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COURSE MAPPING:
Subject: Math
Course: Geometry
Basics of Geometry: 23 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
Assessment
4.3.8.A.1,
4.5.D.5
a) Patterns and Reasoning:
Describing patterns,
Completing patterns,
Conjectures and
counterexamples
Find & describe
patterns.
What is next number or pattern
in sequence? How would you
describe a pattern? What is a
conjecture?
4.2.8.A.1,
4.2.6.A.1
b) Points, Lines, Planes:
Vocab: point, line, segment,
ray, plane, collinear,
coplanar, opposite rays,
intersection
Name, draw, &
visualize intersections
of points, lines &
planes.
When are points collinear?
When are points coplanar?
Where do planes intersect?
How do you draw and label
points, lines, segments & rays?
Quiz - sec a & b:
Patterns and
Reasoning,
Points, Lines,
Planes
4.2.12.C.1,
4.2.12.E.1
c) Segments: Segment
Addition Postulate, Distance
Formula, Pythagorean
Theorem, Congruency of
segments
Use of the Distance
Formula.
What is the Segment Addition
Postulate? What is the
Distance Formula? What is the
Pythagorean Theorem? When
are segments congruent?
Quiz - sec c:
Segments
4.2.8.A.1
d) Angles: Vocab: angle,
vertex, sides, congruent
angles, protractor, measure,
interior of angle, exterior,
angle addition postulate,
Classify as: acute, right,
obtuse or straight
Classification of
angles.
What are parts of angles? How
do you read a protractor?
What is the Angle Addition
Postulate? What are the 4
ways to classify angles?
Co-Curricular
Support Activities/
Experiences
Science patterns:
patterns of the moon,
and doubling period of
bacteria
Relate to pictures &
photos
4.2.8.A.1,
4.2.8.C.1,
4.2.12.C.1
e) Midpoints and Bisectors:
Angle bisectors, Finding
midpoints using formula,
Finding endpoints, Setting
congruent angles equal to
solve algebraically, Finding
angle measures
Find midpoints &
endpoints
What is the midpoint formula?
How do you find a midpoint?
How do you find the next
endpoint? How do you
algebraically solve for
congruent angles?
Pop Quiz - sec e:
Midpoints and
Bisectors
Algebraic problems
with angle measures
4.2.8.A.1,
4.3.12.D.2
f) Angle Pairs: Vertical
angles, Linear Pair,
Complementary angles,
Supplementary angles,
Finding angle measures
Identify particular
angle pairs.
Quiz - sec d,e & f :
Angles, Midpoints
and Bisectors, &
Angle Pairs
Algebraic problems
with angle measures
4.2.8.E.1,
4.3.12.C.1
g) Perimeter & Area:
Formulas for squares,
rectangles, triangles and
circles
Find perimeter & area
using formulas.
What are the special angle
pairs? What is a Linear Pair?
What are Vertical angles?
When are angles
Complementary? When are
angles Supplementary?
What are the formulas for area
and perimeter of special
figures?
Quiz - sec g:
Perimeter & Area
Real-life perimeter &
area problems
Unit Test
Reasoning and Proof: 21 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.2.12.A.4,
4.5.D.5
a) Conditional Statements:
7 postulates on points, lines
and planes, Draw diagrams,
Vocab: conditional,
hypothesis, conclusion,
converse, inverse, negate,
contrapositive, Types of
conditional statements, Give
Counterexamples.
Analyze conditional
statements.
What are the postulates for
points, lines & planes? How do
you write conditional
statements? Which part is the
Hypothesis? Conclusion? What
is the Converse? Inverse?
Contrapositive? What is the
Instance? Counterexample?
b) Reasoning with Algebra
Properties: Properties for
Equality and Congruency:
Reflexive, Symmetric,
Transitive, Addition,
Subtraction, Multiplication,
Division, Substitution,
Congruent Segments,
Segment Addition
Use properties from
Algebra.
4.2.12.A.4,
4.5.D.3,
4.5.D.5
Write postulates using
conditional
statements.
What are the algebraic
properties for Equality and
Congruency?
Assessment
Co-Curricular
Support Activities/
Experiences
English grammar:
correct usage with
if-then statements
Quiz - sec a & b:
Conditional
Statements, 7
postulates, &
Reasoning with
Algebra Properties
4.2.12.A.4,
4.5.D.3,
4.5.D.6
c) Proving Statements about
Segments & Angles = Proofs:
Setting up proofs 2 ways - 2
column table method with
Statements (the steps) &
Reasons (the properties) or a
paragragh giving more of a
logical thinking of the steps to
take (also with the reasons).
Justify statements &
write reasons.
How do you set up a formal
proof? How do you set up a
paragraph proof? Which key
definitions can be reasons?
What other reasons can you
use?
Pop Quiz - sec c:
Proving
Statements about
Segments &
Angles
Quiz - sec c:
Proofs
Group work setting up
proofs
Unit Test
Perpendicular and Parallel Lines: 24 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.2.8.A.1,
4.2.12.A.3
a) Lines, Planes and Angles:
Parallel lines & planes,
Perpendicular lines & planes,
Skew lines, Transversal,
Angles formed by 2 lines with
a transversal, Alternate
Interior ∠s, Consecutive
Interior ∠s, Alternate Exterior
∠s, Corresponding ∠s
Identify relationships
between lines &
angles.
