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Geometry 1-2 Honors 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 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 Geometry 1-2 Honors 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.A.4 + 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-C.B.5 G-CO.D.13 # G-MG.A.1# G-SRT.A.3 G-MG.A.1 # G-GPE.A.1 G-SRT.B.5 # G-SRT.B.4 G-MG.A.2 G-GMD.A.1 # 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 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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 Page 3 of 18 Quarter 2 Triangles (25 dayss) 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 Major Content Supporting Content 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 Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 Unit 1 Foundations and Tools for Geometry Enduring Understandings: Essential Questions: Studying geometry involves learning the basic parts of 1. How can I make formal geometric constructions geometry. Everything is built from points, lines and planes and 2. What are the basic parts of any construction or description in geometry? follows very strict and organized rules. Proofs are a vital 3. Why are proofs important in developing geometric concepts? component for 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, explore different theorems and parallel line, and line segment, based on the undefined parallel line, and line segment based definitions within geometry notions of point, line, distance along a line, and distance on the idea of point, line, and distance Use proofblocks to develop around a circular arc. along a line. critical thinking with proofs Key I can make the following formal Vocabulary constructions using a variety of tools: collinear/Linear C. Prove geometric theorems copying a segment, copying an angle, coplanar/plane G-CO.C.9 Prove theorems about lines and angles. Theorems include: bisecting a segment, bisecting an point vertical angles are congruent; when a transversal crosses angle, constructing perpendicular segment, ray, line parallel lines, alternate interior angles are congruent and lines, constructing perpendicular slope corresponding angles are congruent; points on a bisectors, constructing a line parallel to angle perpendicular bisector of a line segment are exactly those a given line through a given point not segment equidistant from the segment’s endpoints. on the line. perpendicular bisector I can prove the following theorems in linear pair narrative paragraphs, flow diagrams, in complementary/supplementary D. Make geometric constructions vertical angles two column format, and or using G-CO.D.12 Make formal geometric constructions with a variety of tools diagrams without words: parallel/perpendicular/coinciding and methods (compass and straightedge, string, reflective skew vertical angles are congruent, devices, paper folding, dynamic geometric software, etc.). adjacent angles when a transversal crosses parallel Copying a segment; copying an angle; bisecting a segment; lines, and alternate interior angles are midpoint bisecting an angle; constructing perpendicular lines, postulate congruent and corresponding angles including the perpendicular bisector of a line segment; and are congruent. theorem constructing a line parallel to a given line through a point angle bisector I can use coordinates to prove the not on the line. transversal simple geometric theorems. I can disprove false statements using alternate interior angles Page 4 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 B. Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.B.5 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 18 Major Content Supporting Content the properties of the coordinate plane, 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 Honors 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 to G-CO.A.2 Represent transformations in the plane using, e.g., manipulatives. explore transformations transparencies and geometry software; describe and their properties. I can describe a transformation using transformations as functions that take points in the coordinate notation that maps one point onto a Key Vocabulary plane as inputs and give other points as outputs. Transformation, image, unique image point. Compare transformations that preserve distance and pre-image, composition I can compare transformations that preserve angle to those that do not (e.g., translation versus translation distance and angle to those that do not. horizontal stretch). reflection I can demonstrate the rotations and reflections that carry a rectangle, parallelogram, rotation G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular rotational symmetry trapezoid, or regular polygon on to itself. polygon, describe the rotations and reflections that I can make and refine a definition of rotations, reflectional symmetry carry it onto itself. rigid motion reflections, and translations based on the definitions of angles, circles, perpendicular lines, congruent G-CO.A.4 Develop definitions of rotations, reflections, and carry on to itself parallel lines, and line segments. translations in terms of angles, circles, perpendicular carry onto another I can demonstrate and draw transformations lines, parallel lines, and line segments. map onto itself using tools. sequence I can find a sequence of transformations that predict G-CO.A.5 Given a geometric figure and a rotation, reflection, or will carry a shape onto another. vertices translation, draw the transformed figure using, e.g., I can investigate rigid motions and generalize vectors graph paper, tracing paper, or geometry software. their characteristics as preserving congruency. magnitude Specify a sequence of transformations that will carry a I can decide if two shapes are congruent given figure onto another. because of the rigid motions between the two figures. Page 6 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 B. Understand congruence in terms of rigid motions G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Page 7 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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 2. How can proofs help us to develop a deeper and more enduring understanding of triangles? many disciplines such as art, architecture, 3. What is true of the points on a perpendicular bisector? and 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 Use geometry software to G-CO.B.7 Use the definition of congruence in terms of rigid through rigid motions if and only if the verify theorems about motions to show that two triangles are congruent if corresponding pairs of sides and corresponding lines, angles, triangles and and only if corresponding pairs of sides and pairs of angles are congruent. parallelograms. corresponding pairs of angles are congruent. I can explain which series of angles and sides are Key Vocabulary congruence essential in order to show congruence through G-CO.B.8 Explain how the criteria for triangle congruence (ASA, rigid motions. Angle-Side-Angle SAS, and SSS) follow from the definition of Congruence I can prove the following theorems in narrative congruence in terms of rigid motions. paragraphs, flow diagrams, in two column format, Theorem and/or using diagrams without words: points on a Side-Angle-Side C. Prove geometric theorems Congruence Theorem perpendicular bisector of a line segment are G-CO.C.9 Prove theorems about lines and angles. Theorems Side-Side-Side Congruence exactly those equidistant from the segment’s include: vertical angles are congruent; when a Postulate endpoints. transversal crosses parallel lines, alternate interior CPCTC (Congruent Parts of I can prove theorems in narrative paragraphs, angles are congruent and corresponding angles are flow diagrams, in two column format, and or using Congruent Triangles are congruent; points on a perpendicular bisector of a Congruent) diagrams without words. line segment are exactly those equidistant from the perpendicular bisector I can make the following formal constructions segment’s endpoints. using a variety of tools (compass and straightedge circumcenter, equidistant Triangle Sum Theorem and geometric software): constructing interior angles G-CO.C.10 Prove theorems about triangles. Theorems include: perpendicular bisectors. Base Angles Theorem and measures of interior angles of a triangle sum to 180°; I can make the following formal constructions base angles of isosceles triangles are congruent; the using a variety of tools (compass and straightedge its Converse median segment joining midpoints of two sides of a triangle is and geometric software): an equilateral triangle angle bisector parallel to the third side and half the length; the inscribed in a circle. in-center medians of a triangle meet at a point. I can solve problems using congruence criteria Concurrency of Medians of Page 8 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 D. Make geometric constructions G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). 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. G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. for triangles. I can prove relationships in geometric figures using 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. 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 9 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 Unit 4: Quadrilaterals Enduring Understandings: Essential Questions: Polygons can be classified using properties of sides and angles. 1. How are quadrilaterals classified according to sides? Special quadrilaterals are classified based on different 2. What is the difference between concave and convex? properties. 3. What are the properties of quadrilaterals? 4. How are quadrilateral classified? 5. How do you inscribe a triangle, square or regular hexagon into a circle? 6. 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.3 Given a rectangle, parallelogram, trapezoid, or regular the diagonals of a parallelogram bisect each Rectangles polygon, describe the rotations and reflections that carry it other, rectangles are parallelograms with Trapezoid onto itself. Rhombus congruent diagonals. Kite I can make the following formal Squares G-CO.A.5 Given a geometric figure and a rotation, reflection, or constructions using a variety of tools Hexagon translation, draw the transformed figure using, e.g., graph (compass and straightedge and geometric Diagonals paper, tracing paper, or geometry software. Specify a software): an equilateral triangle, a square, Opposite sides sequence of transformations that will carry a given figure a regular hexagon inscribed in a circle. Opposite angles onto another. I can use coordinates to prove properties Bisect C. Prove geometric theorems of quadrilaterals. Equilateral G-CO.C.11 Prove theorems about parallelograms. Theorems include: I can demonstrate the rotations and Inscribed opposite sides are congruent, opposite angles are reflections that carry a rectangle, congruent, the diagonals of a parallelogram bisect each parallelogram, trapezoid, or regular polygon other, and conversely, rectangles are parallelograms with onto itself. congruent diagonals. Page 10 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 D. Make geometric constructions G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 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 11 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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. Unit 5: Similarity Essential Questions: 1. How is similarity defined by transformations? 2. How can I prove two figures are similar? 3. How are trigonometric ratios used to solve problems involving triangles? 4. How can similar figures model real world situations? 5. How are trigonometric ratios used to solve problems involving triangles? 6. What is the relationship between similar right triangles and trigonometric ratios? Standard Learning Targets I can compare transformations that preserve G-CO.A.2 Represent transformations in the plane using, e.g., distance and angle to those that do not. transparencies and geometry software; describe I can prove the Midsegment Theorem (the transformations as functions that take points in the segment joining midpoints of two sides of a plane as inputs and give other points as outputs. triangle is parallel to and half the length of the Compare transformations that preserve distance and third side) in narrative paragraphs, flow angle to those that do not (e.g., translation versus diagrams, in two column format, and or using horizontal stretch). diagrams without words C. Prove geometric theorems I can use the midpoint formula to calculate G-CO.C.10 Prove theorems about triangles. Theorems include: midpoint or endpoint coordinates with various measures of interior angles of a triangle sum to 180°; unknowns (e.g. find the other endpoint, etc.) base angles of isosceles triangles are congruent; the I can verify the following statements by segment joining midpoints of two sides of a triangle is making multiple examples; parallel to the third side and half the length; the medians a. A dilation of a line is parallel to the original of a triangle meet at a point. line if the center of dilation is not on the line A. Understand similarity in terms of similarity transformations and a dilation of a line is coinciding if the center G-SRT.A.1a Verify experimentally the properties of dilations given by is on the line. G-SRT.A.1b a center and a scale factor: b. The dilation of a line segment changes the a. A dilation takes a line not passing through the center length by a ratio given by the scale factor. of the dilation to a parallel line, and leaves a line passing I can extend the properties of dilations to through the center unchanged. polygons. b. The dilation of a line segment is longer or shorter in I can decide if two figures are similar based on the ratio given by the scale factor. A. Experiment with transformations in the plane Page 12 of 18 Major Content Supporting Content Additional Content Technology Standards Use geometry software to verify theorems about lines, angles, triangles and parallelograms Key Vocabulary dilation scale factor similarity similarity transformations center of dilation mid-segment proportional ratio reduction enlargement cofunction Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Page 13 of 18 Major Content Supporting Content similarity transformations (rigid motion followed by a dilation.) I can use similarity transformations to explain the meaning of similar triangles as the equality of all 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. Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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: 1. How are trigonometric ratios used to solve problems involving triangles? 2. How are trigonometric ratios used to solve problems involving triangles? 3. 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 14 of 18 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 Honors PUHSD Curriculum 2014-2015 Unit 7: 2- and 3-dimensional shapes Essential Questions: 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 1. 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 15 of 18 Major Content What is the relationship of the different measures in two and three dimensional objects? 2. How does a change in one dimension of an object affect the other dimensions? 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/~eppstei n/junkyard/model.html Wolfram Mathworld http://mathworld.wolfram.com/t opics/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, Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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. A. Apply geometric concepts in modeling situations G-MG.A.1 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. cylinder 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 16 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors PUHSD Curriculum 2014-2015 Unit 8: Circles Enduring Understandings: Essential Questions: Properties of circles can be explained and applied 1. How are theorems for circles applied and proven? algebraically and geometrically. 2. How are geometric properties of circles embedded in equations? 3. How is proportion used in arc and sector measurements? 4. How are real world situations modeled with circles? Standard Learning Targets Technology Standards A. Understand and apply theorems about circles I can prove all circles are similar to each Geogebra G-C.A.1 Prove that all circles are similar. other based on similarity transformations Key Vocabulary (rigid motion followed by a dilation.) Circle, center G-C.A.2 Identify and describe relationships among inscribed I can identify inscribed angles, radii, and chord angles, radii, and chords. Include the relationship secant, tangent chords. between central, inscribed, and circumscribed minor arc, major arc I can describe relationships between angles; inscribed angles on a diameter are right arc length segment lengths intersecting inside and angles; the radius of a circle is perpendicular to the outside of the circle. inscribed angle/triangle tangent where the radius intersects the circle. circumscribed angle/triangle I can describe relationships between angles central angle G-C.A.3 Construct the inscribed and circumscribed circles of formed inside and outside of the circle. intercepted arc a triangle, and prove properties of angles for a I can construct inscribed and circumscribed diameter quadrilateral inscribed in a circle. circles of a triangle. radius G-C.A.4 (+) Construct a tangent line from a point outside a I can prove the following properties for semi-circle given circle to the circle. quadrilateral ABCD inscribed in a circle; (i.e) point of tangency B. Find arc lengths and areas of sectors of circles ∠A + ∠C =∠B +∠D =180o circumference G-C.B.5 Derive using similarity the fact that the length of I can construct a tangent line from a point area the arc intercepted by an angle is proportional to outside a given circle to the circle. inscribed polygon the radius, and define the radian measure of the I can use similarity to logically arrive at the angle as the constant of proportionality; derive the following; the length of the arc intercepted by locus chord formula for the area of a sector. an angle is proportional to the radius, the inscribed quadrilateral A. Translate between the geometric description and the equation for a definition of radian measure of the angle as conic section the constant of proportionality, the formula G-GPE.A.1 Derive the equation of a circle of given center and for the area of a sector. radius using the Pythagorean Theorem; complete ● I can create the equation of a circle of the the square to find the center and radius of a circle given center and radius based on the given by an equation. definition of a circle. Page 17 of 18 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit Geometry 1-2 Honors 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. Page 18 of 18 Major Content ● I can complete the square in terms of x and y to find the center and radius of a circle. 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. Supporting Content Additional Content Standards in gray are emphasized in a different unit