Explain the Lesson

Educational Epiphany ™ Districtwide PLC Protocol for Mathematics Teacher/Teacher Team: Grade/Course: Geometry Date: Week of February 12, 2024 # Planning Question 1 2 Teacher/Teacher Team Response Geometry Coherence Tool: Access the foundational standards to make connections to previously taught skills during the lesson introduction. Lesson 9.4 – The Tangent Ratio Which state standard is Lesson 9.3 – Similar Right Triangles and Lesson 9.5 – The Sine and Cosine Ratio (Combined) your lesson progression G.SRT.B.3 Use congruence and similarity criteria for triangles to solve G.SRT.C.4 Use side ratios in right triangles to define trigonometric ratios. addressing? problems and to justify relationships in geometric figures. G.SRT.C.4a Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric Foundational Standards: ratios for acute angles. G.SRT.B.3 is an Introductory Standard. G.SRT.C.4b Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.C.5 Solve triangles.* G.SRT.C.5a Know and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.* G.SRT.C.5b Know and use relationships within special right triangles to solve problems in a real-world context.* G.SRT.C.5c Use the Law of Sines and Law of Cosines to solve non-right triangles in a real-world context.* What mathematical concepts are embedded in the state standard? • • Use congruence and similarity criteria for triangles to solve problems. Use congruence and similarity criteria for triangles to justify relationships in geometric figures. Foundational Standards: 7.RP.A.2 is foundational for G.SRT.C.4 8.G.B.4 is foundational for G.SRT.C.5 G.SRT.C.4 • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle. • Understand and apply the definitions of trigonometric ratios for acute angles to find missing sides and angles of a right triangle, given other sides and angles. • Explain and use the relationship between the sine and cosine of complementary angles in right triangles. G.SRT.C.5 • Use the Pythagorean Theorem and trigonometric ratios to solve mathematical and real-world problems. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. • 3 What teacher knowledge, reminders, and misconceptions are assumed in the standard? Knowledge: • Students will need to draw on their understanding of proportional relationships from grade 7 (7.RP.A.2) when solving problems and justifying relationships in similar geometric figures. For example, when determining if a trapezoid is similar to another trapezoid, students will need to verify if the corresponding sides are proportional in addition to establishing that the corresponding angles are congruent. To do this, they may set up a table or a set of equations and test for equivalent ratios (a constant of proportionality), which they can also relate to the scale factor for a dilation. • Students should work with a broad range of situations to make conjectures about geometric relationships using triangle similarity and congruence. As students form connections between pairs of similar and congruent triangles, they should use geometric transformations to explain how the triangles are related and establish the correspondence between them. Many real-world examples can be explored such as finding the height of a building using similar triangles. For example, a person stands in the shadow of the building so that the end of his or her shadow lines up with the end of the shadow of the building. By measuring the length of the shadow of the building and the length of the shadow of the person as well as his or her height, the height of the building can be calculated using the proportional relationship between the similar triangles formed by each object and their shadows. This example utilizes the theorem that states that a line parallel to one side of a triangle divides the other two sides proportionally since the smaller triangle is embedded inside the larger triangle such that they share an angle and have two adjacent sides. Reminders: • In grades 7 and 8, students explored the concepts of similarity and congruence through transformations. Students explored scale drawings and scale factors in grade 7 (7.G.A.1) and the effects of transformations on figures in grade 8 (8.G.A.1). Additionally, in grade 8, students discovered that the corresponding angles of both congruent and similar figures are congruent (8.G.A.1). In this high school course, students solidify their understanding of the criteria for both congruent (G.CO.B.5) and similar (G.SRT.A.2) Use relationships within special right triangles to solve mathematical and real-world problems. • Use the Law of Sines and Law of Cosines to solve mathematical and real-world problems. • Recognize when it is appropriate to use each method to solve mathematical and real-world problems and explain their reasoning. Knowledge: • Trigonometry is one of the most widely used mathematical concepts and serves a variety of applications in many fields, so building the foundational knowledge of