Understanding Trigonometric Ratios

Transcription

1 Understanding Trigonometric Ratios Overview Number of instruction days: 9-11 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Understand definitions of trigonometric ratios for acute angles in right triangles. Understand that side ratios in right triangles are properties of the angles in the triangle because of triangle similarity. Develop and apply relationships between the sine and cosine of complementary angles and use them to solve problems. Solve right triangles in real-world problems using trigonometric ratios, special right triangles, and the Pythagorean Theorem. 1 Make sense of problems and persevere in solving them. Identify what parts of the triangle need to be used in order to find a solution. 4 Model with mathematics. Model with mathematics to solve real-world problems involving trigonometric ratios. 7 Look for and make use of structure. Recognize the significance of an existing line in a geometric figure and use the strategy of drawing an auxiliary line for solving problems. Step back for an overview. Essential Questions Where would you use each of the following to solve a problem: trigonometry, special right triangles, and the Pythagorean Theorem? Why are trigonometric ratios important? What is the relationship of the sine and cosine values of a complementary angle? How are angles of elevation or depression similar and different? Providence Public Schools D-85

2 Understanding Trigonometric Ratios (9-11 days) Standards Common Core State Standards for Mathematical Content Geometry Similarity, Right Triangles, and Trigonometry G-SRT Define trigonometric ratios and solve problems involving right triangles G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to D-86 Providence Public Schools

3 Understanding Trigonometric Ratios (9-11 days) Geometry, Quarter 3, Unit 3.3 solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In Grade 4, students classified shapes according to properties of their angles and worked with word problems involving unknown angle measures. In Grade 5, students classified two-dimensional figures based on their properties. Students worked with ratios and proportional relationships in Grade 6. A major cluster for seventh-grade students was extending their ability to recognize, represent, and analyze proportional relationships. Understanding and applying the Pythagorean Theorem was a major cluster for Grade 8 students, and they also worked informally to establish facts about angle sums and exterior angles of triangles. Current Learning Fluency with triangle congruency and similarity is expected in Geometry. In this unit, students apply their knowledge of similar triangles to develop and then apply three trigonometric ratios: sine, cosine, and tangent. Students add trigonometric ratios to their indirect measurement toolkits, which already include special right triangles and similar triangles. Students will explore and use the relationship between sine and cosine ratios of angles whose sum is 90, the acute angles in a right triangle. Students understand that by the properties of similarity, side ratios in right triangles are properties of the angles in the triangle. They solve real-world problems using these relationships. Defining trigonometric ratios Providence Public Schools D-87

4 Understanding Trigonometric Ratios (9-11 days) and solving problems involving right triangles is major content as defined by the PARCC Model Frameworks for Mathematics. Future Learning Extensions of the study of trigonometry will include the unit circle and graphing and analyzing trigonometric functions and identities in Algebra II. This study is further extended in Precalculus to include solving trigonometric equations. Careers that include use of trigonometry are extensive, including surveying, engineering, construction, physics, navigation, astronomy, etc. Additional Findings This material is challenging for students because they have difficulty distinguishing between the opposite and adjacent sides in a right triangle. According to Principles and Standards of School Mathematics, High school students should develop facility with a broad range of representing geometric ideas... For example, desks can be arranged in a right triangle to physically represent change in opposite and adjacent sides when the reference angle is changed. (p. 309) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Develop a definition of trigonometric ratios using corresponding angles of similar right triangles to show that the relationships of the side ratios are the same. Apply trigonometric ratios and their inverse relationships to determine missing angle measures and side lengths of right triangles in problem situations. Use trigonometric ratios, special right triangles, and the Pythagorean Theorem to solve real world problems. D-88 Providence Public Schools

