Download Geometry -- HCPS3-CCSS Crosswalk (12-19-10)

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