the special relationships in right triangles is crucial for students to make sense of trigonometry and apply trigonometric concepts fluently. In this high school course, they will explore how to find missing side lengths of a right triangle without knowing two side lengths and find an acute angle measure in a right triangle, using trigonometric ratios and their inverses. While the solving is a focus of G.SRT.C.5, the focus of this standard (G.SRT.C.4) is to build a conceptual understanding of the foundation of trigonometric ratios in similarity. It is important to note that for this course, students are only expected to work with sine, cosine, and tangent ratios and their inverses. They are not expected to know or use the reciprocal trig ratios of secant, cosecant, and cotangent. These will be introduced in pre-calculus. • Students should have opportunities to explore different right triangles that have and do not have proportional side lengths (similar and non-similar) to understand that corresponding sides of right triangles with the same acute angle measure will be proportional in length. Students should use this understanding to explore and understand the definitions of trigonometric ratios for acute angles. • Once students have defined the trigonometric ratios in right triangles, they should pay attention to patterns that appear within the ratios. In particular, they should discover a unique relationship between the sine and cosine of the complementary angles, specifically, that the sine of one acute angle is equivalent to the cosine of the other acute angle in any right triangle. This discovery can be extended to recognizing that this relationship exists both within a single right triangle, as well as when complementary angles exist between two different right triangles. Students should be able to explain why this relationship exists and use this concept to solve problems. It is important to make the connection that cosine literally means “Sine of the complement” and the “co” means complement. This will help students remember that cos(x) = sin(90-x). • When asked to solve triangles in G.SRT.C.5, students are expected to find all side lengths and angle measures that are not Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. geometric figures. In this standard (G.SRT.B.3), they apply that understanding to solve problems and justify relationships in geometric figures. Misconceptions: • Students should apply the criteria for similar and congruent triangles to verify properties of other figures. They should be able to provide mathematical justifications explicitly linking similarity and congruence of triangles to other geometric figures. For example, they can use triangle congruency criteria to determine and justify properties of parallelograms and special parallelograms, which supports G.CO.C.10, and they can use triangle similarity criteria to discover that the parts of the intersecting diagonals in a trapezoid are proportional. Additional examples are included in the scope and sequence for this standard. • • • given. This is a modeling standard; therefore, it is essential that students make direct connections between their solution methods and the triangles for which they are solving. In this high school course, students will encounter right triangles to solve that don’t provide two side lengths, and thus cannot be solved using the Pythagorean Theorem. They should recognize that in these cases, as well as when they are working with nonright triangles, there are other methods available to use to find missing measurements including trigonometric ratios, special right triangles, and the Law of Sines and Law of Cosines. They should also develop an understanding of when it is appropriate to use each method. Students should progress from solving right triangles, to solving non-right triangles using the Law of Sines and Law of Cosines. Not only will students need to recognize that these methods apply to non-right triangles, but they will also need to develop a deep understanding of when to use which law when solving a non-right triangle. Instruction should allow students to grapple with the structure of the problems that are given to determine which to use rather than following a given rule. Students should progress from solving right triangles, to solving non-right triangles using the Law of Sines and Law of Cosines. Not only will students need to recognize that these methods apply to non-right triangles, but they will also need to develop a deep understanding of when to use which law when solving a non-right triangle. Instruction should allow students to grapple with the structure of the problems that are given to determine which to use rather than following a given rule. Reminders: • In grade 8 (8.G.B.4), students solved for a missing side of a right triangle given two side lengths using the Pythagorean Theorem. Students can apply their understanding of the criteria of similar triangles (G.SRT.A.2 and