Download geometry common core and mastery objective

Mathematics - Geometry
District Course
Course Description
Open to: Grades 10, 11, 12 One Year Course
Prerequisite: C or better in Algebra 1, or pass Intermediate Algebra 1, Instructor/Counselor approval
Content: Students will use an integrated approach to concepts of algebra, geometry, logic and probability with
emphasis on geometry.
Adopted Materials
Instructional Objectives
H.S. G.CO.1. Know precise definitions of angle, circle, perpendicular
line, parallel line, and line segment, based on the undefined notions of
point, line, distance along a line, and distance around a circular arc
No
Performance Objectives
Resource
Reference
1
Identify vertex, sides, and interior of an
angle
2
Identify the basic parts of a circle (center,
radius, and diameter).
3
Identify parallel vs. perpendicular lines.
4
Compare and contrast lines, rays and
segments
G-CO.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)
No
Performance Objectives
Resource
Reference
1
Manipulate a given figure to represent
the different transformations (rotation,
reflection, translation)
2
Represent a translation as a function in
coordinate notation
3
Compare transformations that preserve
distance and angle to those that do not
Instructional Objectives
G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto
itself.
No
Performance Objectives
Resource
Standard
Reference
None
Assessment
Correlation
Standard
Reference
G.4.3.1
Assessment
Correlation
Standard
Reference
G.4.3.1
Assessment
Reference
No
1
2
3
Given a figure identify the type(s) of
symmetry the figure has. If it has line
symmetry sketch the figure and the lines of
symmetry. If it has rotational symmetry state
the angle of rotation
Instructional Objectives
G-CO.4. Develop definitions of rotations, reflections, and
translations in terms of angles, circles, perpendicular lines, parallel lines,
and line segments
Performance Objectives
Correlation
Standard
Reference
G.4.3.1
Resource
Reference
Assessment
Correlation
Develop a definition for a reflection using
perpendicular lines.
Develop a definition for a translation using
parallel lines
Develop a definition for a rotation using
angles and/or circles
Instructional Objectives
G-CO.5. Given a geometric figure and a rotation, reflection, or
translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence
of transformations that will carry a given figure onto another.
No
Performance Objectives
Resource
Reference
1
Given a geometric figure, draw the new
figure under the given transformation.
2
Given a preimage and an image, specify the
sequence of transformations that will map
the preimage onto the image.
Instructional Objectives
G-CO.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.
No
Performance Objectives
Resource
Reference
1
Determine if two given geometric figures are
congruent in terms of rigid motion.
2
Given two geometric figures transformed by
Standard
Reference
G.4.3.1
Assessment
Correlation
Standard
Reference
G.4.3.1
Assessment
Correlation
rigid motion, determine if the conditions of
congruency have been met.
Instructional Objectives
G-CO.7. Use the definition of congruence in terms of rigid motions
to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
No
Performance Objectives
Resource
Reference
1
Given two triangles transformed by rigid
motion, determine if the conditions of
congruency have been met.
Instructional Objectives
G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms
of rigid motions
No
Performance Objectives
Resource
Reference
Standard
Reference
G.4.3.1
Assessment
Correlation
Standard
Reference
9.4.1.1 Good
10.4.1.1 Good
Assessment
Correlation
Describe why ASA, SAS, and SSS satisfy
the congruency conditions for triangles.
Instructional Objectives
G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
No
Performance Objectives
1
Given two parallel lines and a transversal,
identify the special angle pairs and justify
your answers.
Given two parallel lines and a transversal,
solve for missing angle measures.
Prove the vertical angle theorem, alternate
interior angle theorem.
Solve for lengths of segments on a
perpendicular bisector
2
3
4
Resource
Reference
Standard
Reference
G4.1.3 Strong
(No proofs
included
in Idaho grade
level
standard)
Assessment
Correlation
5
Prove the perpendicular bisector theorem
Resource
Reference
Standard
Reference
G4.1.3 Strong
(Proofs not
included in
Idaho
Grade
Standards)
Assessment
Correlation
Resource
Reference
Standard
Reference
G4.1.1 Good
G4.1.3 Strong
(Proofs not
included in
Idaho
Grade
Standards)
Assessment
Correlation
Instructional Objectives
G-CO.10. Prove theorems about triangles. Theorems include:
measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a triangle meet at a point.
No
Performance Objectives
1
Verify/prove the triangle sum theorem,
isosceles triangle theorem, triangle
midsegment theorem (use a variety of methods).
2
Use angle bisectors and perpendicular
bisectors to solve for segment lengths and
angle measures.
3
Identify the centroid, incenter, orthocenter,
and circumcenter of a triangle,
Instructional Objectives
G-CO.11. Prove theorems about parallelograms. Theorems
include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each
other, and conversely, rectangles are parallelograms with
congruent diagonals
No
Performance Objectives
1
Given the coordinates of the vertices of a
quadrilateral, determine the most precise
name of the quadrilateral.
Justify your answers algebraically.
Compare and contrast properties and
attributes of the differing parallelograms
Under given conditions identify if a given
2
3
4
quadrilateral is a parallelogram.
Instructional Objectives
G-CO.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.
No
Performance Objectives
Standard
Reference
G4.4.1 Strong
(Not included
in
Idaho
Grade
Standards)
Resource
Reference
Assessment
Correlation
Resource
Reference
Standard
Reference
G4.4.1 Strong
(Not
included in
Idaho
Grade
Standards)
Assessment
Correlation
Resource
Standard
Reference
9.4.1.2;
10.4.1.2 strong
G4.3.1 strong
Assessment
1
Create a list of steps needed to construct
congruent segments, angles, bisect segments
and angles, parallel and perpendicular lines
2
Use a compass and straightedge to construct
congruent segments, angles, bisect segments
and angles, parallel and perpendicular lines.
3
Use multiple methods to do the above.
Instructional Objectives
G-CO.13. Construct an equilateral triangle, a square, and a regular
hexagon inscribed in a circle.
No
Performance Objectives
1
Construct an equilateral triangle, a square,
and a regular hexagon inscribed in a circle.
Instructional Objectives
G-SRT.1a. Verify experimentally the properties of dilations
given by a center and a scale factor:
A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
No
Performance Objectives
1
2
3
No
Identify the center of dilation and classify
it as a reduction or enlargement based on
preimage and image
Identify the center of dilation and classify it
as a reduction or enlargement based on
preimage and image.
Compare and contrast the properties of the
preimage and image.
Instructional Objectives
G-SRT.1b.Verify experimentally the properties of dilations given
by a center and a scale factor:
The dilation of a line segment is longer or shorter in the ratio given by the
scale factor
Performance Objectives
Identify the ratio of sides of image to
preimage and relate it to the scale factor of
dilation.
2
Given a preimage and scale factor, determine
the lengths of the sides of the image?
Instructional Objectives
G-SRT.2. 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.
Reference
Correlation
Resource
Reference
Assessment
Correlation
1
Standard
Reference
9.4.1.1;
10.4.1.1 as
both
apply to
similarity
and
congruency
(weak)
9.4.1.2;
10.4.1.2 as
both
apply to
similarity
(strong)
No
Performance Objectives
Resource
Reference
G4.1.2 Good
Assessment
Correlation
Resource
Reference
Standard
Reference
9.4.1.1;
10.4.1.1 as
both
apply to
similarity
and
congruency
(weak)
9.4.1.2;
10.4.1.2 as
both
apply to
similarity
(strong)
G4.1.2 Strong
Assessment
Correlation
1
Determine if two polygons are similar. If so,
write a similarity statement and give the
similarity ratio. If not justify your answer.
Instructional Objectives
G-SRT.3. Use the properties of similarity transformations to
establish the AA criterion for two triangles to be similar
No
Performance Objectives
1
Given limited information about two
triangles (two proportional side lengths, one
proportional side and two congruent angles,
etc.) determine if the triangles are guaranteed
to be similar. What is the least amount of
information needed to guarantee two
triangles are similar?
Instructional Objectives
G-SRT.4. 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.
Standard
Reference
G4.1.3 Strong
(Proofs not
included in
No
Performance Objectives
1
Prove the sidesplitter theorem and its
converse
Prove the Pythagorean theorem
2
Resource
Reference
Idaho
Grade
Standards
Assessment
Correlation
Resource
Reference
Standard
Reference
9.4.1.1;
10.4.1.1 good
G4.1.2 strong
Assessment
Correlation
Resource
Reference
Standard
Reference
9.2.2.1 good
G4.1.4 strong
Assessment
Correlation
Instructional Objectives
G-SRT.5. Use congruence and similarity criteria for triangles to
solve problems and to prove relationships in geometric figures.
No
Performance Objectives
1
Use congruence and similarity criteria to
solve for missing information in triangles.
Use proportions to identify missing
information in similar triangles.
2
Instructional Objectives
G-SRT.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
No
Performance Objectives
1
2
Using the special right triangles, identify the
ratios of sides within the same triangle and
how these relate to a similar triangle.
Given side lengths of a right triangle, write
the sine, cosine, and tangent ratio.
Performance Matter
Teacher:
CORE Geometry
Year: 2012-13
Course: Geometry
Month: September
Review linear equations and inequalities though a study of systems
Introduction to Geometry Vocabulary & Definitions ~ Introduce matrices and apply matrices to solving linear systems
How do variables help you model real-world situations?
What are the undefined terms (building blocks) of geometry?
What is a good definition?
How do points, lines and planes work together to form the other objects in geometry?
Why do we have to use the same symbols and definitions?
What is a good definition?
How do points, lines and planes work together to form the other objects in geometry?
What is a polygon? What shapes are NOT polygons?
What does it mean to be a regular polygon?
What is the definition of a circle?
What is an isometric drawing?
What are some strategies for sketching 3D shapes?
How are 2D nets related to 3D solids?
What is geomtric probability?
Standards
G-CO.1-Know precise definitions
of angle, circle, perpendicular
line, parallel line, and line
segment, based on the
undefined notions of point, line,
distance along a line, and
distance around a circular arc.
Content
Undefined terms
Good definitions
Skills
Vocabulary
Describe the undefined
terms
point
Define geometric terms
plane
Objectives
Resources
Students will be able to
describe the undefined terms of
geometry.
line
definition
collinear
coplanar
congruent
midpoint
Students will be able to use
geometry language to define new
geometric objects.
Students will be able to use
appropriate symbols and
language to refer to or define
geometric objects.
bisects
ray
counterexample
G-GPE.4-Use coordinates to
prove simple geometric
theorems algebraically. For
example, prove or disprove that
a figure defined by four given
points in the coordinate plane is
a rectangle; prove or disprove
that the point (1, ?3) lies on the
circle centered at the origin and
containing the point (0, 2).
G-CO.1-Know precise definitions
of angle, circle, perpendicular
line, parallel line, and line
segment, based on the
undefined notions of point, line,
distance along a line, and
Line segments
Angles
Midpoint on coordinate plane
Angle classifications
Angle pairs
Measure segment length with
a ruler
line segment
Meaure angles in
degrees with a protractor
angle
Classify angles
Use definitions to solve angle
problems
end points
vertex
sides
reflex measure of an angle
Analyze the midpoint formula
on the coordinate plane
angle bisector
Calculate the midpoint of a
segment on the coordinate plane
supplementary angles
complementary angles
vertical angles
linear pair of angles
Students will be able to
measure line segments with a
ruler.
With a protractor, students
will be able to draw angles of a
given measure in degrees.
Students will be able to use
the Midpoint Formula to
calculate the midpoint of a line
segment in the coordinate
plane.
Students will be able to use
appropriate symbols and
language to refer to or define
geometric objects.
distance around a circular arc.
G-CO.1-Know precise definitions
of angle, circle, perpendicular
line, parallel line, and line
segment, based on the undefined
notions of point, line, distance
along a line, and distance around
a circular arc.
Students will be able to use
geometry language to define
new geometric objects.
Polygons
Triangles
Quadrilaterals
Classify polygons based on
polygon names (1-12 sides
Students will be able to use
side lengths, angle measures, and n sides)
geometry language to define
and/or parallel sides
consecutive angles, vertices, new geometric objects.
and sides
Use definitions to solve
Students will be able to use
problems
appropriate symbols and
diagonal
language to refer to or define
convex
geometric objects.
concave
Students will be able to
classify polygons based on side
perimeter
lengths, angle measures, and/or
equilateral, equiangular,
parallel sides.
regular
acute, right, obtuse
scalene, isosceles,
equilateral
G-CO.1-Know precise definitions
of angle, circle, perpendicular
line, parallel line, and line
segment, based on the undefined
notions of point, line, distance
along a line, and distance around
a circular arc.
Circles
Define parts of circles
circle
Identify parts of circles
center
Use definitions to solve
problems
radius
diameter
chord
tangent, point of tangency
Students will be able to use
geometry language to define
new geometric objects.
Students will be able to use
appropriate symbols and
language to refer to or define
geometric objects.
congruent circles
concentric circles
arc, endpoints, minor arc,
major arc
semicircle
arc measure
central angle
G-CO.1-Know precise definitions
of angle, circle, perpendicular
line, parallel line, and line
segment, based on the undefined
notions of point, line, distance
along a line, and distance around
a circular arc.
Isometric drawings
Sketches of 3D shapes
Nets
Draw 3D shapes on isometric
dot paper
Sketch 3D shapes on paper
Identify appropriate nets by
visualization
Isometric drawings
Students will be able to use
geometry language to define
new geometric objects.
Students will be able to use
appropriate symbols and
language to refer to or define
geometric objects.
Students will be able to copy
isometric drawings on isometric
dot paper.
Students will be able to
sketch 3D shapes including
pyramids and prisms with
polygon bases that have more
than 4 sides.
Geometric probability
Analyze word problems
S.MD.6, S.MD.7
Draw pictures to model
probability
Students will be able to
analyze word problems to
calculate geometric probability.
Calculate probabilities
Standards for Mathematics
Standards for Mathematical Practice
Grade High School
Grade High School Standards for Mathematical Practice
The K-12 Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop
in their students. This page gives examples of what the practice standards look like at the specified grade level.
