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Mathematical Models with Applications, Quarter 1, Unit 1.1 Indirect Measurement Overview Number of instruction days: 8 -10 Content to Be Learned (1 day = 53 minutes) Mathematical Practices to Be Integrated Understand relationships in special right triangles. 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. 4 Model with mathematics. 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. Model with mathematics to solve real world problems involving trigonometric ratios. 6 Attend to precision. Use precise language to communicate accurate solutions to trigonometric problems. Why are trigonometric ratios important? Essential Questions Where would you use each of the following to solve a problem: trigonometry, special right triangles or the Pythagorean Theorem? How are angles of elevation or depression similar and different? Providence Public Schools D-1 Math Models, Quarter 1, Unit 1.1 Version 2 Indirect Measurement (8 -10 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.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 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 D-2 Providence Public Schools Indirect Measurement (8 -10 Days) Math Models, Quarter 1, Unit 1.1 Version 2 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. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In Grade 4, students classified shapes by 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 in Grade 7 is extending their ability to recognize, represent, and analyze proportional relationships. In Grade 8 understanding and applying the Pythagorean Theorem is a major cluster. Additionally, Grade 8 students worked informally to establish facts about angle sum and exterior angles of triangles. Fluency with triangle congruency and similarity was expected in Geometry. Students applied their knowledge of similar triangles to three trigonometric ratios: sine, cosine, and tangent. Students added trigonometric ratios to their indirect measurement toolkits. Students explored and used the relationship between sine and cosine ratios of angles whose sum is 90. Students understood that by similarity, side ratios in right triangles are properties of the angles in the triangle. They solved real world problems using these relationships. Current Learning Using the concept of indirect measurement, students who struggled with trigonometric concepts now have an opportunity to extend their previous knowledge through mathematical modeling of the content to be learned in this unit. Fluency with triangle congruency and similarity is expected. Their study of special right triangles begins their study of indirect measurement. 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, adding trigonometric ratios to special right triangles and similar triangles. Students will explore and use the relationship between sine and cosine Providence Public Schools D-3 Math Models, Quarter 1, Unit 1.1 Version 2 Indirect Measurement (8 -10 Days) ratios of angles whose sum is 90°, the acute angles in a right triangle. Students understand that by similarity, side ratios in right triangles are properties of the angles in the triangle. They solve real world problems using these relationships. Future Learning Extensions of the study of trigonometry will include the unit circle, graphing and analyzing trigonometric functions and identities in Algebra 2. This study is 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 to students because they have difficulty distinguishing between the opposite and adjacent sides in a right triangle. According to Principles and Standards of School Mathematics (p. 309), “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. 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 Content Standards and the CCSS Practice Standards should be considered when designing assessments. Standards based mathematics assessment items should vary in difficulty, content and type. The assessment should include 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. Math Models students should be provided with multiple, alternative methods to express their understandings of the concepts that follow: D-4 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. Providence Public Schools Indirect Measurement (8 -10 Days) Math Models, Quarter 1, Unit 1.1 Version 2 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. Solve right triangle problems using angles of elevation and angles of depression. Instruction Learning Objectives Students will be able to: Use the properties of special right triangles and the Pythagorean Theorem to solve problems. Use similar triangles to find distances and heights indirectly. 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. Review and demonstrate knowledge of important concepts and procedures related to indirect measurement. Resources Modeling with Mathematics: A Bridge to Algebra II, W.H. Freeman and Company, 2006 Sections 8.1 through 8.5 (pp. 496 – 513) Online Companion Website: http://bcs.whfreeman.com/bridgetoalgebra2/ Additional Resources located in the Supplementary Unit Materials Section of the Binder: o Graphic Organizer: SOHCAHTOA. o Building Height Activity Instructions o Building Height Activity Sheet Providence Public Schools D-5 Math Models, Quarter 1, Unit 1.1 Version 2 o Clinometer Construction o Water Rockets Indirect Measurement (8 -10 Days) 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, document camera (ELMO), dynamic geometry software; optional – graphic organizer, clinometers, paper clips, straws, 5 by 7 index cards, protractors (preferably those with a center hole at 0°), string or dental floss, scissors, small weights (such as washers or coins), drinking straws, tape. Instructional Considerations Key Vocabulary angle of depression inverse tangent ratio angle of elevation opposite side adjacent side reference angle cosine ratio sine ratio indirect measurement special right triangle inverse cosine ratio tangent ratio inverse sine ratio trigonometric ratio Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: Pythagorean Theorem, hypotenuse, trigonometry, ratio, and similarity. This unit on indirect measurement was purposely placed in the beginning of the year in order to support the Physics curriculum and provide additional support for NECAP preparation. Students who struggle with applications of trigonometric ratios typically do so because they either select the incorrect ratio or they perform the computations incorrectly. Help students understand how trigonometric ratios relate to similar right triangles, students can elaborate on their knowledge by D-6 Providence Public Schools Indirect Measurement (8 -10 Days) Math Models, Quarter 1, Unit 1.1 Version 2 identifying similarities and differences as they compare the ratio of the side lengths and trigonometric values. Create a physical model of a right triangle by arranging the desks to form a right triangle. Label the sides and angles using poster paper to reinforce the appropriate key vocabulary in this unit. Students develop an understanding that opposite and adjacent sides change when the reference angle changes. Indirect measurement is a technique that uses proportions to find a measurement when direct measurement is not possible. A critical resource to make Math Models effective is the use of tables, handouts, and assessments provided by the publisher at http://www.whfreeman.com/Catalog/static/whf/mma/ (or google: “Math Models: A Bridge to Algebra 2”). Also available on this website are power point presentations, lesson plans, assessments and activities. For initial use, you will be prompted to set an an instructors account using an e-mail address as the UserId. You will also be prompted for the following companion website code: BFW41INST. To support students who select incorrect ratios, emphasize the importance of selecting a problem-solving 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. A SOHCAHTOA graphic organizer is available on www.sw-georgia.resa.k12.ga.us/math.html and 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 indirect measurement. The building height activity is provided in the supplementary materials section of this curriculum frameworks binder and is also available on the following NCTM website: http://illuminations.nctm.org/LessonDetail.aspx?ID=L764. In this activity, students use a clinometer (a measuring device built from a protractor) and isosceles right triangles to find the height of a building. The class will compare measurements, talk about the variation results. Detailed instructions for building clinometers are also available on the Water Rockets and Clinometer Construction worksheet included in the supplementary materials section of the binder. It will be important to organize groups and have all the materials prepared before the activity begins in order to maintain a brisk pace for instruction. As you formatively and summatively assess students, a cues, questions, and advance organizers strategy can be used, since students are answering questions about content that is important. Providence Public Schools D-7 Math Models, Quarter 1, Unit 1.1 Version 2 Indirect Measurement (8 -10 Days) Notes D-8 Providence Public Schools