, and use the unit circle to express the values of sine, cosine, and tangent for x, + x, and 2 x in terms of their values for x, where x is any real number. Trigonometric Functions (TF) C. Prove and apply trigonometric identities 8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. Trigonometric Functions (TF) C. Prove and apply trigonometric identities 9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Similarity, Right Triangles, and Trigonometry (SRT) C. Define trigonometric ratios and solve problems involving right triangles 7. Explain and use the relationship between the sine and cosine of complementary angles. Table 1: Common Core content standards addressed by the lesson dents believe. Lastly, and possibly most importantly, the activities require students to engage in mathematical discourse designed to promote cooperation and high-level cognitive thinking. Lesson Component Component Description Timeline Activity 1 Filling the table with data Day 1 Activities 2A, 2B Trigonometric identities Day 2 Activity 3 Making connections Day 3 Activity Determine the function equations Day 1. The Trigonometry Lesson (Sequence of Activities) As designed, the lesson could be implemented at the beginning of a trigonometry unit in high school Algebra II, Integrated Mathematics II or III, or a fourth-year mathematics course (e.g., Precalculus). The specific Common Core content standards addressed by the lesson are presented in table 1. In order to participate productively in the lesson, students should be able to use a function to create a table of values, recall and utilize the Pythagorean theorem and its consequences, and think about angle measure in terms of the measure of a subtended arc. With such an understanding of angle measure, when given an angle, students can imagine an arbitrary circle centered at the angle s vertex. Then, degree is viewed as an arc on the circle whose length Table 2: Lesson Components and Timeline is 1/360 of the circle s circumference and angle measure in degrees is viewed as the length of the arc subtended by the angle, measured in arcs of length 1/360 of the circle s circumference (Dubinsky & Harel, 1992; Thompson et al., 2007). As indicated by Thompson, this way of thinking about angle measure describes the principle by which a protractor works (Thompson, 2008, p. 50). In the following sections, we describe students engagement with the activities over multiple days, and specify how each activity supports the development of significant trigonometric meanings. Although original, the sequence of activities was inspired by Avila s article in which Algebra students used functions to draw engaging artwork (Avila, 2013). Table 2 displays the timeline for the different components of the lesson. 26 Illinois Mathematics Teacher

3 2. Activity 1: Filling the Table with Data Depending on students prior knowledge and skills, the first component of the lesson, completing the trigonometric function table (or Trig Table ), could be done for homework as preparation for subsequent in-school activities. However, for the purposes of this article, we will treat the trigonometric function table component as an inclass activity. Depending on students prior understanding and experience, we sometimes need to demonstrate how to use a compass and protractor. Whereas such demonstration will involve a whole-class discussion, the remaining activities in the lesson are done in small groups of two to three, and managed in ways that promote reflective mathematical discourse (Cobb et al., 1997). The initial activity requires the use of a pencil, graph paper, a compass, and a protractor. The instructions provided to students are displayed in appendix A. Throughout Activity 1, we ask students to imagine a right triangle embedded in the circle so that the triangle s hypotenuse is the circle s radius and one angle is formed by the angle in question (10, 20, etc.). Furthermore, we define sine and cosine as follows: By sine of an angle we mean the percent of the radius length made by the length of the side opposite the origin in the embedded right triangle. By cosine of an angle we mean the percent of the radius length made by the length of the side adjacent the origin.(thompson et al., 2007, p. 17) Such meanings allow the cosine and sine functions to be explored as functions, where the output of the cosine function is the abscissa (x-coordinate) of the terminus of the arc subtended by the angle and the output of the sine function is the ordinate (y-coordinate) of the terminus of the arc subtended by the same angle, with both measured as a fraction of one radius (Moore, 2010, p. 5). Developing Meaning in Trigonometry 3. Activity 2: Trigonometric Identities The second activity requires the use of the completed table of values (Trig Table) from Activity 1. The instructions provided to students for Activity 2A are shown in appendix B. At this stage of the lesson, we bring the groups together and engage the class in a discussion regarding the group results for Activity 2A. Groups are asked to explain their mathematical thinking as they articulate their results. Next, we again partition the class into groups of two to three students (the same groups as for Activity 2A) to begin Activity 2B (see appendix C). At this stage of the lesson, we bring the groups together again and engage the class in a discussion regarding the group results for Activity 2B. Groups are asked to explain their mathematical thinking as they articulate their results.. Activity 3: Making Connections The third activity requires the use of the completed table of values (Trig Table) from Activity 1 and the group results from Activities 2A and 2B. We inform students that they will explore cos(α) and sin(α) in connection with algebra. The instructions provided to students for Activity 3 are shown in appendix D. At this stage of the lesson, we bring the groups together and engage the class in a discussion regarding the group results for Activity 3. Groups are asked to explain their mathematical thinking as they articulate their results. Throughout this discussion, we support students in making connections between the Pythagorean trigonometric identity, the equation of a circle with center (h, k), and meanings for sine and cosine in terms of the embedded right triangle with adjacent and opposite sides relating to the origin. Next, we inform the class that we will continue to think in terms of algebra. Students have used Cartesian coordinates to graph a circle and can recognize certain functions (and function families) by looking at their graphical representation. We inform students that they are to construct a graph of the sine function using graph paper and the given table of values (see figure 1). Illinois Mathematics Teacher 27

