Download 2014-2015. Geometry Honors Curriculum

Geometry 1-2 Honors
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.
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Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
Geometry 1-2 Honors 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.A.4 +
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-C.B.5
G-CO.D.13 #
G-MG.A.1#
G-SRT.A.3
G-MG.A.1 #
G-GPE.A.1
G-SRT.B.5 #
G-SRT.B.4
G-MG.A.2
G-GMD.A.1 #
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 Honors
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

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Quarter 2
Triangles
(25 dayss)
 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
Major Content
Supporting Content
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
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
Unit 1 Foundations and Tools for Geometry
Enduring Understandings:
Essential Questions:
Studying geometry involves learning the basic parts of
1. How can I make formal geometric constructions
geometry. Everything is built from points, lines and planes and 2. What are the basic parts of any construction or description in geometry?
follows very strict and organized rules. Proofs are a vital
3. Why are proofs important in developing geometric concepts?
component for 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,
explore different theorems and
parallel line, and line segment, based on the undefined
parallel line, and line segment based
definitions within geometry
notions of point, line, distance along a line, and distance
on the idea of point, line, and distance  Use proofblocks to develop
around a circular arc.
along a line.
critical thinking with proofs Key
 I can make the following formal
Vocabulary
constructions using a variety of tools:
collinear/Linear
C. Prove geometric theorems
copying a segment, copying an angle,
coplanar/plane
G-CO.C.9
Prove theorems about lines and angles. Theorems include:
bisecting a segment, bisecting an
point
vertical angles are congruent; when a transversal crosses
angle, constructing perpendicular
segment, ray, line
parallel lines, alternate interior angles are congruent and
lines, constructing perpendicular
slope
corresponding angles are congruent; points on a
bisectors, constructing a line parallel to angle
perpendicular bisector of a line segment are exactly those
a given line through a given point not
segment
equidistant from the segment’s endpoints.
on the line.
perpendicular bisector
 I can prove the following theorems in linear pair
narrative paragraphs, flow diagrams, in complementary/supplementary
D. Make geometric constructions
vertical angles
two column format, and or using
G-CO.D.12 Make formal geometric constructions with a variety of tools diagrams without words:
parallel/perpendicular/coinciding
and methods (compass and straightedge, string, reflective
skew
vertical angles are congruent,
devices, paper folding, dynamic geometric software, etc.).
adjacent angles
when a transversal crosses parallel
Copying a segment; copying an angle; bisecting a segment;
lines, and alternate interior angles are midpoint
bisecting an angle; constructing perpendicular lines,
postulate
congruent and corresponding angles
including the perpendicular bisector of a line segment; and are congruent.
theorem
constructing a line parallel to a given line through a point
angle bisector
 I can use coordinates to prove the
not on the line.
transversal
simple geometric theorems.
 I can disprove false statements using alternate interior angles
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Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
B. Use coordinates to prove simple geometric theorems algebraically
G-GPE.B.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, √3) lies on
the circle centered at the origin and containing the point (0,
2).
G-GPE.B.5
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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Major Content
Supporting Content
the properties of the coordinate plane,
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 Honors
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 to
G-CO.A.2
Represent transformations in the plane using, e.g.,
manipulatives.
explore transformations
transparencies and geometry software; describe
and their properties.
 I can describe a transformation using
transformations as functions that take points in the
coordinate notation that maps one point onto a Key Vocabulary
plane as inputs and give other points as outputs.
Transformation, image,
unique image point.