What are the relationships
between lines? Between
planes? What pairs of angles
are formed by a transversal
and 2 lines?
b) Parallel Lines and
Transversals: Angle
relationships when lines are
parallel, Solve Algebraic
equations for angle
relationships, Congruent and
supplementary ∠ equations
c) Proving Lines are Parallel
= Proofs: 3 types of proofs:
a) 2 column (formal) b)
paragraph (informal) c) flow uses boxes, Proofs for
theorems and postulates for
 lines
Prove and use results
about parallel lines
and transversals.
What are the postulates and
theorems of parallel lines and a
transversal? Which ∠s are ≅ ?
Which ∠s are supplementary ?
How do you relate them
algebraically?
Use properties of
parallel lines and
transversals to write
proofs.
What are the 3 types of proofs?
What is a formal proof? What
is an informal proof? What is a
flow proof? How are they
similar? How are they
different? What proof reasons
may be used? Which key
definitions can be reasons?
4.2.8.A.1,
4.2.12.A.3
4.2.12.A.4,
4.5.D.3,
4.5.D.6
Prove results about
perpendicular lines.
Assessment
Co-Curricular
Support Activities/
Experiences
Science: leaf veins &
stems relate to parallel
lines and transversals.
Pop Quiz - sec a &
b: Lines, Planes
and Angles &
Parallel Lines and
Transversals
Quiz - sec a & b
Quiz - sec c:
Proving Lines are
Parallel = Proofs
Algebraic problems
with angle measures
related to parallel lines
and transversals
Group work setting up
proofs
4.2.8.C.1,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.2
4.2.8.C.1,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.1
4.5.B.2,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.2
d) Coordinate Plane &
Graphing: Slopes: visually
uses "rise over run", slope
formula,  and ⊥ slopes,
 and ⊥ equations,
equations in y = mx + b form,
standard form, graph lines
using a table, graph lines
using y = mx + b form,
Graph 2 equations on same
grid & look for point of
intersection
e) Writing equations of lines:
Writing equations - 3 types:
(using y = mx + b twice),
write equation from a slope &
a point (x,y), write equation
from 2 points, write  or ⊥
equations from a point &
equation
Find slopes of lines.
f) Writing tasks on graphing
and equations: Parallel &
Perpendicular slopes,
Parallel & Perpendicular
lines, graphing lines, writing
equations, finding shapes
between lines, use of state
math Rubric for open-ended
questions
Complete open-ended
writing tasks that could
be used as HSPA
Math prompts on lines,
slopes & equations.
Graph lines.
Write equations given
various information.
How do you find slope visually?
From formula? How do you
find slope from an equation?
How do you find  and ⊥
slopes? How do you write 
and ⊥ equations? How do you
graph lines using a table? How
do you graph lines using y =
mx + b form?
Quiz - sec d:
Coordinate Plane
& Graphing
How do you write equations
from a graph? How do you
write equations using y = mx
+b? How do you write
equation from a slope & a point
(x,y)? How do you write
equation from 2 points? How
do you write  or ⊥ equations
from a point & equation?
How do you answer openended questions? How are
open-ended graded with State
Rubric? How do you graph
equations? How do you use
equations to graph shapes on
plane? How to write equations
that are  or ⊥ ?
Quiz - sec e:
Writing equations
of lines
Art: graphic artists use
slopes to create
designs
Pop Quiz - sec d
Project related to
graphs & equations
Unit Test
Quiz - sec f:
Writing tasks on
graphing and
equations
English grammar:
correct usage with
writing tasks on lines
and their relationships
Assessment
Co-Curricular
Support Activities/
Experiences
Congruent Triangles: 15 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.2.8.A.1,
4.2.12.A.3,
4.3.8.D.4
a) Triangle concepts:
Classify by sides, classify by
angles, parts of triangles,
right Δs, isosceles Δs,
equilateral Δs, sum of 3 ∠s of
Δ = 180 °, sum of 2 remote
∠s = exterior ∠
Classify triangles.
How do you classify Δs ?
What are the parts of Δs ?
What is the sum of the 3 ∠s of
the Δ? How do you find the
measure of the exterior ∠?
How do you find ∠ measures in
Δs?
Find angle measures
of triangles.
Algebraic problems
with angle measures
of triangles
4.2.8.A.1,
4.2.12.A.3,
4.3.8.D.4
b) Congruence of Triangles:
Naming ≅ parts of
≅ polygons, apply algebraic
solving to find missing parts
of 2 ≅ Δs
c) Proving Triangles are
Congruent:: Congruency
postulates for ≅ Δs with
drawings, 6 possibilities: 4
work for ≅ Δs: ASA, SSS,
SAS, AAS, and 2 don't work:
AAA & SSA, finding missing
parts that are needed to
prove congruency
d) Proofs for triangles: Use
of ASA, SSS, SAS, and AAS
in proofs, use of CPCTC in
proofs, reflexive prop used
for "common side", use
vertical angles, bisect &
midpoint, perpendicular,
alternate interior angles, etc.
Identify congruent
figures and
corresponding parts.
What are corresponding parts?
How do you name
corresponding parts? What is
CPCTC ?
Quiz - sec a & b:
Triangle concepts
& Congruence of
Triangles
Use of congruence
postulates to prove
triangles are
congruent.
What are the 6 congruency
postulates for ≅ Δs ? Which 4
postulates work? Which 2
don't work? What is needed to
prove 2 Δs ≅? How do you
determine the missing needed
parts?
