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Accelerated GSE Algebra 1/Geometry A TCSS Unit 8 Information Curriculum Map: Similarity, Congruence, and Proofs Content Descriptors: Concept 1: Understand similarity in terms of similarity transformations. Concept 2: Prove theorems involving similarity. Concept 3: Understand congruence in terms of rigid motions. Concept 4: Prove Geometric Theorems. Concept 5: Make geometric constructions. Content from Frameworks: Similarity, Congruence, and Proofs Unit Length: Approximately 26 days 20152016 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A Curriculum Map Unit Rational: Building on standards from Coordinate Algebra and from middle school, students will use transformations and proportional reasoning to develop a formal understanding of similarity and congruence. Students will identify criteria for similarity and congruence of triangles, develop facility with geometric proofs (variety of formats), and use the concepts of similarity and congruence to prove theorems involving lines, angles, triangles, and other polygons. Prerequisites: As identified by the GSE Frameworks Understand and use reflections, translations, and rotations. Define the following terms: circle, bisector, perpendicular and parallel. Solve multi-step equations. Understand angle sum and exterior angle of triangles. Know angles created when parallel lines are cut by a transversal. Know facts about supplementary, complementary, vertical, and adjacent angles. Solve problems involving scale drawings of geometric figures. Draw geometric shapes with given conditions. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Draw polygons in the coordinate plane given coordinates for the vertices. Concept 1 Understand similarity in terms of similarity transformations. GSE Standards MGSE9-12.G.SRT.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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. TCSS Length of Unit Concept 2 Prove theorems involving similarity. GSE Standards MGSE9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally (and its converse); the Pythagorean Theorem using triangle similarity. MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Concept 3 Understand congruence in terms of rigid motions. GSE Standards MGSE9-12.G.CO.6 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. MGSE9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 7/30/2015 26 Days Concept 4 Prove geometric Theorems. GSE Standards MGSE9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Concept 5 Make geometric constructions. GSE Standards MGSE9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and 2 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A MGSE9-12.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. CC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS) MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. constructing a line parallel to a given line through a point not on the line. MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle. MGSE9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or (Focus on quadrilaterals, right triangles, and circles.) Lesson Essential Question Lesson Essential Question What is a dilation and how does this transformation affect a figure in the coordinate plane? What strategies can I use to determine missing side lengths and areas of similar figures? TCSS Lesson Essential Question Under what conditions are similar figures congruent? 7/30/2015 Lesson Essential Question How do I know which method to use to prove two triangles congruent? How do I know which method to use to prove two triangles similar? Lesson Essential Question In what ways can I use congruent triangles to justify many geometric constructions? How do I make geometric 3 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A Vocabulary Dilations Center Scale Factor Parallel lines Line Segments Ratio Similarity Transformations Corresponding angles Corresponding sides Proportionality AA criterion TCSS Vocabulary Adjacent Angles Alternate Exterior Angles Alternate Interior Angles Angle Bisector Centroid Circum center Coincidental Complementary Angles Congruent Congruent Figures Corresponding Angles Corresponding Sides Dilation Parallel Pythagorean Theorem Endpoints Equiangular Similarity Vocabulary Equilateral Exterior Angle of a Polygon In center Intersecting Lines Intersection Line Line Segment or Segment Linear Pair Measure of each Interior Angle of a Regular n-gon: Orthocenter Parallel Lines Perpendicular Lines Plane Point Proportion Ratio Ray Rigid motions Transform Corresponding Angles Corresponding Sides 7/30/2015 How do I prove geometric theorems involving lines, angles, triangles, and parallelograms? Vocabulary Reflection Reflection Line Regular Polygon Remote Interior Angles of a Triangle Rotation Same-Side Interior Angles Same-Side Exterior Angles Scale Factor Similar Figures Skew Lines Sum of the Measures of the Interior Angles of a Convex Polygon Supplementary Angles Transformation Translation Transversal Vertical Angles Alternate interior Perpendicular bisector Equidistant Endpoints Theorems: Interior angle sum Theorem Base angles of Isosceles Triangle Theorem Segments of midpoints of a triangle Theorem Medians of a triangle Theorem Median Isosceles Triangle Midpoints constructions? Vocabulary Construction Segments Angles Bisect Perpendicular lines Perpendicular bisectors Parallel lines Equilateral triangle Regular hexagon inscribed 4 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A Sample Assessment Items Concept 1 Sample Assessment Items Concept 2 Sample Assessment Items Concept 3 Sample Assessment Items Concepts 4 Sample Assessment Items Concepts 5 MGSE9-12.G.SRT.1 MGSE9-12.G.SRT.4 MGSE9-12.G.CO.6 MGSE9-12.G.CO.9 MGSE9-12.G.CO.12 In the coordinate plane segment M’N’ is the result of a dilation of segment MN by a scale factor of ⅓. Which point is the center of dilation? Justify the last two steps of the proof. Parallelogram FGHJ was translated 3 units down to form parallelogram F’G’H’J’. Parallelogram F’G’H’J’ was then rotated 90° counterclockwise about point G’ to obtain parallelogram F”G”H”JJ”. What can be concluded if Which diagram below shows a correct mathematical construction using only a compass and a straightedge to bisect an angle? Given: & Prove: R A. (1, 3) B. (0, 0) C. (– 5 , 0) D. (– 4 , 1) 1 7? a. S T a. t p b. p q c. p q U MGSE9-12.G.SRT.2 Which transformation results in a figure that is similar to the original figure but has a greater perimeter? a. a dilation of scale factor of 0.25 b. a dilation of scale factor of 0.5 c. a dilation of scale factor of 1 d. a dilation of scale factor of 2 1. Given 2. 2. Given 3. 4. 3. 4. by a a. Symmetric Property of ; SSS by a b. Reflexive Property of ; SAS by a c. Reflexive Property of ; SSS by a d. Symmetric Property of ; SAS MGSE9-12.G.SRT.3 Which method would it be possible to prove the triangles below are similar (if TCSS 1. MGSE9-12.G.SRT.5 In the triangle below, what is the approximate value of x? b. d. p t MGSE9-12.G.CO.10 Which statement is true about parallelogram FGHJ and parallelogram F’G’H’J’? a. The figures are similar but not congruent. b. The figures are congruent but not similar. c. The figures are both similar and congruent. d. The figures are neither similar nor congruent. 7/30/2015 Match each statement in the proof with the correct reason. A. Definition of midpoint B. SAS congruence criterion C. Corresponding parts of congruent triangles are congruent. D. SSS congruence criterion E. Definition of perpendicular bisector F. Given G. If a point is on the bisector of an angle, it is equidistant from the sides of the angle. c. d. 5 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A MGSE9-12.G.CO.7 possible)? For which pair of triangles would you use AAS to prove the congruence of the 2 triangles? MGSE9-12.G.CO.13 H. Reflexive property of congruence What is the first step when inscribing a regular hexagon in the circle below? A) a. SSS b. AA c. SAS d Not similar Given: AB AC, M is the midpoint of BC Prove: B C a. b. c. d. 4 in 4.5 in 4.9 in 5.1 in B) ____ 1. AB AC ____ 2. BM MC ____ 3. AM AM C) ____ 4. ABM ACM ____ 5. B C D) Answers: 1.F, 2. A, 3. H, 4.D, 5.C MGSE9-12.G.CO.11 MGSE9-12.G.CO.8 Which statement is true about ABC and FED? a. Set the compass to any distance. Then place the point of the compass at A and draw an arc that passes through any point on the circle. In the accompanying diagram of rhombus ABCD, mCAB = 35° What is mCDA? b. Place the point of the compass at any point on the circle and draw an arc that passes through point A. c. Use the compass to record the radius of the circle. d. Draw a segment from the center of the circle to point A. a. The triangles are similar, but not congruent. b. The triangles are congruent using ASA. c. The triangles are congruent using SAS. TCSS 7/30/2015 A. 35⁰ B. 70⁰ C. 110⁰ D. 140⁰ 6 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A d. The triangles are congruent using SSS. MGSE9-12.G.GPE.4 Given A (2,3), B (5, -1), C (1, 0), D (-4, -1), E (0,2), F (-1,-2) Prove: ∠ABC≅∠DEF Plot the points on a coordinate plane. Use the Distance Formula to find the lengths of the sides of each triangle. Resources – Concept 1 Instructional Strategies and Common Misconceptions Dilations practice (G.SRT.1) Dilation drills G.SRT.1) Similar polygons notes (G.SRT.2) Similar triangles Graphic Organizer (G.SRT.2) TCSS Key Resources – Concept 2 Resources – Concept 3 Instructional Strategies and Common Misconceptions Congruence Graphic Organizer (G.SRT.5) Property/Postulate/T heorem “Cheat Sheet” Properties matching card game (activator) Triangle Bisector notes (G.SRT.5) Instructional Strategies and Common Misconceptions Congruency notes (with proofs) (G.CO.7) Practice problems with application (G.CO.7) 7/30/2015 Resources – Concept 4 Instructional Strategies and Common Misconceptions Properties of Parallelograms notes (G.CO.11) Resources – Concept 5 Instructional Strategies and Common Misconceptions Constructing lines/bisecting angles notes and practice Proving properties of (G.CO.12) parallelograms How to bisect an activity (G.CO.11) angle (G.CO.12) Floor patterns How to copy an extension angle (G.CO.12) activity(G.CO.9-11) How to construct a grading rubric line parallel to 7 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A These tasks were taken from the GSE Frameworks. These tasks were taken from the GSE Frameworks. Shadow Math – extension activity Similar Triangles (G.SRT.2,3,5) Activator/Summarize Pythagorean r (G.SRT.1,2,3) Theorem using Teacher Student Triangle Similarity Proving Similar (G.SRT.4) Triangles – guided notes (G.SRT.3) Teacher Student Textbook Resources Holt McDougal – Explorations in Core Math p 181184(G.SRT.1) Holt McDougal – Explorations in Core Math p 155-165 &168-170(G.SRT.2) Holt McDougal – Explorations in Core Math p 171176(G.SRT.3) Textbook Resources Holt McDougal – Explorations in Core Math p 97-108, 177180 (G.SRT.4) Holt McDougal – Explorations in Core Math p 90,96,102(G.SRT.5) These tasks were taken from the GSE Frameworks. Introducing Congruence activator/summarizer (G.CO.6&7) Teacher Student Proving Triangles are Congruent station activity (G.CO.6-8) Teacher Student Textbook Resources Holt McDougal – Explorations in Core Math p65-70 (G.CO.6) Holt McDougal – Explorations in Core Math p77-82 (G.CO.7) Holt McDougal – Explorations in Core Math p83,8696 (G.CO.8) another line (G.CO.12) These tasks were taken How to construct a from the perpendicular line GSE Frameworks. (G.CO.12) Lunch lines notes/practice (G.CO.9) Constructions worksheet (G.CO.12) Teacher Student Centers of Triangles Constructions practice (G.CO.12) (G.CO.10) Teacher Student How to construct a Proving hexagon and an Quadrilaterals in the equilateral triangle Coordinate Plane inscribed in a circle (G.CO.11) Teacher Student Culminating Task: Geometry Gardens – project (extension activity) Textbook Resources Holt McDougal – Explorations in Core Math p35-54, 60-61, 84-85, 88 (G.CO.9) *not numbers 1-3 on page 37 or 12 on page 38 (G.CO.13) Textbook Resources Holt McDougal – Explorations in Core Math p5-16, 55-59, 62-64 (G.CO.12) Holt McDougal – Explorations in Core Math p115120 (G.CO.13) Holt McDougal – Explorations in Core Math p122-130 (G.CO.10) Holt McDougal – Explorations in Core Math p131-154 (G.CO.11) TCSS 7/30/2015 8 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A Differentiated Activities Concept 1 Dilations in the Coordinate Plane (G.SRT.1&2) Teacher Student Resources recommended for Math Support Interactive Vocabulary Site (differentiate how vocabulary is presented) What is similarity (activator) G.SRT.1 GADOE teacher notes G.SRT.2 GADOE teacher notes G.SRT.3 GADOE teacher notes Differentiated Activities Concept 2 Differentiated Activities Concept 3 Differentiated Activities Concept 4 Differentiated Activities Concept 5 Formalizing Triangle Congruence (G.CO.8) Teacher Student Resources recommended for Math Support Resources recommended for Math Support Resources recommended for Math Support Resources recommended for Math Support Triangle Sum notes Analyzing Congruence Proofs FAL G.CO.6 GADOE teacher notes G.CO.7 GADOE teacher notes These tasks were taken from the GSE Frameworks. G.CO.12 GADOE teacher notes Vocabulary/Postulate Card match activity Proof Scramble These tasks were taken from the GSE Frameworks. Triangle Proportionality Theorem (G.SRT.2-5) Teacher Student Floodlights FAL application (G.SRT.4&5, G.CO.9-11) – extension activity Evaluating Statements about length & area FAL(G.CO.9-11) These tasks were taken from the GSE Frameworks. Intro. activity TCSS 7/30/2015 9 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A At the end of TCSS Unit 5 student’s should be able to say “I can…” Take Away 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. Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the preimage, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor. 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. Use the properties of similarity transformations to develop the criteria for proving similar triangles: AA. Use AA, SAS, SSS similarity theorems to prove triangles are similar. Prove a line parallel to one side of a triangle divides the other two proportionally, and its converse. Prove the Pythagorean Theorem using triangle similarity. Use similarity theorems to prove that two triangles are congruent. Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane. Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea of congruency and develop the definition of congruent. Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent. Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria: ASA, SSS, and SAS. Prove vertical angles are congruent. Prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. Prove points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove the measures of interior angles of a triangle have a sum of 180º. Prove base angles of isosceles triangles are congruent. Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. Prove the medians of a triangle meet at a point. TCSS 7/30/2015 10 TCSS Unit 8 – Accelerated Algebra 1/ Geometry A Prove properties of parallelograms including: opposite sides are congruent, opposite angles are congruent, diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Copy a segment and an angle. Bisect a segment and an angle. Construct perpendicular lines, including the perpendicular bisector of a line segment. Construct a line parallel to a given line through a point not on the line. Construct an equilateral triangle so that each vertex of the equilateral triangle is on the circle. Construct a square so that each vertex of the square is on the circle. Construct a regular hexagon so that each vertex of the regular hexagon is on the circle. TCSS 7/30/2015 11