Standards
Explanations and Examples
Students are expected to:.
1. Make sense of problems and
persevere in solving them.
High school students start to examine problems by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals.
They make conjectures about the form and meaning of the solution and plan a solution pathway rather
than simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. By high school, students can explain correspondences between equations,
verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph
data, and search for regularity or trends. They check their answers to problems using different methods
and continually ask themselves, “Does this make sense?” They can understand the approaches of others
to solving complex problems and identify correspondences between different approaches.
High school students seek to make sense of quantities and their relationships in problem situations. They
abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause
as needed during the manipulation process in order to probe into the referents for the symbols involved.
Students use quantitative reasoning to create coherent representations of the problem at hand; consider
the units involved; attend to the meaning of quantities, not just how to compute them; and know and
flexibly use different properties of operations and objects.
High school students understand and use stated assumptions, definitions, and previously established
results in constructing arguments. They make conjectures and build a logical progression of statements to
explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and
can recognize and use counterexamples. They justify their conclusions, communicate them to others, and
respond to the arguments of others. They reason inductively about data, making plausible arguments that
take into account the context from which the data arose. High school students are also able to compare
the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is
flawed, and—if there is a flaw in an argument—explain what it is. High school students learn to
determine domains to which an argument applies, listen or read the arguments of others, decide whether
they make sense, and ask useful questions to clarify or improve the arguments.
Students are expected to:
2. Reason abstractly and
quantitatively.
Students are expected to:
3. Construct viable arguments
and critique the reasoning of
others.
Continued on next page
Standards
Students are expected to:
4. Model with mathematics.
Students are expected to:
5. Use appropriate tools
strategically.
Students are expected to:
6. Attend to precision
Explanations and Examples
High school students can apply the mathematics they know to solve problems arising in everyday life,
society, and the workplace. By high school, a student might use geometry to solve a design problem or
use a function to describe how one quantity of interest depends on another. High school students making
assumptions and approximations to simplify a complicated ituation, realizing that these may need
revision later. They are able to identify important quantities in a practical situation and map their
relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can
analyze those relationships mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on whether the results make sense,
possibly improving the model if it has not served its purpose.
High school students consider the available tools when solving a mathematical problem. These tools
might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. High school students
should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, high school students analyze graphs of functions and solutions generated using
a graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can enable
them to visualize the results of varying assumptions, explore consequences, and compare predictions with
data. They are able to identify relevant external mathematical resources, such as digital content located
on a website, and use them to pose or solve problems. They are able to use technological tools to explore
and deepen their understanding of concepts
High school students try to communicate precisely to others by using clear definitions in discussion with
others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of
measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate
accurately and efficiently, express numerical answers with a degree of precision appropriate for the
problem context. By the time they reach high school they have learned to examine claims and make
explicit use of definitions.
Continued on next page
Standards
Explanations and Examples
By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14,
older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing
line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective. They can see complicated things, such as
some algebraic expressions, as single objects or as being composed of several objects. For example, they
can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value
cannot be more than 5 for any real numbers x and y. High school students use these patterns to create
equivalent expressions, factor and solve equations, and compose functions, and transform figures.
Students are expected to:
High school students notice if calculations are repeated, and look both for general methods and for
shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x +
8. Look for and express regularity in 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series.
repeated reasoning.
As they work to solve a problem, derive formulas or make generalizations, high school students maintain
oversight of the process, while attending to the details. They continually evaluate the reasonableness of
their intermediate results.
Students are expected to:
7. Look for and make use of
structure.
CT Mathematics Unit Planning Organizers are designed to be a resource for developers of
curriculum. The documents feature standards organized in units with key concepts and skills
identified, and a suggested pacing guide for the unit. The standards for Mathematical Practice are
an integral component of CT Standards (CCSS) and are evident highlighted accordingly in the
units.
The information in the unit planning organizers can easily be placed into the curriculum
model in used at the local level during the revision process. It is expected that local and/or
regional curriculum development teams determine the “Big Ideas” and accompanying
“Essential Questions” as they complete the units with critical vocabulary, suggested
instructional strategies, activities and resources.
Note that all standards are important and are eligible for inclusion on the large scale
assessment to be administered during the 2014-15 school year. The Standards were written to
emphasize correlations and connections within mathematics. The priority and supporting standard
identification process emphasized that coherence. Standards were identified as priority or
supporting based on the critical areas of focus described in the CT Standards, as well as the
connections of the content within and across the K-12 domains and conceptual categories. In some
instances, a standard identified as priority actually functions as a supporting standard in a particular
unit. No stratification or omission of practice or content standards is suggested by the system
of organization utilized in the units.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Pacing: 4 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Experiment with transformations in the plane.
Understand similarity in terms of similarity transformations.
17
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.CO.1 Know precise definitions of angle, circle,
perpendicular line, parallel line, and line segment, based on
the undefined notions of point, line, distance along a line,
and distance around a circular arc.
CC.9-12.G.CO.4 Develop definitions of rotations, reflections,
and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
Students may use geometry software and/or manipulatives to model
transformations. Students may observe patterns and develop definitions of
rotations, reflections, and translations.
CC.9-12.G.CO.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).
Students may use geometry software and/or manipulatives to model and
compare transformations.
CC.9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or
regular polygon, describe the rotations and reflections that carry it
onto itself.
Students may use geometry software and/or manipulatives to model
transformations.
18
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.CO.5 Given a geometric figure and a rotation, Students may use geometry software and/or manipulatives to model
reflection, or translation, draw the transformed figure using, e.g., transformations and demonstrate a sequence of transformations that will carry
graph paper, tracing paper, or geometry software. Specify a a given figure onto another.
sequence of transformations that will carry a given figure onto
another.
CC.9-12.G.GPE.5 Prove the slope criteria for parallel and Lines can be horizontal, vertical, or neither.
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
Students may use a variety of different methods to construct a parallel or
point).
perpendicular line to a given line and calculate the slopes to compare the
relationships.
CC.9-12.G.GPE.4 Use coordinate 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).
Students may use geometric simulation software to model figures and prove
simple geometric theorems.
Example:
● Use slope and distance formula to verify the polygon formed by
connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.
CC.9-12.G.GPE.6 Find the point on a directed line segment
between two given points that partitions the segment in a given
ratio.
Students may use geometric simulation software to model figures or line
segments.
Examples:
19
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Priority and Supporting CCSS
Explanations and Examples*
● Given A(3, 2) and B(6, 11),
o
Find the point that divides the line segment AB two-thirds of the way
from A to B.
The point two-thirds of the way from A to B has x-coordinate twothirds of the way from 3 to 6 and y coordinate two-thirds of the way
from 2 to 11.
So, (5, 8) is the point that is two-thirds from point A to point B.
o
Find the midpoint of line segment AB.
