# Curriculum Map

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10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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Franklin County Community School
Corporation ● Franklin County High School ● Brookville, Indiana

Curriculum Map

Course Title: Geometry

Quarter:

2

2011
-
2012

Essential Questions for this Quarter:

1.

How do you prove that two lines are parallel or perpendicular?

2.

What is the s
um of the measures of the angles of a triangle?

3.

How do you write and equation of a line in the coordinate plane?

4.

How do you indentify corresponding parts of congruent triangles?

5.

How do you show that two triangles are congruent?

6.

How can you tell whether a t
riangle is isosceles or equilateral?

7.

How do you simplify a radical expression

Unit/Time Frame

Standards

Content

Skills

Assessment

Resources

Chapter 3 (Part A)
Parallel and
Perpendicular Lines

3.1 Lines and Angles

3.2 Properties of Parallel
Lines

3.3 Proving Lines Parallel

3.4 Parallel and
Perpendicular Lines

3.5 Parallel Lines and
Triangles

Chapter 3 (Part B)
Parallel and
Perpendicular Lines

3.7 Part 1 Equations of
Lines in the Coordinate
Plane

3.7 Part 2 Equations of
Lines in the Coordinate
Plane

State Standards

G.1.1

G.1.3

G.1.4

G.4.1

G.1.5

G.1.6

G.1.7

G.5.4

G.2.3

G.2.7

G.2.12

G.2.14

Common Core
Standards

CC G.CO.1

CC G.CO.9

CC G.CO.10

CC G.CO.12

CC G.MG.2

CC G.MG.3

CC G.SRT.5

Angle Relationships Given
Parallel Lines

Slope of Parallel and
Perpendicular Lines

Connecting Algebra and
Geometry

Proving Triangles
Congruent

Proving Right Triangles
Congruent

Overlapping Tri
angles

Expressions

To identify relationships between
figures in space
To identify angles formed b two lines
and a transversal

To prove theorems about parallel lines
To use properties

of parallel lines to
find angle measures

To determine whether two lines are
parallel

To relate parallel and perpendicular
lines

To use parallel lines to prove a
To find measures of a
ngles of
triangles

To find slope and graph linear
equations

To write linear equations

T
o relate slope to parallel and
perpendicular lines

To recognize congruent figures and
their corresponding parts

To prove two triangles congruent
using the SSS and SAS Po
stulates

To prove two triangles congruent
usi
n
g the ASA Postulate

Textbook
assignments

Worksheet
assignments

Section Quizzes

Quizzes

Tests

Oral Responses

Observa
tions

Textbook
Prentice
-
Hall
Geometry
Foundation
Series 2011
Edition

Power

point

Presentations

Intro to Geometry
Frank Schaffer
Publications

Teacher
generated
worksheets

Notebooks

Franklin County Community School
Corporation ● Franklin County High School ● Brookville, Indiana

Curriculum Map

Course Title: Geometry

Quarter:

2

2011
-
2012

Essential Questions for this Quarter:

1.

How do you prove that two lines are parallel or perpendicular?

2.

What is the s
um of the measures of the angles of a triangle?

3.

How do you write and equation of a line in the coordinate plane?

4.

How do you indentify corresponding parts of congruent triangles?

5.

How do you show that two triangles are congruent?

6.

How can you tell whether a t
riangle is isosceles or equilateral?

7.

How do you simplify a radical expression

Unit/Time Frame

Standards

Content

Skills

Assessment

Resources

3.8 Slopes of Parallel and
Perpendicular Lines

Congruent Triangles

4.1 Congruent Figures

4.2 Triangle Congruence
by SSS and SAS

4.3 Part 1 Triangle

Congruence by ASA and
AAS

4.3 Part 2 Triangle
Congruence by ASA and
AAS

4
-
4

Using Corresponding
Parts of Congruent
Triangles

4
-
5

Isosceles and
Equilateral Triangles

4
-
6

Congruence in Right
Triangles

Standards for
Mathematical
Practice

SMP1

SMP2

SMP3

SMP4

SMP5

SMP6

SMP7

SMP8

To prove two triangles congruent
using the AAS theorem

To simplify radical expressions

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

State Standards

G.1: Students find lengths and midpoints of line
segments. They describe and
use parallel and perpendicular lines. They find slopes and equations of lines.

G.1.1: Find the lengths and midpoints of line segments in one
-

or two
-
dimensional
coordinate systems.

G.1.2: Construct congruent segments and angles,

angle bisectors, and parallel and
perpendicular lines using a straight edge and compass, explaining and justifying the
process used.

G.1.3: Understand and use the relationships between special pairs of angles formed
by parallel lines and transversals.

G.1
.4: Use coordinate geometry to find slopes, parallel lines, perpendicular lines,
and equations of lines.

G.2: Students identify and describe polygons and measure interior and
exterior angles. They use congruence, similarity, symmetry, tessellations, and
tr
ansformations. They find measures of sides, perimeters, and areas.

G.2.2: Find measures of interior and exterior angles of polygons, justifying the
method used.

G.2.3: Use properties of congruent and similar polygons to solve problems.