Quiz - sec c:
Proving Triangles
are Congruent
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
a) Perpendicular and Angle
Bisectors: Perpendicular
bisector, equidistant from 2
pts, perpendicular bisector
theorem, angle bisector,
angle bisector theorem, find
distance of point to the sides
of angle, conclude if point is
on the bisector, use bisectors
in proofs with triangle
congruency, steps for
congruency of triangles,
circumcenter, incenter
Use properties of
perpendicular
bisectors and angle
bisectors.
What is a perpendicular
bisector? What is an angle
bisector? How do you
represent the shortest distance
of point to line? How do you
know if the point is on the
perpendicular bisector? How
do you know if point is on the
angle bisector? What is the
circumcenter? What is the
incenter?
4.2.8.A.1,
4.2.12.A.3
4.2.12.A.4,
4.5.D.3,
4.5.D.6
Plan & write proofs on
congruent triangles.
How do you set up a proof?
What concepts can be used as
reasons for proofs? Which
definitions can be used as
reasons? What new reasons
can be used for these
triangles? When do you use
CPCTC? When do you use the
Δ ≅ statement?
Properties of Triangles: 20 days
Architecture: triangles
in architecture – use of
Internet
Unit Test
Group work setting up
proofs
Assessment
Co-Curricular
Support Activities/
Experiences
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
b) Altitudes and Medians of
Triangles: Use medians,
altitudes and centroid
relationships: Median,
altitude, centroid,
orthocenter, ratio for the 2
sections of the median
c) Midsegment Theorem:
Use midpoints & midsections
of triangles: Midsegment =
segment connects midpts of
2 sides, midsegment is
parallel and 1/2 the length of
side, Include midpoints,
coordinate grids, slopes,
distance formula, perimeter
of 2 Δs
d) Angle & Side
Relationships of Triangles:
Use relationships of sides
and angles of a triangle:
shortest side of Δ is opposite
the smallest ∠, sum of 2
short sides must be greater
than the long side of Δ, 3rd
side: difference of 2 sides < x
< sum of 2 sides
e) Hinge Theorem: Use the
Hinge Theorem for 2 Δs: If 2
sides of Δ are ≅ to 2 sides of
2nd Δ, then the larger 3rd ∠
is opposite the larger 3rd
side(and converse), as the
angle widens the opposite
side also gets longer
Use properties of
altitudes and medians
of triangles.
What is a median? What is an
altitude? Where do the
medians intersect? What is the
ratio for the 2 sections of the
median? What is the centroid?
What is the orthocenter?
Quiz - sec a & b:
Perpendicular and
Angle Bisectors &
Altitudes and
Medians of
Triangles
Drawings: construct
angle bisectors and
medians to investigate
concurrent lines
Use properties of
midsections of
triangles.
What is the midsection of a
triangle? How does the
midsection relate to the side of
the triangle? What happens
when you connect 3
midsections? How does the
inner small Δ relate to outer
large Δ? Connected
midsections make what type of
triangles?
How do sides of Δ relate to
angles? Which size ∠ is
opposite the largest side?
What must be the relationship
of 2 sides of a triangle to the
third side? How do you find the
limits for the 3rd side of Δ ?
How do you know if the 3 sides
will make a Δ ?
Quiz - sec c:
Midsegment
Theorem
Art & Technology:
Fractals – use Internet
to research
What is the Hinge Theorem for
2 Δs? As the angle widens,
what happens to the opposite
side? What must be true for
the Hinge Theorem to apply?
Quiz - sec d & e:
Angle & Side
Relationships of
Triangles & Hinge
Theorem
Use triangle inequality
to decide which angles
and sides are largest
or smallest.
Use Hinge Theorem
and its converse to
compare side lengths
and angle measures.
Unit Test
Quadrilaterals: 20 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
Assessment
Co-Curricular
Support Activities/
Experiences
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
4.2.8.A.3
4.2.12.C.1
4.2.8.A.3
4.2.12.C.1
4.2.8.A.3
4.2.12.C.1
4.2.8.E.1
4.2.12.E.2
a) Polygons: Definition of
polygon, 3 conditions,
names per # sides, regular &
irregular, convex & concave,
if a triangle's angles total
180° then a quadrilateral =
360°
b) Parallelograms:
Parallelogram Properties:
opp sides are parallel, opp
sides & angles are
congruent, consecutive
angles are supplementary,
diagonals bisect each other,
3 ways to prove if 4 pts are
vertices of a parallelogram
using slopes and distance
formula
c) Rhombuses, Rectangles,
and Squares: Use properties
of special parallelograms:
squares, rectangles &
rhombus
d) Trapezoids and Kites:
Use properties of trapezoids
& kites, midsegment of
trapezoid = 1/2 sum of the 2
bases
e) Areas of Triangles and
Quadrilaterals: Area
Formulas for triangle, square,
parallelogram, rectangle,
rhombus, kite and trapezoid:
Identify and describe
polygons.
Use rule for sum of
angle measures of
polygons.
Use properties of
parallelograms.
Determine if 4 pts are
vertices of a
parallelogram using
slopes and distance
formula.
Use properties of
special parallelograms
and their diagonals.
Use properties of
trapezoids and kites.
Find areas of special
polygons using
formulas.
What is a polygon? What is a
diagonal? What are the names
of special polygons? When is
a polygon regular? Convex?
Concave?
What are the properties of a
parallelogram? What is the
sum of the 4 angles of a
parallelogram? Which angles
are congruent? Which angles
are supplementary? When is a
Quadrilateral a Parallelogram?
How do you prove that 4 points
are actually the vertices of a
parallelogram?
What are the properties of
rectangles? What are the
properties of squares? What
are the properties of a
rhombus?
What are the properties of a
trapezoid? What are the
properties of an isosceles
trapezoid? What are the
properties of a kite?
What is the Area formula for
each special quadrilateral?
Why are there 2 formulas for
the area of the rhombus?
Traffic signs: some
signs are International
warning signs – which
polygon for which
sign?
Quiz - sec a & b:
Polygons &
Parallelograms
Create charts with
properties of all the
special parallelograms
Quiz - sec c & d:
Rhombuses,
Rectangles &
Trapezoids and
Kites
Pop Quiz - sec e:
Areas of Triangles
and Quadrilaterals
Algebraic problems
using properties of
special parallelograms
and quadrilaterals
Algebraic problems
using areas of special
polygons
Unit Test
Transformations: 6 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
a) Symmetry: Lines of
symmetry for reflections,
Rotational symmetry, turns &
degrees, Symmetry in
alphabet
Find either or both
types of symmetry in
objects.
What are the 2 types of
symmetry? Rotate how many
degrees? What are the lines of
symmetry? How many? What
are the 3 types of
transformations?
Assessment
Co-Curricular
Support Activities/
Experiences
Art: designs that show
lines of symmetry
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
b) Translations : Translation
= slide to another position,
add or subtract to
coordinates
c) Reflections: Reflection =
flip over line or an axis, mirror
image,
Identify, describe, and
use translations.
How do coordinates change
when you translate image up?
Down? Left? Right?
Identify, describe, and
use reflections.
How do coordinates change
when you reflect image over X
axis? Over Y axis? Over X =Y?
d) Rotations: Rotation = spin
around point, Clockwise or
counterclockwise, Degrees:
90, 180 or 270, Rotate
around a point or the origin
Identify, describe, and
use rotations.
How do coordinates change
when you rotate image 180° ?
90° ? Around a point on the
image? around the origin?
Writing task Quiz
on sec b - d:
Translations,
Reflections &
Rotations
Writing tasks on
Translations,
Reflections &
Rotations
Project on
Translations,
Reflections &
Rotations – show all 3
transformations on a
cartoon design
More Proofs: 6 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
Assessment
Co-Curricular
Support Activities/
Experiences
4.2.12.A.4,
4.5.D.3,
4.5.D.6
a) Writing formal proofs: List
of possible reasons: Include
triangle congruencies,
definitions, postulates &
theorems
Plan & write proofs on
additional topics using
any definition or
property studied.
How do you set up a proof?
What are the types? What
reasons you can use? Does it
make logical sense? Does it
flow smoothly? Does it prove
what it is supposed to prove?
Quiz - sec a:
Writing formal
proofs
Group work setting up
proofs
Assessment
Co-Curricular
Support Activities/
Experiences
Similarity: 17 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
a) Ratios & Proportions:
Ratios = comparison of 2
items, fraction form or : form,
Solve Proportions by cross
multiply and divide
Find and simplify
ratios.
What are the forms of ratios?
How do you convert units of
ratios? How do you solve a
proportion?
b) Geometric Proportion
Properties: Use the
properties of proportions, use
ratios for parts of triangles,
Cross products, means =
extremes, the geometric
mean: x = square root a x b
Use properties of
proportions.
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
Set up and solve
proportions.
Use the geometric
mean.
What is the Cross Product
Property? What is the
Reciprocal Property? What is
the geometric mean? How do
you find the geometric mean?
Quiz - sec a & b:
Writing formal
proofs &
Geometric
Proportion
Properties
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
4.2.12.E.1
c) Similar Polygons:
Similarity symbol is ~, Angles
are ≅, Sides are proportional
to scale factor, Similar
Polygons have Proportional
perimeters, writing
Proportionality statements,
writing Similarity statements
d) Similar Triangles: Apply
properties of similar triangles:
Use: ≅ ∠s, proportionality
with sides, determining
similarity to solve for x & y,
scale factors, reduced forms
Identify similar
polygons.
e) Proportions and Similar
Triangles: Apply similarity
postulates to triangles:
AA Sim, SSS Sim, SAS Sim,
proportionality theorems with
parallel lines
Apply similarity
postulates to triangles.
Use scale factors to
find to find sides of
similar polygons.
Use properties of
similar triangles.
When are polygons
proportional? What are scale
factors? How do you write
similarity statements? What is
a Proportionality statement?
What is a Similarity statement?
How do you set up proportions
with polygon similarity?
What are the properties of
similar triangles? What are
scale factors used for?
Quiz - sec c:
Similar Polygons
What are the postulates of
similar triangles? When do you
use AA Sim? SSS Sim? SAS
Sim? What are the
proportionality theorems with
parallel lines? What is
relationship of the angle
bisector to the opposite side?
Quiz - sec d & e:
Similar Triangles
& Proportions and
Similar Triangles
Project – Use scale
factors for scale
drawings to calculate
size of real-life objects
Algebraic problems
using similarity to
solve for x & y values
Unit Test
Right Triangles and Trigonometry: 18 days
NJCCC
Standards
Content Topics/ Key Skills
Enduring
Understandings
Essential Questions
4.1.8.B.3,
4.2.12.E.1
a) Operations with square
roots: Review applications of
square roots: simplify
squares and simplify not
squares, add, multiply, &
rationalize the denominator
Simplify radical
expressions.
How do you simplify square
roots? How do you add and
subtract radicals? How do you
rationalize the denominator?
Why do you rationalize the
denominator?
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
b) The Pythagorean
Theorem: a² + b² = c²,
b² = c² - a², finding the
hypotenuse, finding the leg,
Pythagorean triple
Use the Pythagorean
Theorem in real-life
problems.
What is the Pythagorean
Theorem? How do you find the
hypotenuse? How do you
find one of the legs? How do
you apply the Pythagorean
Theorem to Area of Triangles?
Assessment
Co-Curricular
Support Activities/
Experiences
Algebraic problems
using the Pythagorean
Theorem
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
c) Converse of the
Pythagorean Theorem:
c² = a² + b² is right ∠,
c² < a² + b² is acute ∠,
c² > a² + b² is obtuse ∠
Use the Converse of
the Pythagorean
Theorem.
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
d) Similar Right Triangles:
Theorems of similar right Δs:
Use proportions & geometric
means to relate similar Right
Δs, 3 sets of triangle
drawings and proportions
Use the altitude of the
right triangle as a
geometric mean.
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
e) Special Right Triangles:
Draw 30,60,90 & 45,45,90
triangles, relationships of the
angles & legs, 45,45,90
→legs x, x, & x√2 =
hypotenuse, 30,60,90 → legs
are x√3 with hypotenuse =
2x
f) Trigonometric Ratios: Use
of trig ratios and the
calculator transformations:
Sine, Cosine & Tangent, sin
= opp / hyp, cos = adj / hyp,
tan = opp / adj, ratios switch
to decimals, use of calc with
angle degrees
Find the side lengths
of the special right
triangles.
4.2.12.E.1
Use lengths of sides of
triangles to classify
them.
Use trigonometric
ratios to find the angle
of elevation and
indirect
measurements.
How do you classify triangles?
When is it a Right Δ? When is
it an Acute Δ? When is it an
Obtuse Δ? How do you apply
the converse of the
Pythagorean Theorem to
classify Triangles?
Quiz - sec a, b &
c: Operations with
square roots,
Pythagorean
Theorem &
Converse of
Pythagorean
Theorem
What does the altitude form in
a right triangle? How does the
altitude & geometric mean
relate to the hypotenuse? How
does the leg & geometric mean
relate to a right triangle? How
do you set up the proportions
related to these theorems?
What are the special
relationships for 45,45,90 and
30,60,90 triangles? How do
you set up the proportions
related to these theorems?
Quiz - sec d & e:
Similar Right
Triangles &
Special Right
Triangles
What are the 3 trig ratios?
What does SohCahToa mean?
How do you set up trig ratios?
Quiz - sec f:
Trigonometric
Ratios
Unit Test
Construction: Use of
the Pythagorean
Theorem to develop
drawings and
measurements for
construction of
buildings
Algebraic problems
using trigonometric
ratios in real-life
situations
UNIT MAPPING:
Subject: Math
Course: Geometry
Basics of Geometry: 23 days
Units
a) Patterns and
Reasoning
NJCCCS
4.3.8.A.1,
4.5.D.5
Essential Questions
What is next number or pattern in sequence?
How would you describe a pattern? What is a
conjecture?
Content / Skill
Describing patterns, Completing
patterns, Conjectures and
counterexamples
b) Points, Lines,
Planes
4.2.8.A.1,
4.2.6.A.1
When are points collinear? When are points
coplanar? Where do planes intersect? How
do you draw and label points, lines, segments
& rays?
Vocab: point, line, segment, ray, plane,
collinear, coplanar, opposite rays,
intersection
Quiz - sec a & b
c) Segments
4.2.12.C.1,
4.2.12.E.1
What is the Segment Addition Postulate?
What is the Distance Formula? What is the
Pythagorean Theorem? When are segments
congruent?
Segment Addition Postulate, Distance
Formula, Pythagorean Theorem,
Congruency of segments
Quiz - sec c
d) Angles
4.2.8.A.1
What are parts of angles? How do you read a
protractor? What is the Angle Addition
Postulate? What are the 4 ways to classify
angles?
Vocab: angle, vertex, sides, congruent
angles, protractor, measure, interior of
angle, exterior, angle addition postulate,
Classify as: acute, right, obtuse or
straight
e) Midpoints and
Bisectors
4.2.8.A.1,
4.2.8.C.1,
4.2.12.C.1
What is the midpoint formula? How do you find
a midpoint? How do you find the next
endpoint? How do you algebraically solve for
congruent angles?
Angle bisectors, Finding midpoints using
formula, Finding endpoints, Setting
congruent angles equal to solve
algebraically, Finding angle measures
Pop Quiz - sec e
f) Angle Pairs
4.2.8.A.1,
4.3.12.D.2
What are the special angle pairs? What is a
Linear Pair? What are Vertical angles? When
are angles Complementary? When are angles
Supplementary?
Angle pairs: Vertical angles, Linear Pair,
Complementary angles, Supplementary
angles, Finding angle measures
Quiz - sec d,e & f
g) Perimeter & Area
4.2.8.E.1,
4.3.12.C.1
What are the formulas for area and perimeter
of special figures?
Perimeter & Area: Formulas for
squares, rectangles, triangles and
circles
Quiz - sec g
Unit Test
Assessment
Reasoning and Proof: 21 days
Units
a) Conditional
Statements
NJCCCS
4.2.12.A.4,
4.5.D.5
Essential Questions
What are the postulates for points, lines &
planes? How do you write conditional
statements? Which part is the Hypothesis?
Conclusion? What is the Converse? Inverse?
Contrapositive? What is the Instance?
Counterexample?
