Download Honors Geometry - Dublin City Schools

Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
Course Description: Honors Geometry follows the same course of study as Geometry
however this course will have a quickened pace that allows for the mathematical concepts
to be explored with greater depth including a heightened level of critical thinking. This
course integrates the concepts of plane, solid, and coordinate geometry. The course
investigates the properties of plane geometric figures. Using inductive and deductive
methods of reasoning, students present proof of angle and line relationships. A scientific
calculator is required.
I. Content Standard: Number, Number Sense and Operations Standard
Students demonstrate number sense, including an understanding of number systems
and reasonable estimates using paper and pencil, technology-supported and mental
methods.
Benchmarks (Grade 8-10 Band)
Geometry Indicators
A. Connect physical, verbal and
symbolic representations of
integers, rational numbers and
irrational numbers.
1. Connect physical, verbal and symbolic
representation of irrational numbers;
B. Estimate, compute and solve
problems involving scientific
notation, square roots and
numbers with integer exponents.
1.
e.g., construct 2 as a hypotenuse
or on a number line.
Estimate the solutions for problem
situations involving square and cube
roots.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
II. Content Standard: Data Analysis and Probability Standard
Students pose questions and collect, organize, represent, interpret and analyze data to
answer those questions. Students develop and evaluate inferences, predictions and
arguments that are based on data.
Benchmarks (Grade 8-10 Band)
A. Compute probabilities of
compound events, independent
events, and simple dependent
events.
Geometry Indicators
1. Model problems dealing with
uncertainty with area models
(geometric probability).
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
III. Content Standard: Patterns, Functions and Algebra Standard
Students use patterns, relations and functions to model, represent and analyze
problem situations that involve variable quantities. Students analyze, model and solve
problems using various representations such as tables, graphs and equations.
Benchmarks (Grade 8-10 Band)
A. Use algebraic representations,
such as tables, graphs,
expressions, functions and
inequalities, to model and solve
problem situations.
Geometry Indicators
1. Solve equations and formulas for a
specified variable; e.g., express the
base of a triangle in terms of the area
and height.
2. Use algebraic representations and
functions to describe and generalize
geometric properties and
relationships.
3. Dublin Indicator: Describe and apply
inequalities in the context of angles
and sides in triangles.
B.
Solve and graph linear equations
and inequalities.
1. Find linear equations that represent
lines that pass through a given set of
ordered pairs, and find linear
equations that represent lines parallel
or perpendicular to a given line
through a specific point.
C. Solve quadratic equations with
real roots by graphing, formula
and factoring.
1. Graph the quadratic relationship that
defines circles.
D. Describe and interpret rates of
change from graphical and
numerical data.
1. Compute and interpret slope,
midpoint and distance given a set of
ordered pairs.
2. Recognize and explain that the slopes
of parallel lines are equal and the
slopes of perpendicular lines are
negative reciprocals.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
IV. Content Standard: Measurement Standard
Students estimate and measure to a required degree of accuracy and precision by
selecting and using appropriate units, tools and technologies.
Benchmarks
Indicators
A. Use proportional reasoning and
apply indirect measurement
techniques, including right
triangle trigonometry and
properties of similar triangles, to
solve problems involving
measurements and rates.
1. Use unit analysis to check
computations involving
measurement.
2. Use the ratio of lengths in similar
two-dimensional figures or threedimensional objects to calculate the
ratio of their areas or volumes
respectively.
3. Use scale drawings and right triangle
trigonometry to solve problems that
include unknown distances and angle
measures.
4. Solve problems involving unit
conversion for situations involving
distances, areas, volumes and rates
within the same measurement
system.
5. Determine the measures of central
and inscribed angles and their
associated major and minor arcs.
6. Convert rates within the same
measurement system; e.g., miles per
hour to feet per second; kilometers
per hour to meters per second.
B. Estimate and compute areas and
volume in increasingly complex
problem situations..
1. Derive a formula for the surface area
of a cone as a function of its slant
height and the circumference of its
base.
2. Calculate distances, areas, surface
areas and volumes of composite
three-dimensional objects to a
specified number of significant digits.
3. Dublin Indicator: Determine the area
of the sector of a circle.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
V. Content Standard: Geometry and Spatial Sense Standard
Students identify, classify, compare and analyze characteristics, properties and
relationships of one-, two- and three-dimensional geometric figures and objects.
Students use spatial reasoning, properties of geometric objects, and transformations to
analyze mathematical situations and solve problems.
Benchmarks (Grade 8-10 Band)
A. Formally define geometric
figures.
Geometry Indicators
1. Formally define and explain key
aspects of geometric figures,
including:
a.
interior and exterior angles Of
polygons;
b.
segments related to triangles
(median, altitude, midsegment);
c.
points of concurrency related to
triangles (centroid, incenter,
orthocenter, and circumcenter);
d. circles (radius, diameter, chord,
circumference, major arc, minor
arc, sector, segment, inscribed
angle).
2. Recognize and explain the necessity
for certain terms to remain undefined,
such as point, line and plane.
3. Identify the reflection and rotation
symmetries of two- and threedimensional figures.
4. Solve problems involving chords, radii,
and arcs within the same circle.
5. Dublin Indicator: Define and apply the
concept of bisect to shapes.
6. Dublin Indicator: Identify and describe
collinear points and coplanar points,
lines and segments.
7. Dublin Indicator: Describe, draw and
identify polygons and their properties.
8. Dublin Indicator: Classify triangles
based on their side and angle
measures.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
B. Describe and apply the
properties of similar and
congruent figures; and justify
conjectures involving similarity
and congruence.
1. Make and test conjectures about
characteristics and properties (e.g.,
sides, angles, symmetry) of twodimensional figures and threedimensional objects.
2. Use proportions in several forms to
solve problems involving similar
figures (part-to-part, part-to-whole,
corresponding sides between figures).
3. Dublin Indicator: Understand and
apply triangle congruence postulates
and theorems.
C.
Recognize and apply angle
relationships in situations
involving intersecting lines,
perpendicular lines, and parallel
lines.
1. Recognize the angles formed and the
relationship between the angles when
two lines intersect and when parallel
lines are cut by a transversal.
2. Solve problems involving chords, radii,
and arcs within the same circle.
3. Dublin Indicator: Define, describe,
and apply operations involving angles.
D. Use coordinate geometry to
represent and examine the
properties of geometric figures.
1. Make and test conjectures about
characteristics and properties (e.g.,
sides, angles, symmetry) of twodimensional figures and threedimensional objects.
2. Represent and analyze shapes using
coordinate geometry; e.g., given three
vertices and the type of quadrilateral,
find the coordinates of the fourth
vertex.
E.
Draw and construct
representations of two- and
three-dimensional geometric
objects
1. Draw nets for a variety of prisms,
pyramids, cylinders and cones.
2. Construct right triangles, equilateral
triangles, parallelograms, trapezoids,
rectangles, rhombuses, squares and
kites.
3. Construct congruent or similar figures
using tools.
4. Perform reflections and rotations.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
F.
Represent and model
transformations in a coordinate
plane and describe the results.
1. Draw the results of translations,
reflections, rotations and dilations of
objects in the coordinate plane, and
determine properties that remain
fixed; e.g., lengths of sides remain the
same under translations.
2. Derive coordinate rules for
translations, reflections and rotations
of geometric figures in the coordinate
plane.
3. Show and describe the results of
combinations of translations,
reflections and rotations
(compositions); e.g., perform
compositions and specify the result of
a composition as the outcome of a
single motion, when applicable.
G. Prove or disprove conjectures
and solve problems involving
two- and three-dimensional
objects represented within a
coordinate system.
1. Analyze two-dimensional figures in a
coordinate plane; e.g., use slope and
distance formulas to show that a
quadrilateral is a parallelogram.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
H. Establish the validity of
conjectures about geometric
objects, their properties and
relationships by counterexample, inductive and
deductive reasoning, and
critiquing arguments made by
others.
1. Make and test conjectures about
characteristics and properties (e.g.,
sides, angles, symmetry) of twodimensional figures and threedimensional objects.
2. Make, test and establish the validity of
conjectures about geometric
properties and relationships using
counterexample, inductive and
deductive reasoning, and paragraph or
two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle
similarity and congruence;
c. prove theorems involving
properties of lines, angles,
triangles and quadrilaterals;
d. test a conjecture using basic
constructions made with a compass
and straightedge or technology.
3. Solve problems involving chords, radii,
and arcs within the same circle.
4. Dublin Indicator: Identify and describe
conditional statements and determine
the if-then, converse, inverse, and
contrapositive of conditional
statements.
I. Use right triangle trigonometric
relationships to determine
lengths and angle measures.
1. Define the basic trigonometric ratios in
right triangles: sine, cosine and
tangent.
2. Apply proportions and right triangle
trigonometric ratios to solve problems
involving missing lengths and angle
sizes in similar figures.
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Dublin City Schools
February 2008
Dublin City Schools Mathematics Graded Course of Study
HONORS GEOMETRY
VI. Content Standard: Mathematical Processes
The benchmarks for mathematical processes articulate what students should
demonstrate in problem solving, representation, communication, reasoning and
connections at key points in their mathematics program. Specific grade-level
indicators have not been included for the mathematical processes standard because
content and processes should be interconnected at the indicator level. Therefore,
mathematical processes have been embedded within the grade-level indicators for the
five content standards.
By the end of the 8-10 program:
A. Formulate a problem or mathematical model in response to a specific need or
situation, determine information required to solve the problem, choose method for
obtaining this information, and set limits for acceptable solution.
B. Apply mathematical knowledge and skills routinely in other content areas and
practical situations.
C. Recognize and use connections between equivalent representations and related
procedures for a mathematical concept; e.g., zero of a function and the x-intercept of
the graph of the function, apply proportional thinking when measuring, describing
functions, and comparing probabilities.
D. Apply reasoning processes and skills to construct logical verifications or counterexamples to test conjectures and to justify and defend algorithms and solutions.
E. Use a variety of mathematical representations flexibly and appropriately to organize,
record and communicate mathematical ideas.
F. Use precise mathematical language and notations to represent problem situations and
mathematical ideas.
G. Write clearly and coherently about mathematical thinking and ideas.
H. Locate and interpret mathematical information accurately, and communicate ideas,
processes and solutions in a complete and easily understood manner.
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Dublin City Schools
February 2008
Mathematics Targets
Geometry
Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and
transformations are used to analyze mathematical situations and solve problems.
Essential Learning:
Analysis and applications of two- and three-dimensional figures
Target 1
I can…
Problems and statements form, prove and disprove (using counterexamples) conjectures.
involving logic and
geometric figures can be use inductive and deductive reasoning to draw valid conclusions.
analyzed and solved.
identify and describe conditional statements and determine the if-then,
converse, inverse and contrapositive of conditional statements.
describe, compare/contrast and use geometric postulates, theorems, definitions
and corollaries to form logical arguments (proofs).
estimate solutions for problem situations involving square and cube roots.
use algebraic representations and functions to describe and generalize
geometric properties and relationships.
solve equations and formulas for specified variables.
model problems dealing with geometric probability.
Target 2
I can …
Basic geometric figures
and their properties can
be identified,
represented, analyzed,
and applied.
use and apply the concepts of points, lines, planes, segments, rays, and angles.
identify and describe collinear points and coplanar points, lines, and segments.
draw, identify, and apply properties of perpendicular and parallel lines and
segments.
describe, identify, draw and utilize properties of adjacent, vertical,
complementary, supplementary, linear, corresponding, alternate interior,
alternate exterior, and consecutive interior pairs of angles.
