Big Ideas Chapter 4: Transformations

Transcription

1 Geometry Big Ideas Chapter 4: Transformations Students investigate the intersecting lines and their resultant angles. Parallel and/or perpendicular lines are identified using the coordinate plane. Students never lose sight of constructing a logical, and supported, argument as a critical area in Geometry. Adding to their list of proof techniques, students use rigid and non-rigid transformations to define similar or congruent figures. Dynamic software plays an important role, allowing for student explorations and teacher demonstrations. Before: Middle School and Algebra I Draw polygons in the coordinate plane given the vertices, and find lengths of sides. Identify congruent figures and similar figures. Verify the properties of rotations, reflections, and translations. Translate, reflect, stretch, and shrink graphs of functions. Combine transformations of graphs of functions. Use slope to solve real-life problems. During: Geometry Perform translations, reflections, rotations, dilations, and compositions of transformations. Solve real-life problems involving transformations. Identify lines of symmetry and rotational symmetry. Describe and perform congruence transformations and similarity transformations. After: Algebra II Transformations and translations are transferred to functions families in Algebra II. In this course, the parent function is the preimage that will be base for understanding the transformations and translations of every function. Lines, Quadratics, Polynomials, rationals, radicals, trigonometry, exponentials, and logarithmic functions will be interpreted and understood by transforming and translating them into mathematics that is sophisticated and related to real life applications.

2 Geometry CHAPTER 4: Transformations TIME: 2 weeks UNIT NARRATIVE: In Chapter 4, students should have a conceptual understanding of transformations from middle school, where they studied translations, reflections, and rotations. Some may also have been introduced to glide reflections. The focus on plane transformations in the Common Core State Standards makes sense in terms of the continuity from middle school to high school. In middle school, the conditions for triangle congruence were informally explored. In high school, once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. The criteria for triangle similarity are established through similarity transformations. In this chapter, key postulates and theorems relating to rigid motions are presented. Translations, reflections, glide reflections, and rotations are all postulated to be rigid motions. While all could be proven, and established as theorems, we have chosen to treat them as postulates in this book. Dilations are introduced as nonrigid transformations where the scale factor k results in an enlargement (k > 1) or a reduction (0 < k < 1). The composition of a dilation with rigid motions results in a similarity transformation. In the last lesson, similar figures are defined in terms of similarity transformations. The use of dynamic geometry software for the explorations and formal lessons is highly encouraged. This tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students Textbook Correlations: Additional Resources Big Ideas Geometry: Chapter 4 Lesson Tutorials, game closet, student journal, skills review handbook, dynamic classroom, Interactive lessons, lesson planning tool, puzzle time, Practice A and B, Enrichment ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: Initial point, terminal point, horizontal 1. How can you translate a figure in a coordinate plane? component, vertical component, component form, transformation, 2. How can you reflect a figure in a coordinate plane? image, preimage, translation, dilation, congruent figures, center of 3. How can you rotate a figure in a coordinate plane? symmetry, rotational symmetry center of dilation, line of symmetry, center of ration, congruent figures, congruence transformation, rigid 4. What conjectures can you make about a figure reflected in two lines? motions, composition of transformation, reflection, line of reflection, 5. What does it mean to dilate a figure? glide reflection, enlargement, reduction, similar figures, scale factor 6. When a figure is translated, reflected, rotated, or dilated in the plane, is the image always similar to the original figure? CLUSTER HEADING & STANDARDS: Understand congruence in terms of rigid motions 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. Apply geometric concepts in modeling situations MATHEMATICAL PRACTICES MP 1 Make sense of problems and persevere when solving them. G-MG.3, G-CO.4 MP 2 Reason quantitatively and abstractly. G-CO.2, G-CO.3, G-CO.5 MP 3 Construct viable arguments and critique the reasoning of others. G-CO.6, G-CO.3

