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Geometry 1-2 PUHSD Curriculum 2014-2015 PARCC Model Content Frameworks Students bring many geometric experiences with them to high school; in this course, they begin to use more precise definitions and develop careful proofs. Although there are many types of geometry, this course focuses on Euclidean geometry, studied both with and without coordinates. This course begins with an early definition of congruence and similarity with respect to transformations, then moves on through the triangle congruence criteria and other theorems regarding triangles, quadrilaterals and other geometric figures. Students then move on to right triangle trigonometry and the Pythagorean theorem, which they may extend to the Laws of Sines and Cosines (+). An important aspect of the Geometry course is the connection of algebra and geometry when students begin to investigate analytic geometry in the coordinate plane. In addition, students in Geometry work with probability concepts, extending and formalizing their initial work in middle school. They compute probabilities, drawing on area models. Area models for probability can serve to connect this material to the other aims of the course. To summarize, high school Geometry corresponds closely to the Geometry conceptual category in the high school standards. Thus, the course involves working with congruence (G-CO), similarity (G-SRT), right triangle trigonometry (in G-SRG), geometry of circles (G-C), analytic geometry in the coordinate plane (G-GPE), and geometric measurement (G-GMD) and modeling (G-MG). The Standards for Mathematical Practice apply throughout the Geometry course and, when connected meaningfully with the content standards, allow for students to experience mathematics as a coherent, useful and logical subject. Details about the content and practice standards follow in this analysis. Page 1 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Geometry 1-2 Learning Outcomes Unit 1 Foundations and Tools for Geometry G-CO.A.1 Unit 2 Introduction to Transformational Geometry G-CO.A.2 # Unit 3 Triangle Congruence Unit 4 Quadrilaterals Unit 5 Similarity Unit 6 Trigonometry Unit 7 2/3-D Shapes Unit 8 Circles G-CO.B.7 G-GPE.B.4# G-CO.A.2 # G-SRT.C.6 G-CO.A.2 # G-C.A.1 G-CO.D.12 # G-CO.A.3 # G-CO.B.8 G-CO.A.3 # G-CO.C.10 # G-SRT.C.7 G-GPE.B.7 # G-C.A.2 G-CO.C.9# G-CO.A.4# G-CO.C.9 # G-CO.A.5 # G-SRT.A.1a G-SRT.C.8 # G-GMD.A.1 # G-C.A.3 G-GPE.B.4# G-CO.A.5 # G-CO.C.10 # G-CO.C.11 G-SRT.A.1b G-GMD.A.3 G-C.B.5 G-GPE.B.5# G-CO.B.6 G-CO.D.12 # G-CO.D.13 # G-SRT.A.2 G-GMD.B.4 G-GPE.A.1 G-CO.D.13 # G-GPE.B.5# G-SRT.A.3 G-MG.A.1 # G-GMD.A.1 # G-SRT.B.5 # G-MG.A.1# G-SRT.B.4 G-MG.A.2 G-MG.A.3# G-SRT.C.8 # G-SRT.B.5 # G-MG.A.3# G-MG.A.3# G-GPE.B.6 *= standards are addressed in multiple courses #=standards are addressed in multiple units Page 2 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Quarter 1 Introduction to Geometry (19 days) Definitions Constructions Prove Theorems about lines and angles Use coordinate formulas in proofs Different types of proofs two column, flow, paragraph Slope of parallel and perpendicular lines - use with geometric shapes Proofs (justification of thinking) - informal - sequence of logical statements Introduction to transformational Geometry (18 days) Transform Definitions Multiple Transformations Rigid Motion Congruence Quarter 2 Triangles (25 days) Prove triangle theorems Prove triangle congruence ASA, SSS, SAS Pythagorean Theorem Prove lines/angles theorems Constructions involving triangles Proofs involving triangles Quadrilaterals (15 days) Polygons Quadrilaterals Proofs Constructions Quarter 3 Similarity (25 days) Dilations Triangle and Polygon Similarity Midpoint Formula and Section Formula Proofs involving similarity Trigonometry (16 days) Special Right Triangle Relationships Right Triangle Relationships Trigonometry and Inverse Trigonometry Quarter 4 Two and three Dimensional Shapes (19 days) Perimeter/area -Area of sector -Using Coordinates -Changing parameters -Dissection, Argument -Complex polygons Volume -Changing parameters -Cavalieri’s principle Density based on area and volume Design Problems Proofs Circles (16 days) Similarity Parts of Circles Constructions Proportionality Equations Arguments Proofs Constructions Page 3 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 1 Foundations and Tools for Geometry Enduring Understandings: Essential Questions: Studying geometry involves learning the basic parts of geometry. 1. How can I make formal geometric constructions Everything is built from points, lines and planes and follows very 2. What are the basic parts of any construction or description in geometry? strict and organized rules. Proofs are a vital component for 3. Why are proofs important in developing geometric concepts? geometry. 4. How are definitions, postulate and theorems used to write geometric proofs? Standard Learning Targets Technology Standards A. Experiment with transformations in the plane I can define and then identify an Use geometry software to G-CO.A.1 Know precise definitions of angle, circle, perpendicular line, angle, circle, perpendicular line, parallel explore different theorems and parallel line, and line segment, based on the undefined line, and line segment based on the definitions within geometry notions of point, line, distance along a line, and distance idea of point, line, and distance along a Use proofblocks to develop around a circular arc. line. critical thinking I can make the following formal D. Make geometric constructions constructions using a variety of tools: Key Vocabulary G-CO.D.12 Make formal geometric constructions with a variety of tools copying a segment, copying an angle, collinear/Linear and methods (compass and straightedge, string, reflective bisecting a segment, bisecting an angle, coplanar/plane devices, paper folding, dynamic geometric software, etc.). constructing perpendicular lines, point Copying a segment; copying an angle; bisecting a segment; constructing perpendicular bisectors, segment, ray, line bisecting an angle; constructing perpendicular lines, constructing a line parallel to a given slope including the perpendicular bisector of a line segment; and line through a given point not on the angle constructing a line parallel to a given line through a point not line. segment on the line. I can prove the following theorems in perpendicular bisector narrative paragraphs, flow diagrams, in linear pair C. Prove geometric theorems complementary/supplementary two column format, and or using G-CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles diagrams without words: vertical angles are congruent; when a transversal crosses parallel/perpendicular/coinciding vertical angles are congruent, parallel lines, alternate interior angles are congruent and skew when a transversal crosses parallel corresponding angles are congruent; points on a adjacent angles lines, and alternate interior angles are perpendicular bisector of a line segment are exactly those midpoint congruent and corresponding angles equidistant from the segment’s endpoints. postulate are congruent. theorem I can use coordinates to prove the B. Use coordinates to prove simple geometric theorems algebraically angle bisector simple geometric theorems. G-GPE.B.4 Use coordinates to prove simple geometric theorems transversal I can disprove false statements using algebraically. For example, prove or disprove that a figure the properties of the coordinate plane, alternate interior angles Page 4 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum G-GPE.B.5 2014-2015 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). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Page 5 of 17 Major Content Supporting Content e.g. slope and distance. I can generalize the following criteria for parallel and perpendicular lines by investigating multiple examples. I can use the slope criteria for parallel and perpendicular lines to solve geometric problems. I can write the equation of a line parallel or perpendicular to a given a line, passing through a given point. Additional Content alternate exterior angles corresponding angles same side/consecutive interior angles quadrilateral triangle construction proof conjecture counterexample statement , negation inductive reasoning proof, theorem deductive argument paragraph proof informal proof algebraic proof two-column proof formal proof coordinate proof indirect proof (proof by contradiction) flow proofs conclusion Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 2: Introduction to Transformational Geometry Enduring Understandings: Essential Questions: Rotations, reflections and 1. How does each transformation move various objects? translations are examples that 2. How can I define congruence in terms of rigid motions? preserve angles and distances. 3. What are the similarities and differences between the images and pre-images generated by transformations and/or These “rigid motions” can be multiple transformations? used to describe congruence. 4. What is the relationship between the coordinates of the vertices of a figure and the coordinates of the vertices of the figure’s image generated by transformations and/or multiple transformations? 5. How can transformations be applied to real-world situations? Standard Learning Targets Technology Standards A. Experiment with transformations in the plane I can model transformations using Use geometry software G-CO.A.2 Represent transformations in the plane using, e.g., manipulatives. to explore transparencies and geometry software; describe transformations and their I can describe a transformation using transformations as functions that take points in the plane as coordinate notation that maps one point onto a properties. inputs and give other points as outputs. Compare Key Vocabulary unique image point. transformations that preserve distance and angle to those Transformation, image, I can compare transformations that preserve that do not (e.g., translation versus horizontal stretch). pre-image, composition distance and angle to those that do not. G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular translation I can demonstrate the rotations and polygon, describe the rotations and reflections that carry it reflection reflections that carry a rectangle, onto itself. parallelogram, trapezoid, or regular polygon on rotation G-CO.A.4 Develop definitions of rotations, reflections, and translations to itself. rotational symmetry in terms of angles, circles, perpendicular lines, parallel lines, reflectional symmetry I can make and refine a definition of and line segments. rotations, reflections, and translations based on rigid motion G-CO.A.5 Given a geometric figure and a rotation, reflection, or the definitions of angles, circles, perpendicular congruent carry on to itself translation, draw the transformed figure using, e.g., graph lines, parallel lines, and line segments. carry onto another paper, tracing paper, or geometry software. Specify a I can demonstrate and draw transformations map onto itself sequence of transformations that will carry a given figure using tools. onto another. I can find a sequence of transformations that sequence B. Understand congruence in terms of rigid motions predict will carry a shape onto another. G-CO.B.6 Use geometric descriptions of rigid motions to transform I can investigate rigid motions and generalize vertices vectors figures and to predict the effect of a given rigid motion on a their characteristics as preserving congruency. magnitude given figure; given two figures, use the definition of I can decide if two shapes are congruent congruence in terms of rigid motions to decide if they are because of the rigid motions between the two congruent. figures. Page 6 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 3: Triangle Congruence Enduring Understandings: Essential Questions: Triangles are fundamental aesthetic, 1. How do rigid motions lead to an understanding of congruence criteria for triangles? structural elements that are useful in many 2. How can proofs help us to develop a deeper and more enduring understanding of triangles? disciplines such as art, architecture, and 3. What is true of the points on a perpendicular bisector? engineering. 4. How is the Pythagorean Theorem applicable to real-world problems? 5. How can we use properties and theorems about triangles to solve real-world problems? Standard Learning Targets Technology Standards B. Understand congruence in terms of rigid motions I can show that two triangles are congruent through Use geometry software to G-CO.B.7 Use the definition of congruence in terms of rigid rigid motions if and only if the corresponding pairs of verify theorems about lines, motions to show that two triangles are congruent sides and corresponding pairs of angles are angles, triangles and if and only if corresponding pairs of sides and congruent. parallelograms. corresponding pairs of angles are congruent. Key Vocabulary I can explain which series of angles and sides are G-CO.B.8 Explain how the criteria for triangle congruence congruence essential in order to show congruence through rigid (ASA, SAS, and SSS) follow from the definition of Angle-Side-Angle Congruence motions. congruence in terms of rigid motions. Theorem I can prove the following theorems in narrative C. Prove geometric theorems Side-Angle-Side Congruence paragraphs, flow diagrams, in two column format, G-CO.C.9 Prove theorems about lines and angles. Theorems and/or using diagrams without words: points on a Theorem include: vertical angles are congruent; when a Side-Side-Side Congruence perpendicular bisector of a line segment are exactly transversal crosses parallel lines, alternate interior those equidistant from the segment’s endpoints. Postulate angles are congruent and corresponding angles CPCTC (Congruent Parts of I can prove theorems in narrative paragraphs, flow are congruent; points on a perpendicular bisector diagrams, in two column format, and or using Congruent Triangles are of a line segment are exactly those equidistant Congruent) diagrams without words. from the segment’s endpoints. I can make the following formal constructions using perpendicular bisector G-CO.C.10 Prove theorems about triangles. Theorems circumcenter, equidistant a variety of tools (compass and straightedge and include: measures of interior angles of a triangle Triangle Sum Theorem geometric software): constructing perpendicular sum to 180°; base angles of isosceles triangles are bisectors. interior angles congruent; the segment joining midpoints of two I can make the following formal constructions using Base Angles Theorem and its sides of a triangle is parallel to the third side and Converse a variety of tools (compass and straightedge and half the length; the medians of a triangle meet at geometric software): an equilateral triangle inscribed median a point. angle bisector in a circle. D. Make geometric constructions in-center I can solve problems using congruence criteria for G-CO.D.12 Make formal geometric constructions with a Concurrency of Medians of a triangles. variety of tools and methods (compass and Triangle I can prove relationships in geometric figures using Page 7 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum G-CO.D.13 2014-2015 straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. congruence criteria for triangles. I can solve real world problems involving right triangles using the Pythagorean Theorem. I can construct inscribed and circumscribed circles of a triangle centroid scalene triangle isosceles triangle equilateral triangle equiangular triangle acute triangle obtuse triangle right triangle B. 