G.SRT.A.3) to recognize that all right triangles with one congruent acute angle will be similar by the AA similarity postulate, since by definition right angles will always be congruent. Because they are similar, lengths of the corresponding sides will be proportional. Therefore, any right triangle with a given acute angle measure will have a specific ratio between each pair of sides. These specific ratios are defined as trigonometric ratios. To develop a deep conceptual understanding of trigonometric ratios, students should be allowed to discover these properties through a scaffolded task with discourse focused on patterns that they find. • Additionally, students should recognize that they will encounter right triangles to solve that don’t provide two side lengths and Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. cannot be solved using the Pythagorean Theorem. In these cases, as well as when they are working with non-right triangles, there are other methods available to use to find missing measurements including trigonometric ratios, special right triangles, and the Law of Sines and Law of Cosines. They should also develop an understanding of when it is appropriate to use each method. Misconceptions: • A common misunderstanding is that trigonometric ratios apply to any angle in any triangle. Through exploration, students should discover that they only apply to the acute angles in a right triangle. Allowing students to explore what happens when they try to apply trig ratios to the right angle will help them understand why it is not useful. Additionally, further exploration should reveal that non-right similar triangles will have different ratios of the lengths of the sides that are unique to each set of similar triangles. For example, a triangle with angle measures of 32°, 65°, and 83° will have different side length ratios than a triangle with angle measures of 51°, 62°, and 67°. This discovery connects to G.SRT.C.5c and the use of the Law of Sines and Cosines to solve non-right triangles. • To address the common misconception that the Pythagorean Theorem can be applied to any triangle to find missing side lengths, students can explore various types of triangles that are easily measured to test which type triangle it works for. This exploration can also reveal another practical use for the Pythagorean Theorem as a way to classify triangles. 4 What objective(s) must be taught? In what order? Why? PBO: • SWBAT use congruence and similarity criteria for triangles IOT solve problems and to justify relationships in geometric figures. Lesson objectives: • I can explain the Right Triangle Similarity Theorem. • I can find the geometric mean of two numbers. • I can find missing dimensions in right triangles. PBO: • • • • • • SWBAT use side ratios in right triangles IOT define trigonometric ratios. G.SRT.C.4 SWBAT use similarity to determine that side ratios in right triangles are properties of the angles in the triangle IOT develop definitions of trigonometric ratios for acute angles. G.SRT.C.4a SWBAT explain and use the relationship between the sine and cosine of complementary angles IOT solve problems involving triangles. G.SRT.C.4b SWBAT use Pythagorean Theorem and trigonometric ratios IOT solve triangles. G.SRT.C.5 SWBAT know and use the Pythagorean Theorem and trigonometric ratios IOT solve mathematical and real-world problems. G.SRT.C.5a SWBAT know and use relationships within special right triangles IOT solve mathematical and real-world problems. G.SRT.C.5b Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. • SWBAT use the Law of Sines and Law of Cosines IOT solve mathematical and real-world problems. G.SRT.C.5c Lesson objectives: • I can explain the tangent ratio. • I can find tangent ratios. • I can use tangent ratios to solve real-life problems. • I can explain the sine and cosine ratios. • I can find sine and cosine ratios. • I can use sine and cosine ratios to solve real-life problems. 5 What academic language must be taught before the teacher models for students? How will the academic language be taught and assessed? Academic Language: • Use – take, hold, or apply • Slope – the ratio of a line’s vertical change in the coordinate plane to the corresponding horizontal change. • Criteria – a standard on which a judgment or decision may be based • Congruence – having the same shape and size • Similarity – denoting two or more figures that have the same shape but different sizes • Parallel Lines – two lines in a plane that do not intersect • Perpendicular Lines – lines that intersect to form right angles • Solve – to apply an operation(s) in order to find a value; to find an answer • Justify – to prove or show to be correct or reasonable • Relationship – connection between two or more things; the way in which two or more concepts are connected • Geometric – related to geometry Instructional Practice 2: Strategies used to teach unfamiliar words will include: • 30 – 30 – 30 (common math-related