3 G-MG.3Apply 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). Understand similarity in terms of similarity transformations G-SRT.1a 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. G-SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MP 4 Model with mathematics. G-MG.3 MP 5 Use appropriate tools strategically. G-CO.6, G-CO.2, G-CO.3, G-CO.5 MP 6 Attend to precision. G-CO.6, G-CO.3, G-CO.5 MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning Geometry Experiment with transformations in the plane G-CO.2 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. 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. G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO.5 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. Learning Outcomes: Students will perform translations. Perform compositions. Solve real-life problems involving compositions. Perform reflections. Perform glide reflections. Identify lines of symmetry. Solve real-life problems involving reflections. Perform rotations. Perform compositions with rotations. Identify rotational symmetry. Identify congruent figures. Describe congruence transformations. Use theorems about congruence transformations. Identify and perform dilations. Solve real-life problems involving scale factors and dilations. Perform similarity transformations. Describe similarity transformations. Prove that figures are similar. End of the Unit Assessment: AC Driven

4 Revisiting the basic transformations Scaffolding 8 th grade Common Core regarding angle relationships and vocabulary Visual charts using T-Tables Puzzle games Game Closet Video Tutorials Tutor Support Online Skills review handbook Mathematical Proficiency Geometry DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Virtual Technology of real-world ELD Literacy Standards situations. Graphic organizers Use of Rational expressions Highlighting : cloze activities Multiple step with multiple SIOP strategies simplifications. Real-world visuals Multi standard processes Group collaboration Talk Moves Number talks Presentations Student Journal Enrichment Laurie s Notes STEM Videos and Performance Task Vocabulary Concepts Bilingual Vocabulary Small group instruction One on one peer support Smaller size quantities Revisiting the basic operations of fractions Scaffolding 6 th grade Common Core regarding one step equations and inverse properties. Skill base lessons regarding properties and integers. Number lines

5 Big Ideas Chapter 5: Congruent Triangles Geometry In this chapter, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They build upon this familiar foundation of triangle congruence to continue to develop formal proof techniques. Students make conjectures and construct viable arguments using a variety of formats to prove theorems and solve problems about triangles. Using their deductive skills, students explore special segments of triangles and the properties of these segments. Indirect proof is introduced and used to prove several triangle inequality theorems. Before: Middle School and Algebra I Understand that figures are congruent if they can be related by a sequence of translations, reflections, and rotations. Draw triangles with given conditions. Find the distance between points with the same x- or y-coordinate. Solve two-step equations. Write equations in one variable. Solve linear equations that have variables on both sides. Graph points and lines in the coordinate plane. Use properties of radicals to simplify expressions. During: Geometry Identify and use corresponding parts. Use theorems about the angles of a triangle. Use SAS, SSS, HL, ASA, and AAS to prove two triangles congruent. Prove constructions. Write coordinate proofs CHAPTER 5: Congruent Triangles TIME: 3 weeks UNIT NARRATIVE: In this chapter, students will work with a variety of proof formats as they investigate triangle congruence. Methods for establishing triangle congruence (SAS, SSS, ASA, and AAS) are established using rigid motions. The proof of each congruence criteria for triangles is done by composing transformations. This means a sequence of rigid motions maps one triangle onto another triangle. Other proof styles presented in this chapter include the twocolumn proof, the paragraph or narrative proof, and finally, in the last lesson, the coordinate proof. In addition to working with proofs, there are properties of equilateral and isosceles triangles that are proven. The use of dynamic geometry software for the explorations is highly encouraged. This tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students. Textbook Correlations: Additional Resources Big Ideas Geometry: Chapter 5 Lesson Tutorials, game closet, student journal, skills review handbook, dynamic classroom, Interactive lessons, lesson planning tool, puzzle time, Practice A and B, Enrichment

