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Curriculum and Instruction –Mathematics Quarter 3 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 TN 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 TN Standards call for conceptual understanding of key concepts. 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 12/1/16 1 of 19 Curriculum and Instruction –Mathematics Quarter 3 8. Look for and express regularity in repeated reasoning 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. 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 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: 4. Model with mathematics 5. Use appropriate tools strategically 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 12/1/16 2 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY Purpose of the Mathematics Curriculum Maps This curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready (CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and content—the major work of the grade (scope)—and provides a suggested sequencing and pacing and time frames, aligned resources—including sample questions, tasks and other planning tools. Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with colleagues to continuously improve practice and best meet the needs of their students. The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional practice. In fact, our goal is not to merely “cover the curriculum,” but rather to “uncover” it by developing students’ deep understanding of the content and mastery of the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and assessment) of the learners are best-positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected-with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgment aligned to our shared vision of effective instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigor—highquality teaching and learning to grade-level specific standards, including purposeful support of literacy and language learning across the content areas. Additional Instructional Support Shelby County Schools adopted our current math textbooks for grades 9-12 in 2010-2011. The textbook adoption process at that time followed the requirements set forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. We now have new standards; therefore, the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief State Officers (CCSSO) and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of conceptual knowledge development and application of these concepts), of our current materials. The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still incorporating the current materials to which schools have access. Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and external/supplemental resources (e.g., engageny), have been evaluated by district staff to ensure that they meet the IMET criteria. Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 3 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY How to Use the Mathematics Curriculum Maps Overview An overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items. 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 that 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 teacher’s responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard. Content Teachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.). Support for the development of these lesson objectives 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 objectives provide specific outcomes for that standard(s). Best practices tell us that clearly communicating and making objectives measureable leads to greater student mastery. Instructional Support and Resources District and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find instructional/performance tasks and additional resources that align with the standards in that module. 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 12/1/16 4 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY Topics Addressed in Quarter Similarity and Transformations Using Similar Triangles Right Triangles with Trigonometry Properties of Angles and Segments in Circles Overview During the third quarter students formalize their understanding of similarity, which was informally studied prior to geometry. Similarity of polygons and triangles is explored and triangle similarity postulates and theorems are formally proven. The proportionality of corresponding sides of similar figures is applied. Similarity is extended to the side-splitting, proportional medians, altitudes, angle bisectors, and segments theorems. The geometric mean is defined and related to the arithmetic mean. The special right triangles of 30-60-90 and 45-45-90 are also studied. Students are introduced to the right-triangle trigonometric ratios of sine, cosine, and tangent, and their applications. Angles and the sine, cosine, and tangent functions are defined in terms of a rotation of a point on the unit circle. Students will end the quarter by starting their study of circles. They will quickly review circumference and then identify central angles, major and minor arcs, semicircles and find their measures. They will finish the quarter studying inscribed angles and intercepted arcs. Year at a Glance Document Content Standard G-SRT.A.2 Type of Rigor Procedural Skill and Fluency , Conceptual Understanding Foundational Standards 8.G.A.1, 2,3, 4,5 G-SRT.B.4, 5 Procedural Skill and Fluency , Conceptual Understanding & Application 8.G.A.1, 2,3, 4,5 G-SRT.C.6, 7, 8 Procedural Skill and Fluency , Conceptual Understanding & Application Procedural Skill and Fluency , Conceptual Understanding & Application Procedural Skill and Fluency , Conceptual Understanding & Application 8.G.A.1, 2,3, 4,5 G-C.A.1, 2 G-MG.A.3 8.G.A.5; 8.G.B.7 Sample Assessment Items** Illustrative: Are They Similar; Illustrative: Congruent and Similar Triangles; Illustrative: Similar Triangles Illustrative: Joining Two Midpoints of Sides of a Triangle; Illustrative: Pythagorean Theorem; Illustrative: Bank Shot; Illustrative: Points From Directions Math shell: Hopewell Geometry Illustrative: Similar Circles; Illustrative: Neglecting the Curvature of the Earth Illustrative: Ice Cream Cone; Illustrative: Satellite 8.G.A.5; 8.G.B.7 ** 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 12/1/16 5 of 19 Curriculum and Instruction –Mathematics Quarter 3 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 12/1/16 6 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Similarity and Transformations (Allow approximately 3 weeks for instruction, review, and assessment) Domain: G-MG Modeling with Geometry Cluster: Apply geometric concept 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). ★ Enduring Understanding(s) Polygons are similar if and only if corresponding angles are congruent and corresponding sides are proportional. Use the following lesson(s) first to introduce concepts/build conceptual understanding. Better Lesson: Comparing Sequences Essential Question(s) What is the difference between a ratio and a proportion? What operations are used to solve a proportion? Use the textbook resources to address procedural skill and fluency. Lesson 7.1 Ratios and Proportions pp. 457 464 Graphing Technology Lab - Fibonacci Sequence and Ratios p. 464 Objective(s): • Write ratios • Write and solve proportions Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Understand similarity in terms of similarity transformations G-SRT.A.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. Enduring Understanding(s) Polygons are similar if and only if corresponding angles are congruent and corresponding sides are proportional. Essential Question(s) How do you use proportions to find side lengths in similar polygons? How do you identify corresponding parts of similar triangles? Objective(s): • Use proportions to Identify similar polygons • Solve problems using the properties of similar polygons Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic C, Lesson 12 – Similarity Transformations Use the textbook resources to address procedural skill and fluency. Lesson 7.2 Similar Polygons pp.465-473 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. Vocabulary Ratio, extended ratios, proportion, extremes, means, cross products Activity with Discussion Research and Report- The Fibonacci Sequence and the Golden Ratio - what are they, why are they important, and how are they related. Vocabulary Similar polygons, similarity ratio, scale factor Activity with Discussion p. 472 #54 Draw two regular pentagons that are different sizes. Are the pentagon’s similar? Will any two regular polygons with the same number of sides be similar? Explain Writing in Math/Discussion p. 472 #55 Compare and contrast congruent, similar, and equal figures. HS Flip Book with examples of each Standard (Designed as a resource tool to assist teachers in deepening Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 7 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES their understanding of what each standard means in terms of what students must know and be able to do. It outlines only a sample of instructional strategies and examples. Links to conceptual categories and specific standards in the document can be accessed from page 5 Mathematics Standards for High School.) Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Understand similarity in terms of similarity transformations G-SRT.A.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. Enduring Understanding(s) Geometric figures can change size and/or position while maintaining proportional attributes. Essential Question(s) How do you show two triangles are similar? Objective(s): • Identify similarity transformations • Verify similarity after a similarity transformation Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic A, Lesson 2 – Scale Drawings by Ratio Method engageny Geometry Module 2, Topic A, Lesson 3 – Scale Drawings by the Parallel Method engageny Geometry Module 2, Topic B Lesson 6 – Dilations engageny Geometry Module 2, Topic B, Lesson 7 – Do Dilations Map Segments? Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Use the textbook resources to address procedural skill and fluency. Lesson 7.6 Similarity Transformations pp. 505-511 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. G-SRT.C.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. Domain: G-MG Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Major Content Vocabulary dilation, similarity transformation, center of dilation, scale factor of a dilation, enlargement, reduction Activity with Discussion Explain how you can use scale factor to determine whether a transformation is an enlargement, a reduction, or a congruence transformation. HS Flip Book with examples of each Standard Enduring Understanding(s) Geometric figures can change size and/or position while maintaining proportional Supporting Content Use the following lesson(s) first to introduce concepts/build conceptual understanding. Additional Content Vocabulary Scale model, scale drawing, scale Shelby County Schools 2016/2017 Revised 12/1/16 8 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY 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). ★ CONTENT attributes. Essential Question(s) How do you use proportions to find side lengths in similar polygons? Objective(s): • Interpret scale models • Use scale factors to solve problems INSTRUCTIONAL SUPPORT & RESOURCES engageny Geometry Module 2, Topic A, Lesson 1 – Scale Drawings Writing in Math/Discussion Compare and contrast scale and scale factor. Use the textbook resources to address