G.2.4: Apply transfo
rmations (slides, flips, turns, expansions, and contractions) to
polygons in order to determine congruence, similarity, symmetry, and tessellations.
Know that images formed by slides, flips and turns are congruent to the original
shape.

G.2.5: Find and use

measures of sides, perimeters, and areas of polygons, and
relate these measures to each other using formulas.

G.2.6: Use coordinate geometry to prove properties of polygons such as regularity,
congruence, and similarity.

G.3: Students identify and describ
e simple quadrilaterals. They use
congruence and similarity. They find measures of sides, perimeters, and
areas.

G.3.1: Describe, classify, and understand relationships among the quadrilaterals
square, rectangle, rhombus, parallelogram, trapezoid, and kite
.

G.3.2: Use properties of congruent and similar quadrilaterals to solve problems
involving lengths and areas.

G.3.3: Find and use measures of sides, perimeters, and areas of quadrilaterals, and
relate these measures to each other using formulas.

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

G.3.4: Us
e coordinate geometry to prove properties of quadrilaterals such as
regularity, congruence, and similarity.

G.4: Students identify and describe types of triangles. They identify and draw
altitudes, medians, and angle bisectors. They use congruence and simi
larity.
They find measures of sides, perimeters, and areas. They apply inequality
theorems.

G.4.1: Identify and describe triangles that are right, acute, obtuse, scalene,
isosceles, equilateral, and equiangular.

G.4.2: Define, identify, and construct altit
udes, medians, angle bisectors, and
perpendicular bisectors.

G.4.3: Construct triangles congruent to given triangles.

G.4.4: Use properties of congruent and similar triangles to solve problems involving
lengths and areas.

G.4.5: Prove and apply theorems in
volving segments divided proportionally.

G.4.6: Prove that triangles are congruent or similar and use the concept of
corresponding parts of congruent triangles.

G.4.7: Find and use measures of sides, perimeters, and areas of triangles, and
relate these mea
sures to each other using formulas.

G.4.8: Prove, understand, and apply the inequality theorems: triangle inequality,
inequality in one triangle, and hinge theorem.

G.4.9: Use coordinate geometry to prove properties of triangles such as regularity,
congrue
nce, and similarity.

G.5: Students prove the Pythagorean Theorem and use it to solve problems.
They define and apply the trigonometric relations sine, cosine, and tangent.

G.5.1: Prove and use the Pythagorean Theorem.

G.5.2: State and apply the relationshi
ps that exist when the altitude is drawn to the
hypotenuse of a right triangle.

G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cosecant,
secant, cotangent) in terms of angles of right triangles.

G.5.5: Know and use the relationsh
ip sin²x + cos²x = 1.

G.5.6: Solve word problems involving right triangles.

G.6: Students define ideas related to circles: e.g., radius, tangent. They find
measures of angles, lengths, and areas. They prove theorems about circles.
They find equations of ci
rcles.

G.6.2: Define and identify relationships among: radius, diameter, arc, measure of an
arc, chord, secant, and tangent.

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

G.6.3: Prove theorems related to circles.

G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed,
and

intersections of secants and tangents).

G.6.6: Define and identify congruent and concentric circles.

G.6.7: Define, find, and use measures of circumference, arc length, and areas of
circles and sectors. Use these measures to solve problems.

G.6.8: Find th
e equation of a circle in the coordinate plane in terms of its center and

G.7: Students describe and make polyhedra and other solids. They describe
relationships and symmetries, and use congruence and similarity.

G.7.2: Describe the polyhedron that

can be made from a given net (or pattern).
Describe the net for a given polyhedron.

G.7.4: Describe symmetries of geometric solids.

G.7.5: Describe sets of points on spheres: chords, tangents, and great circles.

G.7.6: Identify and know properties of cong
ruent and similar solids.

G.7.7: Find and use measures of sides, volumes of solids, and surface areas of
solids, and relate these measures to each other using formulas.

G.8: Mathematical Reasoning and Problem Solving

G.8.6: Identify and give examples of un
defined terms, axioms, and theorems, and
inductive and deductive proof.

G.8.8: Write geometric proofs, including proofs by contradiction and proofs involving
coordinate geometry. Use and compare a variety of ways to present deductive
proofs, such as flow c
harts, paragraphs, and two
-
column and indirect proofs.

Common Core Standards

Congruence G
-
CO

Experiment with transformations in the plane

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the und
efined notions of point, line, distance along a line, and
distance around a circular arc.

2. Represent transformations in the plane using, e.g., transparencies and geometry
software; describe transformations as functions that take points in the plane as
in
puts and give other points as outputs. Compare transformations that preserve
distance and angle to those that do not (e.g., translation versus horizontal stretch).

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations a
nd reflections that carry it onto itself.

4. Develop definitions of rotations, reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.

5. Given a geometric figure and a rotation, reflection, or tra
nslation, 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.

Understand congruence in terms of rigid motions

6. Use geometric descrip
tions of rigid motions to transform figures and to predict the
effect of a given rigid motion on a given figure; given two figures, use the definition
of congruence in terms of rigid motions to decide if they are congruent.