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. C. 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. A. 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).★ Page 8 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 4: Quadrilaterals Enduring Understandings: Essential Questions: Polygons can be classified using properties of sides and angles. How are quadrilaterals classified according to sides? Special quadrilaterals are classified based on different What is the difference between concave and convex? properties. What are the properties of quadrilaterals? How are quadrilateral classified? How do you inscribe a triangle, square or regular hexagon into a circle? How are polygons used in real- world situations? Standard Learning Targets Technology Standards B. Use coordinates to prove simple geometric theorems algebraically I can demonstrate and draw Geogebra G-GPE.B.4 Use coordinates to prove simple geometric theorems transformations using tools. Key Vocabulary algebraically. For example, prove or disprove that a figure Polygon I can find a sequence of transformations defined by four given points in the coordinate plane is a Convex that will carry a shape onto another. rectangle; prove or disprove that the point (1, √3) lies on Concave I can prove the following theorems in the circle centered at the origin and containing the point (0, narrative paragraphs, flow diagrams, in two Regular polygon 2). Triangles column format, and or using diagrams Quadrilateral without words: opposite sides are A. Experiment with transformations in the plane Parallelograms congruent, opposite angles are congruent, G-CO.A.5 Given a geometric figure and a rotation, reflection, or the diagonals of a parallelogram bisect each Rectangles translation, draw the transformed figure using, e.g., graph Trapezoid other, rectangles are parallelograms with paper, tracing paper, or geometry software. Specify a Rhombus congruent diagonals. sequence of transformations that will carry a given figure Kite I can make the following formal onto another. Squares constructions using a variety of tools Hexagon (compass and straightedge and geometric C. Prove geometric theorems software): an equilateral triangle, a square, Diagonals G-CO.C.11 Prove theorems about parallelograms. Theorems include: Opposite sides a regular hexagon inscribed in a circle. opposite sides are congruent, opposite angles are Opposite angles I can use coordinates to prove properties congruent, the diagonals of a parallelogram bisect each Bisect of quadrilaterals. other, and conversely, rectangles are parallelograms with Equilateral I can demonstrate the rotations and congruent diagonals. Inscribed reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon D. Make geometric constructions onto itself. G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Page 9 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 A. 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). ★ Page 10 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 5: Similarity Enduring Understandings: Essential Questions: Similarity is defines as the result of rigid transformations How is similarity defined by transformations? and dilations. Similar figures have corresponding angles How can I prove two figures are similar? that are congruent and corresponding sides that are How are trigonometric ratios used to solve problems involving triangles? proportional. Trigonometry is a particularly useful How can similar figures model real world situations? application of similar right triangles. How are trigonometric ratios used to solve problems involving triangles? What is the relationship between similar right triangles and trigonometric ratios? Standard Learning Targets Technology Standards A. Experiment with transformations in the plane I can compare transformations that preserve Use geometry software to G-CO.A.2 Represent transformations in the plane using, e.g., distance and angle to those that do not. verify theorems about lines, transparencies and geometry software; describe I can prove the Midsegment Theorem (the segment angles, triangles and transformations as functions that take points in parallelograms joining midpoints of two sides of a triangle is parallel the plane as inputs and give other points as Key Vocabulary to and half the length of the third side) in narrative outputs. Compare transformations that preserve dilation paragraphs, flow diagrams, in two column format, distance and angle to those that do not (e.g., scale factor and or using diagrams without words translation versus horizontal stretch). I can use the midpoint formula to calculate midpoint similarity C. Prove geometric theorems similarity transformations or endpoint coordinates with various unknowns (e.g. G-CO.C.10 Prove theorems about triangles. Theorems center of dilation find the other endpoint, etc.) include: measures of interior angles of a triangle mid-segment I can verify the following statements by making sum to 180°; base angles of isosceles triangles are multiple examples; proportional congruent; the segment joining midpoints of two ratio a. A dilation of a line is parallel to the original line if