6 Geometry ESSENTIAL QUESTIONS: 1. How are the angle measures of a triangle related? 2. Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? 3. What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? 4. What conjectures can you make about the side lengths and angle measures of an isosceles triangle? 5. What can you conclude about two triangles when you know the corresponding sides are congruent? 6. What information is sufficient to determine whether two triangles are congruent? How can you use congruent triangles to make an indirect measurement? 7. How can you use a coordinate plane to write a proof? CLUSTER HEADING & STANDARDS: Understand congruence in terms of rigid motions G-CO.7Use 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. G-CO.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions Prove geometric theorems G-CO.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. Apply geometric concepts in modeling situations G-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ACADEMIC VOCABULARY: interior angles, exterior angles, corollary to a theorem, corresponding parts, legs (of an isosceles triangle), vertex angle (of an isosceles triangle), base (of an isosceles triangle), base angles (of an isosceles triangle), legs (of a right triangle), hypotenuse (of a right triangle), coordinate proof MATHEMATICAL PRACTICES MP 1 Make sense of problems and persevere when solving them. G-MG.1, G-MG.3, G-SRT.5 MP 2 Reason quantitatively and abstractly. G-CO.10, G-GPE.4 MP 3 Construct viable arguments and critique the reasoning of others. G-CO.7, G-CO.10, G-GPE.4 MP 4 Model with mathematics. G-MG.1, G-MG.3, G-SRT.5 MP 5 Use appropriate tools strategically. G-CO.7, G-CO.13 MP 6 Attend to precision. G-CO.7 MP 7 Look for and make use of structure. G-CO.8 MP 8 Look for and express regularity in repeating reasoning

7 G-MG.3Apply 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). Prove theorems involving similarity G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use coordinates to prove simple geometric theorems algebraically 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 disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Make geometric constructions G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry Learning Outcomes: Students will classify triangles by sides and angles. Find interior and exterior angle measures of triangles. Identify and use corresponding parts. Use the Third Angles Theorem. Use the Side-Angle Side (SAS) Congruence Theorem. Solve real-life problems. Use the Base Angles Theorem. Use Isosceles and equilateral triangles. Use the Side-Side-Side (SSS) Congruence Theorem. Use the Hypotenuse-Leg (HL) Congruence Theorem. Use the ASA and AAS Congruence Theorems. Use congruent triangles. Prove constructions. Place figures in a coordinate plane. Write coordinate proofs End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Revisiting the basic transformations Scaffolding 8 th grade Common Core regarding angle relationships and vocabulary Visual charts using T-Tables Puzzle games Game Closet Video Tutorials Tutor Support Online Skills review handbook Mathematical Proficiency Virtual Technology of real-world situations. Use of Rational expressions Multiple step with multiple simplifications. Multi standard processes Talk Moves Presentations Enrichment STEM Videos and Performance Task ELD Literacy Standards Graphic organizers Highlighting : cloze activities SIOP strategies Real-world visuals Group collaboration Number talks Student Journal Laurie s Notes Vocabulary Concepts Bilingual Vocabulary Small group instruction One on one peer support Smaller size quantities Revisiting the basic operations of fractions Scaffolding 6 th grade Common Core regarding one step equations and inverse properties. Skill base lessons regarding properties and integers. Number lines

8 Big Ideas Chapter 6: Relationships with Triangles Geometry In this chapter, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They build upon this familiar foundation of triangle congruence to continue to develop formal proof techniques. Students make conjectures and construct viable arguments using a variety of formats to prove theorems and solve problems about triangles. Using their deductive skills, students explore special segments of triangles and the properties of these segments. Indirect proof is introduced and used to prove several triangle inequality theorems. Before: Middle School and Algebra I Construct geometric figures with given conditions. Draw polygons in the coordinate plane given vertices and find lengths of sides. Establish facts about interior and exterior angles of triangles. Read, write, and evaluate algebraic expressions. Write and solve linear equations in one variable. Use linear equations to solve real-life problems. Graph in the coordinate plane. Find the slope of a line During: Geometry Understand and use angle bisectors and perpendicular bisectors to find measures. Find and use the circumcenter, in center, centroid, and orthocenter of a triangle. Use the Triangle Mid-Segment Theorem and the Triangle Inequality Theorem. Write indirect proofs CHAPTER 6: Relationships with Triangles TIME: 3 weeks UNIT NARRATIVE: This chapter uses the deductive skills developed in the last chapter to explore special segments in a triangle. These segments include perpendicular bisectors, angle bisectors, medians, altitudes, and mid-segments. Students are able to discover special properties of these segments by using dynamic geometry software. To prove these relationships, a variety of proof formats and approaches are used: transformational, synthetic, analytic, and paragraph. The last two lessons in the chapter are about inequalities within one triangle and in two triangles. The indirect proof is introduced and used to prove several of the theorems in these lessons. Triangles have been the geometric structure used to help students develop their deductive reasoning skills. In the next chapter, quadrilaterals and other polygons are studied. Textbook Correlations: Additional Resources Big Ideas Geometry: Chapter 6 Lesson Tutorials, game closet, student journal, skills review handbook, dynamic classroom, Interactive lessons, lesson planning tool, puzzle time, Practice A and B, Enrichment

