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Transcript
Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics HCPS III Code HCPS III Geometry Benchmark Matched Common Core CLUSTER and Standard Match* N.VM.1 (+): Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), |v|, ||v||, v(not bold)). G.1.1 Recognize situations that can be represented by vectors N.VM.3 (+): Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 1 N.VM.2 (+): Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. G.3.1 G.4.1 G.4.2 G.4.3 Use vector addition, subtraction, and scalar multiplication to solve problems N.VM.4 (+): Perform operations on vectors. Add and subtract vectors. 2 N.VM.5 (+): Perform operations on vectors. Multiply a vector by a scalar. Use right triangle trigonometric ratios G.SRT.8: Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. to solve for an unknown length of a side or the measure or an angle Solve problems using the formulas for perimeter, circumference, area, and volume of two- and threedimensional figures and solids G.GMD.3: Explain volume formulas and use them to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* Determine the effect of dimension changes to perimeter, area, and volume for common geometric figures and solids None updated: 12-19-10 Comment Both of the related CC standards are designated as “plus” standards, meaning that these learning expectations go beyond the minimum College and Career Readiness “finish line” for all students. These CC standards will likely not remain in a Geometry course and would likely be placed in a course beyond Algebra II. All three of the related CC standards are designated as “plus” standards, meaning that these learning expectations go beyond the minimum College and Career Readiness “finish line” for all students. These CC standards will likely not remain in a Geometry course and would likely be placed in a course beyond Algebra II. 3 2 This HCPS III benchmark does not map directly to a CC standard for high school mathematics. N/A * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 1 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics HCPS III Code HCPS III Geometry Benchmark Matched Common Core CLUSTER and Standard Match* G.CO.9: Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G.CO.11: Prove geometric theorems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.5.1 G.GMD.1: Explain volume formulas and use them to solve problems. 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. Comment In Principals and Standards for School Mathematics (2000), the National Council of Teachers of Mathematics asserts that “reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions …” (p.342) G.CO.10: Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use inductive and deductive reasoning to create and defend geometric conjectures updated: 12-19-10 2 Both the HCPS III benchmark and the related CC standards concerning reasoning and proof compel instruction that should be purposefully designed to provide learning opportunities that encourage dialogue and questioning, with the aim of enabling all students to “make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs, and select and use various types of reasoning and methods of proofs.” (p. 342) G.GPE.5: Use coordinates to prove simple geometric theorems algebraically. 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). G.SRT.4: Prove theorems involving similarity. 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. * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 2 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics HCPS III Code HCPS III Geometry Benchmark Matched Common Core CLUSTER and Standard Match* G.CO.7: Understand congruence in terms of rigid motions. 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. G.5.2 G.SRT.2: Understand similarity in terms of similarity transformations. 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. Comment The HCPS III benchmark is related to five CC standards concerning the understanding and application of congruent and similar polygons. One significant difference between the HCPS III benchmark and the CC standards is the emphasis on the use of rigid motions in the CC standards. G.CO.8: Understand congruence in terms of rigid motions. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Use the concept of corresponding parts to prove that triangles, and other polygons, are congruent or similar updated: 12-19-10 2 G.SRT.3: Understand similarity in terms of similarity transformations. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.5: Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.5.3 G.5.4 Explain properties and characteristics of angle bisectors, perpendicular bisectors, and parallel lines Use the relationship between pairs of angles (e.g., complementary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties G.CO.9: Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2 G.GPE.5: Use coordinates to prove simple geometric theorems algebraically. 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). G.CO.9: Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 1 The taxonomic levels of the HCPS III benchmark and the CC standard are quite different. Benchmark G.5.4 is an extension of G.5.3. Learning opportunities should include angle measures represented by linear and quadratic expressions so that students can continue to utilize algebraic skills and processes in solving problems in a geometric context. * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 3 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics HCPS III Code G.5.5 HCPS III Geometry Benchmark Apply the concepts of special right triangles to real-world situations Matched Common Core CLUSTER and Standard G.SRT.8: Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Match* G.5.6 G.6.1 Describe three-dimensional figures that are formed by translating twodimensional figures G.7.1 Draw cross-sections, truncations, and compositions/decompositions of three-dimensional objects G.GMD.4: Visualize relationships between two-dimensional and three-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. G.GMD.4: Visualize relationships between two-dimensional and three-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. 2 G.7.2 The taxonomic levels of the HCPS III benchmark and the CC standard are quite different. Learning opportunities should include segment and angle measures represented by linear and quadratic expressions so that students can continue to utilize algebraic skills and processes in solving problems in a geometric context. 2 HCPS III benchmarks G.6.1 and G.7.1 considered together make an excellent match with CC standard G.GMD.4. 2 HCPS III benchmarks G.6.1 and G.7.1 considered together make an excellent match with CC standard G.GMD.4. G.MG.1: Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* Use concrete objects, pictorial representations, computer software, or graphing calculators to solve geometric problems Comment 3 G.C.2: Understand and apply theorems about circles. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use the relationships among properties of circles (e.g., chords, secants, tangents, arcs, G.C.5: Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length circumference, radius, diameter, of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the inscribed polygons) to solve problems angle as the constant of proportionality; derive the formula for the area of a sector. updated: 12-19-10 2 In the CCSS, modeling is both a Standard for Mathematical Practice and a Conceptual Category for High School Mathematics Standards. For information regarding expectations for modeling, refer to page 7 and page 72 in the official CCSS for Mathematics document (a PDF of the document may be downloaded at www.corestandards.org/the-standards). * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 4 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics HCPS III Code HCPS III Geometry Benchmark Matched Common Core CLUSTER and Standard Match* updated: 12-19-10 Comment G.GPE.1: Translate between the geometric description and the equation for a conic section. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.8.1 Use coordinate geometry to produce formulas and prove theorems for the midpoint of a line segment, the distance formula, and forms of equations of lines and circles G.GPE.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, (sqrt3)) lies on the circle centered at the origin and containing the point (0, 2). 2 G.GPE.7: Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* G.CO.2: Experiment with transformations in the plane. 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). G.8.2 Describe the concept of rigid motion on figures in the coordinate plane, including rotation, translation, and reflection G.CO.3: Experiment with transformations in the plane. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 2 G.CO.5: Experiment with transformations in the plane. 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 sequence of transformations that will carry a given figure onto another. * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 5 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics updated: 12-19-10 Below are other Common Core State Standards for grades 9-12 that could be included in a high school geometry course but do not align directly to an HCPS III benchmark. CCSS Code G.CO.1 Common Core CLUSTER and Standard Experiment with transformations in the plane. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.4 Experiment with transformations in the plane. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.6 G.CO.12 Understand congruence in terms of rigid motions. 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. Make geometric constructions. 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.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.GPE.6 Use coordinates to prove simple geometric theorems algebraically. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.SRT.1 G.SRT.6 G.C.1 G.C.3 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Define trigonometric ratios and solve problems involving right triangles. 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. Understand and apply theorems about circles. Prove that all circles are similar. Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT page 6 of 6 Crosswalk between the Geometry Hawaii Content and Performance Standards (HCPS) III and the Common Core State Standards (CCSS) for High School Mathematics * Degree of Match: 1 = WEAK (major aspect of the HCPS III benchmark not addressed in CCSS); 2 = GOOD (minor aspect of the HCPS III benchmark not addressed in CCSS); 3 = EXCELLENT updated: 12-19-10 page 7 of 6