procedural skill and fluency. Lesson 7.7 Scale Drawings and Scale Models pp. 512-517 You can produce a scale model of a certain object by extending each dimension by a constant. What must be true of the shape of the object? Explain your reasoning. Using Similar Triangles (Allow approximately 3 weeks for instruction, review, and assessment) 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. Enduring Understanding(s) Polygons are similar if and only if corresponding angles are congruent and corresponding sides are proportional. Essential Question(s) How do you use proportions to find side lengths in similar polygons? How do you show two triangles are similar? Objective(s): • Identify and prove similar triangles using the AA Similarity Postulate and the SSS and SAS similarity Theorems • Use similar triangles to solve problems Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic C, Lesson 14 – Similarity engageny Geometry Module 2, Topic C, Lesson 15 – AA Similarity engageny Geometry Module 2, Topic C, Lesson 17 – SSS & SAS Similarity engageny Geometry Module 2, Topic C, Lesson 16 – Applying Similar Triangles Writing in Math/Discussion Contrast and compare the triangle congruence theorems with the triangle similarity theorems. Use the textbook resources to address procedural skill and fluency. Lesson 7.3 Similar Triangles pp. 474-483 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to non- Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 9 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES routine problems. HS Flip Book with examples of each Standard 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. Enduring Understanding(s) Use the following lesson(s) first to Polygons are similar if and only if corresponding introduce concepts/build conceptual understanding. angles are congruent and corresponding sides are proportional. engageny Geometry Module 2, Topic A, Lesson 4 – Triangle Side Splitter Theorem engageny Geometry Module 2, Topic B, Essential Question(s) How do you use proportions to find side lengths Lesson 10 – Dividing a Line Segment into Equal Parts in similar polygons? engageny Geometry Module 2, Topic C, Lesson 19 – Parallel Lines and Proportional Segments Objective(s): • Use proportional parts within triangles • Use proportional parts with parallel lines Vocabulary Mid-segment of a triangle Activity with Discussion Use multiple representations to explore angle bisectors and proportions. See p. 492, #47 Use the textbook resources to address procedural skill and fluency. Lesson 7.4 Parallel Lines and Proportional Parts (midsegments was previously covered in unit 2) pp. 484-492 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 10 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Task(s) See Mathematics, Instructional Resources, Geometry, Task Arc: Investigating Coordinate Geometry Partitioning However You Want to Slice It Comparing Shapes Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard (Designed as a resource tool to assist teachers in deepening their understanding of what each standard means in terms of what students must know and be able to do. It outlines only a sample of instructional strategies and examples. Links to conceptual categories and specific standards in the document can be accessed from page 5 Mathematics Standards for High School.) 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. Enduring Understanding(s) • Similar figures map one shape proportionally onto another through nonrigid motions. • Congruence and similarity criteria for triangles are used to solve problems and prove relationships of geometric figures. Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic D, Lesson 21 – Special Relationships within Right Traingles engageny Geometry Module 2, Topic D, Lesson 24 – Prove the Pythagorean Theorem Using Similarity Vocabulary Geometric mean Writing in Math/Discussion What is an arithmetic mean and a geometric mean of two numbers? Are they ever equal? Justify your answer. Domain: G-SRT Similarity, Right Triangles and Trigonometry Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 11 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS CONTENT Cluster: Prove theorems involving similarity GEOMETRY Essential Question(s) Can the geometric mean be used in any triangle? G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Why does geometric mean help us to find the missing sides in a right triangle? Objective(s): • Find the geometric mean between two numbers • Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse 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. Major Content Enduring Understanding(s) • Similar figures map one shape proportionally onto another through nonrigid motions. • Congruence and similarity criteria for triangles are used to solve problems and prove relationships of geometric figures. . Essential Question(s) How might the features of one figure be useful when solving problems about a similar figure? Objective(s): • Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles • Use the Triangle Angle Bisector Theorem Supporting Content INSTRUCTIONAL SUPPORT & RESOURCES Use the textbook resources to address procedural skill and fluency. Lesson 8.1 Geometric Mean pp.531-539 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic C, Lesson 18 – Triangle Angles Bisector Theorem Activity with Discussion Find a counterexample: If the measure of an altitude and side of a triangle are proportional to the corresponding altitude and corresponding side of another triangle, then the triangles are similar Use the textbook resources to address procedural skill and fluency. Lesson 7.5 Parts of Similar Triangles pp.495-503 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 12 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES ACT Practice (sample problems to prepare for the ACT) Glencoe, pp.456-457 Right Triangles and Trigonometry (Allow approximately 1.5 weeks for instruction, review, and assessment)t) Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles G-SRT.C.