9 Geometry ESSENTIAL QUESTIONS: 1. What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle? 2. What conjectures can you make about the perpendicular bisectors and the angle bisectors of a triangle? 3. What conjectures can you make about the medians and altitudes of a triangle? How are the mid-segments of a triangle related to the sides of the triangle? 4. How are the sides related to the angles of a triangle? 5. How are any two sides of a triangle related to the third side? 6. If two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles? CLUSTER HEADING & STANDARDS: Prove geometric theorems G-CO.9Prove 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. G-CO.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. Apply geometric concepts in modeling situations G-MG.1Use 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.3Apply 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). Make geometric constructions 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 ACADEMIC VOCABULARY: Equidistant, concurrent, point of concurrency, circumcenter, in-center, median of triangle, centroid, altitude of triangle, orthocenter, mid-segment of a triangle, indirect proof MATHEMATICAL PRACTICES MP 1 Make sense of problems and persevere when solving them. G-MG.1, G-MG.3 MP 2 Reason quantitatively and abstractly. G-CO.9, G-CO.10 MP 3 Construct viable arguments and critique the reasoning of others. G-CO.9, G-CO.10 MP 4 Model with mathematics. G-MG.1, G-MG.3, MP 5 Use appropriate tools strategically. G-CO.12, G-C.3 MP 6 Attend to precision. MP 7 Look for and make use of structure. G-CO.12, G-C.3 MP 8 Look for and express regularity in repeating reasoning

10 Geometry segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Understand and apply theorems about circles G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Learning Outcomes: Students will use perpendicular bisectors to find measures. Use angle bisectors to find measures and distance relationships. Write equations for perpendicular bisectors. Use and find the circumcenter of a triangle. Use and find the in-center of a triangle. Use medians and find the centroids of triangles. Use altitudes and find the orthocenters of triangles Use mid-segments of triangles in the coordinate plane. Use the Triangle Mid-Segment Theorem to find distances. Write indirect proofs. List sides and angles of a triangle in order by size. Use the Triangle Inequality Theorem to find possible side lengths of triangles. Compare measures in triangles. Solve real-life problems using the Hinge Theorem. End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Revisiting the basic transformations Scaffolding 8 th grade Common Core regarding angle relationships and vocabulary Visual charts using T-Tables Puzzle games Game Closet Video Tutorials Tutor Support Online Skills review handbook Mathematical Proficiency Virtual Technology of real-world situations. Use of Rational expressions Multiple step with multiple simplifications. Multi standard processes Talk Moves Presentations Enrichment STEM Videos and Performance Task ELD Literacy Standards Graphic organizers Highlighting : cloze activities SIOP strategies Real-world visuals Group collaboration Number talks Student Journal Laurie s Notes Vocabulary Concepts Bilingual Vocabulary Small group instruction One on one peer support Smaller size quantities Revisiting the basic operations of fractions Scaffolding 6 th grade Common Core regarding one step equations and inverse properties. Skill base lessons regarding properties and integers. Number lines