Download 2014-2015. Geometry Curriculum

Geometry 1-2
PUHSD Curriculum
2014-2015
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
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Geometry 1-2 Learning Outcomes
Unit 1
Foundations
and Tools for
Geometry
G-CO.A.1
Unit 2
Introduction to
Transformational
Geometry
G-CO.A.2 #
Unit 3
Triangle
Congruence
Unit 4
Quadrilaterals
Unit 5
Similarity
Unit 6
Trigonometry
Unit 7
2/3-D Shapes
Unit 8
Circles
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
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Major Content
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Quarter 1
Introduction to Geometry
(19 days)
 Definitions
 Constructions
 Prove Theorems about lines
and angles
 Use coordinate formulas in
proofs
 Different types of proofs two column, flow, paragraph
 Slope of parallel and
perpendicular lines - use with
geometric shapes
 Proofs (justification of
thinking)
- informal
- sequence of logical
statements
Introduction to transformational
Geometry
(18 days)
 Transform
 Definitions
 Multiple Transformations
 Rigid Motion
 Congruence
Quarter 2
Triangles
(25 days)
 Prove triangle theorems
 Prove triangle congruence
ASA, SSS, SAS
 Pythagorean Theorem
 Prove lines/angles theorems
 Constructions involving
triangles
 Proofs involving triangles
Quadrilaterals
(15 days)
 Polygons
 Quadrilaterals
 Proofs
 Constructions

Quarter 3
Similarity
(25 days)
 Dilations
 Triangle and Polygon Similarity
 Midpoint Formula and Section
Formula
 Proofs involving similarity
Trigonometry
(16 days)
 Special Right Triangle
Relationships
 Right Triangle Relationships
 Trigonometry and Inverse
Trigonometry
Quarter 4
Two and three Dimensional Shapes
(19 days)
 Perimeter/area
-Area of sector
-Using Coordinates
-Changing parameters
-Dissection, Argument
-Complex polygons
 Volume
-Changing parameters
-Cavalieri’s principle
 Density based on area and
volume
 Design Problems
 Proofs
Circles
(16 days)
 Similarity
 Parts of Circles
 Constructions
 Proportionality
 Equations
 Arguments
 Proofs
 Constructions

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Major Content
Supporting Content
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 1 Foundations and Tools for Geometry
Enduring Understandings:
Essential Questions:
Studying geometry involves learning the basic parts of geometry. 1. How can I make formal geometric constructions
Everything is built from points, lines and planes and follows very 2. What are the basic parts of any construction or description in geometry?
strict and organized rules. Proofs are a vital component for
3. Why are proofs important in developing geometric concepts?
geometry.
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
 Use geometry software to
G-CO.A.1
Know precise definitions of angle, circle, perpendicular line,
angle, circle, perpendicular line, parallel explore different theorems and
parallel line, and line segment, based on the undefined
line, and line segment based on the
definitions within geometry
notions of point, line, distance along a line, and distance
idea of point, line, and distance along a  Use proofblocks to develop
around a circular arc.
line.
critical thinking
 I can make the following formal
D. Make geometric constructions
constructions using a variety of tools:
Key Vocabulary
G-CO.D.12 Make formal geometric constructions with a variety of tools copying a segment, copying an angle,
collinear/Linear
and methods (compass and straightedge, string, reflective
bisecting a segment, bisecting an angle, coplanar/plane
devices, paper folding, dynamic geometric software, etc.).
constructing perpendicular lines,
point
Copying a segment; copying an angle; bisecting a segment;
constructing perpendicular bisectors,
segment, ray, line
bisecting an angle; constructing perpendicular lines,
constructing a line parallel to a given
slope
including the perpendicular bisector of a line segment; and
line through a given point not on the
angle
constructing a line parallel to a given line through a point not line.
segment
on the line.
 I can prove the following theorems in perpendicular bisector
narrative paragraphs, flow diagrams, in linear pair
C. Prove geometric theorems
complementary/supplementary
two column format, and or using
G-CO.C.9
Prove theorems about lines and angles. Theorems include:
vertical angles
diagrams without words:
vertical angles are congruent; when a transversal crosses
parallel/perpendicular/coinciding
vertical angles are congruent,
parallel lines, alternate interior angles are congruent and
skew
when a transversal crosses parallel
corresponding angles are congruent; points on a
adjacent angles
lines, and alternate interior angles are
perpendicular bisector of a line segment are exactly those
midpoint
congruent and corresponding angles
equidistant from the segment’s endpoints.
postulate
are congruent.
theorem
 I can use coordinates to prove the
B. Use coordinates to prove simple geometric theorems algebraically
angle bisector
simple geometric theorems.
