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Circles Unit Guide Geometry Big Idea (Cluster): Make geometric constructions Understand and apply theorems about circles Find arc lengths and areas of sectors of circles Translate between the geometric description and the equation for a conic section Use coordinates to prove simple geometric theorems algebraically Explain volume formulas and use them to solve problems Edited 5/22/14 Renton School District Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.C.1 Understand and apply theorems about circles. Prove that all circles are similar. G.C.2 Understand and apply theorems about circles. 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.3 Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.4 (+) Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle. G.C.5 Find arc lengths and areas of sectors of circles. 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.GPE.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.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). G.GMD.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. 2 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Relevant Math Practices and Student Actions MP1 Make sense of problems and persevere in solving them. Perseveres when solving problems Understands what the problem asks and the relationship among the problem’s parts Looks for an entry point into solving the problem Makes a plan for solving a problem Uses multiple strategies and representations Explains why a solution is reasonable MP 2 Reason abstractly and quantitatively. Interprets problems in context Uses representations to make meaning of problems Translates a problem from situation to equation Creates a situation for a given equation Explains connections between equation and situation Understands the meaning of quantities and units Flexibly uses properties of operations and place value MP 3 Construct viable arguments and critique the reasoning of others. Uses definitions and draws on prior mathematical knowledge when constructing arguments Makes conjectures and evaluates their accuracy Justifies conclusions with mathematical evidence and responds to arguments of others Communicates and defends mathematical reasoning using objects, drawings, diagrams, actions, examples and counterexamples Asks clarifying and probing questions to improve argument Attends to, questions, and evaluates the reasoning and arguments of others MP 5 Use appropriate tools strategically. Selects tools strategically for visualizing, exploring, comparing, predicting, and solving problems Uses technological tools and resources to pose problems, solve problems and deepen understanding Justifies tool selection choices, demonstrating understanding of tools’ limitations and error ranges MP 6 Attend to precision. Communicates mathematical thinking accurately both orally and in writing Understands the meaning of mathematical symbols and vocabulary and uses them appropriately Calculates accurately and efficiently Pursues a level of precision appropriate to the context of the problem MP 7 Look for and make use of structure. Looks for, identifies, develops and generalizes patterns and relationships Makes connections to prior mathematical knowledge to solve new problems Breaks down complex problems into manageable parts Views complicated quantities both as single objects or compilations of several objects and uses operations to make sense of problems MP 8 Look for and express regularity in repeated reasoning. Notices repeated calculations and looks for general methods and shortcuts to solve a problem Identifies patterns to develop algorithm, formula, or calculation Evaluates reasonableness of intermediate and final results 3 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. SBAC Required Evidence (Claim 1) All of the standards for this unit do not have an assigned SBAC assessment claim. The standards should be taught and assessed at the classroom level. Vocabulary Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter Balanced assessment items. Angle Annulus Arc Area Bisector Central angle Chord Circumcenter Circumference Concentric Congruent Conjecture Cyclic Diameter Inscribed angle Length of an arc Major arc Minor arc Perpendicular Radius Sector Segment Semicircle Tangent 4 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. See High School Flip Book page 157 for explanations and examples of this standard. Learning Objectives Students will be able to formally construct using a variety of tools and methods a square and a regular hexagon inscribed in a circle. SBAC Claims 1-4 Item Specification Information G.CO.13 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry does NOT address this standard. Must supplement. Connections to Prior Learning 7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics I Module 6 unit (lesson and tasks, OSPI reviewed OER) Engage NY High School Geometry Module 1 – Congruence, Proof and Constructions (OSPI reviewed OER) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) www.betterlesson.com 5 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.C.1 Understand and apply theorems about circles. Prove that all circles are similar. See High School Flip Book pages 171-172 for explanations and examples of this standard. Learning Objectives Students will be able to prove that all circles are similar by using the fact that the ratio of the diameter to circumference is the same for all circles. Students will be able to prove that all circles are similar by showing that for a dilation centered at center of circle the pre-image and image have equal central angle measures. Students will be able to determine a single or sequence of similarity transformations between two circles. SBAC Claims 1-4 Item Specification Information G.C.1 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Prior Learning 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Connections to Curricular Materials Discovering Geometry does NOT address this standard. Must supplement. Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) Missouri DOE Model Curriculum Unit Working with Circles 6 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.C.2 Understand and apply theorems about circles. 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. See High School Flip Book pages 173-174 for explanations and examples of this standard. Learning Objectives Students will be able to identify and describe relationships between central angles, inscribed angles, radii, chords, secants, tangents, and arcs. Students will be able to find angles formed by intersections of chords, tangents and secants. Students will be able to find segment lengths from intersections of chords, tangents and secants. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.C.2 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry sections 6.1, 6.2, and 6.3 addresses this standard. Content missing that will need to be supplemented includes: Find angles formed by intersections of chords, tangents and secants. Find segment lengths from intersections of chords, tangents and secants. Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) www.betterlesson.com Missouri DOE Model Curriculum Unit Working with Circles 7 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.C.3 Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. See High School Flip Book pages 175-176 for explanations and examples of this standard. Learning Objectives Students will be able to use inscribed angle to prove opposite angles of a cyclical quadrilateral are supplementary. Students will be able to apply the arc addition postulate to solve for missing arcs. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.C.3 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Additional Resources/Technology Resources Discovering Geometry sections 6.3, 6.4, and 13.6 address this standard loosely. Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) www.betterlesson.com Missouri DOE Model Curriculum Unit Working with Circles Content missing that will need to be supplemented includes: More problems for this standard are needed. 8 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.C.4 (+) Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle. Learning Objectives Students will be able to formally construct using a variety of tools and methods a tangent line from a point outside a given circle to the circle. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.C.4 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry does NOT address this standard. Must supplement. Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) www.betterlesson.com 9 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.C.5 Find arc lengths and areas of sectors of circles. 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. See High School Flip Book pages 177-178 for explanations and examples of this standard. Learning Objectives Students will be able to derive using similarity the formula for arc length using the fact that the length of the arc intercepted by an angle is proportional to the radius. Students will be able to define radian measure of the angle as the constant of proportionality. Students will be able to explain what a radian is and convert between degrees and radius. Students will be able to derive using similarity the formula for the area of a sector. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.C.5 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry section 6.7 and 8.6 addresses this standard loosely. Content missing that will need to be supplemented includes: Deriving the arc length formula through similarity Define radian measure of the angle as the constant of proportionality Explain what a radian is and convert between degrees and radius. Deriving the area of a sector formula through similarity Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) www.betterlesson.com Missouri DOE Model Curriculum Unit Working with Circles 10 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.GPE.1 Translate between the geometric description and the equation for a conic section. 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. See High School Flip Book pages 179-180 for explanations and examples of this standard. Learning Objectives Students will be able to use the Pythagorean Theorem, the coordinates of a circle center, and the circle’s radius to derive the equation of a circle. Students will be able to find the center and radius of a circle given an equation by completing the square. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.GPE.1 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry section 9.5 addresses this standard. Content missing that will need to be supplemented includes: Find the center and radius of a circle given an equation by completing the square Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 8 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks Geometry Common Core teacher site (objectives, content learning, tasks) www.betterlesson.com Missouri DOE Model Curriculum Unit Working with Circles 11 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). See High School Flip Book pages 182-183 for explanations and examples of this standard. Learning Objectives Students will be able to use distance formula or Pythagorean Theorem to decide if a point is inside the circle, outside the circle or on the circle. SBAC Claims 1-4 Item Specification Information Connections to Prior Learning G.GPE.4 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry does NOT address this standard. Must supplement. Additional Resources/Technology Resources Mathematics Vision Project Secondary Mathematics II Module 7 unit (lesson and tasks, OSPI reviewed OER) Mathematics Vision Project Secondary Mathematics II Module 8 unit (lesson and tasks, OSPI reviewed OER) Geometry Common Core teacher site (objectives, content learning, tasks) Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Missouri DOE Model Curriculum Unit Working with Circles Missouri DOE Model Curriculum Unit Coordinate Geometry Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives) Illustrative Mathematics Geometry tasks www.betterlesson.com 12 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1) Big Idea (Cluster): Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume formulas and use them to solve problems Standard G.GMD.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. See High School Flip Book pages 188-190 for explanations and examples of this standard. Learning Objectives Students will be able to explain the formula for circumference and area of a circle by determining the meaning of each term or factor of the formulas. SBAC Claims 1-4 Item Specification Information G.GMD.1 does not have an assigned SBAC assessment claim. The standard should be taught and assessed at the classroom level. Connections to Curricular Materials Discovering Geometry section 8.5, 6.5, and 6.6 addresses this standard loosely. Content missing that will need to be supplemented includes: The formal arguments for the formulas for the circumference of a circle and area of a circle. Connections to Prior Learning 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Additional Resources/Technology Resources Computing Pi applet Graphing circumference and area of circle ratios (NCTM) Estimating Circumference of a Circle Area of circle informal argument-Rabbi Abraham Area of circle informal argument – Leonardo Da Vinci Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math. The following resources was used to create this curriculum guide: High School Common Core State Standards Flip Book compiled by Trish Hart. A further unpacking of the Geometry Common Core State Standards by Hung-Hsi Wu (2013) titled “Teaching Geometry in Grade 8 and High School According to the Common Core Standards” can be found at http://math.berkeley.edu/~wu/CCSS-Geometry.pdf. This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. 13 14 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.