Compare transformations that preserve distance and
pre-image, composition
 I can compare transformations that preserve
angle to those that do not (e.g., translation versus
translation
distance and angle to those that do not.
horizontal stretch).
reflection
 I can demonstrate the rotations and
reflections that carry a rectangle, parallelogram, rotation
G-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular
rotational symmetry
trapezoid, or regular polygon on to itself.
polygon, describe the rotations and reflections that
 I can make and refine a definition of rotations, reflectional symmetry
carry it onto itself.
rigid motion
reflections, and translations based on the
definitions of angles, circles, perpendicular lines, congruent
G-CO.A.4
Develop definitions of rotations, reflections, and
carry on to itself
parallel lines, and line segments.
translations in terms of angles, circles, perpendicular
carry onto another
 I can demonstrate and draw transformations
lines, parallel lines, and line segments.
map onto itself
using tools.
sequence
 I can find a sequence of transformations that
predict
G-CO.A.5
Given a geometric figure and a rotation, reflection, or
will carry a shape onto another.
vertices
translation, draw the transformed figure using, e.g.,
 I can investigate rigid motions and generalize
vectors
graph paper, tracing paper, or geometry software.
their characteristics as preserving congruency.
magnitude
Specify a sequence of transformations that will carry a
 I can decide if two shapes are congruent
given figure onto another.
because of the rigid motions between the two
figures.
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Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
B. Understand congruence in terms of rigid motions
G-CO.B.6
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.
Page 7 of 18
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Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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
2. How can proofs help us to develop a deeper and more enduring understanding of triangles?
many disciplines such as art, architecture, 3. What is true of the points on a perpendicular bisector?
and 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
 Use geometry software to
G-CO.B.7
Use the definition of congruence in terms of rigid
through rigid motions if and only if the
verify theorems about
motions to show that two triangles are congruent if
corresponding pairs of sides and corresponding
lines, angles, triangles and
and only if corresponding pairs of sides and
pairs of angles are congruent.
parallelograms.
corresponding pairs of angles are congruent.
 I can explain which series of angles and sides are Key Vocabulary
congruence
essential in order to show congruence through
G-CO.B.8
Explain how the criteria for triangle congruence (ASA, rigid motions.
Angle-Side-Angle
SAS, and SSS) follow from the definition of
Congruence
 I can prove the following theorems in narrative
congruence in terms of rigid motions.
paragraphs, flow diagrams, in two column format, Theorem
and/or using diagrams without words: points on a Side-Angle-Side
C. Prove geometric theorems
Congruence Theorem
perpendicular bisector of a line segment are
G-CO.C.9
Prove theorems about lines and angles. Theorems
Side-Side-Side Congruence
exactly those equidistant from the segment’s
include: vertical angles are congruent; when a
Postulate
endpoints.
transversal crosses parallel lines, alternate interior
CPCTC (Congruent Parts of
 I can prove theorems in narrative paragraphs,
angles are congruent and corresponding angles are
flow diagrams, in two column format, and or using Congruent Triangles are
congruent; points on a perpendicular bisector of a
Congruent)
diagrams without words.
line segment are exactly those equidistant from the
perpendicular bisector
 I can make the following formal constructions
segment’s endpoints.
using a variety of tools (compass and straightedge circumcenter, equidistant
Triangle Sum Theorem
and geometric software): constructing
interior angles
G-CO.C.10
Prove theorems about triangles. Theorems include:
perpendicular bisectors.
Base Angles Theorem and
measures of interior angles of a triangle sum to 180°;  I can make the following formal constructions
base angles of isosceles triangles are congruent; the
using a variety of tools (compass and straightedge its Converse
median
segment joining midpoints of two sides of a triangle is and geometric software): an equilateral triangle
angle bisector
parallel to the third side and half the length; the
inscribed in a circle.
in-center
medians of a triangle meet at a point.
 I can solve problems using congruence criteria
Concurrency of Medians of
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Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
D. Make geometric constructions
G-CO.D.12
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.D.13
Construct an equilateral triangle, a square, and a
regular hexagon inscribed in a circle.
for triangles.
 I can prove relationships in geometric figures
using 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.
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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Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
Unit 4: Quadrilaterals
Enduring Understandings:
Essential Questions:
Polygons can be classified using properties of sides and angles. 1. How are quadrilaterals classified according to sides?
Special quadrilaterals are classified based on different
2. What is the difference between concave and convex?
properties.