word parts in the text, problem or objective) • Point of Use Annotation of the Performance Based Objective • Universal Language of Literacy • Word and Definition Walls • Word Parts • Context Clues • Point of Use Annotation of the Text (in Real Time) Academic Language: • Know – be aware of through inquiry, observation, or information • Use – take, hold, or apply • Explain – make clear by describing; to make something clear by describing it in more detail or by revealing relevant facts or ideas • Ratio – the comparison of two quantities by division • Right Triangle – a triangle with one 90-degree angle • Acute Angle – an angle with a measure between zero and 90 degrees • Complementary Angle – two angles which add up to 90 degrees • Define – state meaning; to determine or identify the essential qualities • Trigonometric Ratios – the ratios of the sides of a right triangle • Sine – the ratio of the side opposite a given angle to the hypotenuse in a right triangle • Cosine – the ratio of the side adjacent to an acute angle to the hypotenuse in a right triangle • Determine – to find out something using mathematical processes; to conclude after reasoning; to figure out • Pythagorean Theorem – a theorem that states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs; 𝑎2 + 𝑏 2 = 𝑐 2 • Special Right Triangles – • Law of Sines – • Law of Cosines – • Congruence – having the same shape and size • Similarity – denoting two or more figures that have the same shape but different sizes • Criteria – a standard on which a judgment or decision may be based • Property – a mathematical rule; a character or attribute that something has • Develop – to cause to grow and become more mature, advanced, or elaborate Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. • • • • • Definition – a statement of the exact meaning of a word Solve – to apply an operation(s) in order to find a value; to find an answer Relationship – connection between two or more things; the way in which two or more concepts are connected Geometric – related to geometry Real-world – related to a concrete setting Instructional Practice 2: Strategies used to teach unfamiliar words will include: • 30 – 30 – 30 (common math-related word parts in the text, problem or objective) • Point of Use Annotation of Performance-Based Objective • Universal Language of Literacy • Word-and-Definition Word Walls • Word Parts • Context Clues • Point of Use Annotation of the Texts (In Real Time) 6 What activities/practice problems are you planning to use for Launch the Lesson, Explore It, Examples & SelfAssessment, and Practice portions of the lesson? What did you learn from working the problems in advance of using them in class with students? Launch the Lesson: Launch the Lesson: “You invest $1000. The first year you earn 10%, and the second year you earn 20%. What is your average rate of return over the two years?” • Draw a right triangle and label the side lengths a, b, and c. “How many different ratios can be written using the side lengths? Explain.” 6; a : b, a : c, b : a, b : c, c : a, c : b • Students may incorrectly say 15%. Have them perform the calculations. At the end of the first year, you earn $1000 × 1.1 = • “In this section, you will learn about the tangent ratio, one of $1100. At the end of the second year, you earn $1100 × 1.2 = the six trigonometric ratios.” $1320. • If time permits, share a bit of the history of trigonometry. • “The geometric mean of 1.1 (110%) and 1.2 (120%) is approximately 1.1489, which is the average rate of return over Explore It: the two years.” • The lesson involves finding geometric means. Explore It: Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Examples: Examples: Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Self-Assessment: Self-Assessment: Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Practice: Practice: Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. 7 8 What manipulatives might be integrated into the lesson? What did you learn from using the manipulatives in advance of using them in class with students? Compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, protractor, etc. Compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, protractor, etc. Reference: Interactive Manipulatives • Didax Virtual Manipulatives Reference: Interactive Manipulatives • Didax Virtual Manipulatives What graphic organizer(s) might support students’ conceptual Reference: • Graphic Organizer Templates • Google Drawing Graphic Organizers • Teacher Vision Reference: • Graphic Organizer Templates • Google Drawing Graphic Organizers • Teacher Vision Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide. understanding of the process outlined by the performance-based objective(s)? Additional supporting and prerequisites standards are indicated on the curriculum map. In addition, this is not a comprehensive breakdown of each lesson for this weekly PLC protocol guide.

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