7. Use the definition of congrue
nce 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.

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from
the defi
nition of congruence in terms of rigid motions.

Prove geometric theorems

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 corre
sponding angles are congruent;

points on a perpendicular
bisector of a line segment are exactly those

equidistant from the segment’s

10. Prove theorems about triangles.
Theorems include: measures of interior

angles
of a triangle sum to 180°; bas
e 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.

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.

Make geometric constructions

12. Make formal geometric constructions w
ith a variety of tools and

methods
(compass and straightedge, string, reflective devices,

paper folding, dynamic
geometric software, etc.).
Copying a segment;

copying an angle; bisecting a
segment; bisecting an angle; constructing

perpendicular lines, incl
uding the
perpendicular bisector of a line segment;

and constructing a line parallel to a given line through a point not on the

line.

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

13. Construct an equilateral triangle, a square, and a regular hexagon

inscribed in a
circle.

Similarity, Right Triangles
, and Trigonometry G
-
SRT

Understand similarity in terms of similarity transformations

1. Verify experimentally the properties of dilations given by a center and

a scale
factor:

a.
A dilation takes a line not passing through the center of the

dilation to a
parallel
line, and leaves a line passing through the

center unchanged.

b.
The dilation of a line segment is longer or shorter in the ratio

given by the scale
factor.

2. Given two figures, use the definition of similarity in terms of similarity

transformati
ons to decide if they are similar; explain using similarity

transformations
the meaning of similarity for triangles as the equality

of all corresponding pairs of
angles and the proportionality of all

corresponding pairs of sides.

3. Use the properties of s
imilarity transformations to establish the AA

criterion for
two triangles to be similar.

Prove theorems involving similarity

4. Prove theorems about triangles.
Theorems include: a line parallel to one

side of a
triangle divides the other two proportionally
, and conversely; the

Pythagorean
Theorem proved using triangle similarity.

5. Use congruence and similarity criteria for triangles to solve problems

and to prove
relationships in geometric figures.

Define trigonometric ratios and solve problems involving
right

triangles

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.

7. Explain and use the relationship between the sine and cosine
of

complementary
angles.

8. Use trigonometric ratios and the Pythagorean Theorem to solve right

triangles in
applied problems.

Apply trigonometry to general triangles

9. (+) Derive the formula
A
= 1/2
ab
sin(C) for the area of a triangle by

drawing an
auxi
liary line from a vertex perpendicular to the opposite

side.

10. (+) Prove the Laws of Sines and Cosines and use them to solve

problems.

11. (+) Understand and apply the Law of Sines and the Law of Cosines

to find
unknown measurements in right and non
-
righ
t triangles (e.g.,

surveying problems,
Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

resultant forces).

Circles G
-
C

Understand and apply theorems about circles

1. Prove that all circles are similar.

2. Identify and describe relationships among inscribed angles, radii,

and chords.
Include the relation
ship 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.

3. Construct the inscribed and circumscribed circles
of a triangle, and

prove
properties of angles for a quadrilateral inscribed in a circle.

4. (+) Construct a tangent line from a point outside a given circle to the

circle.

Find arc lengths and areas of sectors of circles

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.

Expressing Geometric Properties with Equations G
-
GPE

Translate between the geometric description and the equation for a

conic section

1. Derive the equation of a circle of given center and radius using the

Pythagorean
Theorem; complete the square to find the center and

radius of a circle given by an
equ
ation.

2. Derive the equation of a parabola given a focus and directrix.

3. (+) Derive the equations of ellipses and hyperbolas given the foci,

using the fact
that the sum or difference of distances from the foci is

constant.

Use coordinates to prove simpl
e geometric theorems algebraically

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,

㌩3li敳

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).

6. Find the point on a directed line segment between two given points

that partitions
the segment in a given ratio.

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS

7. Use coordinates to compute perimeters of polygons and areas of

triangles and
rectangles, e.g.,
using the distance formula.

Geometric Measurement and Dimension G
-
GMD

Explain volume formulas and use them to solve problems

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.

2. (+) Give an informal argument using Cavalieri’s principle for the

formulas for the
volume of a sphere and other solid figures.

3. Use volume formulas for cylinders,

pyramids, cones, and spheres to

solve
problems.

Visualize relationships between two
-
dimensional and three

dimensional

objects

4. Identify the shapes of two
-
dimensional cross
-
sections of three

dimensional

objects, and identify three
-
dimensional objects gen
erated

by rotations of two
-
dimensional objects.

Modeling with Geometry G
-
MG

Apply geometric concepts in modeling situations

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).

2. Apply concepts of density based on area and volume in modeling

situations (e.g.,
persons per square mile, BTUs per cubic foot).

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).

Standards for Mathematical Practice

SMP1. Make sense of problems and persevere in solving them.

SMP2. Reason abstractly and quantitatively.

SMP3. Construct viable arg
uments and critique the reasoning of others.

SMP4. Model with mathematics.

SMP5. Use appropriate tools strategically.

SMP6. Attend to precision

Franklin County Community School Corporation ● Franklin County High School ● Brookville, Indiana

COMMON CORE AND INDIANA ACADEMIC STANDARDS