Content / Skill
7 postulates on points, lines and planes,
Draw diagrams, Vocab: conditional,
hypothesis, conclusion, converse,
inverse, negate, contrapositive, Types
of conditional statements, Give
Counterexamples.
Assessment
b) Reasoning with
Algebra Properties
4.2.12.A.4,
4.5.D.3,
4.5.D.5
What are the algebraic properties for Equality
and Congruency?
Properties for Equality and Congruency:
Reflexive, Symmetric, Transitive,
Addition, Subtraction, Multiplication,
Division, Substitution, Congruent
Segments, Segment Addition
Quiz - sec a & b
c) Proving
Statements about
Segments & Angles =
Proofs
4.2.12.A.4,
4.5.D.3,
4.5.D.6
How do you set up a formal proof? How do
you set up a paragraph proof? Which key
definitions can be reasons? What other
reasons can you use?
Setting up proofs 2 ways - 2 column
table method with Statements (the
steps) & Reasons (the properties) or a
paragraph giving more of a logical
thinking of the steps to take (also with
the reasons).
Pop Quiz - sec c
Quiz - sec c
Unit Test
Perpendicular and Parallel Lines: 24 days
Units
a) Lines, Planes and
Angles
NJCCCS
4.2.8.A.1,
4.2.12.A.3
Essential Questions
What are the relationships between lines?
Between planes? What pairs of angles are
formed by a transversal and 2 lines?
Content / Skill
Parallel lines & planes, Perpendicular
lines & planes, Skew lines, Transversal,
Angles formed by 2 lines with a
transversal, Alternate Interior ∠s,
Consecutive Interior ∠s, Alternate
Exterior ∠s, Corresponding ∠s
b) Parallel Lines and
Transversals
4.2.8.A.1,
4.2.12.A.3
What are the postulates and theorems of
parallel lines and a transversal? Which ∠s are
≅ ? Which ∠s are supplementary ? How do
you relate them algebraically?
Angle relationships when lines are
parallel, Solve Algebraic equations for
angle relationships, Congruent and
supplementary ∠ equations
Pop Quiz - sec a &
b,
Quiz - sec a &
b
c) Proving Lines are
Parallel = Proofs
4.2.12.A.4,
4.5.D.3,
4.5.D.6
What are the 3 types of proofs? What is a
formal proof? What is an informal proof? What
is a flow proof? How are they similar? How
are they different? What proof reasons may be
used? Which key definitions can be reasons?
3 types of proofs: a) 2 column (formal)
b) paragraph (informal) c) flow - uses
boxes, Proofs for theorems and
postulates for  lines
Quiz - sec c
d) Coordinate Plane
& Graphing
4.2.8.C.1,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.2
How do you find slope visually? From formula?
How do you find slope from an equation? How
do you find  and ⊥ slopes? How do you
write  and ⊥ equations? How do you graph
lines using a table? How do you graph lines
using y = mx + b form?
Slopes: visually uses "rise over run",
slope formula,  and ⊥ slopes,  and
⊥ equations, equations in y = mx + b
form, standard form, graph lines using a
table, graph lines using y = mx + b form,
Graph 2 equations on same grid & look
for point of intersection
Quiz - sec d,
Pop Quiz - sec d
e) Writing equations
of lines
4.2.8.C.1,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.1
How do you write equations from a graph?
How do you write equations using y = mx +b?
How do you write equation from a slope & a
point (x,y)? How do you write equation from 2
points? How do you write  or ⊥ equations
from a point & equation?
Writing equations - 3 types: (using y =
mx + b twice), write equation from a
slope & a point (x,y), write equation from
2 points, write  or ⊥ equations from a
point & equation
Quiz - sec e
Unit Test
f) Writing tasks on
graphing and
equations
4.5.B.2,
4.2.12.C.1,
4.3.12.B.1,
4.3.12.B.2
How do you answer open-ended questions?
How are open-ended graded with State
Rubric? How do you graph equations? How
do you use equations to graph shapes on
plane? How to write equations that are  or ⊥
?
Parallel & Perpendicular slopes, Parallel
& Perpendicular lines, graphing lines,
writing equations, finding shapes
between lines, use of state math Rubric
for open-ended questions
Quiz - sec f
Assessment
Congruent Triangles: 15 days
Units
a) Triangle concepts
NJCCCS
4.2.8.A.1,
4.2.12.A.3,
4.3.8.D.4
b) Congruence of
Triangles
4.2.8.A.1,
4.2.12.A.3,
4.3.8.D.4
c) Proving Triangles
are Congruent
d) Proofs for triangles
Essential Questions
How do you classify Δs ? What are the parts
of Δs ? What is the sum of the 3 ∠s of the Δ?
How do you find the measure of the exterior ∠?
How do you find ∠ measures in Δs?
What are corresponding parts? How do you
name corresponding parts? What is CPCTC ?
Content / Skill
Classify by sides, classify by angles,
parts of triangles, right Δs, isosceles Δs,
equilateral Δs, sum of 3 ∠s of Δ = 180 °,
sum of 2 remote ∠s = exterior ∠
Assessment
Naming ≅ parts of ≅ polygons, apply
algebraic solving to find missing parts of
2 ≅ Δs
Quiz - sec a & b
4.2.8.A.1,
4.2.12.A.3
What are the 6 congruency postulates for ≅ Δs
? Which 4 postulates work? Which 2 don't
work? What is needed to prove 2 Δs ≅? How
do you determine the missing needed parts?
Congruency postulates for ≅ Δs with
drawings, 6 possibilities: 4 work for ≅
Δs: ASA, SSS, SAS, AAS, and 2 don't
work: AAA & SSA, finding missing parts
that are needed to prove congruency
Quiz - sec c
4.2.12.A.4,
4.5.D.3,
4.5.D.6
How do you set up a proof? What concepts
can be used as reasons for proofs? Which
definitions can be used as reasons? What new
reasons can be used for these triangles? When
do you use CPCTC? When do you use the Δ
≅ statement?
Use of ASA, SSS, SAS, and AAS in
proofs, use of CPCTC in proofs,
reflexive prop used for "common side",
use vertical angles, bisect & midpoint,
perpendicular, alternate interior angles,
etc.
Unit Test
Properties of Triangles: 20 days
Units
a) Perpendicular and
Angle Bisectors
NJCCCS
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
Essential Questions
What is a perpendicular bisector? What is an
angle bisector? How do you represent the
shortest distance of point to line? How do you
know if the point is on the perpendicular
bisector? How do you know if point is on the
angle bisector? What is the circumcenter?
What is the incenter?
b) Altitudes and
Medians of Triangles
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
What is a median? What is an altitude?
Where do the medians intersect? What is the
ratio for the 2 sections of the median? What is
the centroid? What is the orthocenter?
c) Midsegment
Theorem
4.2.8.A.1
4.2.12.A.3
4.2.8.A.5
What is the midsection of a triangle? How
does the midsection relate to the side of the
triangle? What happens when you connect 3
midsections? How does the inner small Δ
relate to outer large Δ? Connected midsections
make what type of triangles?
d) Angle & Side
Relationships of
Triangles
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
How do sides of Δ relate to angles? Which
size ∠ is opposite the largest side? What must
be the relationship of 2 sides of a triangle to the
third side? How do you find the limits for the
3rd side of Δ ? How do you know if the 3 sides
will make a Δ ?
e) Hinge Theorem
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
What is the Hinge Theorem for 2 Δs? As the
angle widens, what happens to the opposite
side? What must be true for the Hinge
Theorem to apply?
Content / Skill
Perpendicular bisector, equidistant from
2 pts, perpendicular bisector theorem,
angle bisector, angle bisector theorem,
find distance of point to the sides of
angle, conclude if point is on the
bisector, use bisectors in proofs with
triangle congruency, steps for
congruency of triangles, circumcenter,
incenter
Use medians, altitudes and centroid
relationships: Median, altitude, centroid,
orthocenter, ratio for the 2 sections of
the median
Assessment
Quiz - sec a & b
Use midpoints & midsections of
triangles: Midsegment = segment
connects midpts of 2 sides, midsegment
is parallel and 1/2 the length of side,
Include midpoints, coordinate grids,
slopes, distance formula, perimeter of 2
Δs
Use relationships of sides and angles of
a triangle: shortest side of Δ is opposite
the smallest ∠, sum of 2 short sides
must be greater than the long side of Δ,
3rd side: difference of 2 sides < x < sum
of 2 sides
Quiz - sec c
Use the Hinge Theorem for 2 Δs: If 2
sides of Δ are ≅ to 2 sides of 2nd Δ, then
the larger 3rd ∠ is opposite the larger
3rd side(and converse), as the angle
widens the opposite side also gets
longer
Quiz - sec d & e
Unit Test
Quadrilaterals: 20 days
Units
a) Polygons
NJCCCS
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
Essential Questions
What is a polygon? What is a diagonal? What
are the names of special polygons? When is a
polygon regular? Convex? Concave?
Content / Skill
Definition of polygon, 3 conditions,
names per # sides, regular & irregular,
convex & concave, if a triangle's angles
total 180° then a quadrilateral = 360°
Assessment
b) Parallelograms
4.2.8.A.2
4.2.12.A.3
4.2.8.C.1
4.2.12.C.1
What are the properties of a parallelogram?
What is the sum of the 4 angles of a
parallelogram? Which angles are congruent?
Which angles are supplementary? When is a
Quadrilateral a Parallelogram? How do you
prove that 4 points are actually the vertices of a
parallelogram?
Parallelogram Properties: opp sides are
parallel, opp sides & angles are
congruent, consecutive angles are
supplementary, diagonals bisect each
other, 3 ways to prove if 4 pts are
vertices of a parallelogram using slopes
and distance formula
Quiz - sec a & b
c) Rhombuses,
Rectangles, and
Squares
4.2.8.A.3
4.2.12.C.1
What are the properties of rectangles? What
are the properties of squares? What are the
properties of a rhombus?
Use properties of special
parallelograms: squares, rectangles &
rhombus:
d) Trapezoids and
Kites
4.2.8.A.3
4.2.12.C.1
What are the properties of a trapezoid? What
are the properties of an isosceles trapezoid?
What are the properties of a kite?
Use properties of trapezoids & kites,
midsegment of trapezoid = 1/2 sum of
the 2 bases
Quiz - sec c & d
e) Areas of Triangles
and Quadrilaterals
4.2.8.A.3
4.2.12.C.1
4.2.8.E.1
4.2.12.E.2
What is the Area formula for each special
quadrilateral? Why are there 2 formulas for the
area of the rhombus?
Area Formulas for triangle, square,
parallelogram, rectangle, rhombus, kite
and trapezoid:
Pop Quiz - sec e
Unit Test
Transformations: 6 days
Units
a) Symmetry
NJCCCS
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
Essential Questions
What are the 2 types of symmetry? Rotate
how many degrees? What are the lines of
symmetry? How many? What are the 3 types
of transformations?
