Download Geometry 9 - SH - Willmar Public Schools

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Subject Area
Mathematics—Senior High
Course Name
Geometry-9(Prentice Hall Mathematics)
Date
April 20, 2010
Geometry 9 and Geometry parallel each other in content and time. Geometry 9 students are assigned more advanced problems for homework and for formal assessments.
Week
1-2
Content
Tools of Geometry
(Chapter 1)
Standards Addressed
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Solve real-world and mathematical
geometric problems using algebraic
methods.
Skills/Benchmarks
Essential Questions
9.2.2.4(1.1) Express the terms in a
geometric sequence recursively and
by giving an explicit (closed form)
formula, and express the partial
sums of a geometric series
recursively.
How does having a common language,
of basic geometric vocabulary, assist in
further discussions in class and in life?
9.3.2.1 (1.3) Understand the roles
of axioms, definitions, undefined
terms and theorems in logical
arguments.
9.3.2.3 (1.1) Assess the validity of
a logical argument and give
counterexamples to disprove a
statement.
9.3.2.5 (1.1-1.7) Use technology
tools to examine theorems, make
and test conjectures, perform
constructions and develop
mathematical reasoning skills in
multi-step problems. The tools may
include compass and straight edge,
dynamic geometry software, design
software or Internet applets.
9.3.3.1 (1.7) Know and apply
properties of parallel and
perpendicular lines, including
properties of angles formed by a
transversal, to solve problems and
logically justify results.
9.3.3.2 (1.6) Know and apply
properties of angles, including
corresponding, exterior, interior,
vertical, complementary and
supplementary angles, to solve
problems and logically justify
results.
9.3.4.4 (1.5, 1.8) Use coordinate
What are the tools of geometry?
Assessments
Tests and quizzes including
performance assessment questions.
In-class “lab” discovery assignments
using constructions, geo-sketch pad,
and or graphing calculators to reveal
geometric concepts.
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3
Reasoning and Proof
(Chapter 2)
Generate equivalent algebraic
expressions involving polynomials
and radicals; use algebraic properties
to evaluate expressions.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
geometry to represent and analyze
line segments and polygons,
including determining lengths,
midpoints and slopes of line
segments.
9.2.3.7 Justify steps in generating
equivalent expressions by
identifying the properties used. Use
substitution to check the equality of
expressions for some particular
values of the variables; recognize
that checking with substitution does
not guarantee equality of
expressions for all values of the
variables.
9.3.2.2 (2.1 - 2.2) Accurately
interpret and use words and phrases
such as "if…then," "if and only if,"
"all," and "not." Recognize the
logical relationships between an
"if…then" statement and its
inverse, converse and
contrapositive.
9.3.2.3 (2.1) Assess the validity of
a logical argument and give
counterexamples to disprove a
statement.
9.3.2.4 (2.5) Construct logical
arguments and write proofs of
theorems and other results in
geometry, including proofs by
contradiction. Express proofs in a
form that clearly justifies the
reasoning, such as two-column
proofs, paragraph proofs, flow
charts or illustrations.
9.3.3.2 (2.5) Know and apply
properties of angles, including
corresponding, exterior, interior,
vertical, complementary and
supplementary angles, to solve
problems and logically justify
results.
How do we use patterns to study
geometry?
Tests and quizzes including
performance assessment questions.
How are statements known as
conditional, biconditional, and various
definitions used for geometric proof?
Do group proof. Use cooperative
learning to first understand proofs/
Partner proofs, then individual
proofs.
How can doing proofs help with
making other logical connections
throughout geometry and life?
Patty paper investigations to
discover geometric concepts.
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3-4
Parallel and
Perpendicular lines
(Chapter 3)
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.2.1.6 (3.6) Identify intercepts,
zeros, maxima, minima and
intervals of increase and decrease
from the graph of a function.
9.2.2.3 (3.6) Sketch graphs of
linear, quadratic and exponential
functions, and translate between
graphs, tables and symbolic
representations. Know how to use
graphing technology to graph these
functions.
9.3.2.4 (3.1) Construct logical
arguments and write proofs of
theorems and other results in
geometry, including proofs by
contradiction. Express proofs in a
form that clearly justifies the
reasoning, such as two-column
proofs, paragraph proofs, flow
charts or illustrations.
9.3.2.5 (p. 126) Use technology
tools to examine theorems, make
and test conjectures, perform
constructions and develop
mathematical reasoning skills in
multi-step problems. The tools may
include compass and straight edge,
dynamic geometry software, design
software or Internet applets.
9.3.3.1 (3.1) Know and apply
properties of parallel and
perpendicular lines, including
properties of angles formed by a
transversal, to solve problems and
logically justify results.