4 Valerie May, Scott Courtney Value of angle x (radians) Value of sin x Figure 1: Table for graphing the sine function. This figure displays the table of values that students are requested to complete, and then use to graph the sine function. Once groups have completed the graph of the sine function, they are asked to explain any patterns that they notice ( What does the function look like? ), and to explain the pattern of the sine function in term of angles around the circle. Students will need to explain how points on the unit circle correspond to points on the sine graph, and describe how the pattern of the sine function emerges through continuously evaluating the ordinate (y-coordinate) of the terminus of the arc subtended by the angle as one moves around the unit circle with increasing angle values. 5. Activity : Determine the Function Equations Activity Examine the following picture and find a series of function (equations) that, when graphed, generate the picture. Use an online graphing calculator tool (such as desmos.com) to graph your function equations. You should utilize those functions that we have discussed in class or that you have experienced in your prior courses. You may need to restrict the domains and/or ranges of certain functions. Please print a screen shot of your function equations and the picture that they generate. The culminating activity is designed to be hands-on and challenging, yet enjoyable, especially for students who like to use technology to manipulate graphical representations of functions. For this activity, students are required to employ their prior knowledge and skills, including recalling the graphical representations of several families of functions and determining the domain and range of a function (particularly, functions with restricted domains and ranges). In addition, students will learn that distinct function equations can generate similar graphical representations. Finally, the activity is designed to motivate students to utilize online graphing calculator technology to explore functions and quickly ascertain how they might need to modify their functions to create their diagrams. For example, students will need to construct two circular functions in order to draw the moon. The directions and diagram (or picture ) given to the students are shown in figure 2. The final activity is followed by a full class discussion in which students compare pictures, the function equations students used to re-create Figure 2: Sailing under the moon. This figure illustrates a sample diagram (created using desmos.com) that students are required to re-create by determining the function equations whose graphs generate the diagram. For a resource to help you use Desmos to create graphics, see, e.g., Ebert s recent article (Ebert, 2015). 28 Illinois Mathematics Teacher