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. Enduring Understanding(s) Use the following lesson(s) first to Trigonometry can be used to measure sides and introduce concepts/build conceptual understanding. angles indirectly in right triangles. engageny Geometry Module 2, Topic D, Lesson 24 - Prove the Pythagorean Theorem Using Essential Question(s) Similarity How do you find a side length or angle measure in a right triangle? Use the textbook resources to address procedural skill and fluency. Objective(s): Lesson 8.3 Special Right Triangles pp.552-559 • Identify and apply side ratios in 45-4590 right triangles. Use the following resources to deepen • Identify and apply side ratios in 30-60students' conceptual understanding of 90 right triangles mathematical content and develop their Activity with Discussion p.559 #50 Explain how you can find the lengths of two legs of a 30-60-90 triangle in radical form if you are given the length of the hypotenuse. ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Task(s) Discovering Special Right Triangles Learning Task Finding Right Triangles in your Environment Learning Task Create your own triangles Learning Task Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 13 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Enduring Understanding(s) Use the following lesson(s) first to Trigonometry can be used to measure sides and introduce concepts/build conceptual understanding. angles indirectly in right triangles. engageny Geometry Module 2, Topic E Lesson 25: Incredibly Useful Ratios Essential Question(s) How do you find a side length or angle measure Lesson 26: The Definition of Sine, Cosine, and Tangent in a right triangle? How do trigonometric ratios relate to similar right Lesson 27: Sine and Cosine of Complementary Angles and Special Angles triangles? Lesson 28: Solving Problems Using Sine and Cosine Objective(s): Lesson 29: Applying Tangents Lesson 30: • Define trigonometric ratios for acute Trigonometry and the Pythagorean Theorem angles in right triangles • Use trigonometric rations and Pythagorean Use the textbook resources to address Theorem to solve right triangles procedural skill and fluency. • Use the relationship between the sine and Lesson 8.4 Trigonometry pp.562-271 cosine of complementary angles. G-SRT.C.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. Vocabulary Trigonometry, trigonometry ratio, sine, cosine, tangent, inverse sine, inverse cosine, inverse tangent Activity with Discussion p.570 #65 Explain how you can use ratios of the side lengths to find the angle measures of the acute angles in a right triangle. Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Task(s) Discovering Trigonometric Ratio Relationships learning task p.22 Domain: G-SRT Similarity, Right Triangles and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Major Content Enduring Understanding(s) Use the following lesson(s) first to Trigonometry can be used to measure sides and introduce concepts/build conceptual understanding. angles indirectly in right triangles. engageny Geometry Module 2, Topic D, Lesson Supporting Content Additional Content Vocabulary Angle of elevation, angle of depression Shelby County Schools 2016/2017 Revised 12/1/16 14 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Lesson 31: Using Trigonometry to Determine Area Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Lesson 33: Applying the Laws of Sines and Cosines Lesson 34: Unknown Angles Objective(s): Use the textbook resources to address • Solve problems involving angles of elevation. procedural skill and fluency. • Solve problems involving angles of Lesson 8.5 – Angles of Elevation and Depression depression. pp.574-581 Essential Question(s) How do you find a side length or angle measure in a right triangle? How do trigonometric ratios relate to similar right triangles? Writing in Math/Discussion p.580 #25 Classify the statement below as true or false. Explain. “As a person moves closer to an object he or she is sighting, the angle of elevation increases” Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Task(s) Find that Side or Angle Task Edutoolbox: Interstate Task ACT Practice (sample problems to prepare for the ACT) Glencoe, pp.618-619 Properties of Angles and Segments in Circles (Allow approximately 1.5 weeks for instruction, review, and assessment)t) Domain: G-C Circles Cluster: Understand and apply theorems about circles Major Content Enduring Understanding(s) The concept of similarity as it relates to circles can be extended with proof. Supporting Content Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 3, Topic A, Lesson Additional Content Vocabulary Circle, center, radius, chord, diameter, congruent circles, concentric circles, circumference, pi, Shelby County Schools 2016/2017 Revised 12/1/16 15 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY TN STATE STANDARDS CONTENT Essential Question(s) What role do circles play in modeling the word Domain: G-CO Congruence Cluster: Experiment with transformations in the around us? plane Objective(s): G-CO.A.1 Know precise definitions of • Give an argument to justify the formula angle, circle, perpendicular line, parallel for the circumference of a circle. line, and line segment, based on the undefined notions of point, line, distance • Prove that all circles are similar. along a line, and distance around a circular arc. G-C.A.1 Prove that all circles are similar. Domain: G-GMD Geometric Measurement and Dimension Cluster: Explain volume formulas and use them to solve problems G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. inscribed, circumscribed Use the textbook resources to address procedural skill and fluency. Lesson 10.1 – Circles and Circumference pp.683691 Writing in Math/Discussion p.690 #54 Research and write about the history of pi and its importance to the study of geometry. Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. Task(s) Illustrative Math: Similar Circles Task All Circles are Similar Task Enduring Understanding(s) • Relationships between angles, radii and chords will be investigated. • Similarities will be applied to derive an arc length and a sector area. Essential Question(s) When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments? Objective(s): Major Content 4 – Proving the Area of a Disk HS Flip Book with examples of each Standard G-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Domain: G-C Circles Cluster: Understand and apply theorems about circles INSTRUCTIONAL SUPPORT & RESOURCES Supporting Content Use the following lesson(s) first to introduce concepts/build conceptual understanding. . engageny Geometry Module 5, Topic A, Lesson 4 – Explore Relationships between Inscribed Angles, Central Angles and their Intercepted Arcs Use the textbook resources to address procedural skill and fluency. Lesson 10.2 Measuring Angles and Arcs pp.692- Additional Content Vocabulary Central angle, arc, minor arc, major arc, semicircle, congruent arcs, adjacent arcs, arc length Writing in Math/Discussion p.699 #62 Describe the three different types of arcs in a circle and the method for finding the measure of each one. Shelby County Schools 2016/2017 Revised 12/1/16 16 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT • Identify central angles, major arcs, minor arcs, and semicircles and find their measures. INSTRUCTIONAL SUPPORT & RESOURCES 700 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. HS Flip Book with examples of each Standard Task(s) Circles and their Relationships among Central Angles, Arcs and Chords (p. 15) Investigating Angle Relationships in Circles (pp. 46 & 52) Domain: G-C Circles Cluster: Understand and apply theorems about circles G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Enduring Understanding(s) • Relationships between angles, radii and chords will be investigated. • Similarities will be applied to derive an arc length and a sector area. Use the following lesson(s) first to introduce concepts/build conceptual understanding. Engageny Geometry Module 5, Topic A, Lesson 5 – Prove Inscribed Angle Theorem Essential Question(s) When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments? Use the textbook resources to address procedural skill and fluency. Lesson 10.4 Inscribed Angles pp.709-716 Objective(s): • Identify and describe relationships involving inscribed angles. • Prove properties of angles for a quadrilateral inscribed in a circle. Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to nonroutine problems. Vocabulary Inscribed angle, intercepted arc Writing in Math/Discussion p.715 #50 Compare and contrast inscribed angles and central angles of a circle. If they intercept the same arc, how are they related? HS Flip Book with examples of each Standard Task(s) Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 17 of 19 Curriculum and Instruction –Mathematics Quarter 3 TN STATE STANDARDS GEOMETRY CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Illustrative Math: Opposite angles in a cyclic quadrilateral Major Content Supporting Content Additional Content Shelby County Schools 2016/2017 Revised 12/1/16 18 of 19 Curriculum and Instruction –Mathematics Quarter 3 GEOMETRY RESOURCE TOOLBOX The Resource Toolbox provides additional support for comprehension and mastery of subject-level skills and concepts. While some of these resources are embedded in the map, the use of these categorized materials can assist educators with maximizing their instructional practices to meet the needs of all students. 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 HS Flip Book with Examples of each Standard (The Flip Book is designed as a resource tool to assist teachers in deepening their understanding of what each standard means in terms of what students must know and be able to do. It outlines only a sample of instructional strategies and examples. Links to conceptual categories and specific standards in the document can be accessed from page 5 Mathematics Standards for High School.) 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 Major Content Geometry Model Curriculum http://www.ccsstoolbox.org/ http://insidemathematics.org/index.php/high-school-geometry 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 Utah Electronic School - Geometry Ohio Common Core Resources Chicago Public Schools Framework and Tasks Videos Math TV Videos The Teaching Channel Khan Academy Videos (Geometry) Tasks Edutoolbox (formerly TNCore) Tasks Inside Math Tasks Mars Tasks Dan Meyer's ThreeNYC tasks Illustrative Math Tasks UT Dana Center GSE Analytic Geometry Unit 1: Similarity, Congruence and Proofs NWEA MAP Resources:https://teach.mapnwea.org/assist/help_map/Applic ationHelp.htm#UsingTestResults/MAPReportsFinder.htm - Sig n 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. ACT TN ACT Information & Resources ACT College & Career Readiness Mathematics Standards Interactive Manipulatives Literacy Resources GeoGebra – Free software for dynamic math and science learning Literacy Skills and Strategies for Content Area Teachers (Math, p. 22) 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 TINspire. Supporting Content Additional Content Glencoe Reading & Writing in the Mathematics Classroom Graphic Organizers (9-12) (teachervision.com) Shelby County Schools 2016/2017 Revised 12/1/16 19 of 19