G-GPE.B.4
Use coordinates to prove simple geometric theorems
transversal
 I can disprove false statements using
algebraically. For example, prove or disprove that a figure
the properties of the coordinate plane, alternate interior angles
Page 4 of 17
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Geometry 1-2
PUHSD Curriculum
G-GPE.B.5
2014-2015
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).
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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.
Additional Content
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
Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 2: Introduction to Transformational Geometry
Enduring Understandings:
Essential Questions:
Rotations, reflections and
1. How does each transformation move various objects?
translations are examples that 2. How can I define congruence in terms of rigid motions?
preserve angles and distances. 3. What are the similarities and differences between the images and pre-images generated by transformations and/or
These “rigid motions” can be
multiple transformations?
used to describe congruence.
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.,
manipulatives.
to explore
transparencies and geometry software; describe
transformations and their
 I can describe a transformation using
transformations as functions that take points in the plane as
coordinate notation that maps one point onto a properties.
inputs and give other points as outputs. Compare
Key Vocabulary
unique image point.
transformations that preserve distance and angle to those
Transformation, image,
 I can compare transformations that preserve
that do not (e.g., translation versus horizontal stretch).
pre-image, composition
distance and angle to those that do not.
G-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular
translation
 I can demonstrate the rotations and
polygon, describe the rotations and reflections that carry it
reflection
reflections that carry a rectangle,
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,
reflectional symmetry
 I can make and refine a definition of
and line segments.
rotations, reflections, and translations based on rigid motion
G-CO.A.5
Given a geometric figure and a rotation, reflection, or
the definitions of angles, circles, perpendicular congruent
carry on to itself
translation, draw the transformed figure using, e.g., graph
lines, parallel lines, and line segments.
carry onto another
paper, tracing paper, or geometry software. Specify a
 I can demonstrate and draw transformations
map onto itself
sequence of transformations that will carry a given figure
using tools.
onto another.
 I can find a sequence of transformations that sequence
B. Understand congruence in terms of rigid motions
predict
will carry a shape onto another.
G-CO.B.6
Use geometric descriptions of rigid motions to transform
 I can investigate rigid motions and generalize vertices
vectors
figures and to predict the effect of a given rigid motion on a
their characteristics as preserving congruency.
magnitude
given figure; given two figures, use the definition of
 I can decide if two shapes are congruent
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
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Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 3: Triangle Congruence
Enduring Understandings:
Essential Questions:
Triangles are fundamental aesthetic,
1. How do rigid motions lead to an understanding of congruence criteria for triangles?
structural elements that are useful in many 2. How can proofs help us to develop a deeper and more enduring understanding of triangles?
disciplines such as art, architecture, and
3. What is true of the points on a perpendicular bisector?
engineering.
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
rigid motions if and only if the corresponding pairs of
verify theorems about lines,
motions to show that two triangles are congruent sides and corresponding pairs of angles are
angles, triangles and
if and only if corresponding pairs of sides and
congruent.
parallelograms.
corresponding pairs of angles are congruent.
Key Vocabulary
 I can explain which series of angles and sides are
G-CO.B.8
Explain how the criteria for triangle congruence
congruence
essential in order to show congruence through rigid
(ASA, SAS, and SSS) follow from the definition of
Angle-Side-Angle Congruence
motions.
congruence in terms of rigid motions.
Theorem
 I can prove the following theorems in narrative
C. Prove geometric theorems
Side-Angle-Side Congruence
paragraphs, flow diagrams, in two column format,
G-CO.C.9
Prove theorems about lines and angles. Theorems and/or using diagrams without words: points on a
Theorem
include: vertical angles are congruent; when a
Side-Side-Side Congruence
perpendicular bisector of a line segment are exactly
transversal crosses parallel lines, alternate interior those equidistant from the segment’s endpoints.
Postulate
angles are congruent and corresponding angles
CPCTC (Congruent Parts of
 I can prove theorems in narrative paragraphs, flow
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
Congruent)
diagrams without words.
from the segment’s endpoints.
 I can make the following formal constructions using perpendicular bisector
G-CO.C.10
Prove theorems about triangles. Theorems
circumcenter, equidistant
a variety of tools (compass and straightedge and
include: measures of interior angles of a triangle
Triangle Sum Theorem
geometric software): constructing perpendicular
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
Converse
a variety of tools (compass and straightedge and
half the length; the medians of a triangle meet at
geometric software): an equilateral triangle inscribed median
a point.
angle bisector
in a circle.
D. Make geometric constructions
in-center
 I can solve problems using congruence criteria for
G-CO.D.12 Make formal geometric constructions with a
Concurrency of Medians of a
triangles.
variety of tools and methods (compass and
Triangle
 I can prove relationships in geometric figures using
Page 7 of 17
Major Content
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
G-CO.D.13
2014-2015
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).★
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Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 4: Quadrilaterals
Enduring Understandings:
Essential Questions:
Polygons can be classified using properties of sides and angles.
How are quadrilaterals classified according to sides?
Special quadrilaterals are classified based on different
What is the difference between concave and convex?
properties.
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
transformations using tools.
Key Vocabulary
algebraically. For example, prove or disprove that a figure
Polygon
 I can find a sequence of transformations
defined by four given points in the coordinate plane is a
Convex
that will carry a shape onto another.
rectangle; prove or disprove that the point (1, √3) lies on
Concave
 I can prove the following theorems in
the circle centered at the origin and containing the point (0, narrative paragraphs, flow diagrams, in two Regular polygon
2).
Triangles
column format, and or using diagrams
Quadrilateral
without words: opposite sides are
A. Experiment with transformations in the plane
Parallelograms
congruent, opposite angles are congruent,
G-CO.A.5
Given a geometric figure and a rotation, reflection, or
the diagonals of a parallelogram bisect each Rectangles
translation, draw the transformed figure using, e.g., graph
Trapezoid
other, rectangles are parallelograms with
paper, tracing paper, or geometry software. Specify a
Rhombus
congruent diagonals.
sequence of transformations that will carry a given figure
Kite
 I can make the following formal
onto another.
Squares
constructions using a variety of tools
Hexagon
(compass and straightedge and geometric
C. Prove geometric theorems
software): an equilateral triangle, a square, Diagonals
G-CO.C.11
Prove theorems about parallelograms. Theorems include:
Opposite sides
a regular hexagon inscribed in a circle.
opposite sides are congruent, opposite angles are
Opposite angles
 I can use coordinates to prove properties
congruent, the diagonals of a parallelogram bisect each
Bisect
of quadrilaterals.
other, and conversely, rectangles are parallelograms with
Equilateral
 I can demonstrate the rotations and
congruent diagonals.
Inscribed
reflections that carry a rectangle,
parallelogram, trapezoid, or regular polygon
D. Make geometric constructions
onto itself.
G-CO.D.13
Construct an equilateral triangle, a square, and a regular
hexagon inscribed in a circle.
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Geometry 1-2
PUHSD Curriculum
2014-2015
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
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Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 5: Similarity
Enduring Understandings:
Essential Questions:
Similarity is defines as the result of rigid transformations
How is similarity defined by transformations?
and dilations. Similar figures have corresponding angles
How can I prove two figures are similar?
that are congruent and corresponding sides that are
How are trigonometric ratios used to solve problems involving triangles?
proportional. Trigonometry is a particularly useful
How can similar figures model real world situations?
application of similar right 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
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., distance and angle to those that do not.
verify theorems about lines,
transparencies and geometry software; describe
 I can prove the Midsegment Theorem (the segment angles, triangles and
transformations as functions that take points in
parallelograms
joining midpoints of two sides of a triangle is parallel
the plane as inputs and give other points as
Key Vocabulary
to and half the length of the third side) in narrative
outputs. Compare transformations that preserve
dilation
paragraphs, flow diagrams, in two column format,
distance and angle to those that do not (e.g.,
scale factor
and or using diagrams without words
translation versus horizontal stretch).
 I can use the midpoint formula to calculate midpoint similarity
C. Prove geometric theorems
similarity transformations
or endpoint coordinates with various unknowns (e.g.
G-CO.C.10
Prove theorems about triangles. Theorems
center of dilation
find the other endpoint, etc.)
include: measures of interior angles of a triangle
mid-segment
I can verify the following statements by making
sum to 180°; base angles of isosceles triangles are multiple examples;
proportional
congruent; the segment joining midpoints of two
ratio
 a. A dilation of a line is parallel to the original line if
sides of a triangle is parallel to the third side and
reduction
the center of dilation is not on the line and a dilation
half the length; the medians of a triangle meet at
enlargement
of a line is coinciding if the center is on the line.