5 Understanding Trigonometric Ratios (9-11 days) Geometry, Quarter 3, Unit 3.3 Solve right triangle problems using angles of elevation and angles of depression. Apply the relationship between the sine and cosine values of complementary angles to solve problems. Instruction Learning Objectives Students will be able to: Use the properties of special right triangles to solve problems. Compare side and angle measurements in similar right triangles to develop three trigonometric ratios. Apply the trigonometric ratios of sine, cosine, and tangent to determine missing side lengths of right triangles. Apply inverse relationships and trigonometric ratios to determine missing angle measures in right triangles from problem situations. Apply trigonometric ratios, angles of depression, and angles of elevation to solve real-world problems. Explore and use the relationship between the sine and cosine values of complementary angles. Review and demonstrate knowledge of important concepts and procedures related to trigonometric ratios. Resources Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions Section 8-3 (pp ) Section 8-4 (pp ) Section 8-5 (pp ) 493) Glencoe McGraw-Hill Online Teaching with Manipulatives Providence Public Schools D-89

6 Understanding Trigonometric Ratios (9-11 days) Problem Solving Guide (p. 19) Geometry Lab Transparency Master Trigonometry (p. 115) Geometry Lab Trigonometry (pp Chapter 8 Resource Masters (p. 29) Interactive Classroom CD (PowerPoint Presentations) Teacher Works CD-ROM TI-Nspire Teacher Software Exam View Assessment Suite Graphic Organizer: SOHCAHTOA. See the Supplementary Unit Materials section of this binder for notes for this graphic organizer. Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery section below for specific recommendations. Materials Ruler, protractor, meter sticks or tape measures (1 for each pair of students), colored pencils, TI-Nspire graphing calculator, calculator viewscreen, dynamic geometry software; optional graphic organizer, clinometers, paper clips, straws, 5 by 7 index cards, and kite string. Instructional Considerations Key Vocabulary angle of depression angle of elevation cosine sine tangent trigonometric ratio Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: ratio and special right triangle. Students who struggle with applications of trigonometric ratios typically do so because they either select the incorrect ratio or they perform the computations incorrectly. In Section 8-4, to help students understand how trigonometric ratios relate to similar right triangles, students can elaborate on their D-90 Providence Public Schools

7 Understanding Trigonometric Ratios (9-11 days) Geometry, Quarter 3, Unit 3.3 knowledge by identifying similarities and differences as they compare the ratio of the side lengths and trigonometric values. This strategy is also helpful as students look for patterns that exist in the trigonometric values of complementary angles (i.e., sin A = cos B when A + B = 90 ). Refer to the Differentiated Instruction on page 560 and extend this activity to include sine and cosine trigonometric ratios. To support students who select incorrect ratios, emphasize the importance of selecting a problemsolving strategy, such as drawing a diagram to help solve the problem. Use colored pencils to color code the sides of a right triangle. Mnemonic devices such as SOHCAHTOA help students remember the side lengths involved in each of the three ratios. The four-step problem-solving model and nonlinguistic representations such as graphic organizers also help struggling students, including English language learners and students with special needs, organize their knowledge. One resource is Teaching with Manipulatives on page 19 and another is The SOHCAHTOA graphic organizer is also provided in the supplementary materials section of this curriculum frameworks binder. To help increase students computational fluency, provide multiple opportunities for students to practice using the graphing calculator to compute trigonometric ratios in the context of solving a problem. Using proportional reasoning to solve equations generated using trigonometric ratios also helps increase the accuracy of students computations, as it connects the idea of solving equations with trigonometric ratios to the familiar knowledge of solving proportions. Use real world problem situations to increase the relevance of problems involving angles of elevation and depression. The Differentiated Instruction for Kinesthetic Learners on page 576 of the Teacher Edition provides multiple examples of classroom applications. Modeling can also be done with a clinometer as a tool to help students understand angles of elevation and depression. Numerous resources on the web reference the integration of clinometers. The following example guides students through a series of reading and math activities to help them understand how the Northern Lights work, what causes them, and how to observe them: Another opportunity for the integration of a physical models as a nonlinguistic representation is provided in the Trigonometry Lab, detailed in the Teaching with Manipulatives resource book. In this activity, students use hypsometers and indirect measurement to calculate measurements of real world objects. As you assess students using the 5-minute check transparencies, a cues, questions, and advance organizers strategy is being used, since students are answering questions about content that is important. Some of the questions help students review prior knowledge, and these should be used at the beginning. Providence Public Schools D-91

8 Understanding Trigonometric Ratios (9-11 days) Notes D-92 Providence Public Schools