Grade 10 Unit # 3 Pacing 6-8 weeks (MP 3)

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1 Montclair Public Schools CCSS Geometry Honors Unit: Marshall A.b.G Subject Geometry Honors Grade 10 Unit # 3 Pacing 6-8 weeks (MP 3) Unit Name Similarity, Trigonometry, and Transformations Overview Unit 3 takes previously learned geometric figures and transforms them (rotations, reflections, translations, dilations) throughout the coordinate plane. Students should be familiar recognizing the transformations as well as perform indicated transformations. Based on knowledge of dilations and other transformations, students will begin to understand similarity of figures. Students will reuse previous information from congruence in Unit 2 to develop similarity statements and formal proofs for similar triangles. Students will use the idea of similarity to explore what happens in the case of a right triangle and be able to find geometric mean. Right triangles will be studied in depth. Pythagorean Theorem, trigonometric ratios, Law of Sines/Cosines and finding area of non-right triangles will be explored. Standard # Standard, SC, or AC SLO # Student Learning Objectives Depth of Knowledge G.SRT.4 Prove theorems about right triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using similarity. 1 Apply the following theorems about triangles: a) The Side-Side-Side Similarity Theorem b) The Side-Angle-Side Similarity Theorem c) The Triangle Proportionality Theorem and its converse. d) Triangle Midsegment Theorem e) The Angle Bisector Theorem 2 3 Explore similar triangle s proportional perimeters Prove the following theorems about right triangles. When an altitude is drawn from the right angle to the hypotenuse: a) the two triangles formed are similar to the given triangle, and to each other. The altitude 1: Clarke

2 drawn is the geometric mean between the measures of the two segments of the hypotenuse. b) The sides are geometric means between the measures of the hypotenuse and the segment of the hypotenuse adjacent to it. c) the Pythagorean Theorem and its converse. 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. 4 Prove the following theorems about special triangles: a) In a triangle, the hypotenuse is sqrt2 times as long as the leg. b) In a triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is sqrt3 times as long as the shorter leg. 5 Define Sine, Cosine, Tangent as the ratio of side lengths in right triangles, using the concepts of similar triangles and scale factors. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. 6 Demonstrate using the definitions of the three basic trigonometric ratios, that the Sine of one of the acute angles in a right triangle references the same side in the same ratio as the Cosine of the other acute angle. G.SRT.8 2: Clarke Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 7 Construct and solve appropriate trigonometric ratios in right triangles to find specific side lengths

3 or angle measures Define angles of elevation and angles of depressions Construct right triangles and label them appropriately, to represent relationships in a descriptive scenario Model and solve a series of descriptive scenarios, using right triangle models G.SRT.9 Derive the formula A = ½ ab sin C for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side SC 11 Derive the area of a triangle, using the formula A = ½ bh and trigonometric ratios. Level 3 G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems. SC 12 Solve triangles using the Law of Sines/Cosines, or a combination of the two laws. G.SRT.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) 13 Model and solve a series of descriptive scenarios and special projects requiring the application of the Law of Sines and the Law of Cosines 3: Clarke

4 G.SRT.1 G.SRT.2 G.SRT.3 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. B. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 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. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar SC 14 Justify 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 b) The dilation of a line segment is longer or shorter in the ratio given by the scale factor Use the definition of similarity in terms of similarity transformations to decide if two figures 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. 17 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Level 3 G.GPE.7 G.CO.2 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. 18 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles and explore the relationship in regards to similar figures 19 Develop and perform rigid transformations that include reflections, rotations, translations, and dilations using geometric software, graph paper, tracing paper, and geometric tools and compare them to non-rigid transformations. 4: Clarke

5 Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 20 Describe the transformations (rotations and reflections) that would carry the given polygon onto itself. G.CO.4 G.CO.5 G.CO.6 Mathematical Practice # MP.1 MP.3 MP.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 21 Apply the definitions of angles, circles, parallel lines, perpendicular lines and line segments to describe rotations, reflections, and translations. 22 Draw given geometric figures and their images using graph paper, tracing paper, or geometric software. Level 3 Level 3 23 Identify a sequence of transformations that will carry a given figure onto another. 24 Use rigid transformations to determine, explain the congruence of geometric figures. Selected Opportunities for Connections to Mathematical Practices Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics 5: Clarke

6 MP.5 Use appropriate tools strategically. Big Ideas Triangles can be proved similar by AA, ASA, SSS, SAS, AAS Similar figures have angles that are congruent and sides in proportion. Sin/Cos/Tan are ratios of the sides of the triangle, and can be used to solve for missing side lengths or angle measures. Transformations (reflections, rotations, and translations) will change a figure s position, but not the size or shape. A dilation will change the size, but not shape, of the figure. Essential Questions How do you prove triangle similarity? What are the differences between similar and congruent figures? How is trigonometry used to find unknown measures? How do you change a figure s position, without changing its size or shape? How you do change a figure s size, without changing its shape? Assessments Required: Unit 3 Interim Assessment Recommended: Quizzes Transformations, Similarity (proportions), Trig Ratios (sin/cos/tan) Tests Transformations, Similarity (include finding perimeters), Trig Ratios (included angle of depression/elevation, word problems) Key Vocabulary Transformation Similarity Sine Cosine Tangent Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards) Chapter 7 Sections: 7-2, 7-3 6: Clarke

7 Prentice Hall Geometry Textbook Chapter 8 Sections: All Chapter 9 Sections: 9-1, 9-2, 9-3, 9-5 Chapter 12 Sections: All Learning Experiences (last area to be completed) Instructional Focus Student Learning Objectives Assessments What can I do to make the work maximally engaging and effective? What content should we cover? What content needs to be uncovered? List SLOs that are addressed via instructional focus All SLOs should be addressed; if listed in unit then they should be taught How will you assess these learning events? What types of assessments will you use to check for understanding? When should the basics come first? When should they be on a need to know basis? When should I teach, when should I coach, and when should I facilitate student discovery? How do I know who and where the learners are? In order to truly meet the standard, what should they be able to do independently (transfer)? What should I be doing to make them more independent and able to transfer? What events will help students practice & get feedback in transfer using the learning in realistic ways? What mathematical practices will my students engage in to make connections to the content? 7: Clarke