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Unit 5 Similarity and Trigonometry Geometry Unit Goals – Stage 1 Number of Days: 34 days 2/21/17 – 4/7/17 Unit Description: In Unit 5, students are introduced to trigonometry through a revisit to similarity. Unlike congruence, here triangles are shown to be similar with sides and angles being proportional, not congruent. Previously seen in middle school, the students will use the Pythagorean Theorem and their knowledge of similar triangles to investigate the relationships found in the special right triangles. Using these relationships, students will solve for parts of right triangles and apply these skills to real-life situations. Materials: Patty paper, straight edges, compasses, calculators, dynamic software (Desmos, Geogebra) Standards for Mathematical Transfer Goals Practice Students will be able to independently use their learning to… SMP 1 Make sense of problems • Make sense of never-before-seen problems and persevere in solving them. and persevere in solving • Construct viable arguments and critique the reasoning of others. them. SMP 2 Reason abstractly and Making Meaning quantitatively. UNDERSTANDINGS ESSENTIAL SMP 3 Construct viable QUESTIONS Students will understand that… arguments and critique Students will keep • Ratios and proportions are necessary to the study of similarity. the reasoning of others. considering… • The properties of similarity hold for all polygons. SMP 4 Model with mathematics. • How can you use • Dilation can be used to show that figures are similar. SMP 5 Use appropriate tools ratios and • In similar figures, all angles are congruent and corresponding side lengths are strategically. proportions to prove proportional. SMP 6 Attend to precision. that triangles are • Using similarity, one can solve for missing side lengths and angle measures in SMP 7 Look for and make use of similar? applied situations. structure. • How does knowing • Trigonometric functions are a ratio of a set of given sides. If the angle is held SMP 8 Look for and express that triangles are constant, a trigonometric ratio will remain the same, no matter the size of the triangle. regularity in repeated similar lead us to • Sine, cosine and tangent can be used to solve for side lengths and angle measures of reasoning. trigonometry? right triangles. Standards for Mathematical Content Clusters Addressed [m] G-SRT.A Understand similarity in terms of similarity transformations. [m] G-SRT.B Prove theorems involving similarity. Acquisition SKILLS Students will be skilled at and/or be able to… • Use ratios to determine if figures are similar. • Prove whether or not corresponding sides or angles are congruent. • Use dilations to show that figures are similar. • Apply AA, SSS, and SAS to prove that triangles are similar. • Identify the opposite side, adjacent side, and hypotenuse of a right triangle given a specified angle. • Solve for a missing side using the Triangle Proportionality Theorem. KNOWLEDGE Students will know… • The properties and vocabulary of dilation. • All circles are similar. • Triangle similarity shortcuts: AA, SSS, and SAS. • The Triangle Proportionality Theorem. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 1 Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Unit Goals – Stage 1 [m] G-SRT.C Define trigonometric ratios and solve problems involving right triangles. • Sine, Cosine and Tangent are defined ratios that can be used to solve for a missing measure in any right triangle. • The definitions of: complementary angles, angles of elevation and angles of depression. • Use the Triangle Proportionality Theorem to prove that sides are parallel to each other. • Apply knowledge of similar triangles and rectangles to solve problems. • Use the coordinate plane to represent dilations. • Use similarity criteria for triangles to solve for missing sides and to prove relationships in geometric figures. • Prove the Pythagorean Theorem using triangle similarity. • Construct a tangent, sine or cosine ratio to solve for a missing side or angle of a right triangle. • Solve for an angle, when given two sides of a right triangle, by using an inverse trigonometric ratio. [m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. [m] G-MG.A Apply geometric concepts in modeling situations. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 2 Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] G-SRT.A Understand similarity in terms of similarity transformations. G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. [m] G-SRT.B Prove theorems involving similarity. G-SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [m] G-SRT.C 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. G-SRT.8.1 Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90° and 45°, 45°, 90°). CA [m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean theorem] G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [m] G-MG.A Apply geometric concepts in modeling situations. G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G-MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Key: [m] = major clusters; [s] = supporting clusters, [a] = additional clusters Indicates a modeling standard linking mathematics to everyday life, work, and decision-making CA Indicates a California-only standard LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 3 Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Evidence of Learning – Stage 2 Assessment Evidence Unit Assessment Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: [m] G-SRT.A • Students will use the definition of similarity in terms of similarity transformations to decide if two triangles are similar. • Students will explain the meaning of similarity for triangles as the proportionality of all corresponding pairs of sides using similarity transformations. • Students will use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. [m] G-SRT.B • Students will use similarity to prove theorems about triangles. • Students will use similarity criteria for triangles to solve problems. • Students will use similarity criteria to prove relationships in geometric figures. [m] G-SRT.C • Students will use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. • Students will show, that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to the definitions of trigonometric ratios for acute angles. • Students will explain and use the relationship between the sine and cosine of complementary angles. [m] G-GPE.B • Students will use the slope criteria to solve geometric problems. • Students will find the point on a directed line segment between two given points that partition the segment in a given ratio. [m] G-MG.A • Students will use geometric shapes, measures and properties to solve real-world problems. • Students will apply geometric methods to solve design problems. Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: • G-SRT.C Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: • G-SRT.A • G-SRT.B Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: • G-MG.A Other Evidence Formative Assessment Opportunities • Opening Tasks • Tasks • Informal teacher observations • Formative Assessment Lessons (FAL) • Checking for understanding using active participation strategies • Quizzes / Chapter Tests • Exit slips/summaries • Big Ideas Math Performance Tasks • Modeling Lessons (SMP 4) • SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 4 Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Learning Plan – Stage 3 Days 2-3 days Learning Target I will solve for the height of a building, or other tall object, using similar triangles. Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Geometry Expectations (Activities and Lessons) Opening Task – How Tall Is That “Building”? Students will use ratios and proportions to solve for the heights of buildings on campus. This is a review of previous learning about ratios and proportions, and a lead-in to the new idea of similar triangles and proportions. Later in this unit, students will be able to use trigonometry to repeat this activity via the tangent ratio and an inclinometer. Curriculum Intranet Applications: • How Tall Is That Building? This task is a gateway into the entire unit on similar triangles and trigonometry. I investigate polygon similarity by… 2-4 days • • • • • • • Using similarity statements. Finding corresponding lengths in similar polygons. Finding scale factors. Finding perimeters and areas of similar polygons. Deciding whether polygons are similar. Using similarity theorems to solve real-life problems. Answering questions such as… o How are a congruency transformation and a similarity transformation alike? Different? o Are congruent figures similar? o Is there an advantage to simplifying a ratio in a proportion? o If polygons have corresponding sides that are proportional, are the polygons similar? • Section 8.1 • STEM Video: Scale Model of a Pool Conceptual Understanding: • Illustrative Mathematics: Similarity • Illustrative Mathematics: Similar Quadrilaterals • Which One Doesn’t Belong: Similarity Procedural Skills and Fluency: • MathOpenRef: Similar Triangles and Polygons Applications: • STEM Video Performance Task: Pool Maintenance LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 5 Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Learning Plan – Stage 3 Days Learning Target I will prove triangles similar by… 4-5 days I will investigate proportionality theorems by… 2-3 days Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Geometry Expectations (Activities and Lessons) • Section 8.2 • Using the Angle-Angle Similarity Theorem to solve problems. • Section 8.3 • Using the Side-Side-Side Similarity Theorem to solve problems. • • • • Using the Side-Angle-Side Similarity Theorem to solve problems.. Using similarity theorems to solve real-life problems. Proving slope criteria using similar triangles. Answering questions such as… o Can you assume that corresponding sides and corresponding angles of any two similar triangles are congruent? o What is the minimum amount of information you need about two triangles to claim that they