Content / Skill
Lines of symmetry for reflections,
Rotational symmetry, turns & degrees,
Symmetry in alphabet
b) Translations
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
How do coordinates change when you
translate image up? Down? Left? Right?
Translation = slide to another position,
add or subtract to coordinates
c) Reflections
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
How do coordinates change when you reflect
image over X axis? Over Y axis? Over X =Y?
Reflection = flip over line or an axis,
mirror image,
d) Rotations
4.2.8.B.1
4.2.8.C.2
4.2.12.B.1
How do coordinates change when you rotate
image 180° ? 90° ? Around a point on the
image? around the origin?
Rotation = spin around point, Clockwise
or counterclockwise, Degrees: 90, 180
or 270, Rotate around a point or the
origin
Writing task Quiz
on sec b - d
Project on sec b - d
NJCCCS
4.2.12.A.4,
4.5.D.3,
4.5.D.6
Essential Questions
How do you set up a proof? What are the
types? What reasons you can use? Does it
make logical sense? Does it flow smoothly?
Does it prove what it is supposed to prove?
Content / Skill
List of possible reasons: Include triangle
congruencies, definitions, postulates &
theorems
Assessment
Quiz - sec a
Assessment
More Proofs: 6 days
Units
a) Writing formal
proofs
Similarity: 17 days
Units
a) Ratios and
Proportions
NJCCCS
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
Essential Questions
What are the forms of ratios? How do you
convert units of ratios? How do you solve a
proportion?
Content / Skill
Ratios = comparison of 2 items, fraction
form or : form, Solve Proportions by
cross multiply and divide
b) Geometric
Proportion Properties
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
What is the Cross Product Property? What is
the Reciprocal Property? What is the
geometric mean? How do you find the
geometric mean?
Use the properties of proportions, use
ratios for parts of triangles, Cross
products, means = extremes, the
geometric mean: x = square root a x b
Quiz - sec a & b
c) Similar Polygons
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
When are polygons proportional? What are
scale factors? How do you write similarity
statements? What is a Proportionality
statement? What is a Similarity statement?
How do you set up proportions with polygon
similarity?
Similar Polygons: Similarity symbol is ~,
Angles are ≅, Sides are proportional to
scale factor, Similar Polygons have
Proportional perimeters, writing
Proportionality statements, writing
Similarity statements
Quiz - sec c
d) Similar Triangles
4.1.8.A.3,
4.2.8.A.4,
4.2.12.E.1
What are the properties of similar triangles?
What are scale factors used for?
Apply properties of similar triangles:
Use: ≅ ∠s, proportionality with sides,
determining similarity to solve for x & y,
scale factors, reduced forms
e) Proportions and
Similar Triangles
4.2.12.E.1
What are the postulates of similar triangles?
When do you use AA Sim? SSS Sim? SAS
Sim? What are the proportionality theorems
with parallel lines? What is relationship of the
angle bisector to the opposite side?
Apply similarity postulates to triangles:
AA Sim, SSS Sim, SAS Sim,
proportionality theorems with parallel
lines
Assessment
Quiz - sec d & e
Unit Test
Right Triangles and Trigonometry: 18 days
Units
a) Operations with
square roots
NJCCCS
4.1.8.B.3,
4.2.12.E.1
Essential Questions
How do you simplify square roots? How do
you add and subtract radicals? How do you
rationalize the denominator? Why do you
rationalize the denominator?
Content / Skill
Review applications of square roots:
simplify squares and simplify not
squares, add, multiply, & rationalize the
denominator
b) The Pythagorean
Theorem
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
What is the Pythagorean Theorem? How do
you find the hypotenuse? How do you find
one of the legs? How do you apply the
Pythagorean Theorem to Area of Triangles?
Pythagorean Theorem is a² + b² = c², b²
= c² - a², finding the hypotenuse, finding
the leg, Pythagorean triple
c) Converse of the
Pythagorean
Theorem
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
Converse of Pythagorean Theorem: c² =
a² + b² is right ∠, c² < a² + b² is acute
∠, c² > a² + b² is obtuse ∠
d) Similar Right
Triangles
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
How do you classify triangles? When is it a
Right Δ? When is it an Acute Δ? When is it an
Obtuse Δ? How do you apply the converse of
the Pythagorean Theorem to classify
Triangles?
What does the altitude form in a right triangle?
How does the altitude & geometric mean relate
to the hypotenuse? How does the leg &
geometric mean relate to a right triangle? How
do you set up the proportions related to these
theorems?
e) Special Right
Triangles
4.1.8.B.3,
4.2.8.A.2,
4.2.12.E.1
What are the special relationships for 45,45,90
and 30,60,90 triangles? How do you set up the
proportions related to these theorems?
Draw 30,60,90 & 45,45,90 triangles,
relationships of the angles & legs,
45,45, 90 →legs x, x, & x√2 =
hypotenuse, 30,60,90 → legs are x√3
with hypotenuse = 2x
Quiz - sec d & e
f) Trigonometric
Ratios
4.2.12.E.1
What are the 3 trig ratios? What does
SohCahToa mean? How do you set up trig
ratios?
Use of trig ratios and the calculator
transformations: Sine, Cosine &
Tangent, sin = opp / hyp, cos = adj /
hyp, tan = opp / adj, ratios switch to
decimals, use of calc with angle
degrees
Quiz - sec f
Unit Test
Assessment
Quiz - sec a, b & c
Theorems of similar right Δs: Use
proportions & geometric means to relate
similar Right Δs, 3 sets of triangle
drawings and proportions