ITHS
2013 – 2014
Geometry: MGS-42
MATHEMATICS
Curriculum Map
Common Core State Standards
Pierre-Max Foucault-Room 340
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Common Core State Standards
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them. (MP.1)
Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process
which sometimes requires perseverance, flexibility, and a bit of ingenuity.
2. Reason abstractly and quantitatively. (MP.2)
The concrete and the abstract can complement each other in the development of mathematical understanding:
representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete
context can help make sense of abstract symbols.
3. Construct viable arguments and critique the reasoning of others. (MP.3)
A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and
supporting evidence.
4. Model with mathematics. (MP.4)
Many everyday problems can be solved by modeling the situation with mathematics.
5. Use appropriate tools strategically. (MP.5)
Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen
mathematical understanding.
6. Attend to precision. (MP.6)
Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical
explanations.
7. Look for and make use of structure. (MP.7)
Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.
8. Look for and express regularity in repeated reasoning. (MP.8)
Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results
more quickly and efficiently.
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Geometry: Common Core State Standards
The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students
explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments.
Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are
emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school
CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience
mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized
into five units are as follows.
Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have
prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to
be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle
congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems
about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of
similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle
trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order
to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are
able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles.
Extending to Three Dimensions: Students’experience with two-dimensional and three-dimensional objects is extended to include informal explanations
of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of crosssections and the result of rotating a two-dimensional object about a line.
Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances,
students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of
parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the
geometric and algebraic definitions of the parabola.
Circles With and Without Coordinates: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius,
inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures.They study
relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the
distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the
graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine
intersections between lines and circles or parabolas and between two circles.
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Unit 7- Properties of Polygons and Quadrilaterals
G-CO.3.11
G-GPE.2.4
G-GPE.2.5
SSA
Unit 8- Similarity
G-SRT.1.2
G-SRT.1.3
Unit 9- Right Triangles and Trigonometry
G-SRT.3.6
G-SRT.3.7
G-SRT.3.8
G-SRT.4.10
G-SRT.4.11
Unit 10- Transformational Geometry
G-CO.1.4
G-CO.1.5
G-CO.1.2
G-CO.1.3
G-CO.2.6
G-SRT.1.1
Unit 11- Two-Dimensional Measurements
G-MG.1.1
G-MG.1.2
G-MG.1.3
G-GPE.2.7
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Unit 12- Properties of Circles
G-GPE.1.1
G-C.1.1
G-C.1.2
G-C.1.3
G-C.1.4
G-C.2.5
The following English Language Arts CCSS should be taught throughout the course:
RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing
tasks, attending to special cases or exceptions defined in the text.
RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in
context and topics.
RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information
expressed visually or mathematically into words.
L.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners.
L.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of
each source.
L.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or
exaggerated or distorted evidence.
SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can
follow the line of reasoning.
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
WHST.1.1: Write arguments focused on discipline-specific content.
WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task,
purpose, and audience.
WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.
Unit 7-Properties of Quadrilaterals
Standard
The students will:
G-CO.3.11
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.)
MP #2, #3
Mathematics Department
ITHS
Essential Question(s):
How are quadrilaterals precisely classified?
Learning Goals
I can:
 apply theorems, postulates, or definitions to
prove theorems about parallelograms,
including:
a) Prove opposite sides of a parallelogram
are congruent;
b) Prove opposite angles of a parallelogram
are congruent;
c) Prove the diagonals of a parallelogram
bisect each other;
d) Prove that rectangles are parallelograms
with congruent diagonals.
CCSS
Remarks
The definition of a
parallelogram includes
two pairs of opposite
sides parallel and
congruent.
This includes
rectangles, squares,
rhombi, kites, and
trapezoids.
Resources
http://www.shmoop.com/com
mon-corestandards/handouts/gco_worksheet_11.pdf
http://www.shmoop.com/com
mon-corestandards/handouts/gco_worksheet_11_ans.pdf
http://ccssmath.org/?page_id
=2311
Geometry Curriculum Map
G-GPE.2.4
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).)
MP #3, #7



represent the vertices of a figure in the
coordinate plane using variables.
connect a property of a figure to the tool
needed to verify the property.
use coordinates and the right tool to prove or
disprove a claim about a figure. For example:
a) Use slope to determine if sides are
parallel, intersecting, or perpendicular;
b) Use the distance formula to determine if
sides are congruent.
c) Use the midpoint formula or the distance
formula to decide if a side has been
bisected.
