Geometry PUHSD Curriculum

Transcription

1 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

2 Unit 1 Foundations and Tools for Geometry Unit 2 Introduction to Transformational Geometry Unit 3 Triangle Congruence Geometry 1-2 Learning Outcomes Unit 4 Quadrilaterals Unit 5 Similarity Unit 6 Trigonometry Unit 7 2/3-D Shapes Unit 8 Circles G-CO.A.1 G-CO.A.2 # 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

3 Quarter 1 Quarter 2 Quarter 3 Quarter 4 Introduction to Geometry (19 days) Triangles (25 days) Similarity (25 days) Two and three Dimensional Shapes (19 days) Definitions Prove triangle theorems Dilations Perimeter/area Constructions Prove triangle congruence Triangle and Polygon Similarity -Area of sector Prove Theorems about lines and angles ASA, SSS, SAS Pythagorean Theorem Midpoint Formula and Section Formula -Using Coordinates -Changing parameters Use coordinate formulas in proofs Prove lines/angles theorems Constructions involving Proofs involving similarity -Dissection, Argument -Complex polygons Different types of proofs - triangles Trigonometry Volume two column, flow, paragraph Proofs involving triangles (16 days) -Changing parameters Slope of parallel and Special Right Triangle -Cavalieri s principle perpendicular lines - use with Relationships Density based on area and geometric shapes Right Triangle Relationships volume Proofs (justification of Trigonometry and Inverse Design Problems thinking) Quadrilaterals Trigonometry Proofs - informal (15 days) - sequence of logical Polygons statements Quadrilaterals Circles Proofs (16 days) Introduction to transformational Geometry (18 days) Constructions Similarity Parts of Circles Constructions Transform Proportionality Definitions Equations Multiple Transformations Arguments Rigid Motion Proofs Congruence Constructions Page 3 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

4 Unit 1 Foundations and Tools for Geometry Enduring Understandings: Studying geometry involves learning the basic parts of geometry. Everything is built from points, lines and planes and follows very strict and organized rules. Proofs are a vital component for geometry. Essential Questions: 1. How can I make formal geometric constructions 2. What are the basic parts of any construction or description in geometry? 3. Why are proofs important in developing geometric concepts? 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 G-CO.A.1 Know precise definitions of angle, circle, perpendicular line, angle, circle, perpendicular line, parallel parallel line, and line segment, based on the undefined line, and line segment based on the notions of point, line, distance along a line, and distance idea of point, line, and distance along a around a circular arc. line. I can make the following formal D. Make geometric constructions constructions using a variety of tools: G-CO.D.12 Make formal geometric constructions with a variety of tools copying a segment, copying an angle, and methods (compass and straightedge, string, reflective bisecting a segment, bisecting an angle, devices, paper folding, dynamic geometric software, etc.). constructing perpendicular lines, Copying a segment; copying an angle; bisecting a segment; constructing perpendicular bisectors, bisecting an angle; constructing perpendicular lines, constructing a line parallel to a given including the perpendicular bisector of a line segment; and line through a given point not on the constructing a line parallel to a given line through a point not line. on the line. I can prove the following theorems in narrative paragraphs, flow diagrams, in C. Prove geometric theorems two column format, and or using G-CO.C.9 Prove theorems about lines and angles. Theorems include: diagrams without words: vertical angles are congruent; when a transversal crosses vertical angles are congruent, parallel lines, alternate interior angles are congruent and when a transversal crosses parallel corresponding angles are congruent; points on a lines, and alternate interior angles are perpendicular bisector of a line segment are exactly those congruent and corresponding angles equidistant from the segment s endpoints. are congruent. I can use coordinates to prove the B. Use coordinates to prove simple geometric theorems algebraically simple geometric theorems. G-GPE.B.4 Use coordinates to prove simple geometric theorems I can disprove false statements using algebraically. For example, prove or disprove that a figure the properties of the coordinate plane, Use geometry software to explore different theorems and definitions within geometry Use proofblocks to develop critical thinking Key Vocabulary collinear/linear coplanar/plane point segment, ray, line slope angle segment perpendicular bisector linear pair complementary/supplementary vertical angles parallel/perpendicular/coinciding skew adjacent angles midpoint postulate theorem angle bisector transversal alternate interior angles Page 4 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

5 G-GPE.B.5 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). 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. 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 Page 5 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

