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Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations High School: Geometry Mathematical Content Standard: 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Featured Mathematical Practice: Correlated WA Standard: G.5.B MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. Standard Clarification/Example: Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged. Task Analysis: Describe the effect of dilations on two-dimensional figures using coordinates. Construct a dilation of a line not passing through the center. Construct a dilation of a line passing through the center. Compare/Contrast dilations of a line not passing and passing through the center. Predict the outcome of a dilation on a line based on the relationship to the center. Vocabulary: Prior Dilation Center of Dilation Parallel Pre-Image/Image Segment Ratio Scale Factor Explicit Introductory Pacing: Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations High School: Geometry Mathematical Content Standard: 1. Verify experimentally the properties of dilations given by a center and a scale factor: b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Featured Mathematical Practice: Correlated WA Standard: G.5.B MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. Standard Clarification/Example: Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the pre-image, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor. Task Analysis: Describe the effect of dilations on two-dimensional figures using coordinates. Construct dilations of a line segment given a center and various scale factors. Compare/Contrast dilations of a line segment given a center and various scale factors. Predict the lengths of line segments based on the ratio given by the scale factor. Vocabulary: Prior Prior Vocabulary from G-SRT.1a Explicit Introductory Pacing: Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations High School: Geometry Mathematical Content Standard: 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. Featured Mathematical Practice: Correlated WA Standard: G.3.B MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.8 Look for and express regularity in repeated reasoning. Standard Clarification/Example: Use the idea of dilation transformations to develop the definition of similarity. Given two figures determine whether they are similar and explain their similarity based on the equality of corresponding angles and the proportionality of corresponding sides. Task Analysis: Pacing: Understand that a two-dimensional figure is similar to another if the second can be obtained from a sequence of rigid motions and a dilation. Construct various similarity transformations on two-dimensional figures to develop the concept/definition of similarity. Given two-dimensional figures with minimal information, deduce the equality of corresponding angles and the proportionality of corresponding sides to determine similarity. Vocabulary: Prior Explicit Vocabulary from GSRT.1b Corresponding Angles Corresponding Sides Proportionality Sample Assessment Item: Definition of Similarity Introductory Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations High School: Geometry Mathematical Content Standard: 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Featured Mathematical Practice: Correlated WA Standard: G.3.B MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.8 Look for and express regularity in repeated reasoning. Standard Clarification/Example: Pacing: Use the properties of similarity transformations to develop the criteria for proving similar triangles; AA . Task Analysis: Understand that a two-dimensional figure is similar to another if the second can be obtained from a sequence of rigid motions and a dilation. Determine if two triangles are similar by using the properties of similarity transformations. Make observations to discover the minimal information required to show that two triangles are similar. Use the properties of similarity transformations to develop the criteria for proving similar triangles; AA. Vocabulary: Prior Vocabulary from GSRT.2 Explicit Angle-Angle Similarity (AA) Introductory Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Prove theorems involving similarity High School: Geometry Mathematical Content Standard: 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. Featured Mathematical Practice: Correlated WA Standard: G.3.D MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.7 Look for and make use of structure. Standard Clarification/Example: Task Analysis: Pacing: Use AA, SAS, and SSS similarity theorems to prove triangles are similar. Use triangle similarity to prove other theorems about triangles. Prove a line parallel to one side of a triangle divides the other two proportionally, and it’s converse. Prove the Pythagorean Theorem using triangle similarity. Determine and use AA, SAS, and SSS similarity theorems to prove triangles are similar. Use triangle similarity to prove Midsegment Theorem, Triangle Proportionality Theorem and its converse. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles. Use two similar right triangles to prove the Pythagorean Theorem. Vocabulary: Prior Midsegment Proportionally Transversal Bisect Pythagorean Theorem Explicit Side-Side-Side Similarity (SSS) Side-Angle-Side Sample Assessment Item: Similarity (SAS) Midsegment Theorem Triangle Proportionality Theorem Converse of Triangle Proportionality Theorem Introductory Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Prove theorems involving similarity High School: Geometry Mathematical Content Standard: 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Featured Mathematical Practice: Correlated WA Standard: G.3.B MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.7 Look for and make use of structure. Standard Clarification/Example: Pacing: Solve real world problems using congruence and similarity criteria for triangles. Prove geometric figures, other than triangles, are similar and/or congruent. Task Analysis: Solve real world problems using congruence and similarity criteria for triangles. Show geometric figures are similar by using the definition of similarity. Show geometric figures are congruent by using the definition of congruency. Prove geometric figures, other than triangles, are similar and/or congruent. Vocabulary: Prior Definition of Congruency Definition of Similarity Explicit Introductory Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Define trigonometric ratios and solve problems involving right triangles High School: Geometry Mathematical Content Standard: 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. Featured Mathematical Practice: Correlated WA Standard: G.3.C MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure. Standard Clarification/Example: Task Analysis: Pacing: Using a corresponding angle of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles. Explore the trigonometric ratios in a variety of similar and special right triangles. Determine the relationship of the side ratios for corresponding acute angles are the same. Define the trigonometric ratios for acute angles. Vocabulary: Prior Acute Angle Right Angle Corresponding Angle Hypotenuse Leg Ratio Explicit Trigonometric Ratios Sine Cosine Tangent Sample Assessment Item: Opposite Adjacent Special Right Triangles (30 -60 -90 ) Triangle (45 -45 -90 ) Triangle Introductory Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Define trigonometric ratios and solve problems involving right triangles High School: Geometry Mathematical Content Standard: 7. Explain and use the relationship between the sine and cosine of complementary angles. Featured Mathematical Practice: Correlated WA Standard: G.3.E MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Standard Clarification/Example: Explore the sine of an acute angle and the cosine of its complement and determine their relationship. Task Analysis: Use facts about complementary angles in a multi-step problem. (7th grade math) Identify the trigonometric ratios for the acute angles of a triangle. Explore the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Vocabulary: Prior Complementary Angles Prior Vocabulary from G-SRT.6 Explicit Introductory Pacing: Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Define trigonometric ratios and solve problems involving right triangles High School: Geometry Mathematical Content Standard: 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Featured Mathematical Practice: Correlated WA Standard: MP.2 Reason abstractly and quantitatively. G.3.E G.3.D MP.3 Construct viable arguments and critique the reasoning of others. MP. 4 Model with mathematics. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Standard Clarification/Example: Apply both trigonometric ratios and Pythagorean Theorem to solve application problems involving right triangles. Task Analysis: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles. Apply trigonometric ratios to determine unknown angle measures in right triangles. Apply Pythagorean Theorem and trigonometric ratios to solve for right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Vocabulary: Prior Vocabulary from GSRT.7 Explicit Solve a Right Triangle Introductory Pacing: Sample Assessment Item: Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Apply trigonometry to general triangles High School: Geometry Mathematical Content Standard: 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by Featured Mathematical Practice: drawing an auxiliary line from a vertex perpendicular to the opposite side. MP.1 Make sense of problems and persevere in solving them. Correlated WA Standard: NONE MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. Standard Clarification/Example: For a triangle that is not a right triangle, draw an auxiliary line from a vertex, perpendicular to the opposite side and derive the formula, A=½ ab sin (C), for the area of a triangle, using the fact that the height of the triangle is, h=a sin(C). Task Analysis: Vocabulary: (7th Solve mathematical problems involving area of a triangle. grade math) Draw the auxiliary line from a vertex, perpendicular to the opposite side and find the height in terms of the trigonometric ratio. Apply the trigonometric ratios of the height to the formula for the area of a triangle. Prior Height Base Area of Triangle Perpendicular Vertex Vocabulary from GSRT.8 Explicit Auxiliary Line Pacing: Sample Assessment Item: Introductory Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Apply trigonometry to general triangles High School: Geometry Mathematical Content Standard: 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Featured Mathematical Practice: Correlated WA Standard: NONE MP1. Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. Standard Clarification/Example: Task Analysis: Pacing: Using trigonometry and the relationship among sides and angles of any triangle, such as h= a sin(C), to prove the Law of Sines. Using trigonometry and the relationship among sides and angles of any triangle and the Pythagorean Theorem to prove the Law of Cosines. Use the Laws of Sines to solve problems. Use the Laws of Cosines to solve problems. Vocabulary: Use trigonometric ratios and the Pythagorean Theorem to solve right Prior triangles in applied problems. Vocabulary from GDraw the auxiliary line from a vertex, perpendicular to the opposite side SRT.9 and find the height in terms of the trigonometric ratio. Using trigonometry and the relationship among sides and angles of any Explicit triangle, such as h= a sin(C), to prove the Law of Sines. Law of Sines Using trigonometry and the relationship among sides and angles of any Law of Cosines triangle and the Pythagorean Theorem to prove the Law of Cosines. Trigonometric Use the Laws of Sines to solve problems. Identity Use the Laws of Cosines to solve problems. Introductory Sample Assessment Item: (Theta) Domain: Cluster: Level: Similarity, Right Triangles, and Trigonometry G-SRT Apply trigonometry to general triangles High School: Geometry Mathematical Content Standard: 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Featured Mathematical Practice: Correlated WA Standard: NONE MP.1 Make sense of problems and persevere in solving them. MP.4 Model with mathematics. MP.6 Attend to precision. MP.7 Look for and make use of structure. Standard Clarification/Example: Task Analysis: Pacing: Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right triangles. Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in nonright triangles. Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in right triangles. Understand and apply the Law of Sines and the Law of Cosines to find unknown measures in non-right triangles Vocabulary: Prior Vocabulary from GSRT.10 Explicit Introductory Sample Assessment Item: