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Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY Introduction In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025, 80% of our students will graduate from high school college or career ready 90% of students will graduate on time 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, college and career ready aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and career readiness is rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor. Focus Coherence Rigor • The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. • For geometry, the major clusters, account for 65% of time spent on instruction. • Supporting Content - information that supports the understanding and implementation of the major work of the grade. • Additional Content - content that does not explicitly connect to the major work of the grade yet it is required for proficiency. • Thinking across grades: • The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding on to foundations built in previous years. Each standard is not a new event, but an extension of previous learning. • Linking to major topics: • Instead of allowing additional or supporting topics to detract from the focus of the grade, these concepts serve the grade level focus. For example, instead of data displays as an end in themselves, they are an opportunity to do grade-level word problems. • Conceptual understanding: • The Standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures. • Procedural skill and fluency: • The Standards call for speed and accuracy in calculation. While high school standards for math do not list high school fluencies, there are fluency standards for algebra 1, geometry, and algebra 2.. • Application: • The Standards call for students to use math flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and access content. Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 1 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 8. Look for and express regularity in repeated reasoning 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quatitatively Mathematical Practices(MP) 7. Look for and make use of structure GEOMETRY 6. Attend to precision 3. Construct viable arguments and crituqe the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically The Standards for Mathematical Practice describe varieties of expertise, habits of minds and productive dispositions that mathematics educators at all levels should seek to develop in their students. These practices rest on important National Council of Teachers of Mathematics (NCTM) “processes and proficiencies” with longstanding importance in mathematics education. Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice. This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that, ultimately our students, can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their practice so that it is in alignment with the three mathematics instructional shifts. Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around the content standards and mathematical practice standards that teachers should consistently access: The TN Mathematics Standards The Tennessee Mathematics Standards: Teachers can access the Tennessee State standards, which are featured https://www.tn.gov/education/article/mathematics-standards throughout this curriculum map and represent college and career ready learning at reach respective grade level. Standards for Mathematical Practice Mathematical Practice Standards Teachers can access the Mathematical Practice Standards, which are https://drive.google.com/file/d/0B926oAMrdzI4RUpMd1pGdEJTYkE/view featured throughout this curriculum map. This link contains more a more detailed explanation of each practice along with implications for instructions. Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 2 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY Purpose of the Mathematics Curriculum Maps The Shelby County Schools curriculum maps are intended to guide planning, pacing, and sequencing, reinforcing the major work of the grade/subject. Curriculum maps are NOT meant to replace teacher preparation or judgment; however, it does serve as a resource for good first teaching and making instructional decisions based on best practices, and student learning needs and progress. Teachers should consistently use student data differentiate and scaffold instruction to meet the needs of students. The curriculum maps should be referenced each week as you plan your daily lessons, as well as daily when instructional support and resources are needed to adjust instruction based on the needs of your students. How to Use the Mathematics Curriculum Maps Tennessee State Standards The TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards the supports student’s learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teachers' responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard. Content Weekly and daily objectives/learning targets should be included in your plan. These can be found under the column titled content. The enduring understandings will help clarify the “big picture” of the standard. The essential questions break that picture down into smaller questions and the learning targets/objectives provide specific outcomes for that standard(s). Best practices tell us that making objectives measureable increases student mastery. Instructional Support and Resources District and web-based resources have been provided in the Instructional Support and Resources column. The additional resources provided are supplementary and should be used as needed for content support and differentiation. Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 3 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY Topics Addressed in Quarter Triangle Congruence with Applications Properties of Triangles Special Segments in Triangles Properties of Quadrilaterals with Coordinate Proof Overview During the second quarter, students will continue to work with the concept of rigid motion and congruency. They will determine if two triangles are congruent by SSS, SAS, ASA, AAS, or HL and then provide appropriate reasoning for why they are congruent. They also will gain a deeper insight into constructing two-column, paragraph, and coordinate proofs. Students will classify triangles based on its’ angles and side measures and determine whether a triangle exists given three side measures and find the range of the third side when given two side measures. Students will compare the sides or angles of a given triangle and apply the Hinge theorem. Students will learn how to find missing angles in triangles both interior and exterior angles. They will investigate the special segments of a triangle; altitude, angle bisector, perpendicular bisector, and median. They will also practice with the points of concurrency; orthocenter, incenter, circumcenter, and centroid. Identifying quadrilaterals using given properties concludes the second quarter. Students should be able to solve equations to find various missing parts of the quadrilaterals as well as write two-column, paragraph and coordinate proofs using definitions and properties. Content Standard G-CO.A.1,2,3,4,5 Type of Rigor Procedural Skill and Fluency , Conceptual Understanding & Application Foundational Standards 8.G.A.1, 2,3, 4,5 G-CO.B.6, 7, 8 G-CO.C.9, 10 G-CO.D.12 G-GPE.B.4, 5 Conceptual Understanding & Application Conceptual Understanding & Application Conceptual Understanding & Application Procedural Skill and Fluency 8.G.A.1, 2,3, 4,5 8.G.A.1, 2,3, 4,5 8.G.A.5; 8.EE.B.6 8.EE.B.6 Sample Assessment Items** Defining Parallel Lines; Defining Perpendicular Lines; Fixed Points of Rigid Motion; C-CO.A.4 Tasks; GCO.A.5 Tasks Hexagon Art; Parallelogram G-CO.C.9 Tasks; G-CO.C.10 Tasks G-CO.C.12 Tasks Lucio’s Ride ** TN Tasks are available at http://www.edutoolbox.org/ and can be accessed by Tennessee educators with a login and password. Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 4 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY Fluency The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebra can help students get past the need to manage computational and algebraic manipulation details so that they can observe structure and patterns in problems. Such fluency can also allow for smooth progress toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to help students gain fluency. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency. The fluency recommendations for geometry listed below should be incorporated throughout your instruction over the course of the school year. G-SRT.B.5 Fluency with the triangle congruence and similarity criteria G-GPE.B.4,5,7 Fluency with the use of coordinates G-CO.D.12 Fluency with the use of construction tools References: http://www.tn.gov/education/article/mathematics-standards http://www.corestandards.org/ http://www.nctm.org/ http://achievethecore.org/ Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 5 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Properties of Triangles and Triangle Congruence with Applications (Allow 3 weeks for instruction, review, and assessment) Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 Prove theorems about triangles. Domain: G-CO Congruence Cluster: Make geometric constructions G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 Prove theorems about triangles. Enduring Understanding(s) The properties of triangles create a basis for understanding and reasoning that extends to other geometric figures. Lesson 4.1 Classifying Triangles, pp.235-242 Essential Question(s) How do the properties of triangles contribute to the geometric understanding of the world around us? Activity with Discussion Pair the categories of classifications of sides of triangles with the categories of classifications of angles to determine which combinations can exist and which ones cannot exist. Explain why certain combinations cannot exist. (Example, can a right equilateral triangle exist?) Error Analysis pg. 241, #56 (H.O.T. Problem) Objective(s): • Students will identify and classify triangles by angle measure • Students will identify and classify triangles by side measure Enduring Understanding(s) Many relationships exist between the measure of the angles of a triangle and the measure of the sides of the triangle. Lesson 4.2 Angles of Triangles, pp. 243-252 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Essential Question(s) Task(s) What can you say about the interior and exterior Geometry Lab: Angles of Triangles p. 243 angles of a triangle and other polygons? Domain: G-CO Congruence Cluster: Make geometric constructions G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic Objective(s): geometric software, etc.). • Students will apply the Triangle Angle Sum Theorem • Students will prove the measures of interior angles of a triangle have a sum Major Content Supporting Content Vocabulary acute triangle, equiangular triangle, obtuse triangle, right triangle, equilateral triangle isosceles triangle, scalene triangle Additional Content Vocabulary Auxiliary line, exterior angle, remote interior angles, flow proof, corollary Writing in Math Explain in words how to find the measure of a missing angle of a triangle if you know two of the angles. (Have students write this as if they were explaining it to someone who has never taken geometry before.) Shelby County Schools 2016/2017 Revised 9/2/16 6 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES of 180º. Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Prove theorems involving similarity G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Domain: G-CO Congruence Cluster: Understand congruence in terms of rigid motions G-CO.B.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. Domain: G-CO Congruence Cluster: Understand congruence in terms of rigid motions Enduring Understanding(s) There exist methods for proving triangles congruent. Essential Question(s) What does the SAS Triangle Congruence Theorem tell you about triangles? What does the SSS Triangle Congruence Theorem tell you about triangles? Objective(s): • Students will use the SSS Postulate to test for triangle congruence. • Students will use the SAS Postulate to test for triangle congruence. • Students will write two-column proofs to show that two triangles are congruent by SSS or SAS. G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. p. 269 #30, (H.O.T. Problems) Investigating Congruence in Terms of Rigid Motion (TN Task Arc) Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Proving properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties(Students learn why any two triangles that satisfy the SAS congruence criter. G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Major Content Select appropriate tasks from GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs Additional Lesson(s): Engageny Geometry Module 1, Topic D, Lesson 22 – Triangle Congruence Domain: G-CO Congruence Cluster: Make geometric constructions Lesson 4.4 Proving Triangles Congruent – SSS, Vocabulary SAS, pp. 262-271 Included angle Lesson 4.4 Extension – Geometry Lab: Proving Writing in Math Constructions p. 271 Create a chart for triangle congruence theorems (theorem, definition, and picture) Choose from the following resources to highlighting the sides and angles that are ensure that the intended outcome and level congruent in each pair of triangles. Compare of rigor (mainly conceptual understanding and contrast the theorems in your own and application) of the standards are met. words. Be sure to include both similarities and differences between the theorems. Task(s) Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 7 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Prove theorems involving similarity G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Domain: G-CO Congruence Cluster: Understand congruence in terms of rigid motions G-CO.B.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. Domain: G-CO Congruence Cluster: Understand congruence in terms of rigid motions G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. GEOMETRY CONTENT Enduring Understanding(s) There exist methods for proving triangles congruent. Essential Question(s) What does the ASA Triangle Congruence Theorem tell you about triangles? What does the AAS Triangle Congruence Theorem tell you about triangles? What does the HL Triangle Congruence Theorem tell you about two triangles? Objective(s): • Students will use the ASA Postulate to test for triangle congruence. • Students will use the AAS Postulate to test for triangle congruence. • Students will explore congruence in right triangles. • Students will write formal proofs to show that two triangles are congruent by AAS, ASA or HL. INSTRUCTIONAL SUPPORT & RESOURCES Lesson 4.5 Proving Triangles Congruent – ASA, AAS. Pp.273-280 Lesson 4.5 ext Congruence in Right Triangles p.281-282 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Select appropriate tasks from GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs Vocabulary Included side Writing in Math Explain why identifying two pairs of congruent angles with their included sides congruent is enough to prove that two triangles are congruent. Analyzing Congruency Proofs Are the Triangles Congruent? Domain: G-CO Congruence Cluster: Make geometric constructions G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 8 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 Prove theorems about triangles. Domain: G-CO Congruence Cluster: Make geometric constructions G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Prove theorems involving similarity G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. GEOMETRY CONTENT Enduring Understanding(s) Many relationships exist between the measure of the angles of a triangle and the measure of the sides of the triangle. Essential Question(s) What are the special relationships among angles and sides in isosceles and equilateral triangles? Objective(s): • Students will use properties of isosceles triangles. • Students will use properties of equilateral triangles. • Students will prove base angles of isosceles triangles are congruent. INSTRUCTIONAL SUPPORT & RESOURCES Lesson 4.6 Isosceles and Equilateral Triangles, pp. Vocabulary 283-291 Pythagorean triple Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Writing in Math p. 290 #45 Challenge – proof (H.O.T. problem) Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Additional Lesson(s): Engageny Geometry Module 1, Topic D, Lesson 23 – Isosceles Triangles Special Segments in Triangles (Allow 2.5 weeks for instruction, review, and assessment) Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 Prove theorems about triangles. Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations G-MG.A.3 Apply geometric methods Major Content Enduring Understanding(s) The properties of triangles create a basis for understanding and reasoning that extends to other geometric figures. Lesson 5.1 Bisectors of Triangles pp. 321-331 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Essential Question(s) Task(s) How can you use perpendicular bisectors to find the point that is equidistant from all the vertices Centers of Triangles of a triangle? Centers of Triangles Solutions How can you use angle bisectors to find the Supporting Content Additional Content Vocabulary Perpendicular bisector, concurrent lines, point of concurrency, circumcenter, incenter Writing in Math Compare and contrast the perpendicular bisectors and angle bisectors of a triangle. Be sure to include their points of concurrency. Shelby County Schools 2016/2017 Revised 9/2/16 9 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS 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). ★ Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations G-MG.A.3 Apply 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). ★ Major Content GEOMETRY CONTENT point that is equidistant from all the sides of a triangle? INSTRUCTIONAL SUPPORT & RESOURCES Hospital Locator Dividing a Town into Pizza Delivery Regions Geometry Lab - Constructing Bisectors p. 321 Why are the points of concurrency called incenter for angle bisectors of triangles and circumcenter for the perpendicular bisectors? Enduring Understanding(s) The properties of triangles create a basis for understanding and reasoning that extends to other geometric figures. Lesson 5.2 Medians and Altitudes of Triangles pp. 332-341 Vocabulary Median, centroid, altitude, orthocenter Essential Question(s) How can you find the balance point or center of gravity of a triangle? Additional Lesson(s): Engageny Geometry Module 1, Topic E, Lesson 30 – Medians of Triangles Objective(s): • Students will identify and use perpendicular bisectors in triangles • Students will identify and use angle bisectors in triangles. • Students will construct the special segments (perpendicular bisectors and angle bisectors) in acute, right and obtuse triangles. • Students will prove the perpendicular bisectors and the angle bisectors of a triangle meet at a point. Objective(s): • Students will identify and use medians in triangles • Students will identify and use altitudes in triangles. • Students will construct the special segments (medians and altitudes) in acute, right and obtuse triangles. Supporting Content Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Writing in Math Summarize the special segments of a triangle including their names, properties and diagrams into a chart or booklet. Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Select appropriate tasks from GSE Analytic Geometry Unit 1: Similarity, Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 10 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS CONTENT • Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 triangles. Prove theorems about Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations G-MG.A.3 Apply 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). ★ Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.10 Prove theorems about triangles. Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Major Content GEOMETRY Students will prove the medians and the altitudes of a triangle meet at a point. Enduring Understanding(s) Many relationships exist between the measure of the angles of a triangle and the measure of the sides of the triangle. Essential Question(s) How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? Objective(s): • Students will recognize and apply properties of inequalities to the measures of the angles of a triangle. • Students will recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle. Enduring Understanding(s) Many relationships exist between the measure of the angles of a triangle and the measure of the sides of the triangle. Essential Question(s) In what ways can congruence be useful? Supporting Content INSTRUCTIONAL SUPPORT & RESOURCES Congruence and Proofs Geometry Lab - Constructing Medians and Altitudes p. 332 The Centroid of a Triangle Balancing Act Exploring the Centroid of a Triangle Lesson 5.3 Inequalities in one triangle pp. 342-349 Writing in Math Lesson 5.5 The Triangle Inequality Theorem p. 348 #43 & 48 (H.O.T. Problems) pp.359-366 p. 365 #45 & 48 (H.O.T. Problems) Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Graphing Technology Lab - The Triangle Inequality p. 359 Triangle Inequality Task Triangle Inequalities Lesson 5.6 Inequalities in Two Triangles pp. 367376 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Inequalities in Two Triangles Activity Additional Content Writing in Math Compare and contrast the Hinge Theorem to the SAS Postulate for Triangle Congruence. Shelby County Schools 2016/2017 Revised 9/2/16 11 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS G-MG.A.3 Apply 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). ★ Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Prove theorems involving similarity G-SRT.B.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. Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Prove theorems involving similarity G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Objective(s): • Students will apply the Hinge Theorem or its converse to make comparisons in two triangles • Prove triangle relationships using the hinge theorem or its converse Enduring Understanding(s) Many relationships exist between the measure of the angles of a triangle and the measure of the sides of the same triangle. Essential Question(s) How are the segments that join the midpoints of a triangle’s sides related to the triangle’s sides? Objective(s): • Students will use proportional parts within triangles. • Students will use proportional parts with parallel lines. • Students will prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. Lesson 7.4 Parallel Lines and Proportional Parts (mid-segments of triangles) pp. 484-493 Vocabulary mid-segment of a triangle Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) TN Geometry Task: Midpoint Madness See Mathematics, Instructional Resources, Geometry Writing in Math Draw all of the mid-segments of one triangle. Explain what you see. Give as much detail as possible. Research and report on Sierpinski's triangle TN Task Arc: How Should We Divide This See Mathematics, Instructional Resources, Geometry, Task Arc: Investigating Coordinate Geometry Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Additional Lesson(s): Engageny Geometry Module 1, Topic E, Lesson 29 – Mid-segments of Triangles Properties of Quadrilaterals and Coordinate Proof (Allow 2.5 weeks for instruction, review, and assessment) Domain: G-MG Modeling with Geometry Major Content Enduring Understanding(s) Supporting Content Lesson 6.1 Angles of Polygons pp. 389-398 Additional Content Vocabulary Shelby County Schools 2016/2017 Revised 9/2/16 12 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS Cluster: Apply geometric concepts in modeling situations G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★ GEOMETRY CONTENT There exist certain patterns in the angle measures of polygons. Essential Question(s) Is there a limit to the sum of the interior/exterior angles of a polygon why or why not? INSTRUCTIONAL SUPPORT & RESOURCES Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Angle Sums Spreadsheet Lab p. 398 diagonal Lesson 6.2 Parallelograms, pp. 399-408 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Vocabulary parallelogram Objective(s): • Students will find and use the sum of the measures of the interior angles of a polygon • Find and use the sum of the measures of the exterior angles of a polygon Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.11 Prove theorems about parallelograms. Domain: G-GPE Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically G-GPE.B.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). Major Content Enduring Understanding(s) The properties of quadrilaterals help you to categorize quadrilaterals. Essential Question(s) What can you conclude about the sides, angles, and diagonals of a parallelogram? Objective(s): • Students will recognize and apply properties of the sides and angles of parallelograms • Students will recognize and apply properties of parallelograms Task(s) Select appropriate tasks from GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs TN Task: Expanding Triangles See Mathematics, Instructional Resources, Geometry TN Task: Parallelograms Writing in Math p. 396 #52 Open ended - Sketch a polygon and find the sum of its interior angles. How many sides does a polygon with twice this interior angles sum have? Justify your answer Writing in Math p. 406 # 43 Open ended - Provide a counterexample to show that parallelograms are not always congruent if their corresponding sides are congruent. (H.O.T. Problem) Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 13 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Additional Lesson(s): Geometry Module 1, Topic E, Lesson 28 – Properties of Parallelograms Engageny Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.11 Prove theorems about parallelograms. Domain: G-GPE Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Enduring Understanding(s) The properties of quadrilaterals help you to categorize quadrilaterals Essential Question(s) What criteria can you use to prove that a quadrilateral is a parallelogram? Objective(s): G-GPE.B.4 Use coordinates to prove • Students will recognize the conditions simple geometric theorems algebraically. that ensure a quadrilateral is a For example, prove or disprove that a parallelogram. figure defined by four given points in the • Students will prove that a set of points coordinate plane is a rectangle; prove or forms a parallelogram in the coordinate disprove that the point (1, 3) lies on the plane. circle centered at the origin and containing the point (0, 2). Lesson 6.3 Tests for Parallelograms pp.409-417 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) Select appropriate tasks from GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs Graphing Technology Lab - Parallelograms p. 408 Whitebeard's Treasure Task TN Task: Park City Similarity, Congruence & Proofs Writing in Math Journal Question: Are two parallelograms congruent if they both have four congruent angles? Justify your answer Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Additional Lesson(s): Geometry Module 4, Topic D, Lesson 13 – Coordinate Proof Engageny Domain: G-CO Congruence Cluster: Prove geometric theorems G-CO.C.11 Prove theorems about Major Content Enduring Understanding(s) The properties of quadrilaterals help you to categorize quadrilaterals Supporting Content Lesson 6.4 Rectangles Lesson 6.5 Rhombi and Squares Additional Content Vocabulary rectangle, rhombi, and square. Shelby County Schools 2016/2017 Revised 9/2/16 14 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 TN STATE STANDARDS GEOMETRY CONTENT parallelograms. Domain: G-GPE Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the Objective(s): coordinate plane is a rectangle; prove or • Students will recognize and use the disprove that the point (1, 3) lies on the properties of rectangles circle centered at the origin and containing the point (0, 2). • Students will determine whether parallelograms are rectangles • Students will recognize and apply the properties of rhombi and squares. • Students will determine whether quadrilaterals are rectangles, rhombi, or squares. Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Essential Question(s) How are the properties of rectangles, rhombi, and squares used to classify quadrilaterals? How can you use given conditions to prove that a quadrilateral is a rectangle, rhombus or square? G-MG.A.3 Apply 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). ★. Major Content INSTRUCTIONAL SUPPORT & RESOURCES Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Task(s) TN Task: Getting in Shape TN Task: Lucio’s Ride Use the following Engageny Lessons to introduce the concepts/build conceptual understanding. If used, these lessons should be used before the lessons from the textbooks. Additional Lesson(s): Geometry Module 1, Topic D, Lesson 33 – Review of the Assumptions 1 Engageny Engageny Geometry Module 1, Topic D, Lesson 34– Review of the Assumptions 2 Enduring Understanding(s) The properties of quadrilaterals help you to categorize quadrilaterals Lesson 6.6 Trapezoids and Kites, pp.435-446 Choose from the following resources to ensure that the intended outcome and level of rigor (mainly conceptual understanding and application) of the standards are met. Essential Question(s) Task(s) What are the properties of kites and trapezoids? TN Task: Go Fly a Kite Objective(s): • Students will apply properties of trapezoids • Students will apply properties of kites Supporting Content Writing in Math See Engageny lessons Additional Content Vocabulary trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid Graphic Organizer Use a Venn Diagram to show the relationship of the quadrilaterals you studied in Chapter 6 Shelby County Schools 2016/2017 Revised 9/2/16 15 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 GEOMETRY RESOURCE TOOLBOX Textbook Resources Standards ConnectED Site - Textbook and Resources Glencoe Video Lessons Hotmath - solutions to odd problems Common Core Standards - Mathematics Common Core Standards - Mathematics Appendix A TN Core CCSS Flip Book with Examples of each Standard Geometry Model Curriculum http://www.ccsstoolbox.org/ http://insidemathematics.org/index.php/high-school-geometry http://www.azed.gov/azcommoncore/mathstandards/hsmath/ http://learnzillion.com/common_core/math/hs http://www.livebinders.com/play/play/454480 https://www.livebinders.com/play/play?id=464831 http://www.livebinders.com/play/play?id=571735 North Carolina – Unpacking Common Core http://thegeometryteacher.wordpress.com/the-geometry-course/ http://mathtermind.blogspot.com/2012/07/common-core- geometry.html Utah Electronic School - Geometry Ohio Common Core Resources Chicago Public Schools Framework and Tasks Mathy McMatherson Blog - Geometry in Common Core Comprehensive Geometry Help: Online Math Learning (Geometry) I LOVE MATH NCTM Illuminations New Jersey Center for Teaching & Learning (Geometry) Calculator Finding Your Way Around TI-83+ & TI-84+ (mathbits.com) Texas Instruments Calculator Activity Exchange Texas Instruments Math Nspired STEM Resources Casio Education for Teachers *Graphing Calculator Note: TI tutorials are available through Atomic Learning and also at the following link: Math Bits graphing calculator steps Some activities require calculator programs and/or applications. Use the following link to access FREE software for your MAC. This will enable your computer and TI Calculator to communicate: Free TI calculator downloads Tasks Edutoolbox (formerly TNCore) Tasks Inside Math Tasks Mars Tasks Dan Meyer's Three-Act Math Tasks NYC tasks Illustrative Math Tasks UT Dana Center SCS Math Tasks GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs Major Content Interactive Manipulatives GeoGebra – Free software for dynamic math and science learning NCTM Core Math Tools http://www.keycurriculum.com/products/sketchpad (Not free) Any activity using Geometer’s Sketchpad can also be done with any software that allows construction of figures and measurement, such as Cabri, Cabri Jr. on the TI-83 or 84 Plus,TI-92 Plus, or TI-Nspire Videos Math TV Videos The Teaching Channel Khan Academy Videos (Geometry) NWEA MAP Resources:https://teach.mapnwea.org/assist/help_ map/ApplicationHelp.htm#UsingTestResults/MAPRe portsFinder.htm - Sign in and Click the Learning Continuum Tab – this resources will help as you plan for intervention, and differentiating small group instruction on the skill you are currently teaching. (Four Ways to Impact Teaching with the Learning Continuum) https://support.nwea.org/khanrit - These Khan Academy lessons are aligned to RIT scores. Literacy Resources Literacy Skills and Strategies for Content Area Teachers (Math, p. 22) Glencoe Reading & Writing in the Mathematics Classroom Graphic Organizers (9-12) (teachervision.com) Others TN Ready Geometry Blueprint State ACT Resources Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 16 of 17 Curriculum and Instruction – Office of Mathematics Quarter 2 Major Content GEOMETRY Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 9/2/16 17 of 17