6 Unit 2: Introduction to Transformational Geometry Enduring Understandings: Rotations, reflections and translations are examples that preserve angles and distances. These rigid motions can be used to describe congruence. Essential Questions: 1. How does each transformation move various objects? 2. How can I define congruence in terms of rigid motions? 3. What are the similarities and differences between the images and pre-images generated by transformations and/or multiple transformations? 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., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare manipulatives. I can describe a transformation using coordinate notation that maps one point onto a unique image point. to explore transformations and their properties. Key Vocabulary transformations that preserve distance and angle to those I can compare transformations that preserve Transformation, image, that do not (e.g., translation versus horizontal stretch). distance and angle to those that do not. pre-image, composition G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular I can demonstrate the rotations and translation polygon, describe the rotations and reflections that carry it reflections that carry a rectangle, reflection 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, I can make and refine a definition of reflectional symmetry and line segments. rotations, reflections, and translations based on rigid motion G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph the definitions of angles, circles, perpendicular lines, parallel lines, and line segments. congruent carry on to itself paper, tracing paper, or geometry software. Specify a I can demonstrate and draw transformations carry onto another sequence of transformations that will carry a given figure using tools. map onto itself onto another. I can find a sequence of transformations that sequence B. Understand congruence in terms of rigid motions will carry a shape onto another. predict G-CO.B.6 Use geometric descriptions of rigid motions to transform I can investigate rigid motions and generalize vertices figures and to predict the effect of a given rigid motion on a their characteristics as preserving congruency. vectors given figure; given two figures, use the definition of I can decide if two shapes are congruent magnitude 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

7 Unit 3: Triangle Congruence Enduring Understandings: Triangles are fundamental aesthetic, structural elements that are useful in many disciplines such as art, architecture, and engineering. Essential Questions: 1. How do rigid motions lead to an understanding of congruence criteria for triangles? 2. How can proofs help us to develop a deeper and more enduring understanding of triangles? 3. What is true of the points on a perpendicular bisector? 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 motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. rigid motions if and only if the corresponding pairs of sides and corresponding pairs of angles are congruent. I can explain which series of angles and sides are verify theorems about lines, angles, triangles and parallelograms. Key Vocabulary G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. essential in order to show congruence through rigid motions. I can prove the following theorems in narrative congruence Angle-Side-Angle Congruence Theorem C. Prove geometric theorems paragraphs, flow diagrams, in two column format, Side-Angle-Side Congruence G-CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior and/or using diagrams without words: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. Theorem Side-Side-Side Congruence Postulate angles are congruent and corresponding angles I can prove theorems in narrative paragraphs, flow CPCTC (Congruent Parts of 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 diagrams without words. Congruent) from the segment s endpoints. I can make the following formal constructions using perpendicular bisector G-CO.C.10 Prove theorems about triangles. Theorems a variety of tools (compass and straightedge and circumcenter, equidistant include: measures of interior angles of a triangle geometric software): constructing perpendicular Triangle Sum Theorem 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 a variety of tools (compass and straightedge and Converse half the length; the medians of a triangle meet at geometric software): an equilateral triangle inscribed median a point. in a circle. angle bisector D. Make geometric constructions I can solve problems using congruence criteria for in-center G-CO.D.12 Make formal geometric constructions with a triangles. Concurrency of Medians of a variety of tools and methods (compass and I can prove relationships in geometric figures using Triangle Page 7 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

8 G-CO.D.13 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

9 Unit 4: Quadrilaterals Enduring Understandings: Polygons can be classified using properties of sides and angles. Special quadrilaterals are classified based on different properties. Essential Questions: How are quadrilaterals classified according to sides? What is the difference between concave and convex? 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 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). transformations using tools. I can find a sequence of transformations that will carry a shape onto another. I can prove the following theorems in narrative paragraphs, flow diagrams, in two column format, and or using diagrams without words: opposite sides are Key Vocabulary Polygon Convex Concave Regular polygon Triangles Quadrilateral A. Experiment with transformations in the plane congruent, opposite angles are congruent, Parallelograms G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a the diagonals of a parallelogram bisect each other, rectangles are parallelograms with congruent diagonals. Rectangles Trapezoid Rhombus sequence of transformations that will carry a given figure I can make the following formal Kite onto another. constructions using a variety of tools Squares (compass and straightedge and geometric Hexagon C. Prove geometric theorems software): an equilateral triangle, a square, Diagonals G-CO.C.11 Prove theorems about parallelograms. Theorems include: a regular hexagon inscribed in a circle. Opposite sides opposite sides are congruent, opposite angles are I can use coordinates to prove properties Opposite angles congruent, the diagonals of a parallelogram bisect each of quadrilaterals. Bisect other, and conversely, rectangles are parallelograms with I can demonstrate the rotations and Equilateral congruent diagonals. reflections that carry a rectangle, Inscribed D. Make geometric constructions parallelogram, trapezoid, or regular polygon G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. onto itself. Page 9 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