CC.9-12.G.GPE.7 Use coordinates to compute perimeters of Students may use geometric simulation software to model figures.
polygons and areas of triangles and rectangles, e.g., using the
distance formula.*
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
20
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Bloom’s Taxonomy Levels
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
● precise definitions
Bloom’s Taxonomy Levels
● KNOW
1
● DEVELOP
3
● definition of similarity
● USE
3
● meaning of similarity
● EXPLAIN
2
● similarity transformations
● USE
3
● slope criteria for
● PROVE and USE
5 and 3
o
angle
o
circle
o
perpendicular line
o
parallel line
o
line segment
● definitions
o
rotations
o
reflections
o
translations
o
parallel lines
21
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
o
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
Bloom’s Taxonomy Levels
perpendicular lines
● geometric problems
● SOLVE
3
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
Consortium (SBAC) and has input into the development of the assessment.
22
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 1: Transformations and the Coordinate Plane
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
23
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 2: Congruence, Proof and Constructions
Pacing: 5 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Understand congruence in terms of rigid motions.
Prove geometric theorems.
24
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 2: Congruence, Proof and Constructions
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.CO.7 Use the definition of congruence in terms of A rigid motion is a transformation of points in space consisting of a sequence
rigid motions to show that two triangles are congruent if and of one or more translations, reflections, and/or rotations. Rigid motions are
only if corresponding pairs of sides and corresponding pairs assumed to preserve distances and angle measures.
of angles are congruent.
Congruence of triangles
Two triangles are said to be congruent if one can be exactly superimposed on
the other by a rigid motion, and the congruence theorems specify the
conditions under which this can occur.
CC.9-12.G.CO.6 Use geometric descriptions of rigid motions to A rigid motion is a transformation of points in space consisting of a sequence
transform figures and to predict the effect of a given rigid motion of one or more translations, reflections, and/or rotations. Rigid motions are
on a given figure; given two figures, use the definition of assumed to preserve distances and angle measures.
congruence in terms of rigid motions to decide if they are
congruent.
Students may use geometric software to explore the effects of rigid motion on
a figure(s).
CC.9-12.G.CO.8 Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from the definition
of congruence in terms of rigid motions
25
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 2: Congruence, Proof and Constructions
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.CO.9 Prove theorems about lines and angles.
Theorems include: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles
are congruent and corresponding angles are congruent;
points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment's endpoints.
Students may use geometric simulations (computer software or graphing
calculator) to explore theorems about lines and angles.
CC.9-12.G.CO.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
Students may use geometric software to make geometric constructions.
Examples:
● Construct a triangle given the lengths of two sides and the measure of the
angle between the two sides.
● Construct the circumcenter of a given triangle.
26
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 2: Congruence, Proof and Constructions
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
Bloom’s Taxonomy Levels
● definition of congruence
● USE
3
● two triangles are congruent if and only if corresponding pairs
● SHOW
2
● EXPLAIN
2
of sides and corresponding pairs of angles are congruent
● criteria for triangle congruence
o
ASA
o
SAS
o
SSS
Essential Questions
Corresponding Big Ideas
27
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 2: Congruence, Proof and Constructions
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
Consortium (SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
28
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 3: Polygons
Pacing: 4 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Prove geometric theorems.
Make geometric constructions.
29
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 3: Polygons
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.CO.10 Prove theorems about triangles. Theorems
include: measures of interior angles of a triangle sum to 180
degrees; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a
triangle meet at a point.
Students may use geometric simulations (computer software or graphing
calculator) to explore theorems about triangles.
CC.9-12.G.CO.13 Construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle.
Students may use geometric software to make geometric constructions.
CC.9-12.G.CO.11 Prove theorems about parallelograms.
Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms
with congruent diagonals.
Students may use geometric simulations (computer software or graphing
calculator) to explore theorems about parallelograms.
CC.9-12.G.CO.13 Construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle.
Students may use geometric software to make geometric constructions.
30
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 3: Polygons
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
Bloom’s Taxonomy Levels
● theorems about triangles
● PROVE
5
● theorems about parallelograms
● PROVE
5
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
31
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 3: Polygons
Consortium (SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
32
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Pacing: 5 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Prove theorems involving similarity.
Define trigonometric ratios and solve problems involving right triangles.
33
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Priority and Supporting CCSS
CC.9-12.G.SRT.5 Use congruence and similarity criteria for
triangles to solve problems and to prove relationships in
geometric figures.
Explanations and Examples*
Similarity postulates include SSS, SAS, and AA.
Congruence postulates include SSS, SAS, ASA, AAS, and H-L.
Students may use geometric simulation software to model transformations and
demonstrate a sequence of transformations to show congruence or similarity of
figures.
CC.9-12.G.SRT.1 Verify experimentally the properties of dilations
given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the
dilation to a parallel line, and leaves a line passing through the
center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio
given by the scale factor.
A dilation is a transformation that moves each point along the ray through the
point emanating from a fixed center, and multiplies distances from the center
by a common scale factor.
Students may use geometric simulation software to model transformations.
Students may observe patterns and verify experimentally the properties of
dilations.
34
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Priority and Supporting CCSS
CC.9-12.G.SRT.2 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
Explanations and Examples*
A similarity transformation is a rigid motion followed by a dilation.
Students may use geometric simulation software to model transformations and
demonstrate a sequence of transformations to show congruence or similarity of
figures.
CC.9-12.G.SRT.3 Use the properties of similarity transformations
to establish the AA criterion for two triangles to be similar.
CC.9-12.G.SRT.4 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.
Students may use geometric simulation software to model transformations and
demonstrate a sequence of transformations to show congruence or similarity of
figures.
CC.9-12.G.SRT.8 Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles in applied
problems.*
Students may use graphing calculators or programs, tables, spreadsheets, or
computer algebra systems to solve right triangle problems.
Example:
● Find the height of a tree to the nearest tenth if the angle of elevation of
the sun is 28° and the shadow of the tree is 50 ft.
35
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Priority and Supporting CCSS
Explanations and Examples*
36
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Priority and Supporting CCSS
CC.9-12.G.SRT.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.
Explanations and Examples*
Students may use applets to explore the range of values of the trigonometric
ratios as θ ranges from 0 to 90 degrees.
37
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.SRT.7 Explain and use the relationship between the
sine and cosine of complementary angles.
Geometric simulation software, applets, and graphing calculators can be used
to explore the relationship between sine and cosine.
CC.9-12.G.MG.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).*
Students may use simulation software and modeling software to explore which
model best describes a set of data or situation.
38
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
● congruence / similarity criteria for triangles
Bloom’s Taxonomy Levels
● USE
3
● SOLVE (problems)
3
● PROVE (relationships in geometric
5
figures)
● trigonometric ratios
● USE
3
● SOLVE
3
● Pythagorean Theorem
● right triangles in applied problems
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
39
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 4: Similarity, Proof and Trigonometry
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
Consortium (SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
40
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Pacing: 4 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Understand and apply theorems about circles.
Translate between the geometric description and the equation for a conic section.
41
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Priority and Supporting CCSS
CC.9-12.G.C.2 Identify and describe relationships among
inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the radius of
a circle is perpendicular to the tangent where the radius
intersects the circle.
Explanations and Examples*
Examples:
Given the circle below with radius of 10 and chord length of 12, find the
distance from the chord to the center of the circle.
Find the unknown length in the picture below.
42
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.C.3 Construct the inscribed and circumscribed circles
of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
Students may use geometric simulation software to make geometric
constructions.