are similar? o What are ways to use corresponding sides of two triangles to determine that triangles are similar? o Why isn’t there a SSSS Similarity Theorem for quadrilaterals? o What are the key steps in proving the SSS Similarity Theorem? The SAS Similarity Theorem? o Why is there no Side-Side-Angle Similarity Theorem? • Using the Triangle Proportionality Theorem and its converse to solve problems. • Using other proportionality theorems to solve problems. • Answering questions such as… o What are methods you can use to construct a series of parallel segments? o How is the Triangle Midsegment Theorem related to the Triangle Proportionality Theorem? LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 6 Curriculum Intranet Conceptual Understanding: • Geogebra Applet: AA Similarity • Geogebra Applet: Triangle Similarity Shortcuts Procedural Skills and Fluency: • MathOpenRef: Similar Triangles and Polygons • Section 8.4 Conceptual Understanding: • Illustrative Mathematics: Finding Triangle Coordinates • Geogebra Applet: Proportionality Theorems Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Learning Plan – Stage 3 Days Learning Target I will investigate the relationships in right triangles by… • 5-8 days I will solve for parts of a right triangle by… 5-6 days • • • • • • • • • • • • • • • Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Geometry Expectations (Activities and Lessons) • Section 9.1 Using the Pythagorean Theorem. • Section 9.2 Using the Converse of the Pythagorean Theorem. • Section 9.3 Classifying triangles. • STEM Video: Finding the side lengths in special right triangles. Height of a Rock Solving real-life problems using special right triangles. Wall Identifying similar triangles. Using geometric means. Solving real-life problems using special right triangles and similar triangles. Answering questions such as: o How would YOU prove the Pythagorean Theorem? If p, q, and r satisfy the Pythagorean Theorem, will o o o o o o Is the converse of every true statement true? What are techniques to simplify radical expressions? What is the difference between opposite and adjacent? In your own words, what is the geometric mean? How does a geometric mean apply to the real world? Illustrate a situation that involves angle of elevation. Where is the right angle in this situation? Using the tangent, sine and cosine ratios. Solving real-life problems involving the tangent, sine and cosine ratios. Finding the sine and cosine angle measures in special right triangles. Using inverse trigonometric ratios. Solving right triangles. Defining the trigonometric ratios for the acute angles in a right triangle. Answering questions such as: o How does the tangent of an acute angle in a right triangle change as the angle measure increases? o Why is it not possible to find the tangent of a right triangle or an obtuse angle? o How do you know which side of the triangle is “opposite”? “Adjacent”? 7 Conceptual Understanding: • Geogebra Applet: Special Right Triangles Applications: • STEM Video Performance Task: Challenging the Rock Wall p q r , , and ? 2 2 2 o LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 Curriculum Intranet Section 9.4 Section 9.5 Section 9.6 Conceptual Understanding: • Illustrative Mathematics: Defining Trigonometric Ratios • Illustrative Mathematics: Tangent of Acute Angles • Geogebra Applet: Ratio of Sides of a Right Triangle Reposted 2/3/17 Unit 5 Similarity and Trigonometry Geometry Learning Plan – Stage 3 Days Learning Target o o o o o o o o o 2-3 days 2-3 days I will check my ability to use similar triangles to solve real world problems by participating in the FAL. I will prepare for the unit assessment on similarity and trigonometry by... Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Geometry Expectations (Activities and Lessons) How are the sine and cosine ratios the same? Different? Will the sine and cosine ratios ever be greater than 1? After solving for a missing side length, how can you check to see if your answer is reasonable? What do you have to know about a triangle to use the sine ratio or the cosine ratio? When you are solving a problem, how do you know whether to use the sine, cosine or tangent ratio? How would you use trigonometric ratios to find the missing angle measures in a triangle? How can you find the trigonometric ratios without a calculator? Why are trig ratios important? Where/when can they be used? Explain the difference between using the calculator to solve for the trig ratio and using the calculator to solve for the inverse trig ratio. Curriculum Intranet • Which One Doesn’t Belong: Right Triangle Trigonometry Procedural Skills and Fluency: • MathOpenRef: Trigonometry Functions Applications: • 3-Act Lesson: Boat on a River • FAL: Deducing Relationships – Floodlight Shadows FORMATIVE ASSESSMENT LESSON Deducing Relationships – Floodlight Shadows Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. 1-2 days LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 Unit Assessment (LBUSD Math Intranet, Assessment) 8 Reposted 2/3/17