10 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

11 Unit 5: Similarity 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. Essential Questions: How is similarity defined by transformations? How can I prove two figures are similar? How are trigonometric ratios used to solve problems involving triangles? How can similar figures model real world situations? 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., 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). distance and angle to those that do not. I can prove the Midsegment Theorem (the segment joining midpoints of two sides of a triangle is parallel to and half the length of the third side) in narrative paragraphs, flow diagrams, in two column format, and or using diagrams without words I can use the midpoint formula to calculate midpoint verify theorems about lines, angles, triangles and parallelograms Key Vocabulary dilation scale factor similarity C. Prove geometric theorems or endpoint coordinates with various unknowns (e.g. similarity transformations G-CO.C.10 Prove theorems about triangles. Theorems find the other endpoint, etc.) center of dilation include: measures of interior angles of a triangle I can verify the following statements by making mid-segment sum to 180 ; base angles of isosceles triangles are multiple examples; proportional congruent; the segment joining midpoints of two a. A dilation of a line is parallel to the original line if ratio sides of a triangle is parallel to the third side and the center of dilation is not on the line and a dilation reduction half the length; the medians of a triangle meet at of a line is coinciding if the center is on the line. enlargement 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

12 G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT.A.3 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 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. G-SRT.B.5 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. 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. Page 12 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

13 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 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? Standard Learning Targets Technology Standards C. Define trigonometric ratios and solve problems involving right I can: Use geometry software to triangles Discover the relationship between the sides and verify theorems about lines, G-SRT.C.6 Understand that by similarity, side ratios in right angles of a right triangle and be able to state the angles, triangles and triangles are properties of the angles in the triangle, sine, cosine, or tangent of a reference angle given parallelograms leading to definitions of trigonometric ratios for acute a right triangle. Key Vocabulary angles. Be able to find the three basic trig ratios given a triangle triangle. Understand the sine and cosines of triangle trigonometry G-SRT.C.7 Explain and use the relationship between the sine complementary angles are equal. trigonometric ratios: and cosine of complementary angles. Use a trig table. sine Have a basic understanding of how to use trig to cosine solve a real world problem. tangent G-SRT.C.8 Set up a trig equation and solve for a missing inverse trigonometry Use trigonometric ratios and the Pythagorean side. Theorem to solve right triangles in applied problems. 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 triangle and use the pattern to find the missing sides of a triangle. Discover the pattern of a triangle and use the pattern to find the missing sides of a triangle Page 13 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

14 Enduring Understandings: Area, surface area, and volume have many real life applications. Many polygons and polyhedron have common features based on their common characteristics. 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? Standard Learning Targets Technology Standards A. Experiment with transformations in the plane I can explore the effect of altering 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). 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, B. Use coordinates to prove simple geometric theorems algebraically volume of a cylinder, pyramid, and cone by G-GPE.B.7 Use coordinates to compute perimeters of using: polygons and areas of triangles and rectangles, e.g., -Dissection arguments, separating a shape into using the distance formula. two or more shapes. -Cavalieri s principle, if two solids have the same height and the same cross-sectional area A. Explain volume formulas and use them to solve problems at every level, then they have the same G-GMD.A.1 Give an informal argument for the formulas for the volume. circumference of a circle, area of a circle, volume of -Informal Limit arguments, find the area and a cylinder, pyramid, and cone. Use dissection volume of curved shapes using an infinite arguments, Cavalieri s principle, and informal limit number of rectangles and prisms. arguments. I can use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G-GMD.A.3 Use volume formulas for cylinders, pyramids, I can identify the shapes of two-dimensional cross sections of three-dimensional objects. cones, and spheres to solve problems. I can identify three-dimensional objects generated by rotations of two-dimensional 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 unkyard/model.html Wolfram Mathworld 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 Page 14 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

15 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 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 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. G-MG.A.2 G-MG.A.3 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 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

16 Unit 9: Circles Enduring Understandings: Properties of circles can be explained and applied algebraically and geometrically. Essential Questions: How are theorems for circles applied and proven? 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 Technology Standards A. Understand and apply theorems about circles I can prove all circles are similar to each other G-C.A.1 G-C.A.2 Prove that all circles are similar. 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 based on similarity transformations (rigid motion followed by a dilation.) I can identify inscribed angles, radii, and chords. I can describe relationships between segment lengths intersecting inside and outside of the 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 circles of a triangle. I can prove the following properties for Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 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. quadrilateral ABCD inscribed in a circle; (i.e) A + C = B + D =180 o 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. 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 Page 16 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit

17 A. Explain volume formulas and use them to solve problems I can explain the formulas for the 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 grid systems based on ratios). 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. Page 17 of 17 Major Content Supporting Content Additional Content Standards in gray are emphasized in a different unit