CC.9-12.G.C.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.
Students can use geometric simulation software to explore angle and radian
measures and derive the formula for the area of a sector.
CC.9-12.G.C.1 Prove that all circles are similar.
Students may use geometric simulation software to model transformations and
demonstrate a sequence of transformations to show congruence or similarity of
figures.
CC.9-12.G.GPE.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.
Students may use geometric simulation software to explore the connection
between circles and the Pythagorean Theorem.
Examples:
● Write an equation for a circle with a radius of 2 units and center at (1, 3).
● Write an equation for a circle given that the endpoints of the diameter are
(-2, 7) and (4, -8).
43
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Priority and Supporting CCSS
Explanations and Examples*
● Find the center and radius of the circle 4x2 + 4y2 - 4x + 2y – 1 = 0.
CC.9-12.G.GPE.2 Derive the equation of a parabola given a focus
and directrix.
Students may use geometric simulation software to explore parabolas.
Examples:
● Write and graph an equation for a parabola with focus (2, 3) and directrix
y = 1.
CC.9-12.G.GPE.4 Use coordinate 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).
Students may use geometric simulation software to model figures and prove
simple geometric theorems.
Example:
● Use slope and distance formula to verify the polygon formed by
connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.
44
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
● relationships among
Bloom’s Taxonomy Levels
● IDENTIFY
1
● DESCRIBE
1
● DERIVE (using similarity)
5
● DEFINE
1
● DERIVE
5
● equation of circle
● DERIVE
5
● center and radius of a circle
● FIND (complete the square)
3
o
central angles
o
inscribed angles
o
circumscribed angles
o
radii
o
chords
o
tangent
● length of the arc intercepted by an angle is proportional to
the radius
● the radian measure of the angle
● formula for the area of a sector
45
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 5: Circles and other Conic Sections
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
Consortium (SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
46
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Pacing: 4 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards Overview
Explain volume formulas and use them to solve problems.
Apply geometric concepts in modeling situations.
47
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Priority and Supporting CCSS
Explanations and Examples*
CC.9-12.G.GMD.3 Use volume formulas for cylinders,
pyramids, cones, and spheres to solve problems.*
Missing measures can include but are not limited to slant height, altitude,
height, diagonal of a prism, edge length, and radius.
CC.9-12.G.GMD.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.
Cavalieri’s principle is if two solids have the same height and the same crosssectional area at every level, then they have the same volume.
CC.9-12.G.GMD.4 Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional
objects.
Students may use geometric simulation software to model figures and create
cross sectional views.
Example:
● Identify the shape of the vertical, horizontal, and other cross sections of a
cylinder.
CC.9-12.G.MG.1 Use geometric shapes, their measures, and their Students may use simulation software and modeling software to explore which
properties to describe objects (e.g., modeling a tree trunk or a
model best describes a set of data or situation.
human torso as a cylinder).*
48
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Priority and Supporting CCSS
CC.9-12.G.MG.2 Apply concepts of density based on area and
volume in modeling situations (e.g., persons per square mile,
BTUs per cubic foot).*
Explanations and Examples*
Students may use simulation software and modeling software to explore which
model best describes a set of data or situation.
49
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
● Volume formulas for
o
Cylinders
o
Pyramids
o
Cones
o
Spheres
● USE (to solve problems)
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
50
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Bloom’s Taxonomy Levels
3
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the Smarter Balanced Assessment
Consortium (SBAC) and has input into the development of the assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed appropriate by the teacher.
51
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Pacing: 3 weeks + 1 week for reteaching/enrichment
Mathematical Practices
Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and lea
old are to be emphasized in the unit.
e of problems and persevere in solving them.
stractly and quantitatively.
viable arguments and critique the reasoning of others.
mathematics.
priate tools strategically.
recision.
nd make use of structure.
nd express regularity in repeated reasoning.
Standards Overview
d independence and conditional probability and use them to interpret data.
rules of probability to compute probabilities of compound events ina uniform probability model.
52
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Priority and Supporting CCSS
Explanations and Examples*
CP.1 Describe events as subsets of a sample Intersection: The intersection of two sets A and B is the set of elemen
set of outcomes) using characteristics (or are common to both set A and set B. It is denoted by A ∩ B and is rea
of the outcomes, or as unions, intersections, or intersection B’.
ts of other events ("or," "and," "not").
● A ∩ B in the diagram is {1, 5}
● this means: BOTH/AND
Union: The union of two sets A and B is the set of elements, which are
in B or in both. It is denoted by A ∪ B and is read ‘A union B’.
● A ∪ B in the diagram is {1, 2, 3, 4, 5, 7}
● this means: EITHER/OR/ANY
● could be both
Complement: The complement of the set A ∪B is the set of elements
members of the universal set U but are not in A ∪B. It is denoted by (A
53
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Priority and Supporting CCSS
Explanations and Examples*
(A ∪ B )’ in the diagram is {8}
CP.3 Understand the conditional probability of A
P(A and B)/P(B), and interpret independence of A
ying that the conditional probability of A given B
as the probability of A, and the conditional
of B given A is the same as the probability of B.
CP.2 Understand that two events A and B are
if the probability of A and B occurring together is the
eir probabilities, and use this characterization to
they are independent.
CP.4 Construct and interpret two-way frequency tables
n two categories are associated with each object
ied. Use the two-way table as a sample space to
nts are independent and to approximate conditional
For example, collect data from a random sample of
our school on their favorite subject among math,
English. Estimate the probability that a randomly
dent from your school will favor science given that the
tenth grade. Do the same for other subjects and
results.
Students may use spreadsheets, graphing calculators, and simulations
create frequency tables and conduct analyses to determine if events a
independent or determine approximate conditional probabilities.
CP.5 Recognize and explain the concepts of
robability and independence in everyday language
y situations. For example, compare the chance of
cancer if you are a smoker with the chance of being a
u have lung cancer.
Examples:
● What is the probability of drawing a heart from a standard deck o
on a second draw, given that a heart was drawn on the first draw
replaced? Are these events independent or dependent?
● At Johnson Middle School, the probability that a student takes co
science and French is 0.062. The probability that a student takes
computer science is 0.43. What is the probability that a student ta
French given that the student is taking computer science?
54
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Priority and Supporting CCSS
Explanations and Examples*
CP.6 Find the conditional probability of A given B
on of B's outcomes that also belong to A, and
e answer in terms of the model.
Students could use graphing calculators, simulations, or applets to mo
probability experiments and interpret the outcomes.
CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B)
, and interpret the answer in terms of the model.
Students could use graphing calculators, simulations, or applets to mo
probability experiments and interpret the outcomes.
Example:
● In a math class of 32 students, 18 are boys and 14 are girls. On a
test, 5 boys and 7 girls made an A grade. If a student is chosen a
random from the class, what is the probability of choosing a girl o
student?
55
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
Concepts
Skills
What Students Need to Know
What Students Need To Be Able To Do
Bloom’s Taxonomy
● DESCRIBE (events)
1
onal probability
● UNDERSTAND
2
ndent events
● INTERPRET
2
onal probability
● FIND
1
● INTERPRET (in terms of the model)
1
s
space
ctions
ments
Essential Questions
Corresponding Big Ideas
Standardized Assessment Correlations
(State, College and Career)
Expectations for Learning (in development)
This information will be included as it is developed at the national level. CT is a governing member of the
Smarter Balanced Assessment Consortium (SBAC) and has input into the development of the
56
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
assessment.
Unit Assessments
The items developed for this section can be used during the course of instruction when deemed
appropriate by the teacher.
GEOMETRY
Congruence
Experiment with transformations in the plane.
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc. [G-CO1]
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). [GCO2]
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry
it onto itself. [G-CO3]
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments. [G-CO4]
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure
onto another. [G-CO5]
Core
What Does It Mean?
Standard
Know precise definitions
1
of angle, circle,
perpendicular line, parallel
line, and line segment,
based on the undefined
Pre-Requisite Skills
What Does Mastery
Look Like?
-Exposure to (recognize -Students will be
picture of or be able to able to define,
draw a picture of) an
draw, and
angle, circle,
recognize picture of
perpendicular line,
selected definitions
57
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Future Math
Conics
Unit Circle
Systems of equations
(graphing )
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
2
3
notions of point, line,
distance along a line, and
distance around a circular
arc.
parallel line, line
segment, point, line,
distance along a line,
and distance around a
circular arc.
Represent transformations
using a variety of methods
(ELMO, Geometer’s
Sketchpad); using the
coordinates of two figures,
describe the change in x
and y values; compare
distance and angles of
both figures
Given a rectangle,
parallelogram, trapezoid,
or regular polygon,
describe the rotations and
reflections that carry it
onto itself.
-Vocabulary
(transformation,
coordinate, distance,
angles, vertices)
-Coordinate system (xaxis, y-axis, origin, plot,
point)
-Introduction to
Geometer’s Sketchpad
-Vocabulary (rectangle,
parallelogram,
trapezoid, regular
polygon, rotations, and
reflections)
-Coordinate system (xaxis, y-axis, origin, plot,
point)
-Basic degree changes
-Know how to describe
changes in the x and y
values
-Vocabulary (angles,
circles, perpendicular
lines, parallel lines, and
line segments)
-Recognize examples
of rotations,
translations, and
reflections
-Recognize figures of
angles, circles,
perpendicular lines,
parallel lines, and line
segments
4
Develop definitions of
rotations, reflections, and
translations in terms of
angles, circles,
perpendicular lines,
parallel lines, and line
segments.
5
Given a geometric figure
and a rotation, reflection,
-Recognize geometric
figures.
58
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
(an angle, circle,
perpendicular line,
parallel line, line
segment, point,
line, distance along
a line, and distance
around a circular
arc).
-Students should be Vectors
able to complete a Graph transformations
representation of a Matrices
transformation and
compare distance
and angles of both
figures
-Students should be
able to describe
rotations and
reflections on the
coordinate plane.
Calculus rotations ( about
the axes )
Pre-Cal/algebra graph
transformations
Matrices
-Students should be
able to identify
examples of
rotations,
reflections, and
translations and
figures of angles,
circles,
perpendicular lines,
parallel lines, and
line segments to
develop their own
definitions.
-Students should be
able to draw a
Calculus rotations ( about
the axes )
Pre-Cal/algebra graph
transformations
Matrices
Calculus rotations ( about
the axes )
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
or translation, draw the
transformed figure using,
e.g., graph paper, tracing
paper, or geometry
software. Specify a
sequence of
transformations that will
carry a given figure onto
another.
-Recognize rotation,
reflection, and
translation.
-Coordinate system (xaxis, y-axis, origin, plot,
point)
transformed figure Pre-Cal/algebra graph
using graph paper, transformations
tracing paper, or
Matrices
geometry software.
-Students should be
able to describe the
order of
transformations
that will carry a
given figure onto
another on the
coordinate plane.
Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for
development of concept of geometric proof)
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. [G-CO6]
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent. [G-CO7]
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence
in terms of rigid motions. [G-CO8]
Core
What Does It Mean?
Standard
6
Use translation, rotation,
reflection
7
Show that a triangle that
has been rotated,
reflected, and translated is
still congruent to the
original because they have
corresponding pairs of
sides and angles.
8
Use rotation, reflection,
and translation to prove
triangle congruence (ASA,
SAS, and SSS).
Pre-Requisite Skills
What Does Mastery
Look Like?
-translation
Students will
-rotation
demonstrate that
-reflection
they can take a figure
-Congruence
and apply a
transformation.
-Vocabulary (rotation, -Students should be
reflection, translation, able to explain why a
angles, corresponding triangle that has
sides, corresponding
been rotated,
angles, congruency)
reflected, or
translated is still
congruent to the
original.
-Vocabulary (rotation, -Students should be
reflection, translation, able to prove a
congruency, angle,
triangle that has
side)
been rotated,
59
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Future Math
Pre-Engineering
Computer Science
and Animation
Pre-Engineering
Computer Science
and Animation
Pre-Engineering
Computer Science
and Animation
Trigonometry (
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
-Coordinate system (x- reflected, or
axis, y-axis, origin,
translated is
plot, point)
congruent to the
original by ASA, SAS,
or SSS.
understanding law
of sine, cosine)
Unit Circle
Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing
proofs)
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal
crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on
a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [G-CO9]
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; base
angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point. [G-CO10]
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are
congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with
congruent diagonals. [G-CO11]
Core
What Does It Mean?
Pre-Requisite Skills
Standard
9
Recognize angle pairs and -Vocabulary
what is true when the lines Vertical angles
are parallel. Use theorems Transversal
and definition of bisectors.
Parallel lines
Congruent
Alternate interior
angles
Corresponding
angles
Perpendicular
Bisector
Segment
Equidistant
End point
Same-Side interior
angles
Alternate exterior
angles
Supplementary
-Developing
equations based
on known
60
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
What Does Mastery
Look Like?
Students will be
able to develop and
solve algebraic
equations based on
the relationships
that exist between
the angles /
segments
Students will be
able to prove the
alternate exterior
angle theorem by
using the
corresponding
angles postulate
and vertical angles.
Students should be
able to use the
bisector definition
and bisector
Future math
Architecture
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
relationships
10
11
12
Properties of triangles will
be discovered through
logical reasoning. This
includes the fact that the
sum of the interior angles
of a triangle equal 180
degrees, the base angles
of an isosceles triangle are
congruent, and a
midsegment is parallel to
the third side and half its
length.
Prove theorems about
parallelograms
Use a compass and
straight edge, paper
folding, reflectors,
software (Geometer’s
Sketchpad) to create
geometric constructions
-Vocabulary
Interior angles
Base angles
Isosceles triangles
Midsegment
Medians
-Vocabulary
Parallelogram
Congruent
Opposite angles
Diagonals
Bisect
Rectangles
Opposite Sides
-Triangle
congruence
theorems AAS, SAS
etc
- CPCTC
-Vocabulary
Circle Center
Arc
Angle
Perpendicular
Bisector
Parallel
Segment
Radius
-Skills
Use a compass
and straight edge
61
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
theorems in related
proofs.
Students will be
able to use more
than one method to
prove the triangle
sum theorem;
Students will be
able to prove the
base angles
theorem; prove the
midsegment
theorem
Unit Circle
Trigonometry
Word problems
Engineering
Architecture
Art
Students will be
able to use a variety
of methods to
prove relationships
about
parallelograms
Word problems
Engineering
Architecture
Art
Students will be
able to accurately
do the following
constructions:
- Copy a
segment
- Copy an
angle
- Bisect a
segment
- Bisect an
angle
- Perpendicul
ar lines
- Parallel lines
( through a
Engineering
CAD
Architecture
Construction/design
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
13
Construct an equilateral
triangle, square, and
regular hexagon inscribed
in a circle
14
Verify experimentally the
properties of dilations
given by a center and a
scale factor
15
Determine if two figures
are similar, explain why
two triangles are similar.
16
Use the properties of
-Vocabulary
Equilateral
triangle
Square
Regular
Inscribed
Circle
Hexagon
-Vocabulary
Scale factor
Dilation
Center of dilation
Parallel Line
Ratio
-Vocabulary
Similarity
Similarity
transformation
Similar
Proportional
Corresponding
angles
Corresponding
sides
Using cross
products to solve
proportions
-Vocabulary
62
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
pt not on
the line)
- Perpendicul
ar bisector
of a segment
Students will be
Unit circle
able to accurately
Calculus
construct an
Engineering
equilateral triangle,
square, and regular
hexagon inscribed
in a circle.
Students will be
able to draw and
describe dilations (
properties) using
various tools (
software/compass
& straight edge)
Engineering
CAD
Architecture
Graphing
Transformations
Students will be
able to determine
and explain why
two triangles are
similar using
transformations
and corresponding
parts.
Daily life
(proportion
concept)
All other Math
classes
Students will be
CAD/ Engineering
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
17
18
similarity transformations
to establish the AA
criterion for two triangles
to be similar
Prove theorems about
triangles
Similar
Transformation
-sum of angles of a
triangle
SAS, AAS, SAS, SSS,
HL
Vocabulary
-Hypotenuse
-Angle
-Side
-Congruent
-proportion
-similarity
-Pythagorean
Theorem
Using the fact that two
-proving triangles
triangles are congruent or are congruent
similar to prove other facts -Using CPCTC
about various geometric
-Recognizing
figures.
properties of other
geometric figures
19
Trigonometric ratio
definitions can be derived
from the similarity of the
side ratios in right
triangles as properties of
the angles.
-right triangle
-sine,cosine,
tangent
-opposite,
adjacent,
hypotenuse
-ratio
-similarity
20
Explain and use the
relationship between the
sine and cosine of
complementary angles
-sine
-cosine
-complementary
-opposite,
adjacent,
hypotenuse
21
*Use the trigonometric
ratios and the
Pythagorean theorem to
-sine
-cosine
-tangent
63
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
able to discover,
recognize, and use
the AA criterion for
triangle similarity.
Students will be
able to prove
theorems about
triangles including
but not limited to
the Triangle angle
bisector theorem
and the
Pythagorean
theorem using
triangle similarity.
Students will be
able to produce a
proof in which
several steps are
required including
using congruent
and / or similar
triangles.
Students will be
able to
demonstrate the
connection
between the
trigonometric ratios
and the side ratios
in right triangles as
a property of the
acute angles.
Students will
demonstrate that
the sine of one of
the two acute
angles is the cosine
of the other and
vice versa.
Students will be
able to use the
trigonometric ratios
Design
College Algebra
Proof concepts used
in all future math
courses/life
Unit Circle
Calculus
Trigonometry
Physics
Calculus
Proof concepts used
in all future math
courses
Algebra II
Precalculus
Calculus
Probability/Stats
Alg. Connections
Trigonometry
Unit Circle
Trigonometry
Calculus
Algebra II
Alg. Connections
Unit Circle
Trigonometry
Calculus
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
solve right triangles in
applied problems.
-acute angles
-right triangle
22
+Derive the formula
A=1/2abSin(C) by drawing
an auxiliary line from a
vertex perpendicular to
the opposite side
-vocabulary
-sine
-auxiliary line
-vertex
-perpendicular
-opposite side
-derive
23
+Prove the laws of Sines
and Cosines and use them
to solve problems
-sine
-cosine
-acute angle
24
+Understand and apply
the Law of Sines and
Cosines to find unknown
measurements in any
triangle.
-Law of Sines
-Law of Cosines
-Acute angles
25
Prove that all circles are
similar
26
Identify and describe
relationships among
inscribed angles, radii, and
chords
-circle
-radius
-diameter
-circumference
-pi
-radius
-chord
-circle
-inscribed
-circumscribed
-central angle
-right angle
64
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
to solve real world
problems such as
measuring ‘tall’
objects, finding a
distance, etc.
Students will be
able to draw an
auxiliary line in a
triangle in such a
way that the
formula
A=1/2abSin(C) is
derived.
Students will be
able to prove the
Law of Sines and
Cosines.
Additionally, they
will be able to use
them to solve
problems.
Students will be
able to use these
Laws to solve
problems such as
surveying problems,
resultant forces,
etc. The student
should be able to
use any kind of
triangle.
Students will be
able to
demonstrate or
explain why all
circles are similar.
Students will be
able to describe and
use the relationship
between inscribed,
central, and
circumscribed
angles, show that
Algebra II
Alg. Connections
Algebra II
Precalculus
Calculus
Trigonmetry
Physics
Algebra II
Precalculus
Calculus
Trigonometry
Physics
Algebra II
Precalculus
Calculus
Trigonometry
Physics
Conics
Trigonometry
PreCalculus
Engineering
Computer
Animations
Vectors
Unit Circle
Calculus
Trigonometry
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
-perpendicular
-tangent
-intersection
27
Construct the inscribed
and circumscribed circles
of a triangle and prove
properties of angles for a
quadrilateral inscribed in a
circle.
-inscribed
-circumscribed
-quadrilateral
-opposite,
consecutive angles
28
+Construct a tangent line
from a point outside a
given circle to the circle
-tangent
-circle
29
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.
-similarity
-sector ( Area )
-arc ( length )
-circle
-proportion
-radius
-intercepted arc
-radius
-radian measure
-constant
30
Derive the equation of a
circle given a center and
radius using the
Pythagorean theorem.
Complete the square to
find the center and radius
of a circle given by an
-circle equation
-completing the
square
-Pythagorean
theorem
-radius
-circle center
65
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
angles inscribed in a
semicircle are right
angles (inscribed on
a diameter), radius
is tangent to a
tangent at the
intersection point.
Students will be
able to use either
compass and
straight edge or
software to prove
properties of
inscribed and
circumscribed
triangles.
Students will use a
compass and
straight edge or
software to
construct the
tangent to a circle
from a point
outside the circle.
Students will be
able to show that
the length of the arc
is proportional to
the radius and the
radian measure is
the constant of
proportionality.
Students should be
able to derive the
formula for the area
of a sector.
Students will be
able to derive the
equation of a circle
with given radius
and center using
the Pythagorean
theorem. Students
Physics
Engineering
CAD
Architecture
Calculus
PreCalculus
College Algebra
Unit Circle
Algebra II
PreCalculus
Conics
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
equation.
-diameter
31
Use coordinates to prove
simple geometric
theorems algebraically.
32
Prove the slope criteria for
parallel and perpendicular
lines and use them to
solve geometric problems.
-solving equations
-coordinate proofs
-slope
-parallel &
perpendicular lines
-Pythagorean
Theorem
-distance formula (
between points)
-quadrilateral
definitions
-quadrilateral
properties
-graphing circles
-slope
-parallel
-perpendicular
-graphing lines
-equations of lines
33
Find the point on a
directed line segment
between two given points
that partitions the
segment in a given ratio.
Use coordinates to
compute perimeters of
polygons and areas of
triangles and rectangles.
-line segment
-point
-ratio
-proportion
Determine areas and
perimeters of regular
polygons including
-perimeter
-area
-inscribed
34
35
-distance formula
-perimeter
-polygon
-area
-triangle
-rectangle
66
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
will be able to
complete the
square to find the
center and radius of
a circle.
Students should be
able to use algebra
and algebraic
formulas to prove
or disprove
properties of circles
and quadrilaterals.
Students will be
able to determine
and use the
relationship
between the slopes
of parallel and
perpendicular lines.
Students will be
able to locate a
point on a segment
using a ratios and
proportions
Students will be
able to calculate
perimeters of
polygons and areas
of rectangles and
triangles by using
the distance
formula for
example.
Students will be
able to calculate the
perimeter and areas
Algebra II
Precalculus
All future math
courses
Algebra II
All future math
courses
All future math
courses
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
inscribed and
circumscribed polygons,
given the coordinates of
vertices or other
characteristics
Give an informal argument
for the formulas for the
circumference of a circle,
area of a circle, volume of
a cylinder, pyramid, and
cone.
-circumscribed
-polygon
-vertex
of regular polygons.
-circle
-circumference
-area
-volume
-cylinder
-pyramid
-cone
Algebra II
Physics
Calculus
PreCalculus
37
Use volume formulas for
cylinders, pyramids, cones,
and spheres to solve
problems
-volume
-cylinder
-cone
-sphere
38
Determine the relationship
between surface areas of
similar figures and
volumes of similar figures
39
Identify the shapes of twodimensional cross-sections
of three-dimensional
objects, and identify 3-D
objects generated by
rotations of twodimensional objects.
40
Use geometric shapes,
their measures, and their
properties to describe
objects
-similarity
-area
-volume
-ratio
-proportion
-circle
-ellipse
-parabola
-hyperbola
-square
-rectangle
-parallelogram
-triangle
-various and
sundry 3-D objects
-solid geometry
shapes
-volume
-area
-measuring
Students will be
able to use
dissection
arguments,
cavalieri’s principle,
and informal limit
arguments to derive
or defend the
formulas for
circumference,
area, and volume.
Students will be
able to use the
volume formulas for
cylinders, cones,
and spheres to
solve problems.
Students will be
able to use the fact
that two figures are
similar to calculate
volume or area.
Students will be
able to determine
the names of cross
sections of 3-D
objects.
Students will be
able to use
composite figures
to model and
estimate area and
volume of various
CAD
Computer Science
Art
Architecture
36
67
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Calculus
PreCalculus
Calculus
Construction
Algebra II
Word Problems
Architecture
Computer
Animation
Computer Science
CAD
Airforce/military
tests
Conics
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
41
Apply concepts of density
based on area and volume
in modeling situations (
persons per square miles,
BTUs per cubic feet)
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).*
-density
-ratios
-proportions
-area
-volume
( possible )
-area
-perimeter
-minimum
-maximum
-distance
-measurements
43
Understand the
conditional probability of
A given B as P(A and
B)/P(B), and interpret
independence of A and B
as saying that the
conditional probability of
A given B is the same as
the probability of A, and
the conditional probability
of B given A is the same as
the probability of B.
-Probability
-independent
variable
-dependent
variable
44
Construct and interpret
two-way frequency tables
of data when two
categories are associated
with each object being
classified. Use the twoway table as a sample
space to decide if events
are independent and to
approximate conditional
probabilities.
Recognize and explain the
42
45
3-D objects.
Students will be
able to model
density using
geometric concepts.
Physics
Chemistry
Precalculus
Calculus
Students will be
able to use
geometric concepts
to design solutions
to real life
problems. This
would be a good
opportunity to use
graphing
calculators,
spreadsheets, or
GSP
Students will be
able to calculate the
probability of
independent of
events.
Physics
Architecture
Engineering
CAD
Design/planning
Real life budget
management
-frequency table
-tally marks
-probability
-independent
variable
-dependent
variable
-random
Students will be
able to collect, tally,
and interpret
statistical data.
Statistics
Economics/Govern
ment/Social Studies
-probability
Students should be
Statistics
68
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Algebra II
PreCalculus
Probability/Stats
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
concepts of conditional
probability and
independence in everyday
language and everyday
situations.
-independent
variable
-dependent
variable
Find the conditional
probability of A given B as
the fraction of B's
outcomes that also belong
to A, and interpret the
answer in terms of the
model.
Apply the Addition Rule,
P(A or B) = P(A) + P(B) - P(A
and B), and interpret the
answer in terms of the
model.
-conditional
probability
-model/modeling
-fractions
-outcome
(+) Apply the general
Multiplication Rule in a
uniform probability model,
P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret
the answer in terms of the
model.
(+) Use permutations and
combinations to compute
probabilities of compound
events and solve
problems.
-probability
-independent
events
-dependent events
-fractions
50
(+) Use probabilities to
make fair decisions (e.g.,
drawing by lots, using a
random number
generator).
-probability
-random
51
(+) Analyze decisions and
-probability
strategies using probability
concepts (e.g., product
testing, medical testing,
pulling a hockey goalie at
46
47
48
49
-probability
-independent
events
-dependent events
-fractions
-combination
-permutation
-probability
69
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
able to compare
and analyze
statistical data to
determine if events
are dependent or
independent.
Students should be
able to find the
probability of
dependent events.
Business
Probability
Economics
Students will be
able to model and
also show
mathematically the
probability of
events A or B.
Students will be
able to model and
also show
mathematically the
probability of
events A and B.
Statistics
Business
Probability
Economics
Students should be
able to calculate a
combination and a
permutation and
solve related
problems
Students should be
able to use
probability to make
a fair decision in a
given random
situation.
Students will be
able to use
probability to
determine
appropriate
Computer Science (
Gaming industry)
Statistics
Probability
Business
Statistics
Business
Probability
Economics
Statistics
Business
Probability
Economics
Government
Statistics
Economics
Probability
Government
Statistics
Economics/
Marketing
Probability
Connecticut Curriculum Design Unit Planning Organizer
Geometry
Unit 6: Extend to Three Dimensions
the end of a game).
strategies.
70
Adapted from The Leadership and Learning Center “Rigorous Curriculum Design” model.
*Adapted from the Arizona Academic Content Standards.
Sports and/or
Medicine