9.3.3.2 (3.1,3.2) Know and apply
properties of angles, including
corresponding, exterior, interior,
vertical, complementary and
supplementary angles, to solve
problems and logically justify
results.
9.3.3.3 (3.4) Know and apply
properties of equilateral, isosceles
and scalene triangles to solve
What are the relationships of parallel
and perpendicular lines and how can on
apply these concepts to everyday life?
How are parallel and perpendicular
lines used with triangles and other
polygons with or without a coordinate
plane?
How can using parallel lines and a
transversal help to determine angle
measures in a real-life setting?
How can doing proofs help with
making other logical connections
throughout geometry and life?
Tests and quizzes including
performance assessment questions.
In class activity with geo-sketch pad
with exploration style proofs and
construction tools. p. 126 and p.156
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problems and logically justify
results.
9.3.3.7 (3.7) Use properties of
polygons—including quadrilaterals
and regular polygons—to define
them, classify them, solve problems
and logically justify results.
4-6
Congruent Triangles
(chapter 4)
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.2.4 (4.3) Construct logical
arguments and write proofs of
theorems and other results in
geometry, including proofs by
contradiction. Express proofs in a
form that clearly justifies the
reasoning, such as two-column
proofs, paragraph proofs, flow
charts or illustrations.
9.3.2.5 (Pre 4.4) Use technology
tools to examine theorems, make
and test conjectures, perform
constructions and develop
mathematical reasoning skills in
multi-step problems. The tools may
include compass and straight edge,
dynamic geometry software, design
software or Internet applets.
What makes shapes congruent?
Where are congruent figures and their
properties used in real life?
How do you prove two triangles are
congruent without show all part are
congruent?
Tests and quizzes including
performance assessment questions.
In class activity with geo-sketch pad
and construction tools. (p.220)
In class discovery with patty paper.
(p. 227)
What impact does the triangle
congruence properties have on proving
triangular shapes are congruent in every
day situations?
Flow chart proofs. (p.247)
What are the properties of a triangles
and how does “if…then” vs. “if and
only if” affect the properties?
Tests and quizzes including
performance assessment questions.
9.3.3.3 (4.5) Know and apply
properties of equilateral, isosceles
and scalene triangles to solve
problems and logically justify
results.
9.3.3.6 (4.6-4.7) Know and apply
properties of congruent and similar
figures to solve problems and
logically justify results.
6-7
Relationship within
triangles
(chapter 5)
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
9.3.2.2 (5.4) Accurately interpret
and use words and phrases such as
"if…then," "if and only if," "all,"
and "not." Recognize the logical
relationships between an "if…then"
statement and its inverse, converse
and contrapositive.
9.3.2.4 (5.4) Construct logical
arguments and write proofs of
Where are concurrent lines used in real
life?
How can inductive reasoning be used
for determining unknown concepts in
geometry and in the real world?
In-class activity with geo-sketch pad
and construction tool. ( P. 258, 271)
Paper folding activities to help
discover concepts.
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theorems and other results in
geometry, including proofs by
contradiction. Express proofs in a
form that clearly justifies the
reasoning, such as two-column
proofs, paragraph proofs, flow
charts or illustrations.
9.3.2.5 (pre5.1) Use technology
tools to examine theorems, make
and test conjectures, perform
constructions and develop
mathematical reasoning skills in
multi-step problems. The tools may
include compass and straight edge,
dynamic geometry software, design
software or Internet applets.
8
9 - 10
Quadrilaterals
(chapter 6)
Similarity
(Chapter 7)
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
9.3.3.7 (6.1, 6.3) Use properties of
polygons—including quadrilaterals
and regular polygons—to define
them, classify them, solve problems
and logically justify results.
What are the different types of
quadrilaterals?
Tests and quizzes including
performance assessment questions.
How do the properties of various
quadrilaterals compare?
Geo-sketch pad lesson (p. 342)
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.4.4 (6.1, 6.7) Use coordinate
geometry to represent and analyze
line segments and polygons,
including determining lengths,
midpoints and slopes of line
segments.
How do the properties help in proving
coordinate geometry proofs?
9.3.3.6 (7.2, 7.4) Know and apply
properties of congruent and similar
figures to solve problems and
logically justify results.
What is the relationship of similar
triangles?
Solve real-world and mathematical
geometric problems using algebraic
methods.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Solve real-world and mathematical
geometric problems using algebraic
methods.
11 - 12
Right triangles and
trigonometry
(chapter 8)
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.4.7 (7.3, 7.4) Use algebra to
solve geometric problems unrelated
to coordinate geometry, such as
solving for an unknown length in a
figure involving similar triangles,
or using the Pythagorean Theorem
to obtain a quadratic equation for a
length in a geometric figure.
9.3.3.4 (8.1) Apply the Pythagorean
Theorem and its converse to solve
problems and logically justify
results.
How can students apply these
properties of congruent figures to those
that are just similar?
Tests and quizzes including
performance assessment questions.
Why are ratios and proportions needed
to find unknown lengths and angles?
How can we apply the concept of
similarity to right triangles and the
basic trigonometric ratios?
How can similarity and scale changes
be useful for everyday life situations?
What is the Pythagorean theorem and
where can it be used for real world
problems?
Tests and quizzes including
performance assessment questions.
Hands on investigations dealing with
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Solve real-world and mathematical
geometric problems using algebraic
methods.
How do the basic trigonometric ratios
apply to geometry and where can they
be used in real-life applications?
the disco very of how Pythagorean
theorem is created.
9.2.1.9 (All chapter 9) Determine
how translations affect the
symbolic and graphical forms of a
function. Know how to use
graphing technology to examine
translations.
What are some types of transformations
and where are they found in the real
world?
Tests and quizzes including
performance assessment questions.
9.3.1.4 (9.5) Understand and apply
the fact that the effect of a scale
factor k on length, area and volume
is to multiply each by k, k2 and k3,
respectively.
Where are transformations used in art?
9.3.3.5 (8.2) Know and apply
properties of right triangles,
including properties of 45-45-90
and 30-60-90 triangles, to solve
problems and logically justify
results.
9.3.4.1 (8.4) Understand how the
properties of similar right triangles
allow the trigonometric ratios to be
defined, and determine the sine,
cosine and tangent of an acute
angle in a right triangle.
9.3.4.2 (8.4 – 8.6) Apply the
trigonometric ratios sine, cosine
and tangent to solve problems, such
as determining lengths and areas in
right triangles and in figures that
can be decomposed into right
triangles. Know how to use
calculators, tables or other
technology to evaluate
trigonometric ratios.
9.3.4.3 (8.5) Use calculators, tables
or other technologies in connection
with the trigonometric ratios to find
angle measures in right triangles in
various contexts.
13
Transformations
(chapter 9)
Understand the concept of function,
and identify important features of
functions and other relations using
symbolic and graphical methods
where appropriate.
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
Solve real-world and mathematical
geometric problems using algebraic
methods.
9.3.4.6 (9.1-9.3, 9.5) Use numeric,
graphic and symbolic
representations of transformations
in two dimensions, such as
reflections, translations, scale
changes and rotations about the
How are transformations used in real
life?
Tessellation Project which includes
at least three different styles of
transformations with the template.
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origin by multiples of 90˚, to solve
problems involving figures on a
coordinate grid.
14 – 15
Area
(chapter 10)
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.1.2 (10.1 to 10.3) Compose and
decompose two- and threedimensional figures; use
decomposition to determine the
perimeter, area, surface area and
volume of various figures.
9.3.1.3 (throughout chapter 10)
Understand that quantities
associated with physical
measurements must be assigned
units; apply such units correctly in
expressions, equations and problem
solutions that involve
measurements; and convert
between measurement systems.
How are areas and perimeters
calculated?
Tests and quizzes including
performance assessment questions.
How can area be applied to congruent
and similar figures?
In-class assignment of 574.
When can area and perimeter be used
throughout life?
How does area apply to finding the
volume of a three-dimensional figure?
9.3.3.8 (10.6) Know and apply
properties of a circle to solve
problems and logically justify
results.
16 – 17
Surface Area and
Volume
(chapter 11)
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
9.3.1.1 (11.3, 11.5, 11.6) Determine
the surface area and volume of
pyramids, cones and spheres. Use
measuring devices or formulas as
appropriate.
What is the relationship between
length, area and volume?
Tests and quizzes including
performance assessment questions.
What types of figures have volume and
surface area?
Geo-labs on finding the relationship
between volumes of a cone/pyramid
to a cylinder/prism.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
9.3.1.2 (11.2-11.6) Compose and
decompose two- and threedimensional figures; use
decomposition to determine the
perimeter, area, surface area and
volume of various figures.
How are length, area, and volume
different in real life situations?
Solve real-world and mathematical
geometric problems using algebraic
methods.
18
Circles
(chapter 12)
9.3.1.4 (11.7) Understand and
apply the fact that the effect of a
scale factor k on length, area and
volume is to multiply each by k, k2
and k3, respectively.
9.3.3.8 (12.1 – 12.3) Know and
apply properties of a circle to solve
problems and logically justify
results.
What are the basic definitions
associated with circles and the lines that
intersect circles?
How do arcs and angles relate within a
Tests and quizzes including
performance assessment questions.
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9.3.4.5 (12.5) Know the equation
for the graph of a circle with radius
r and center (h, k), (x – h)2 + (y –
k)2 = r2, and justify this equation
using the Pythagorean Theorem
and properties of translations.
circle?
How can properties of circles help with
future math courses?