5 the pictures, the required domain and/or range restrictions, and the strategies students employed to help them develop their series of function equations. 6. Conclusion As indicated in this article, the teacher plays a significant role throughout the sequence of activities in managing student thinking and discourse to focus on meanings and the coherence of meanings. The main difficulty that we have encountered when engaging students in these activities involve those students lacking in experiences that asked them to consider what it is they are measuring when they measure an angle. For such students, when pushed to consider such a query, a common reply is the amount the terminal side is rotated from the initial side in a triangle. As described by Thompson, et al., for such students, Angles are measured in fractions of a rotation [and] trigonometry is about solving triangles (Thompson et al., 2007, p. 30). Such understandings and ways of thinking can be quite resistant to attempts at their modification. One of the more productive outcomes that we have noted in using these activities is that students exhibit an increased level of comfort in articulating their meanings, thinking, and reasoning, and critiquing the thinking and reasoning of others including their teacher. Furthermore, the sequencing of the activities allows students to start with the conceptions they have developed, even those typically deleterious to connecting right triangle trigonometry with the trigonometry of periodic functions. Finally, we have noticed that students who have developed meanings for angle measure, degree, sine of an angle, and cosine of an angle, as presented here, lack the difficulties their classmates encounter when trying to construct an image of cos(sin(15 )); that is, when attempting to conceive of sin(15 ) as an argument to cosine. Developing Meaning in Trigonometry References Avila, C. L. (2013). Graphing art revisited: The evolution of a good idea. Ohio Journal of School Mathematics, 67, Brown, S. A. (2006). The trigonometric connection: students understanding of sine and cosine. In Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (p. 228). Prague: International Group for the Psychology of Mathematics Education. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for research in mathematics education, 28 (3), Common Core State Standards Initiative and others (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Dubinsky, E., & Harel, G. (1992). The nature of the process concept of function. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp ). Volume 25 of MAA Notes. Ebert, D. (2015). Graphing projects with Desmos. Mathematics Teacher, 108 (5), Moore, K. C. (2010). The role of quantitative and covariational reasoning in developing precalculus students images of angle measure and central concepts of trigonometry. In Proceedings for the Thirteenth SIGMAA on Research in Undergraduate Mathematics Education Conference. Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (pp. 5 6). Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The design of tasks in support of teachers development of coherent mathematical meanings. Journal of Mathematics Teacher Education, 10 (), Weber, K. (2005). Students understanding of trigonometric functions. Mathematics Education Research Journal, 17 (3), Valerie May WADSWORTH HIGH SCHOOL 625 BROAD STREET WADSWORTH, OH wadc vmay@wadsworthschools.org Scott Courtney KENT STATE UNIVERSITY 01 WHITE HALL P.O. BOX 5190 KENT, OH scourtn5@kent.edu Illinois Mathematics Teacher 29

6 Valerie May, Scott Courtney Appendix A Instructions for Activity 1 Please follow and complete the directions below using a pencil, graph paper (preferably 1 cm grid), a compass, and a protractor: 1. Draw a Cartesian coordinate system. Make certain to label the x- and y-axes. 2. Construct a quarter circle (in the first quadrant) of radius 10 cm (or 1 dm). Consider the radius of the circle to be 1 unit. 3. Use your protractor to construct an angle of 10 degrees, that is an arc of length 10/360 the circle s circumference. Use the origin (O) as the vertex and the x-axis for one side of the angle.. Label as P the point that intersects the angle segment and the unit circle. 5. Determine the x- and y-coordinates for P and insert these values in the appropriate row and column of the table. 6. The x-value for P represents the value for the cosine of 10 degrees, i.e., cos(10 ); the y-value for P represents the value for the sine of 10 degrees, i.e., sin(10 ). 7. Determine the remaining values in the table for sine and cosine by repeating the process (parts 3-6 above) for angles of measure 20, 30, etc. 8. Determine the value for the slope of the segment OP, for every angle in the table. 9. Construct a tangent line to the (quarter) circle at point P, for every angle in the table; determine the slope of each tangent line. Trig Table Angle measure (arc length, in degrees) Cosine (x-coordinate) Sine (y-coordinate) Slope of triangle hypotenuse slope of segment OP Slope of tangent line slope of line tangent to circle at the point (x, y) Illinois Mathematics Teacher

7 Appendix B Instructions for Activity 2A Developing Meaning in Trigonometry In small groups of 2 3, please follow and complete the directions below. You will need to use your completed Trig Table from Activity Examine the four quantities obtained for the 5 angle. 2. Discuss any observations (relationships, patterns) that your group makes regarding data for the 5 angle. 3. Describe any relationships that your group identifies between the slope of the hypotenuse and the slope of the tangent line for the 5 angle. Use complete sentences.. Examine the quantities obtained for sine (ordinate, y-coordinate), cosine (abscissa, x-coordinate) and the slope of the tangent line at the point (x, y) for the 5 angle. 5. Describe any relationships that your group identifies between sine, cosine, and tangent (the slope of the tangent line at the point (x, y)) for the 5 angle. 6. Determine whether the relationship identified in part 5 holds true for any other angle measures. Which angles, if any? 7. Examine the four quantities obtained for the 60 angle. 8. Discuss any observations (relationships, patterns) that your group makes regarding data for the 60 angle. 9. Describe any relationships that your group identifies between the slope of the hypotenuse and the slope of the tangent line for the 60 angle. Use complete sentences. 10. Examine the quantities obtained for sine (ordinate, y-coordinate), cosine (abscissa, x-coordinate) and the slope of the tangent line at the point (x, y) for the 60 angle. 11. Describe any relationships that your group identifies between sine, cosine, and tangent (the slope of the tangent line at the point (x, y)) for the 60 angle. 12. Determine whether the relationship identified in part 11 holds true for any other angle measures. Which angles, if any? 13. Can you generalize any relationships between sine, cosine, and tangent regardless of the angle measure? Explain. 1. From Activity 1, recall the value for the slope of the segment OP. What type of triangle do you see? Describe any relationships that your group identifies between sine, cosine, and the hypotenuse OP. Explain. 15. Share your results with another group. Illinois Mathematics Teacher 31

8 Valerie May, Scott Courtney Appendix C Instructions for Activity 2B In small groups of 2 3, please follow and complete the directions below. You will need to use your completed Trig Table from Activity 1 and your work from Activity 2A. 1. Choose an arbitrary angle θ (measured in degrees) from your Trig Table. Consider the values for sine and cosine for angle θ. 2. Consider the values that are obtained for cos(90 θ) and sin(90 θ). What can you conclude? 3. Describe how and why your conclusion from part 2 makes sense. Make certain to explain your mathematical thinking and reasoning. Use complete sentences where appropriate.. Think of the angle 90 as being formed from the sum 90 = Examine the value of cos( ) in relation to the values of cos(60 ), sin(60 ), cos(30 ), and sin(30 ). 6. Discuss any relationships that you identify between the values cos( ), cos(60 ), sin(60 ), cos(30 ), and sin(30 ). What can you conclude? Explain. 7. Describe how and why your conclusion from part 6 makes sense. Make certain to explain your mathematical thinking. Use complete sentences where appropriate. 8. Formulate a similar, and more general conjecture, for cos(α + β), where α and β are each angles measured in degrees. Check your conjecture with other angle values from your Trig Table. 9. Repeat the process described in parts 7 above for sin( ). 10. Formulate a general conjecture for sin(α + β), where α and β are each angles measured in degrees. Check your conjecture with other angle values from your Trig Table. 11. Share your results with another group. 32 Illinois Mathematics Teacher

9 Appendix D Instructions for Activity 3 Developing Meaning in Trigonometry In small groups of 2 3, please follow and complete the directions below. You will need to use your completed Trig Table from Activity 1 and your work from Activities 2A (#1) and 2B. 1. Make a table of values for the relation x 2 + y 2 = 1; that is, make a table of x- and y-values that make the relation true. 2. On a separate piece of graph paper, graph the relation x 2 + y 2 = 1. What geometric object does the graph of the relation represent? 3. Make a table of values for the relation x 2 + y 2 = ; that is make a table of x- and y-values that make the relation true.. On the same piece of graph paper used in part 2 above, graph the relation x 2 + y 2 =. What geometric object does the graph of the relation represent? 5. Imagine a graph of the relation (x 3) 2 + (y 2) 2 =. Describe the geometric object that this relation represents. Make certain to explain your mathematical thinking. Use complete sentences where appropriate. 6. Share your results with another group. Illinois Mathematics Teacher 33