3. What are the properties of quadrilaterals?
4. How are quadrilateral classified?
5. How do you inscribe a triangle, square or regular hexagon into a circle?
6. 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.3
Given a rectangle, parallelogram, trapezoid, or regular
the diagonals of a parallelogram bisect each Rectangles
polygon, describe the rotations and reflections that carry it other, rectangles are parallelograms with
Trapezoid
onto itself.
Rhombus
congruent diagonals.
Kite
 I can make the following formal
Squares
G-CO.A.5
Given a geometric figure and a rotation, reflection, or
constructions using a variety of tools
Hexagon
translation, draw the transformed figure using, e.g., graph
(compass and straightedge and geometric
Diagonals
paper, tracing paper, or geometry software. Specify a
software): an equilateral triangle, a square,
Opposite sides
sequence of transformations that will carry a given figure
a regular hexagon inscribed in a circle.
Opposite angles
onto another.
 I can use coordinates to prove properties
Bisect
C. Prove geometric theorems
of quadrilaterals.
Equilateral
G-CO.C.11
Prove theorems about parallelograms. Theorems include:
 I can demonstrate the rotations and
Inscribed
opposite sides are congruent, opposite angles are
reflections that carry a rectangle,
congruent, the diagonals of a parallelogram bisect each
parallelogram, trapezoid, or regular polygon
other, and conversely, rectangles are parallelograms with
onto itself.
congruent diagonals.
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Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
D. Make geometric constructions
G-CO.D.13
Construct an equilateral triangle, a square, and a regular
hexagon inscribed in a circle.
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 11 of 18
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Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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.
Unit 5: Similarity
Essential Questions:
1. How is similarity defined by transformations?
2. How can I prove two figures are similar?
3. How are trigonometric ratios used to solve problems involving triangles?
4. How can similar figures model real world situations?
5. How are trigonometric ratios used to solve problems involving triangles?
6. What is the relationship between similar right triangles and trigonometric ratios?
Standard
Learning Targets
 I can compare transformations that preserve
G-CO.A.2
Represent transformations in the plane using, e.g.,
distance and angle to those that do not.
transparencies and geometry software; describe
 I can prove the Midsegment Theorem (the
transformations as functions that take points in the
segment joining midpoints of two sides of a
plane as inputs and give other points as outputs.
triangle is parallel to and half the length of the
Compare transformations that preserve distance and
third side) in narrative paragraphs, flow
angle to those that do not (e.g., translation versus
diagrams, in two column format, and or using
horizontal stretch).
diagrams without words
C. Prove geometric theorems
 I can use the midpoint formula to calculate
G-CO.C.10
Prove theorems about triangles. Theorems include:
midpoint or endpoint coordinates with various
measures of interior angles of a triangle sum to 180°;
unknowns (e.g. find the other endpoint, etc.)
base angles of isosceles triangles are congruent; the
I can verify the following statements by
segment joining midpoints of two sides of a triangle is
making multiple examples;
parallel to the third side and half the length; the medians  a. A dilation of a line is parallel to the original
of a triangle meet at a point.
line if the center of dilation is not on the line
A. Understand similarity in terms of similarity transformations
and a dilation of a line is coinciding if the center
G-SRT.A.1a Verify experimentally the properties of dilations given by is on the line.
G-SRT.A.1b a center and a scale factor:
 b. The dilation of a line segment changes the
a. A dilation takes a line not passing through the center
length by a ratio given by the scale factor.
of the dilation to a parallel line, and leaves a line passing  I can extend the properties of dilations to
through the center unchanged.
polygons.
b. The dilation of a line segment is longer or shorter in
 I can decide if two figures are similar based on
the ratio given by the scale factor.
A. Experiment with transformations in the plane
Page 12 of 18
Major Content
Supporting Content
Additional Content
Technology Standards
 Use geometry software
to verify theorems about
lines, angles, triangles and
parallelograms
Key Vocabulary
dilation
scale factor
similarity
similarity transformations
center of dilation
mid-segment
proportional
ratio
reduction
enlargement
cofunction
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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
Find the point on a directed line segment between two
given points that partitions the segment in a given ratio.