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
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
G-SRT.A.2
G-SRT.A.3
2014-2015
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.
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
G-SRT.B.5
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.
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
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.
Find the point on a directed line segment between
two given points that partitions the segment in a
given ratio.
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Major Content
Supporting Content
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Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
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.
Standard
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?
C. Define trigonometric ratios and solve problems involving right
triangles
G-SRT.C.6
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.
G-SRT.C.7
Explain and use the relationship between the sine
and cosine of complementary angles.
G-SRT.C.8
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied problems.
★
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Major Content
Learning Targets
I can:
Discover the relationship between the sides and
angles of a right triangle and be able to state the
sine, cosine, or tangent of a reference angle given
a right triangle.
Be able to find the three basic trig ratios given a
triangle.
Understand the sine and cosines of
complementary angles are equal.
Use a trig table.
Have a basic understanding of how to use trig to
solve a real world problem.
Set up a trig equation and solve for a missing
side.
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 45-45-90 triangle and
use the pattern to find the missing sides of a
triangle.
Discover the pattern of a 30-60-90 triangle and
use the pattern to find the missing sides of a
triangle
Supporting Content
Additional Content
Technology Standards
 Use geometry software to
verify theorems about lines,
angles, triangles and
parallelograms
Key Vocabulary
45-45-90 triangle
30-60-90 triangle
trigonometry
trigonometric ratios:
 sine
 cosine
 tangent
inverse trigonometry
Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Enduring Understandings:
Area, surface area, and volume have many real life
applications. Many polygons and polyhedron have
common features based on their common characteristics.
Standard
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?
A. Experiment with transformations in the plane
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).
B. Use coordinates to prove simple geometric theorems algebraically
G-GPE.B.7
Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles, e.g.,
using the distance formula.★
A. Explain volume formulas and use them to solve problems
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.
G-GMD.A.3
Use volume formulas for cylinders, pyramids,
cones, and spheres to solve problems. ★
Page 14 of 17
Major Content
Learning Targets
I can explore the effect of altering
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,
volume of a cylinder, pyramid, and cone by
using:
-Dissection arguments, separating a shape into
two or more shapes.
-Cavalieri’s principle, if two solids have the
same height and the same cross-sectional area
at every level, then they have the same
volume.
-Informal Limit arguments, find the area and
volume of curved shapes using an infinite
number of rectangles and prisms.
 I can use volume formulas for cylinders,
pyramids, cones, and spheres to solve
problems.
 I can identify the shapes of two-dimensional
cross sections of three-dimensional objects.
 I can identify three-dimensional objects
generated by rotations of two-dimensional
Supporting Content
Additional Content
Technology Standards
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
http://www.ics.uci.edu/~eppstein/j
unkyard/model.html
 Wolfram Mathworld
http://mathworld.wolfram.com/topi
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
Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
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.
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.
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). ★
G-MG.A.2
Apply concepts of density based on area and
volume in modeling situations (e.g., persons per
square mile, BTUs per cubic foot). ★
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 15 of 17
Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
Unit 9: Circles
Enduring Understandings:
Essential Questions:
Properties of circles can be explained and
How are theorems for circles applied and proven?
applied algebraically and geometrically.
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
A. Understand and apply theorems about circles
 I can prove all circles are similar to each other
G-C.A.1
Prove that all circles are similar.
based on similarity transformations (rigid
motion followed by a dilation.)
 I can identify inscribed angles, radii, and
G-C.A.2
Identify and describe relationships among inscribed angles, chords.
radii, and chords. Include the relationship between central,  I can describe relationships between segment
inscribed, and circumscribed angles; inscribed angles on a
lengths intersecting inside and outside of the
diameter are right angles; the radius of a circle is
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
Construct the inscribed and circumscribed circles of a
circles of a triangle.
triangle, and prove properties of angles for a quadrilateral I can prove the following properties for
inscribed in a circle.
quadrilateral ABCD inscribed in a circle; (i.e) ∠A
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.
Page 16 of 17
Major Content
Supporting Content
+ ∠C =∠B +∠D =180o
 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.
Additional Content
Technology Standards
 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
Standards in gray are emphasized in a different unit
Geometry 1-2
PUHSD Curriculum
2014-2015
A. Explain volume formulas and use them to solve problems
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
 I can explain the formulas for the
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.
grid systems based on ratios).★
Page 17 of 17
Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit