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Unit Title and
Time Frame
January
Regents/
Midterm Exams
End of Fall
Term—2/1
Intro to
Geometry
2/2 - 2/12
Essential
Questions
Concepts/Topics
(# of Regents ??s)
# of Days
Focused on
Topic
5
Major
Assessments
Primary
Sources/
Materials
o Defining point,
plane, and line
(building blocks of
rest of geometry)
o Defining
congruent,
collinear, coplanar,
line segment, ray,
etc.
o Labeling line
segments and
angles, and
indentifying
congruent angles
and segments
o Finding the slopes
of lines parallel
and perpendiculr
to given lines
o Writing equations
of lines parallel
and perpendicular
to a given line that
pass through a
specific point
Unit 1 Exam
Mini Quiz after
week 1
JMAP
Regents
questions,
TFA ISAT
questions
1
What are the
building blocks of
geometry?
o Basic geometric terms
o Parallel and perpendicular
o Writing equations
8
How can I name
and label
geometric
shapes?
What is parallel
and
perpendicular?
How can I use
slopes to
determine if two
lines are parallel,
perpendicular or
neither?
Midwinter
Recess—2/132/21
Skills
9
Parallel Lines
and
Quadrilaterals
2/22 – 3/5
How can I prove
that two lines
that look parallel
really are
parallel?
What are the
characteristics of
the figures that
parallel lines
form?
How can I justify
that a
quadrilateral is a
parallelogram,
rhombus,
rectangle, square,
or trapezoid?
Triangle
Congruency
3/8 – 3/19
**will be very
tight. May need
to add in flex
days here.
What does it
mean for
triangles to be
congruent?
How can I prove
that two
triangles are
G.35: Determine if two lines cut by a
transversal are parallel, based on the
measure of given pairs of angles
formed by the transversal and the
lines
G.38 Investigate, justify, and apply
theorems about parallelograms
involving their angles, sides and
diagonals
G.39 Investigate, justify, and apply
theorems about special
parallelograms (rectangles,
rhombuses, squares) involving their
angles, sides, and diagonals
G.40 Investigate, justify and apply
theorems about trapezoids
(including isosceles trapezoids)
involving their angles, sides,
medians, and diagonals
G.41 Justify that some quadrilaterals
are parallelograms, rhombuses,
rectangles, squares, or trapezoids
G.69 Investigate, justify, and apply
the properties of triangles and quads
in the coordinate plane, using
distance, midpoint and slope
formulas (JUST FOR QUADS.
TRIANGLES LATER)
G.28 Determine the congruence of
two triangles using one of the five
congruence techniques (SSS, SAS,
ASA, AAS, HL) given sufficient
information about the sides and/or
angles of two congruent triangles
G.29 Identify corresponding parts of
congruent triangles
10 (Feb 22 –
March 5)
**IA 4
MARCH 5
INSTEAD OF
NEXT
WEEK??**
10 days
o Identify and label
angles (alternate
interior, alternate
exterior,
complementary,
supplementary,
etc.)
o straight line has
180 degrees
o Vocabulary
o Distance formula
Unit 2 exam.
Project –
design a fabric
or stained glass
window using
quads
JMAP
Regents
questions,
TFA ISAT
questions
(1.) Proof is a
justification that is
logically valid and
based on definitions,
postulates, and
Unit 3 examcombo of
formal and
informal proofs
JMAP
Regents
questions,
TFA ISAT
questions
theorems.
(2.) Logical
congruent?
What are some of
the defining
characteristics of
a triangle?
What
characterizes
different types of
triangles?
G.30 Investigate, justify, and apply
theorems about the sum of the
measures of the angles of a triangle
G.32 Investigate, justify, and apply
theorems about geometric
inequalities, using the exterior angle
theorem
G.31 Investigate, justify and apply
the isosceles triangle theorem and its
converse
G.33 Investigate, justify and apply
the triangle inequality theorem
G.34 Determine the longest side of a
triangle given the three angle
measures or the largest angle given
the lengths of three sides of a triangle
arguments consist of
a set of premises or
hypotheses and a
conclusion.
(3.) Reasoning and
proofs are
fundamental to
mathematics and
help us prove or
disprove various
conjectures.
(4.) Only 1
counterexample is
necessary to prove a
statement false, but 1
example does not
prove the truth value
of a statement.
(1.) If two triangles
are congruent, then
their corresponding
sides have equal
lengths as one
proceeds around
both triangles (2.) If
two triangles are
congruent, their
corresponding
interior angles (or
angles opposite to the
corresponding sides)
have equal measures
as you move around
the two triangles.
(1.) The sum of the
angles for all
triangles—obtuse,
isosceles, or right—is
180 degrees. (2.)
Isosceles triangles
has two equal sides
or angles (3.) In an
isosceles triangle, the
angles opposite
congruent sides are
congruent. (4.) The
relative length of a
side corresponds to
the relative measure
of the angle opposite
it. That is, the largest
side is opposite the
largest angle; the
smallest side is
opposite the smallest
angle, etc
(1.) In any triangle, if
one of the sides is
produced, then the
exterior angle is
greater than either of
the opposite interior
angles. The exterior
angle is also equal to
the sum of the 2
opposite interior
angles.
(2.) In order to form a
triangle, the sum of
the 2 smallest sides
must be greater than
the length of the 3rd
side.
Seminar week –
3/22 – 3/26
Spring Break—
3/27-4/6
Similarity
4/7 – 4/21
5
11
What does it
mean for
geometric figures
to be similar?
How can I prove
similar triangles?
What are the
characteristics of
similar triangles?
G.42 Investigate, justify, and apply
theorems about geometric
relationships, based on the
properties of the line segment joining
the midpoints of two sides of the
triangle
G.44 Establish similarity of triangles,
using the following theorems: AA,
SAS, and SSS
G.45 Investigate, justify, and apply
theorems about similar triangles
G.46 Investigate, justify, and apply
theorems about proportional
relationships among the segments of
the sides of the triangle, given one or
more lines parallel to one side of a
triangle and intersecting the other
two sides of the triangle
G.47 Investigate, justify, and apply
theorems about mean
proportionality: (1)the altitude to the
hypotenuse of a right triangle is the
10 days
(1.) If we know that a
line joins the
midpoints of 2
sides in a
triangle, we can
make
generalizations
about the
relationship of
this line to the 3rd
side.
(1.) Triangles are
similar if and only if
their sides are in
proportion (2.)
Corresponding angles
will still be congruent
(as opposed to in
proportion) in similar
triangles (3.) AAA
only proves
JMAP
Regents
questions,
TFA ISAT
questions
mean proportional between the two
segments along the hypotenuse (2)
the altitude to the hypotenuse of a
right triangle divides the hypotenuse
so that either leg of the right triangle
is the mean proportional between
the hypotenuse and segment of the
hypotenuse adjacent to that leg
similarity, not
congruence.
(1.) If a line is parallel
to one side of a
triangle and
intersects the other
sides in two distinct
points, then it
separates these sides
into segments of
proportional lengths
(1.) The altitude to
the hypotenuse of a
right triangle forms
two triangles that are
similar to each other
and to the original
triangle
(2.) The altitude to
the hypotenuse of a
right triangle is the
mean proportional
between the
segments into which
it divides the
hypotenuse.
Transformations
4/22 – 5/3
How can I
transform
geometric figures
in the coordinate
grid?
G.54 Define, investigate, justify, and
apply isometries in the plane
(rotations, reflections, translations,
glide reflections) Note: Use proper
function notation.
G.55
8 days
(1.) An isometry of
the coordinate plane
is a linear
transformation which
preserves length. Put
JMAP
Regents
questions,
TFA ISAT
questions
How do the
different types of
transformations
affect the figures?
Investigate, justify, and apply the
properties that remain invariant
under translations, rotations,
reflections, and glide reflections
G.56 Identify specific isometries by
observing orientation, numbers of
invariant points, and/or parallelism
G.57 Justify geometric relationships
(perpendicularity, parallelism,
congruence) using transformational
techniques (translations, rotations,
reflections) G.58 Define, investigate,
justify, and apply similarities
(dilations and the composition of
dilations and isometries)
G.59
Investigate, justify, and apply the
properties that remain invariant
under similarities G.60 Identify
specific similarities by observing
orientation, numbers of invariant
points, and/or parallelism G.61
Investigate, justify, and apply the
analytical representations for
translations, rotations about the
origin of 90º and 180º, reflections
over the lines x=0, y=0 , and y=x ,
and dilations centered at the origin
another way,
isometries produce
congruent figures.
(1.) We can use our
existing knowledge of
perpendicularity,
parallelism, and
congruence to make
generalizations about
the results of various
transformations.
(1.) Not all
transformations are
isometries. Dilations
produce figures that
are similar, but not
congruent.
(2.) It is possible to
perform more than
one transformation
on the same figure to
obtain the image
from the preimage
Circles and Loci
5/4 – 5/14
What is a circle?
How can I graph
a circle on a
coordinate grid?
G.71 Write the equation of a circle,
given its center and radius or given
the endpoints of a diameter
G.72 Write the equation of a circle,
given its graph
Note: The center is an ordered pair of
integers and the radius is an integer.
G.73 Find the center and radius of a
circle, given the equation of the circle
in center-radius form
G.74 Graph circles of the form
( x  h)2  ( y  k )2  r 2
G.22 Solve problems using
compound loci
G.23 Graph and solve compound loci
10 days
(1.) Reflections result
in symmetry about a
point or line.
(1.) A circle includes
all points equidistant,
r, from the center (h,
k).
The equation of a
circle includes the
center point and
radius
(1.) We can figure out
the location of the
center and the length
of the radius of a
circle simply by
looking at the
equation of a circle.
(2.) The center (h, k)
Exam &
possible
presentation
JMAP
Regents
questions,
TFA ISAT
questions
in the coordinate plane
has the opposite sign
from how it appears
in the equation
(3.) The use of square
roots allows us to
find the center of the
circle.
(1.) Loci are sets of
points that satisfy 1
given condition,
usually involving a
distance from
another object. There
are 5 main scenarios
that loci problems
involve.
(2.) Compound loci
are sets of points that
satisfy more than 1
given condition.
Flex Days
IA 4: 5/18
Regents Review
5/19 – 6/14
1
Practice Regents
Regents review
packet covering
all previous units
Additional packet
mini lessons on
smaller topics
not formally
New topics in Review Packet (for
students who finish review early):
G.70 solving system of one linear and
one quadratic equation (already
know from IA)
G.25 – G.26 Formal logic
(conjunction, disjunction,
conditional, biconditional, inverse,
Lateral Area and
Volume
After School!!
--- in order to
cover this unit,
which is
essentially just
memorizing
formulas and
plugging and
chugging, and
leave enough
room for Regents
Review and flex
days, students
will come after
school or before
school for 1 hour,
once a week, for 2
months. This
creates another
week of class for
them, which is
enough time to
cover this unit
taught, for
students who
finish review
packet early
What is the
difference
between lateral
area and volume?
What is a prism,
a pyramid, a
cylinder, a cone,
and a sphere?
How can I find
their volumes
and lateral areas?
converse, contrapositive)
G.G.10 Know and apply that the
lateral edges of a prism are
congruent and parallel
G.G.11 Know and apply that two
prisms have equal volumes if their
bases have equal areas and their
altitudes are equal
G.G.12 Know and apply that the
volume of a prism is the product of
the area of the base and the altitude
G.G.13 Apply the properties of a
regular pyramid, including:
o
lateral edges are congruent
o
lateral faces are congruent
isosceles triangles
o
volume of a pyramid equals
one-third the product of the area of
the base and the altitude
G.G.14 Apply the properties of a
cylinder, including:
o
bases are congruent
o
volume equals the product
of the area of the base and the
altitude
o
lateral area of a right
-manipulating literal
equations to solve for
a different variable
than normal (ex:
given volume and h,
find area of base)
(1.) Volume is a
measure of 3dimensional space
like the amount of
liquid that a
container could hold.
(2.) Volume of a
prism is the area of
the base times the
height (or altitude) (v
= l * w * h)
(1.) A pyramid has a
square or rectangular
base and four slanted,
triangular sides. At a
minimum, the
opposing faces are
congruent
(2.) Lateral area is
the area of the lateral
faces. Surface area is
lateral area plus the
area of the base.
circular cylinder equals the product
of an altitude and the
circumference of the base
G.G.15 Apply the properties of a
right circular cone, including:
o
lateral area equals one-half
the product of the slant height and
the circumference of its base
o
volume is one-third the
product of the area of its base and its
altitude
G.G.16 Apply the properties of a
sphere, including:
o
the intersection of a plane
and a sphere is a circle
o
a great circle is the largest
circle that can be drawn on a sphere
o
two planes equidistant from
the center of the sphere and
intersecting the sphere do so in
congruent circles
o
surface area is
o
volume is
4 r 2
4 3
r
3
(3.) “Height” refers to
the perpendicular
distance from the
center of the base to
the common vertex
where all lateral faces
meet.
(4.) “Slant height”
refers to the distance
along a lateral face
from the base to the
common vertex
where all lateral faces
meet.
(1.) Volume of a
cylinder is the area of
the base times the
height (or altitude) (v
= ∏r2 * h)
(1.) A cone is made of
two surface pieces—
a circular base and an
additional lateral
surface
(1.) The cross section
of a sphere is a circle
(2.) A sphere consists
of all the points
whose distance from
the center is less than
or equal to the radius
June Regents/
Final Exams
5