Page 13 of 18
Major Content
Supporting Content
similarity transformations (rigid motion
followed by a dilation.)
 I can use similarity transformations to explain
the meaning of similar triangles as the equality
of all 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.
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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:
1. How are trigonometric ratios used to solve problems involving triangles?
2. How are trigonometric ratios used to solve problems involving triangles?
3. 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 Honors
PUHSD Curriculum
2014-2015
Unit 7: 2- and 3-dimensional shapes
Essential Questions:
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
1.
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 15 of 18
Major Content
What is the relationship of the different measures in two and three dimensional objects?
2. How does a change in one dimension of an object affect the other dimensions?
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/~eppstei
n/junkyard/model.html
 Wolfram Mathworld
http://mathworld.wolfram.com/t
opics/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,
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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.
A. Apply geometric concepts in modeling situations
G-MG.A.1
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.
cylinder
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 16 of 18
Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
PUHSD Curriculum
2014-2015
Unit 8: Circles
Enduring Understandings:
Essential Questions:
Properties of circles can be explained and applied
1. How are theorems for circles applied and proven?
algebraically and geometrically.
2. How are geometric properties of circles embedded in equations?
3. How is proportion used in arc and sector measurements?
4. How are real world situations modeled with circles?
Standard
Learning Targets
Technology Standards
A. Understand and apply theorems about circles
 I can prove all circles are similar to each
 Geogebra
G-C.A.1
Prove that all circles are similar.
other based on similarity transformations
Key Vocabulary
(rigid motion followed by a dilation.)
Circle, center
G-C.A.2
Identify and describe relationships among inscribed  I can identify inscribed angles, radii, and
chord
angles, radii, and chords. Include the relationship
secant, tangent
chords.
between central, inscribed, and circumscribed
minor arc, major arc
 I can describe relationships between
angles; inscribed angles on a diameter are right
arc length
segment lengths intersecting inside and
angles; the radius of a circle is perpendicular to the outside of the circle.
inscribed angle/triangle
tangent where the radius intersects the circle.
circumscribed angle/triangle
I can describe relationships between angles
central angle
G-C.A.3
Construct the inscribed and circumscribed circles of formed inside and outside of the circle.
intercepted arc
a triangle, and prove properties of angles for a
 I can construct inscribed and circumscribed
diameter
quadrilateral inscribed in a circle.
circles of a triangle.
radius
G-C.A.4
(+) Construct a tangent line from a point outside a
I can prove the following properties for
semi-circle
given circle to the circle.
quadrilateral ABCD inscribed in a circle; (i.e)
point of tangency
B. Find arc lengths and areas of sectors of circles
∠A + ∠C =∠B +∠D =180o
circumference
G-C.B.5
Derive using similarity the fact that the length of
I can construct a tangent line from a point
area
the arc intercepted by an angle is proportional to
outside a given circle to the circle.
inscribed polygon
the radius, and define the radian measure of the
 I can use similarity to logically arrive at the
angle as the constant of proportionality; derive the following; the length of the arc intercepted by locus
chord
formula for the area of a sector.
an angle is proportional to the radius, the
inscribed quadrilateral
A. Translate between the geometric description and the equation for a
definition of radian measure of the angle as
conic section
the constant of proportionality, the formula
G-GPE.A.1
Derive the equation of a circle of given center and
for the area of a sector.
radius using the Pythagorean Theorem; complete
● I can create the equation of a circle of the
the square to find the center and radius of a circle
given center and radius based on the
given by an equation.
definition of a circle.
Page 17 of 18
Major Content
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit
Geometry 1-2 Honors
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.
Page 18 of 18
Major Content
● I can complete the square in terms of x and
y to find the center and radius of a circle.
 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.
Supporting Content
Additional Content
Standards in gray are emphasized in a different unit