sides of a triangle is parallel to the third side and reduction the center of dilation is not on the line and a dilation half the length; the medians of a triangle meet at enlargement of a line is coinciding if the center is on the line. a point. b. The dilation of a line segment changes the length cofunction A. Understand similarity in terms of similarity transformations by a ratio given by the scale factor. G-SRT.A.1a Verify experimentally the properties of dilations I can extend the properties of dilations to polygons. G-SRT.A.1b given by a center and a scale factor: I can decide if two figures are similar based on a. A dilation takes a line not passing through the similarity transformations (rigid motion followed by a center of the dilation to a parallel line, and leaves dilation.) a line passing through the center unchanged. I can use similarity transformations to explain the b. The dilation of a line segment is longer or meaning of similar triangles as the equality of all shorter in the ratio given by the scale factor. Page 11 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum G-SRT.A.2 G-SRT.A.3 2014-2015 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. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. B. Prove theorems involving similarity G-SRT.B.4 G-SRT.B.5 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. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. B. Use coordinates to prove simple geometric theorems algebraically G-GPE.B.6 corresponding pairs of angles and the proportionality of all corresponding pairs of sides. I can establish the AA criterion by looking at multiple examples using similarity transformations of triangles. I can prove the following theorems in narrative paragraphs, flow diagrams, in two column format, and or using diagrams without words: A line parallel to one side of a triangle divides the other two proportionally, and conversely. Pythagorean Theorem proved using triangle similarity. I can solve problems using similarity criteria for triangles. I can prove relationships in geometric figures using similarity criteria for triangles. I can prove the following theorems in narrative paragraphs, flow diagrams, in two column format, and or using diagrams without words: A line parallel to one side of a triangle divides the other two proportionally, and conversely. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Page 12 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Enduring Understandings: Similarity is defines as the result of rigid transformations and dilations. Similar figures have corresponding angles that are congruent and corresponding sides that are proportional. Trigonometry is a particularly useful application of similar right triangles. Standard Unit 6: Trigonometry Essential Questions: How are trigonometric ratios used to solve problems involving triangles? How are trigonometric ratios used to solve problems involving triangles? What is the relationship between similar right triangles and trigonometric ratios? C. 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. 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. ★ Page 13 of 17 Major Content Learning Targets I can: Discover the relationship between the sides and angles of a right triangle and be able to state the sine, cosine, or tangent of a reference angle given a right triangle. Be able to find the three basic trig ratios given a triangle. Understand the sine and cosines of complementary angles are equal. Use a trig table. Have a basic understanding of how to use trig to solve a real world problem. Set up a trig equation and solve for a missing side. Read an application problem, set up a trig equation, and solve for a missing side length. Simplify a square root and rationalize the denominator with a square root. Discover the pattern of a 45-45-90 triangle and use the pattern to find the missing sides of a triangle. Discover the pattern of a 30-60-90 triangle and use the pattern to find the missing sides of a triangle Supporting Content Additional Content Technology Standards Use geometry software to verify theorems about lines, angles, triangles and parallelograms Key Vocabulary 45-45-90 triangle 30-60-90 triangle trigonometry trigonometric ratios: sine cosine tangent inverse trigonometry Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Enduring Understandings: Area, surface area, and volume have many real life applications. Many polygons and polyhedron have common features based on their common characteristics. Standard Unit 7: 2- and 3-dimensional shapes Essential Questions: What is the relationship of the different measures in two and three dimensional objects? How does a change in one dimension of an object affect the other dimensions? A. Experiment with transformations in the plane G-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). B. Use coordinates to prove simple geometric theorems algebraically G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ A. Explain volume formulas and use them to solve problems 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 problems. ★ Page 14 of 17 Major Content Learning Targets I can explore the effect of altering dimensions on the surface area and volume of a