Dublin City Schools
August 2007
Mathematics Targets
Geometry
Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations
are used to analyze mathematical situations and solve problems.
Essential Learning:
Analysis and applications of two- and three-dimensional figures
Target 3
I can …
Polygons and their properties
can be identified, represented,
analyzed, and applied.
describe, draw and identify polygons and their properties.
classify triangles based on their side and angle measures.
generalize and apply relationships of interior and exterior angles of
polygons, including regular polygons.
understand and apply triangle congruence postulates and theorems.
describe, identify, draw, and apply properties of medians, altitudes, angle
bisectors and perpendicular bisectors in triangles.
define and explain points of concurrency related to triangles.
describe and apply inequalities in the context of angles and sides in
triangles.
identify, draw, and apply properties of the following parallelograms,
rectangles, rhombi, squares and trapezoids.
Dublin City Schools
August 2007
Mathematics Targets
Geometry
Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations
are used to analyze mathematical situations and solve problems.
Essential Learning:
Analysis and applications of two- and three-dimensional figures
Target 4
I can…
Circles and their properties can
be identified, represented,
analyzed, and applied.
define, draw, and apply the properties of tangents, secants, radii, diameters
and chords.
determine the measures of central and inscribed angles and their associated
major and minor arcs.
determine arc length.
determine the area of the sector of a circle.
Target 5
I can…
Three-dimensional figures and
their properties can be
identified, represented analyzed
and applied.
identify, name, and draw representations of prisms, cylinders, cones,
pyramids and spheres.
draw nets for three-dimensional figures.
Dublin City Schools
August 2007
Mathematics Targets
Geometry
Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and
transformations are used to analyze mathematical situations and solve problems.
Essential Learning:
Relating geometric properties and figures to the coordinate plane
Target 1
I can…
Properties and
measurements of lines
and line segments can
be determined
find the length of a segment using the distance formula.
find the length of a segment using the Pythagorean Theorem.
define, describe and apply operations using a midpoint of a segment on the
coordinate plane.
apply the concept of slope to parallel and perpendicular lines
Target 2
I can …
Figures on the
coordinate plane can be
classified and analyzed.
classify shapes when given the coordinates of its vertices.
apply concepts of slope and distance formula to analyze shapes on the
coordinate plane.
graph a circle and write its equation.
Target 3
I can…
Geometric figures can
be modified through
transformations.
define, describe, and perform rotations, translations, dilations, and reflections..
explain transformation effects on coordinates and shapes (including
combinations of transformations).
Dublin City Schools
August 2007
Mathematics Targets
Geometry
Big Idea: Measurement involves using appropriate units, tools, and technology to a defined
degree of estimation or accuracy.
Essential Learning:
Measurement of geometric figures
Dublin City Schools
August 2007
Target 1
I can…
Measures of angles and
segments can be
represented, analyzed,
and applied.
define, describe and apply operations involving angles.
Target 2
I can …
Measures of twodimensional figures can
be represented, analyzed
and applied.
apply the concept of perimeter and area to polygons, including regular, and
circles.
describe and apply the idea of congruence to segments and angles.
describe and apply the idea of congruence to shapes.
define and apply the concept of bisect to shapes.
state, understand and apply the sine, cosine, and tangent ratios in right
triangles.
identify, draw and apply relationships of 30-60 and 45-45 right triangles.
use the concepts of ratio and proportion to solve problems, including converting
units between systems and checking computations with unit analysis.
identify and solve problems involving similar figures, including working with
perimeter, area, volume and corresponding parts within similar figures.
Target 3
I can…
Measures of threedimensional figures can
be represented and
analyzed.
determine volume of prisms, cylinders, cones, pyramids, and spheres.
determine surface area of prisms, cylinders, cones, pyramids, and spheres
Dublin City Schools
August 2007