Important formulas for
coordinate geometry
include distance
formula, slope formula,
midpoint formula, and
definitions of
quadrilaterals.
https://www.cohs.com/editor/
userUploads/file/Meyn/321%2
0Student%20Workbook.pdf
http://www.shmoop.com/com
mon-corestandards/handouts/g-gpeworksheet_4.pdf
http://www.shmoop.com/com
mon-corestandards/handouts/g-gpeworksheet_4_ans.pdf
Unit 7 – Properties of Quadrilaterals
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Standard
The students will:
G-GPE.2.5
Prove the slope criteria for parallel
and perpendicular lines and use
them to solve geometric problems
(e.g., find the equation of a line
parallel or perpendicular to a given
line that passes through a given
point).
MP #3, #8
Essential Question(s):
How are quadrilaterals precisely classified?
Learning Goals
I can:
 determine if lines are parallel or
perpendicular using their slopes.
Remarks
Apply these concepts to
the characteristics of
special quadrilaterals.
Resources
http://www.google.com/url?sa
=t&rct=j&q=ggpe.4&source=web&cd=1&ca
d=rja&ved=0CDIQFjAA&url=h
ttp%3A%2F%2Flearnzillion.c
om%2Flessons%2F286prove-whether-a-point-is-onacircle&ei=V01kUcPTMI629gS
GkYD4Cg&usg=AFQjCNEdTl
gXqxqTVyWyaIKtVmFbRE9Xg&b
vm=bv.44990110,d.eWU
http://neaportal.k12.ar.us/inde
x.php/2012/02/perpendicularand-parallel-lines/
http://map.mathshell.org/mate
rials/download.php?fileid=703
Unit 8 - Similarity
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Course: Geometry
Unit 8- Similarity
Essential Question(s): How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
Resources
The students will:
G-SRT.1.2
Students think of similarity
Tasks and TI-nspire
 define similarity as a composition of rigid
Given two figures, use the
and
congruence
as
separate
lessons:
motions followed by dilations in which angle
definition of similarity in terms of
and distinct categories.
http://ccssmath.org/?pa
measure is preserved and side length is
similarity transformations to
Remind students that
ge_id=2275
proportional.
decide if they are similar; explain
congruent figures are just
 identify corresponding sides and
using similarity transformations
similar figures with a scale
“How Tall is the
corresponding angles of similar triangles.
the meaning of similarity for
factor
of
1:1.
School’s Flagpole”:
 demonstrate that in a pair of similar
triangles as the equality of all
http://alex.state.al.us/les
triangles, corresponding angles are
corresponding pairs of angles and
son_view.php?id=1669
congruent (angle measure is preserved) and
the proportionality of all
corresponding sides are proportional.
corresponding pairs of sides.
Similarity in Right
 determine that two figures are similar by
SMP #3
Triangles task:
verifying that angle measure is preserved
http://alex.state.al.us/les
and corresponding sides are proportional.
son_view.php?id=26341
G-SRT.1.3
Students often confuse the
 show and explain that when two angle
Use the properties of similarity
triangle congruence theorems
measures are known (AA), the third angle
Videos, practice and
transformations to establish the
with the triangle similarity
measure is also known (Third Angle
assessments:
AA criterion for two triangles to be
theorems. It may be
Theorem).
http://fabienneriesen.co
similar.
 conclude and explain that AA similarity is a necessary to go back and
m/Geometry%3ASMP #3
review SSS, SAS, ASA and
sufficient condition for two triangles to be
the HL theorem to distinguish tutorials---chaptersimilar.
11.php
them from AA~, SAS~ and
SSS~.
“Solving Geometry
G-SRT.2.4
Students often cannot
 use theorems, postulates, or definitions to
Prove theorems about triangles.
visualize corresponding parts Problems: Floodlights”:
prove theorems about triangles, including:
http://map.mathshell.org
(Theorems include: a line parallel
of similar overlapping
a) A line parallel to one side of a triangle
/materials/download.php
to one side of a triangle divides
triangles. Have them
divides the other two proportionally.
?fileid=1257
the other two proportionally, and
separate and redraw the
b) If a line divides two sides of a triangle
conversely; the Pythagorean
triangles.
proportionally, then it is parallel to the
Theorem proved using triangle
third side.
similarity.)
c) The Pythagorean Theorem proved using
SMP #3
triangle similarity.
Mathematics Department
CCSS
Geometry Curriculum Map
ITHS
Course: Geometry
Unit 9- Right Triangles and Trigonometry
Standard
The students will:
G-SRT.3.6
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.
MP #2, 7
G-SRT.3.7
Explain and use the
relationship between the
sine and cosine of
complementary angles.
MP #2
G-SRT.3.8
Use trigonometric ratios
and the Pythagorean
Theorem to solve right
triangles in applied
problems.
MP #1, #4
Mathematics Department
ITHS
Essential Question(s):
How can right triangles be used to solve application problems?
Learning Goals
I can:
Remarks
Resources
http://ccssmath.org/?p
Some students believe that right
 demonstrate that within a right triangle, line
age_id=2283
triangles must be oriented a
segments parallel to a leg create similar triangles by
particular way or they do not realize “Trig River”:
angle-angle similarity.
http://www.teachengin
that opposite and adjacent sides
 use characteristics of similar figures to justify the
eering.org/view_activit
need to be identified with reference
trigonometric ratios.
y.php?url=http://www.t
to a particular acute angle in a right eachengineering.org/c
 define the following trigonometric ratios for acute
triangle.
angles in a right triangle: sine, cosine, and tangent.
ollection/cub_/activitie
Extension: Use division and the
s/cub_navigation/cub_
Pythagorean Theorem (a2 + b2 =
navigation_lesson03_
c2) to prove that sin2A + cos2A = 1.
activity2.xml
 define complementary angles.
http://fabienneriesen.c
 calculate sine and cosine ratios for acute angles in a
om/Geometry.php
right triangle when given two side lengths.
“Temple Geometry”:
 use a diagram of a right triangle to explain that for a
http://map.mathshell.o
pair of complementary angles A and B, the sine of
rg/materials/tasks.php
angle A is equal to the cosine of angle B and the
?taskid=261&subpage
cosine of angle A is equal to the sine of angle B.
=apprentice
“Pythagorean
 use angle measures to estimate side lengths and vice Includes the Triangle Inequality
Theorem”:
Theorems and Hinge Theorem
versa.
http://map.mathshell.o
Some
students
believe
that
the
 solve right triangles by finding the measures of all
rg/materials/lessons.p
trigonometric
ratios
defined
in
this
sides and angles in the triangles using Pythagorean
hp?taskid=419&subpa
cluster apply to all triangles, but
Theorem and/or trigonometric ratios and their
ge=concept
they are only defined for acute
inverses.
angles in right triangles.
 draw right triangles that describe real world problems
and label the sides and angles with their given
measures.
 solve application problems involving right triangles,
including angle of elevation and depression,
navigation, and surveying.
CCSS
Geometry Curriculum Map
Course: Geometry
Unit 9- Right Triangles and Trigonometry (cont)
Standard
The students will:
G-SRT.4.10
Prove the Laws of Sines
and Cosines and use
them to solve problems.
MP #1, #2, #7
G-SRT.4.11
Understand and apply the
Law of Sines and the Law
of Cosines to find
unknown measurements
in right and non-right
triangles (e.g., surveying
problems, resultant
forces).
MP #1, #4
Essential Question(s):
How can right triangles be used to solve application problems?
Learning Goals
I can:
Remarks
 derive the Law of Sines by drawing an altitude in a triangle,
using the sine function to find two expressions for the length
of the attitude and simplifying the equation that results from
setting these expressions equal.
 derive the Law of Cosines using the Pythagorean Theorem,
two right triangles formed by drawing an altitude, and
substitution.
 generalize the Law of Cosines to apply to each included
angle.
 apply the Law of Sines and Cosines to solve real world
problems.
 use the triangle inequality and side/angle relationships to
estimate the measure of unknown sides and angles.
 distinguish between situations that require the Law of Sines
and Law of Cosines.
 apply and use the Law of Sines and Cosines to find
unknown side lengths and unknown angle measures in right
and non-right triangles.
 represent real world problems with diagrams of right and
non-right triangles and use them to solve for unknown side
lengths and angle measures.
Unit 10 – Transformational Geometry
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Resources
Course: Geometry
Unit 10- Transformational Geometry
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
Remarks
I can:
The students will:
The terms “mapping” and
G-CO.1.4
 construct the definition of reflection, translation, and
“under” are used in special
Develop definitions of
rotation.
rotations, reflections, and
ways when studying
 construct the reflection definition by connecting any point
translations in terms of
on the preimage to its corresponding point on the reflected transformations.
angles, circles, perpendicular
image and describing the line segment’s relationship to the Students sometimes confuse
lines, parallel lines, and line
line of reflection.
the terms “transformation”
segments.
 construct the translation definition by connecting any
and “translation.”
MP #6
point on the preimage to its corresponding point on the
Remind students that that
translated image, and connecting a second point on the
corresponding vertices have
preimage to its corresponding point on the translated
to be listed in order so that
image, and describing how the two segments are equal in
corresponding sides and
length, point in the same direction, and are parallel.
angles can be easily
 construct the rotation definition by connecting the center
identified and that included
of rotation to any point on the preimage and to its
sides or angles are
corresponding point on the rotated image, and describing
apparent.
the measure of the angle formed and the equal measures
of the segments that formed the angle as part of the
definition.
Mathematics Department
ITHS
CCSS
Resources
“Translations,
Reflections and
Rotations” task:
http://www.shodor.org/int
eractivate/lessons/Transl
ationsReflectionsRotatio
ns/
Tessellation based Quilt
design:
http://alex.state.al.us/les
son_view.php?id=29240
Exploring
Transformations on a TI84 graphing calculator:
http://alex.state.al.us/les
son_view.php?id=29240
Dilations Applet:
Geometry Curriculum Map
G-CO.1.5
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.
MP #5
Mathematics Department
ITHS


Course: Geometry
Unit 10- Transformational Geometry
draw a specific transformation when given a geometric
figure and a rotation, reflection, or translation.
predict and verify the sequence of transformations (a
composition) that will map a figure onto another.
CCSS
Students may confuse
rotations and reflections and
be unable to differentiate the
two. Allowing them the
opportunity to physically
manipulate the shapes (such
as with cut-outs or patty
paper) can clear up
misconceptions.
http://nlvm.usu.edu/en/n
av/frames_asid_296_g_
4_t_3.html
Geometry Curriculum Map
Course: Geometry
Unit 10- Transformational Geometry
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
The students will:
G-CO.1.2
Rigid transformations
 draw transformations of reflections, rotations, translations,
Represent transformations in the
preserve distance and
and combinations of these using graph paper,
plane using, e.g., transparencies
angle measure
transparencies, and/or geometry software.
and geometry software; describe
(reflections, rotations,
 determine the coordinates for the image (output) of a
transformations as functions that
figure when a transformation rule is applied to the preimage translations, or
take points in the plane as inputs
combinations of those).
(input).
and give other points as outputs.
 distinguish between transformations that are rigid and
Compare transformations that
those that are not.
preserve distance and angle to
those that do not (e.g., translation
versus horizontal stretch).
MP #6
G-CO.1.3
This is a discussion of
 describe and illustrate how a rectangle, parallelogram,
Given a rectangle, parallelogram,
and isosceles trapezoid are mapped onto themselves using symmetry.
trapezoid, or regular polygon,
transformations.
describe the rotations and
 calculate the number of lines of reflection symmetry and
reflections that carry it onto itself.
the degree of rotational symmetry of any regular polygon.
MP #7
G-CO.2.6
Students may believe
 define rigid motions as reflections, rotations, translations
Use geometric descriptions of rigid
that all transformations,
and combinations of these, all of which preserve distance
motions to transform figures and to
including dilations, are
and angle measure.
predict the effect of a given rigid
rigid motions or that any
 define congruent figures as figures that have the same
motion on a given figure; given two
two figures that have
shape and size and state that a composition of rigid
figures, use the definition of
the same area
motions will map one congruent figure onto the other.
congruence in terms of rigid
represent a rigid
 predict the composition of transformations that will map a
motions to decide if they are
transformation. Provide
figure onto a congruent figure.
congruent.
counterexamples.
 determine if two figures are congruent by determining if
MP #3
rigid motions will turn one figure into the other.
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Resources
Tessellation Applet:
http://www.shodor.
org/interactivate/act
ivities/Tessellate/
Online
Transformation
Games:
http://www.onlinem
athlearning.com/tra
nsformationgame.html
Tasks and TInspire activities:
http://ccssmath.org/
?page_id=2245
Course: Geometry
Unit 10- Transformational Geometry (cont)
Essential Question(s):
In what ways can congruence be useful?
How might the features of one figure be useful when solving problems about a similar figure?
Standard
Learning Goals
I can:
Remarks
The students will:
G-SRT.1.1
Some students have prior
 define dilation.
Verify experimentally the properties  perform a dilation with a given center and scale factor on a knowledge of dilations from
the concept of “dilated
figure in the coordinate plane.
of dilations given by a center and a
pupils” which may lead them
 verify that when a side passes through the center of
scale factor:
to believe that dilation refers
dilation, the side and its image lie on the same line.
a. A dilation takes a line not
only to objects getting larger.

verify
that
corresponding
sides
of
the
preimage
and
passing through the center of
Similarly, they may have
images are parallel.
the dilation to a parallel line,
difficulty figuring out when to
 verify that a side length of the image is equal to the scale
and leaves a line passing
multiply versus divide by the
factor multiplied by the corresponding side length of the
through the center unchanged.
scale factor.
preimage.
b. The dilation of a line segment is
longer or shorter in the ratio
given by the scale factor.
MP #6, #8
Unit 11 – Two-Dimensional Measurements
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Resources
Course: Geometry
Unit 11- Two-Dimensional Measurements
Standard
The students will:
G-GPE.2.7
Use coordinates to
compute perimeters of
polygons and areas of
triangles and rectangles,
e.g., using the distance
formula.
MP #1
G-MG.1.1
Use geometric shapes,
their measures, and their
properties to describe
objects (e.g., modeling a
tree trunk or a human torso
as a cylinder).
MP #4
G-MG.1.2
Apply concepts of density
based on area and volume
in modeling situations
(e.g., persons per square
mile, BTUs per cubic foot).
MP #1, #4
Mathematics Department
ITHS
Essential Question(s):
In what ways can geometric figures be used to understand real-world situations?
Learning Goals
I can:
Remarks
Graphing the given coordinates of
 use the distance formula to compute the
the vertices may help students
perimeter and area given the coordinates of
visualize the polygon in order to find
vertices of a polygon.
the perimeter and area.







represent real-world objects as geometric
figures.
estimate measures (circumference, area,
perimeter, volume) of real-world objects using
comparable geometric shapes or threedimensional figures.
apply the properties of geometric figures to
comparable real-world objects.
Example: The spokes of a wheel of
a bicycle are equal lengths because
they represent the radii of a circle.
Students may have issues with
estimating (rounding, conceptual,
not being exact, etc.).
decide whether it is best to calculate or
estimate the area or volume of a geometric
figure and perform the calculation or
estimation.
break composite geometric figures into
manageable pieces.
convert units of measure.
apply area and volume to situations involving
density.
Example: Determine the population
in an area.
Students have difficulty converting
units of area and volume due to the
difference in scale factors.
CCSS
Resources
Patchwork Task:
http://map.mathshell
.org/materials/downl
oad.php?fileid=754
Square Task:
http://map.mathshell
.org/materials/downl
oad.php?fileid=792
Security Camera
Task:
http://map.mathshell
.org/materials/downl
oad.php?fileid=798
Location, Location,
Location Task:
http://illuminations.n
ctm.org/LessonDetai
l.aspx?id=L660
Geometry Curriculum Map
Course: Geometry
Unit 11- Two-Dimensional Measurements (cont)
Standard
The students will:
G-MG.1.3
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).
MP #1, #4
Essential Question(s):
In what ways can geometric figures be used to understand real-world situations?
Learning Goals
I can:
Remarks
 create a visual representation of a design problem. Mathematical modeling involves solving
problems in which the path to the
 solve design problems using a geometric model
solution is not obvious. A challenge for
(graph, equation, table, formula).
teaching modeling is finding problems
 interpret the results and make conclusions based
that are interesting and relevant to high
on the geometric model.
school students and, at the same time,
solvable with the mathematical tools at
the students’ disposal.
Resources
Definition of pi
investigation:
http://illuminations.
nctm.org/LessonD
etail.aspx?id=L575
Circles and
Triangles:
http://map.mathsh
ell.org/materials/le
ssons.php?taskid=
222&subpage=pro
blem
-
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Course: Geometry
Unit 11- Two-Dimensional Measurements (cont)
Standard
The students will:
G-GMD.1.1
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.)
MP #3, #7
Essential Question(s):
In what ways can geometric figures be used to understand real-world situations?
Learning Goals
I can:
Remarks
Circumference of a Circle
An informal survey of students
from elementary school through
 define π (pi) as the ratio of a circle’s circumference to its
college showed the number pi
diameter.
to be the mathematical idea
 use algebra to demonstrate that because of the definition of
about which more students
π (pi), the formula for a circumference must be C = π d.
were curious than any other.
Area of a Circle
There are at least three facets
 inscribe a regular polygon in a circle and break it into many
to this curiosity: the symbol π
congruent triangles to find its area.
itself, the number 3.14159…,
 explain and use the dissection method on regular polygons
and the formula for the area of
to generate an area formula for regular polygons A = ½ 
a circle. All of these facets can
apothem perimeter (A = aP).
be addressed here, at least
briefly.
 calculate the area of a regular polygon A = aP.


G-CO.4.13
Construct an equilateral
triangle, a square, and a
regular hexagon inscribed
in a circle.
MP #5, #6
Mathematics Department
ITHS





use pictures to explain that a regular polygon with many
sides is nearly a circle, its perimeter is nearly the
circumference of a circle, and that its apothem is nearly the
radius of a circle.
substitute the “nearly” values of a many sided regular
polygon into A = aP to show that the formula for the area of
a circle is A = πr2.
define inscribed polygons (the vertices of the figure must be
points on the circle).
construct an equilateral triangle inscribed in a circle.
construct a square inscribed in a circle.
construct a hexagon inscribed in a circle.
explain the steps to constructing an equilateral triangle, a
square, and a regular hexagon inscribed in a circle.
CCSS
Emphasize the need for
precision and accuracy when
doing constructions.
Stress the idea that a compass
and straightedge are identical
to a protractor and ruler.
Explain the difference between
measurement and construction.
Geometry Curriculum Map
Resources
Rolling Cup
Lesson:
http://map.mat
hshell.org/mat
erials/downloa
d.php?fileid=1
254
Evaluating
Statements
about Lengths
and Areas:
http://map.mat
hshell.org/mat
erials/downloa
d.php?fileid=6
75
Course: Geometry
Unit 12 – Properties of Circles
Essential Question(s):
How can the properties of circles be useful when solving geometric problems?
Standard
Learning Goals
I can:
Remarks
The students will:
G-GPE.1.1
Students
will need to learn
 identify the center and radius of a circle
Derive the equation of a circle of
how to complete the
given its equation.
given center and radius using the
square.
 draw a right triangle with a horizontal leg, a
Pythagorean Theorem; complete
Students forget to change
vertical leg, and the radius of a circle as its
the square to find the center and
the sign of (h,k) to find the
hypotenuse.
radius of a circle given by an
coordinates of the center.
 use the distance formula (Pythagorean
equation.
Theorem), the coordinates of a circle’s
MP #2, #3, #7
center, and the circle’s radius to write the
equation of a circle.
 convert an equation of a circle in general
(quadratic) form to standard form by
completing the square.
G-C.1.1
The definition of similarity
 prove that all circles are similar by showing
Prove that all circles are similar.
and dilation will need to be
that for a dilation centered at the center of a
MP #3
reviewed with students.
circle, the preimage and the image have
Online applets can be
equal central angle measures.
helpful in seeing this
relationship.
G-C.1.2
Students may think they
 identify central angles, inscribed angles,
Identify and describe relationships
can tell by inspection
circumscribed angles, diameters, radii,
among inscribed angles, radii, and
whether a line intersects a
chords, and tangents.
chords. (Include the relationship
circle in exactly one point.
 describe the relationship between a central
between central, inscribed, and
It may be beneficial to
angle, inscribed angle, or circumscribed
circumscribed angles; inscribed
formally define a tangent
angle and the arc it intercepts.
angles on a diameter are right
line as the line
 recognize that an inscribed angle whose
angles; the radius of a circle is
perpendicular to a radius
sides intersect the endpoints of the diameter
perpendicular to the tangent where
at the point where the
of a circle is a right angle and that the radius
the radius intersects the circle.)
radius intersects the circle.
of a circle is perpendicular to the tangent
MP #1, #6
where the radius intersects the circle.
Mathematics Department
ITHS
CCSS
Resources
Sectors of Circles Task:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=441&subpage=concept
Deriving equations of
Circles: Part 1:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=406&subpage=concept
Part 2:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=425&subpage=concept
Inscribed and
Circumscribed Circles
Task:
http://map.mathshell.org/m
aterials/download.php?filei
d=1194
Geometry Curriculum Map
Standard
The students will:
G-C.1.3
Construct the inscribed and
circumscribed circles of a
triangle, and prove
properties of angles for a
quadrilateral inscribed in a
circle.
MP #5
G-C.1.4
Construct a tangent line
from a point outside a given
circle to the circle.
MP #5
G-C.2.5
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.
MP #6, #7
Mathematics Department
ITHS
Essential Question(s):
How can the properties of circles be useful when solving geometric problems?
Learning Goals
I can:
Remarks
Students
sometimes
confuse
 define the terms inscribed, circumscribed, angle bisector,
inscribed
angles
and
central angles.
and perpendicular bisector.
For
example
they
will
assume that
 construct the inscribed circle whose center is the point of
the inscribed angle is equal to the
intersection of the angle bisectors (the incenter)and
arc like a central angle.
circumscribed circle whose center is the point of
intersection of the perpendicular bisectors of each side of
the triangle (the circumcenter).
 apply the Arc Addition Postulate to solve for missing arc
measures.
 prove that opposite angles in an inscribed quadrilateral
are supplementary.
 define and identify a tangent line.
 construct a tangent line from a point outside the circle to
the circle using construction tools or computer software.







define similarity as rigid motions with dilations, which
preserves angle measures and makes lengths
proportional.
use similarity to calculate the length of an arc.
define and calculate the radian measure of an angle as
the ratio of an arc length to its radius.
convert degrees to radians using the constant of
proportionality.
calculate the area of a circle.
define a sector of a circle.
calculate the area of a sector using the ratio of the
intercepted arc measure and 360multiplied by the area
of the circle.
CCSS
Constant of proportionality for radian
measures: 2π angle measure / 360.
The formulas for converting radians
to degrees and vice versa are easily
confused. Knowing that the degree
measure of given angle is always a
number larger than the radian
measure can help students use the
correct unit.
Sectors and segments are often
used interchangeably in everyday
conversation. Care should be taken
to distinguish these two geometric
concepts.
Geometry Curriculum Map
Resources
Essential Question(s):
How can the properties of circles be useful when solving geometric problems?
Standard
Learning Goals
I can:
Remarks
The students will:
G-GPE.1.1
Students will need to learn
 identify the center and radius of a circle
Derive the equation of a circle of
how to complete the
given its equation.
given center and radius using the
square.
 draw a right triangle with a horizontal leg, a
Pythagorean Theorem; complete
Students forget to change
vertical leg, and the radius of a circle as its
the square to find the center and
the sign of (h,k) to find the
hypotenuse.
radius of a circle given by an
coordinates of the center.
 use the distance formula (Pythagorean
equation.
Theorem), the coordinates of a circle’s
MP #2, #3, #7
center, and the circle’s radius to write the
equation of a circle.
 convert an equation of a circle in general
(quadratic) form to standard form by
completing the square.
G-C.1.1
The definition of similarity
 prove that all circles are similar by showing
Prove that all circles are similar.
and dilation will need to be
that for a dilation centered at the center of a
MP #3
reviewed with students.
circle, the preimage and the image have
Online applets can be
equal central angle measures.
helpful in seeing this
relationship.
Mathematics Department
ITHS
CCSS
Resources
Sectors of Circles Task:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=441&subpage=concept
Deriving equations of
Circles: Part 1:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=406&subpage=concept
Part 2:
http://map.mathshell.org/m
aterials/lessons.php?taski
d=425&subpage=concept
Inscribed and
Circumscribed Circles
Task:
http://map.mathshell.org/m
aterials/download.php?filei
Geometry Curriculum Map
G-C.1.2
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.)
MP #1, #6
Mathematics Department
ITHS



identify central angles, inscribed angles,
circumscribed angles, diameters, radii,
chords, and tangents.
describe the relationship between a central
angle, inscribed angle, or circumscribed
angle and the arc it intercepts.
recognize that an inscribed angle whose
sides intersect the endpoints of the diameter
of a circle is a right angle and that the radius
of a circle is perpendicular to the tangent
where the radius intersects the circle.
CCSS
Students may think they
can tell by inspection
whether a line intersects a
circle in exactly one point.
It may be beneficial to
formally define a tangent
line as the line
perpendicular to a radius
at the point where the
radius intersects the circle.
d=1194
Geometry Curriculum Map
Mathematics Department
ITHS
CCSS
Geometry Curric