MGS41 Geometry Term 1 Curriculum Map

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2013 – 2014
Geometry=>MGS-41
MATHEMATICS
Curriculum Map
ITHS
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.
Geometry: Common Core State Standards At A Glance
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
September
Unit 1- Geometry Fundamentals
G-GPE.2.6
G-GPE.2.7
G-CO.4.12
G-CO.4.13
October
Unit 2- Geometric Reasoning
D.6.2
D.6.3
D.6.4
G-CO.3.9
November
Unit 3- Parallel and Perpendicular Lines
G-CO.1.1
G-CO.3.9
G-GPE.2.5
December/January
Unit 4- Triangle Fundamentals
G-CO.3.10
Mathematics Department
ITHS
CCSS
Geometry Curriculum Map
Unit 5- Triangle Congruence
G-CO.2.7
G-CO.2.8
G-CO.3.10
Unit 6- Triangle Properties and Attributes
G-CO.3.10
G-SRT.2.4
G-SRT.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.
Course: Geometry
Unit 1- Geometry Fundamentals
Essential Question(s):
How can algebra be useful when expressing geometric properties?
Standard
Learning Goals
Remarks
I can:
The students will:
G-GPE.2.6
 calculate the point(s) on a directed This includes the midpoint formula.
Find the point on a directed line
Find a missing midpoint given an
line segment whose endpoints are
segment between two given points
(x1, y1) and (x2, y2) that partitions
endpoint and the midpoint of the
that partitions the segment in a given
segment.
the line segment into a given ratio.
ratio.
In addition to using the formula,
MP #7
students may find the midpoint
graphically using slope.
G-GPE.2.7
This standard provides practice
 use the distance formula to
Use coordinates to compute
compute segment length given two with the distance formula and its
perimeters of polygons and areas of
connection with the Pythagorean
coordinates
triangles and rectangles, e.g., using
Theorem.
the distance formula.
MP #1
Mathematics Department
ITHS
CCSS
Resources
Partitioning Segments:
http://learni.st/users/S3357
2/boards/3128-partitioningsegments-into-a-particularratio-common-corestandard-9-12-g-gpe-6
TI-Nspire Lessons:
http://ccssmath.org/?page_i
d=2315
Describing Constructions
Worksheet:
http://www.shmoop.com/co
mmon-corestandards/handouts/g-coworksheet_12.pdf
Geometry Curriculum Map
G-CO.4.12
Make formal geometric constructions
with a variety of tools and methods
(compass and straightedge, string,
reflective devices, paper folding,
dynamic geometric software, etc.
This includes: copying a segment;
copying an angle; bisecting a




identify the tools used in formal
constructions.
use tools and methods to precisely
copy a segment
use tools and methods to precisely
copy an angle
use tools and methods to precisely
bisect a segment
Students should be able to
informally perform the constructions
listed using string, reflective
devices, paper folding, and/or
dynamic geometric software.
Some students may believe that a
construction is the same as a sketch
or drawing. Emphasize the need for
First to Finish Task (toward
the bottom of this webpage)
https://commoncoregeomet
ry.wikispaces.hcpss.org/Uni
t+1
Midpoint resources:
http://www.shmoop.com/co
mmon-core-standards/ccsshs-g-gpe-6.html
Course: Geometry
Unit 1- Geometry Fundamentals
Essential Question(s):
How can algebra be useful when expressing geometric properties?
Standard
Learning Goals
I can:
Remarks
The students will:
G-GPE.2.6
 calculate the point(s) on a directed This includes the midpoint formula.
Find the point on a directed line
Find a missing midpoint given an
line segment whose endpoints are
segment between two given points
endpoint and the midpoint of the
(x1, y1) and (x2, y2) that partitions
that partitions the segment in a given
segment.
the line segment into a given ratio.
ratio.
In addition to using the formula,
MP #7
students may find the midpoint
graphically using slope.
Mathematics Department
ITHS
CCSS
Resources
Partitioning Segments:
http://learni.st/users/S3357
2/boards/3128-partitioningsegments-into-a-particularratio-common-corestandard-9-12-g-gpe-6
TI-Nspire Lessons:
Geometry Curriculum Map
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

use the distance formula to
compute segment length given two
coordinates
This standard provides practice
with the distance formula and its
connection with the Pythagorean
Theorem.
G-CO.4.12
Make formal geometric constructions
with a variety of tools and methods
(compass and straightedge, string,
reflective devices, paper folding,
dynamic geometric software, etc.
This includes: copying a segment;
copying an angle; bisecting a

identify the tools used in formal
constructions.
use tools and methods to precisely
copy a segment
use tools and methods to precisely
copy an angle
use tools and methods to precisely
bisect a segment
Students should be able to
informally perform the constructions
listed using string, reflective
devices, paper folding, and/or
dynamic geometric software.
Some students may believe that a
construction is the same as a sketch
or drawing. Emphasize the need for
Mathematics Department
ITHS



CCSS
http://ccssmath.org/?page_i
d=2315
Describing Constructions
Worksheet:
http://www.shmoop.com/co
mmon-corestandards/handouts/g-coworksheet_12.pdf
First to Finish Task (toward
the bottom of this webpage)
https://commoncoregeomet
ry.wikispaces.hcpss.org/Uni
t+1
Midpoint resources:
http://www.shmoop.com/co
mmon-core-standards/ccsshs-g-gpe-6.html
Geometry Curriculum Map
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.)
MP #5, #6
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 tools and methods to precisely
bisect an angle
construct perpendicular lines and
bisectors
construct a line parallel to a given
line through a point not on the line.
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.
As the Crow Flies Task:
http://www.nctm.org/upload
edFiles/Journals_and_Book
s/Books/FHSM/RSMTask/Crow.pdf
define inscribed polygons.
construct an equilateral triangle, a
square, a hexagon inscribed in a
circle.
explain the steps to constructing
and equilateral triangle, a square,
and a regular hexagon inscribed in
a circle.
CCSS
Geometry Curriculum Map
Unit 2-Geometric Reasoning
Essential Question(s): How can inductive and deductive reasoning help to construct and defend logical arguments?
Standard
Learning Goals
I can:
Remarks
The students will:
D.6.2
Emphasize all forms of a conditional
 identify and write a conditional
Find the converse, inverse, and
statement.
statement and its converse, inverse,
contrapositive of a statement.
and contrapositive.
MP #2, #3
D.6.3
Determine whether two propositions
are logically equivalent.
MP #2, #3

D.6.4
Use methods of direct and indirect
proof and determine whether a short
proof is logically valid.
MP #2, #3
G-CO.3.9
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.)
MP #2, #3
Mathematics Department
ITHS
determine whether two statements are
logically equivalent.
identify a bi-conditional statement.
Conditional and Contrapositive/Inverse
and Converse are logically equivalent.
If conditional and converse are both
true, then the statement can be written
as a biconditional statement.

use methods of proof to validate a
postulate, theorem, corollary, definition,
or property.

identify and use the properties of
congruence and equality (reflexive,
symmetric, transitive) in my proofs.
order statements based on the Law of
Syllogism when constructing my proof.
interpret geometric diagrams and
identifying what can and cannot be
assumed.
use theorems, postulates, or definitions
to prove theorems about lines and
angles
Common misconceptions:
students believe that exceptions to a
proven generalization exist
one counterexample is not sufficient to
prove a conjecture false
the converse of a statement is true
a conjecture is true because it worked
in all examples that were explored
Theorems/postulates students may
use in their proofs include, but are not
limited to:
a. Vertical angles are congruent.
b. Ruler Postulate
c. Segment Addition Postulate
d. Protractor Postulate
e. Angle Addition Postulate
f. Definitions from Unit 1




CCSS
Geometry Curriculum Map
Resources
http://www.youtub
e.com/watch?v=B
Xd9SWVZEzo&sa
fety_mode=true&p
ersist_safety_mod
e=1&safe=active
Unit 3- Parallel and Perpendicular Lines
Essential Question(s):
In what ways can parallel and perpendicular lines be used to solve real world problems?
Standard
Learning Goals
I can:
Remarks
The students will:
G-CO.1.1
 identify the undefined notions used in
Know precise definitions of angle,
geometry (point, line, plane, distance) and
circle, perpendicular line, parallel
describe their characteristics
line, and line segment, based on the
 identify angles, perpendicular lines, parallel
undefined notions of point, line,
lines, rays, and line segments.
distance along a line, and distance
 define angles, perpendicular lines, parallel
around a circular arc.
lines, rays, and line segments precisely using
MP #6
the undefined terms and “if-then” and “if-andonly-if” statements.
G-CO.3.9
 identify and use the properties of congruence Theorems include: Angle
Prove theorems about lines and
pairs formed by parallel
and equality (reflexive, symmetric, transitive)
angles. (Theorems include: vertical
lines cut by a transversal
in my proofs.
angles are congruent; when a
(alternate interior angles,
 order statements based on the Law of
transversal crosses parallel lines,
corresponding angles,
Syllogism when constructing my proof.
alternate interior angles are
alternate exterior angles
 interpret geometric diagrams and identifying
congruent and corresponding angles
are congruent; same-side
what can and cannot be assumed.
are congruent; points on a
interior angles are
 use theorems, postulates, or definitions to
perpendicular bisector of a line
supplementary)
prove theorems about lines and angles.
segment are exactly those
Vertical angles are
equidistant from the segment’s
congruent; linear pairs are
endpoints.)
supplementary.
MP #2, #3
Mathematics Department
ITHS
CCSS
Resources
http://www.purplemath.c
om/modules/slope2.htm
http://www.mathsisfun.c
om/geometry/parallellines.html
http://mathforum.org/mat
htools/cell/g,10.5,ALL,AL
L/
http://interactmath.com/
Exercises.aspx?chapterI
d=3&sectionId=2&objecti
veId=1
http://www.shodor.org/int
eractivate/activities/Slop
eSlider/
http://www.shodor.org/int
eractivate/activities/Slop
eSlider/
Geometry Curriculum Map
Unit 3- Parallel and Perpendicular Lines (cont)
Essential Question(s):
In what ways can parallel and perpendicular lines be used to solve real world problems?
Standard
Learning Goals
I can:
Remarks
The students will:
G-GPE.2.5
May
need
to review
 state that parallel lines have the same slope.
Prove the slope criteria for parallel
graphing coordinates and
determine if lines are parallel using their slopes.
and perpendicular lines and use
finding slope before using it
 state that parallel lines have the opposite
them to solve geometric problems
to determine if lines are
reciprocal slopes.
(e.g., find the equation of a line
parallel or perpendicular.
 determine if lines are perpendicular using their
parallel or perpendicular to a given
slopes.
line that passes through a given
point).
MP #3, #8
Mathematics Department
ITHS
CCSS
Resources
Geometry Curriculum Map
Unit 4- Triangle Fundamentals
Standard
The students will:
G-CO.3.10
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.)
MP #2, #3
Mathematics Department
ITHS
Essential Question(s):
What are the properties of a triangle?
Learning Goals
I can:
 interpret geometric diagrams (what
can and cannot be assumed).
 apply theorems, postulates, or
definitions to prove theorems about
triangles, including:
a) Measures of interior angles of a
triangle sum to 180;
b) Base angles of isosceles triangles
are congruent;
c) An exterior angle of a triangle is
equal to the sum of the measures
of the remote interior angles.
CCSS
Remarks
Classifying triangles should
be reviewed in this unit.
Resources
http://teachers.henrico.k
12.va.us/math/IGO/03Tr
ianglesPolygons/3_1.ht
ml
http://www.mathsisfun.c
om/triangle.html
Geometry Curriculum Map
Unit 5- Triangle Congruence
Standard
The students will:
G-CO.2.7
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.
MP #3
G-CO.2.8
Explain how the criteria for
triangle congruence (ASA, SAS,
and SSS) follow from the
definition of congruence in terms
of rigid motions.
MP #2, #3
Mathematics Department
ITHS
Essential Question(s):
In what ways can congruence be useful in the real world?
Learning Goals
I can:
Remarks
Some
students
may believe:
 identify corresponding sides and
Combinations such as SSA or AAA
corresponding angles of congruent
are also a congruence criterion for
triangles.
 explain that in a pair of congruent triangles, triangles.
That all transformations, including
corresponding sides are congruent
dilations, are rigid motions
(distance is preserved) and corresponding
That any two figures that have the
angles are congruent (angle measure is
same area represent a rigid
preserved).
transformation.
 demonstrate that when distance is
preserved (corresponding sides are
congruent) and angle measure is preserved
(corresponding angles are congruent) the
triangles must also be congruent.
Include AAS and HL as a criterion
 list the sufficient conditions to prove
for congruence.
triangles are congruent.
Once you prove your triangles are
 map a triangle with one of the sufficient
congruent, the "left-over" pieces
conditions (e.g., SSS) onto the original
that were not used in your method
triangle and show that corresponding sides
of proof are also congruent.
and corresponding angles are congruent.
Remember, congruent triangles
have 6 sets of congruent pieces.
We now have a "follow-up"
theorem to be used AFTER the
triangles are known to be
congruent: CPCTC.
CCSS
Resources
http://illuminations.
nctm.org/ActivityDe
tail.aspx?id=4
http://nlvm.usu.edu/
ennav/frames_asid
_165_g_1_t_3.html
?open=instructions
&from=topic_t_3.ht
ml
http://ccssmath.org/
?page_id=2261
Geometry Curriculum Map
Unit 5- Triangle Congruence (cont)
Standard
The students will:
G-CO.3.10
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.)
MP #2, #3
Mathematics Department
ITHS
Essential Question(s):
In what ways can congruence be useful in the real world?
Learning Goals
I can:
Remarks
Include
proofs
of triangle
 interpret geometric diagrams (what can and
congruence.
cannot be assumed).
 apply theorems, postulates, or definitions to
prove theorems about triangles.
CCSS
Resources
Geometry Curriculum Map
Unit 6- Triangle Properties and Attributes
Essential Question(s):
What conclusions can be drawn from triangles and their congruency?
Standard
Learning Goals
I can:
Remarks
The students will:
G-CO.3.10
Include
perpendicular
 interpret geometric diagrams (what can and
Prove theorems about triangles.
bisectors, altitudes,
cannot be assumed).
(Theorems include: measures of
medians, angle bisectors
 apply theorems, postulates, or definitions to
interior angles of a triangle sum to
and their points of
prove theorems about triangles, including:
180°; base angles of isosceles
concurrency.
a) The segment joining midpoints of two
triangles are congruent; the segment
sides of a triangle is parallel to the third
joining midpoints of two sides of a
side and half the length;
triangle is parallel to the third side and
b) The medians of a triangle meet at a point.
half the length; the medians of a
triangle meet at a point.)
MP #2, #3
G-SRT.2.4
Some students may
 apply theorems, postulates, or definitions to
Prove theorems about triangles.
confuse the alternate
prove theorems about triangles, including:
(Theorems include: a line parallel to
interior angle theorem and
a) A line parallel to one side of a triangle
one side of a triangle divides the other
its converse as well as the
divides the triangle proportionally;
two proportionally, and conversely; the
Pythagorean theorem and
b) If a line divides two sides of a triangle
Pythagorean Theorem proved using
proportionally, then it is parallel to the third its converse.
triangle similarity.)
side;
MP #3
c) The Pythagorean Theorem proved using
triangle similarity.
G-SRT.2.5
Use congruence and similarity criteria
for triangles to solve problems and to
prove relationships in geometric
figures.
MP #1
Mathematics Department
ITHS


use triangle congruence and triangle similarity
to solve problems (e.g., indirect measure,
missing sides/angle measures, side splitting).
use triangle congruence and triangle similarity
to prove relationships in geometric figures.
CCSS
Resources
http://www.showme.
com/sh/?h=Epf0Rk0
https://ccgps.org/GSRT_G6QQ.html
http://staff.argyll.eps
b.ca/jreed/math9/str
and3/triangle_congr
uent.htm
https://docs.google.c
om/viewer?a=v&pid
=sites&srcid=ZGVm
YXVsdGRvbWFpbn
xnZW9tZXRyeXdpd
Ghtb3N0ZWxsZXJ8
Z3g6MmQ5MTRmN
GNkMzk3Y2E3OQ
Some students may
assume that segments,
angles, and triangles are
congruent/ similar without
given that information.
Geometry Curriculum Map
Mathematics Department
ITHS
CCSS
Geometry Curric
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