three-dimensional figure (similar figures and non-similar solids). I can use the distance formula to compute perimeters of polygons and areas of triangles and rectangles. I can explain the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone by using: -Dissection arguments, separating a shape into two or more shapes. -Cavalieri’s principle, if two solids have the same height and the same cross-sectional area at every level, then they have the same volume. -Informal Limit arguments, find the area and volume of curved shapes using an infinite number of rectangles and prisms. I can use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. I can identify the shapes of two-dimensional cross sections of three-dimensional objects. I can identify three-dimensional objects generated by rotations of two-dimensional Supporting Content Additional Content Technology Standards Sample lessons from education.ti.com Exploring Cavalier’s Principle (TI Nspire) Minimizing Surface Area of a Cylinder Given a Fixed Volume (TI Nspire) Illustrate geometric models. Some examples are: The Geometry Junkyard http://www.ics.uci.edu/~eppstein/j unkyard/model.html Wolfram Mathworld http://mathworld.wolfram.com/topi cs/SolidGeometry.html Key Vocabulary Area , perimeter Population density Cavalier’s principle Semi-circle, circle Surface area Volume Cross-section Rotation Two-dimensional Three-dimensional Density Base, height, radius, prism, cylinder Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 B. Visualize relationships between two-dimensional and three dimensional objects G-GMD.B.4 Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. objects. I can model real objects with geometric shapes. I can use the concept of density in the process of modeling a situation. I can use geometric properties to solve real world problems. A. 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). ★ 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-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). ★ Page 15 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 Unit 9: Circles Enduring Understandings: Essential Questions: Properties of circles can be explained and How are theorems for circles applied and proven? applied algebraically and geometrically. How are geometric properties of circles embedded in equations? How is proportion used in arc and sector measurements? How are real world situations modeled with circles? Standard Learning Targets A. Understand and apply theorems about circles I can prove all circles are similar to each other G-C.A.1 Prove that all circles are similar. based on similarity transformations (rigid motion followed by a dilation.) I can identify inscribed angles, radii, and G-C.A.2 Identify and describe relationships among inscribed angles, chords. radii, and chords. Include the relationship between central, I can describe relationships between segment inscribed, and circumscribed angles; inscribed angles on a lengths intersecting inside and outside of the diameter are right angles; the radius of a circle is circle. perpendicular to the tangent where the radius intersects I can describe relationships between angles the circle. formed inside and outside of the circle. I can construct inscribed and circumscribed G-C.A.3 Construct the inscribed and circumscribed circles of a circles of a triangle. triangle, and prove properties of angles for a quadrilateral I can prove the following properties for inscribed in a circle. quadrilateral ABCD inscribed in a circle; (i.e) ∠A B. Find arc lengths and areas of sectors of circles 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. A. Translate between the geometric description and the equation for a conic section 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. Page 16 of 17 Major Content Supporting Content + ∠C =∠B +∠D =180o I can use similarity to logically arrive at the following; the length of the arc intercepted by an angle is proportional to the radius, the definition of radian measure of the angle as the constant of proportionality, the formula for the area of a sector. ● I can create the equation of a circle of the given center and radius based on the definition of a circle. ● I can complete the square in terms of x and y to find the center and radius of a circle. Additional Content Technology Standards Geogebra Key Vocabulary Circle, center chord secant, tangent minor arc, major arc arc length inscribed angle/triangle circumscribed angle/triangle central angle intercepted arc diameter radius semi-circle point of tangency circumference area inscribed polygon locus chord inscribed quadrilateral Standards in gray are emphasized in a different unit Geometry 1-2 PUHSD Curriculum 2014-2015 A. Explain volume formulas and use them to solve problems 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. A. 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 I can explain the formulas for the circumference of a circle and area of a circle by using: -Dissection arguments, separating a shape into two or more shapes. -Informal Limit arguments, find the area of curved shapes using an infinite number of rectangle. grid systems based on ratios).★ Page 17 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit