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Sequence of Concepts Trigonometry Identifying sides, defining ratios in terms of similarity, solving right triangles Variety of tasks in applying trigonometry Geometry Year-Long Curriculum Guide Third Quarter Rationale for Sequence Nearly three weeks are allowed for development of trigonometry. Activity 22 is supplemented by rich tasks in applied contexts (angles of elevation/depression, indirect measurement, etc.). Law of Sines and Law of Cosines are not included in the TNCore Geometry standards. Area of plane figures builds on prior knowledge in the middle grades and will be followed by three dimensional geometry. Prior Knowledge Math 8: Pythagorean Theorem Math 6: Area of triangles/quads/composite figures Area of Plane Figures Math 7: Circle formulas, area problem solving Area of triangles, quadrilaterals, and circles (includes sector area, arc length, and radian measure) Regular polygons: area, interior/exterior angles Circles unit will be studied in Quarter Four. Math 6: Prism volume Three Dimensional Geometry Book order is otherwise followed up to Math 7: 3-D problem solving, composite solids, Introducing prisms, cylinders, cones, pyramids, omitting spherical geometry lesson (36-3). cross sections and spheres: including nets, cross sections and Supplemental tasks allow for deeper Math 8: Cone, cylinder, sphere volume formulae revolution application in real-world problem solving. Volume, surface area of each solid: explaining and using formulas, applying density Effects on area and volume from changing dimensions HCDE Math Website: Task Arcs, Tasks, and other teacher resources referenced on the pacing guide may be found at www.hcde.org. Website Directions: 1) Log in using username and password in top right hand corner of webpage (HCDE email & password). 2) Select 6-HS Math/Teacher Resources/Geometry Springboard Login: Textbook resources can be accessed online at http://hamiltontn.springboardonline.org Website Directions: 1) Log in Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 1 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 2) Select Algebra 2 Teacher Notes: Tasks, CRAs, and Assessments that previously appeared on TNCore are now located on EduTOOLBOX.org, which is a free website. Many of these tasks do not require a user account to access. You just go to EduTOOLBOX.org. For tasks or resources in this pacing guide which show an * beside them, a user account is necessary. To create a user account: 1. Go to EduTOOLBOX.org 2. Click on Login in the right hand corner 3. Select the Tab: Create a New Account 4. Complete the information which includes a username that you create, a password of your choice, and other additional information. 5. Once you log in with your username and password, resources with the * will be unlocked and available to the user Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 2 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 Dates 1/4 – 1/6 (3 days) Tennessee State Standards 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. G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ 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). ★ Core Material L 22-1, L 22-2 Additional Resources Useful Ratios 1/9 – 1/13 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. G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ L 22-2, L 22-3, L 22-4, Skate Park Task* Proving Pythagorean Th. 1/17 – 1/20 (4 days) 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. G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles TV Size Task* Unknown Angles in applied problems. ★ Grade of Interstate Task L 30-1, L 30-2, L 30-3 1/23-1/27 1/30-2/3 G-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ 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). ★ G-MG.A.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★ G-CO.C.10: Prove 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. Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 3 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Making Right Triangles Evaluating Statements about Length and Area Composite Area L 31-1, L 31-2, L 31-3 Revised Area of Octagon and Circle 5/29/16 Dates 2/6-2/10 Tennessee State Standards G-CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G-C.A.1: Prove that all circles are similar. G-C.B.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 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. G-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and Core Material L 32-1, L 32-2, L 32-3, EA 1Unit 5 p. 477 Additional Resources Circle Pattern Gerrymandering rectangles, e.g., using the distance formula. ★ 2/13-2/17 G-GMD.B.4: Identify the shapes of two‐dimensional cross‐sections of three dimensional objects, and identify three‐dimensional objects generated by rotations of two‐dimensional objects. 2/21-2/24 (4 days) 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. G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve Review and Reteach Day, L 33-1, L 33-2, L 33-3 L 34-1, L 34-2, L 35-1 Popcorn Anyone? Fruit Boxes Prisms and Cyl problems. ★ G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects 2/27-3/3 (e.g., modeling a tree trunk or a human torso as a cylinder). ★ 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). 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. G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve L 35-2, L 35-3, EA 2 Unit 5 p. 549, L 36-1, L 36-2 Fishing for the Best Prism Pyramid and Cone problems. ★ G-GMD.B.4: Identify the shapes of two‐dimensional cross‐sections of three dimensional objects, and identify three‐dimensional objects generated by rotations of two‐dimensional objects. Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 4 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 Dates Tennessee State Standards G-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects Core Material Additional Resources (e.g., modeling a tree trunk or a human torso as a cylinder). ★ G-MG.A.2: Apply concepts of density based on area and volume in modeling situations 3/6-3/10 3/13-3/16 (4 days) (e.g., persons per square mile, BTUs per cubic foot). ★ 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). 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). ★ G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ L 37-1, L 37-2, EA 3 Unit 5 p. 549, Glasses (accompanying lesson) 3D in 2D, Propane Tank Modeling with Geometry Ice Cream Puddle End of Third Quarter Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 5 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 Geometry Year-Long Curriculum Guide Fourth Quarter Rationale for Sequence Return to Activity 24 for circles unit to close out the year. Suggest revisiting 32-3 for arc length/sector area/radians. Activity 26 is omitted given its focus on lines. Activity 28 is omitted as parabolas are not a geometry standard. Lessons 29-1 and 29-2 were addressed in Quarter 1. Probability (Unit 6) is not a tested geometry standard. Sequence of Concepts Prior Knowledge Math 7: Circle circumference formula Circles Algebra I: Completing the square, vertex form Basic vocabulary of circles’ lines and segments and their relationships Relationships among lines, arcs, and angles: central, inscribed, and circumscribed angles, inscribed triangles and quadrilaterals Equations of circles on the coordinate plane, including completing the square for standard form Constructing the incircle and circumcircle of a triangle Input from Algebra II teachers on key concepts Revisiting Algebra 1 concepts in preparation for Algebra II Preparation motivated discussion on which gap topics to Algebra 2. Simplifying radicals, including rational emphasize during post-EOC time. exponents Solving a linear system of equations Solving a quadratic equation (factoring, completing the square, graphing) HCDE Math Website: Task Arcs, Tasks, and other teacher resources referenced on the pacing guide may be found at www.hcde.org. Website Directions: 1) Log in using username and password in top right hand corner of webpage (HCDE email & password). 2) Select 6-HS Math/Teacher Resources/Geometry Springboard Login: Textbook resources can be accessed online at http://hamiltontn.springboardonline.org Website Directions: 1) Log in 2) Select Algebra 2 Teacher Notes: Tasks, CRAs, and Assessments that previously appeared on TNCore are now located on EduTOOLBOX.org, which is a free website. Many of these tasks do not require a user account to access. You just go to EduTOOLBOX.org. For tasks or resources in this pacing guide which show an * beside them, a user account is necessary. To create a user account: Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 6 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 1. 2. 3. 4. 5. Go to EduTOOLBOX.org Click on Login in the right hand corner Select the Tab: Create a New Account Complete the information which includes a username that you create, a password of your choice, and other additional information. Once you log in with your username and password, resources with the * will be unlocked and available to the user Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 7 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Revised 5/29/16 Dates 3/20-3/24 3/27-3/31 4/3-4/7 4/10-4/13 (4 days) 4/17-4/21 4/24-4/28 5/1-5/5 Tennessee State Standards Spring Break G-CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 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. Core Material Spring Break L 24-1, L 24-2, L 24-3 L 25-1, L 25-2 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. G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G-CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G-GPE.A.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G-GPE.B.4: Use coordinates to prove simple geometric theorems algebraically: prove or disprove that the point (1,√3 ) lies on the circle centered at the origin and containing the point (0, 2). L 25-3, L 25-4, EA 1 Unit 4 p. 371 Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 8 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) L 32-3 (arc/sector/radian) Secants and Tangents (Lessons 11-16) 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. G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Find arc lengths and areas of sectors of circles. G-CO.D.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Algebra II Prep: Topics include: - simplifying radicals including rational exponents Additional Resources Circles in Triangles Folding Circles Central and Inscribed Angles Arc Length/Sector Cyclic Quadrilaterals L 27-1, L 27-2 Writing the Equation of a Circle Comp the Square L 29-3, EA 2 Unit 4 p.429, Review and Reteach Day Revised Equations of Circles and Their Tangents Hospital Locator 5/29/16 Dates 5/8-5/12 5/15-5/19 5/22-5/26 Tennessee State Standards - solving a linear system of equations solving quadratic equations (by factoring, by completing the square, and by graphing) Alg II Prep continued Alg II Prep continued Alg II Prep continued Hamilton County Department of Education 2016-2017 Geometry Year-Long Curriculum Guide Page 9 Key: L – Lesson (SpringBoard) ST – Supplemental Task EA – Embedded Assessment (SpringBoard) Core Material Additional Resources End of Fourth Quarter Revised 5/29/16