GEOMETRY SYLLABUS Geometry Unit Descriptions

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GEOMETRY SYLLABUS



Geometry Unit Descriptions

Mathematical reasoning skills developed in previous courses, provide a solid
foundation for understanding proof and logic in Geometry. A study of analytical
geometry is weaved throughout this course, highlig
hted by a study of
transformations in the plane. The concept of area is also weaved throughout,
with a discussion of the areas of many different geometric objects. Previous
knowledge of angles and angle pairs is used to prove important results about the
c
ongruence of angles. A construction of the angle bisector is explored.
Triangles, quadrilaterals, and polygons are classified and important results are
proven. The triangle congruence theorems are explored in depth

and used to prove two triangles congr
uent. The concepts of ratio and proportion
are applied in the study of similar polygons. This leads to a discussion of special
right triangles, and the Pythagorean theorem. Different proofs of the
Pythagorean theorem are explored. The concept of a rati
o is necessary for the
study of the trigonometric ratios and their relationship to right triangles.
Connections among shapes and their properties are made in the study of circles,
including an exploration of how polygons, lines, and circles interact. Fin
ally, the
concept of shape is expanded to three dimensions, with special focus on the
study of prisms, pyramids, and cylinders. The concept of function is revisited in
the study of the formulas for volume and surface area.


Unit 1: Logic and Reasoning

T
his unit lays the foundation for proving geometric relationships by developing
the effective thinking needed to construct mathematically accurate logical
arguments and proofs. Different forms of logical statements are explored,
including the converse, neg
ation, contrapositive, and biconditional statements.
Counterexamples are used to show that a logical statement is false. Logical
statements are then used in defining basic geometric objects (i.e. points, lines,
and planes), postulates (i.e. addition, sub
traction, substitution and partition
postulates), and properties (i.e. reflexive, symmetric, transitive). Deductive
reasoning is used to prove basic geometric statements. Two column proofs are
introduced and a connection is made between this type of proof
, and written
justification of steps used in an algebra problem. Multiple types of proofs are
used to prove statements about segment congruence, including two
-

column
proofs and paragraph proofs. As an introduction to coordinate geometry, the

distance f
ormula and the midpoint are derived and organized as proofs. The
definition of equidistant is given and illustrated in the coordinate plane using the
distance formula.



High Priority Standards:

Geometry: 1.0, 2.0, 3.0, 17.0




Supporting Standards:

Geom
etry: 4.0


Unit 2: Angles and Angle Relationships


This unit begins with a discussion of congruence properties of angles, and how
they are used to prove that all right angles are congruent. A linear pair is then
defined and used to illustrate the conce
pt of supplementary angles. The vertical
angle theorem is then introduced and a proof given. The perpendicular postulate
is presented and illustrated using a compass and straightedge construction.
Important results about perpendicular angles are proven a
nd used to find the
measure of missing angles. The parallel postulate is then introduced and
important results are proven. The definition of a transversal is given, as well as
definitions for the different types of angels formed by two parallel lines and

a
transversal. Congruence theorems are proven for corresponding angles,
alternate interior angles, and alternate exterior angles. Theorems are also

proven for same side interior angles and same side exterior angles. The concept
of an angle bisector is

introduced, and illustrated using constructions. Angle
bisector theorems, including the perpendicular bisector theorem, and their
converses are proven. Throughout this unit, important theorems and results are
used to make logical arguments, find measure
s of missing angles, and solve
problems in a real world context. Continuing the study of coordinate geometry,
parallel and perpendicular lines, and their properties, are explored in the
coordinate plane. Translation, reflection, and rotation of angles in

the coordinate
plane are introduced. It is clearly illustrated that these motions preserves the
measure of angles, and the congruence of angles.



High Priority Standards:

Geometry: 2.0, 4.0, 7.0, 16.0, 17.0




Supporting Standards:

Geometry: 1.0, 3.0,
22.0


Unit 3: Triangles and Triangle Congruence

This unit begins with proofs of the triangle sum theorem and the exterior angle
theorem. Triangles are then classified by side, and by angle. Important
vocabulary is defined, including special vocabulary

for right and isosceles
triangles. The base angle theorem for isosceles triangles is proven, as well as
results about equilateral triangles. Side relationships of triangles are studied,
and the proof of the triangle inequality theorem is given. The mid
-
segment
theorem and its proof are discussed and illustrated in the coordinate plane.
Corresponding parts of congruent triangles are identified and basic congruence
properties of triangles are given. Various postulates and theorems that are

used to prov
e the congruence of two triangles are explored, including the Side
-
Side
-
Side postulate, Side
-
Angle
-
Side postulate, Angle
-
Side
-
Angle postulate, and
the Angle
-

Angle
-
Side theorem. The Hypotenuse
-
Leg Theorem is then proven
as a method for proving two right
triangles are congruent. Throughout this unit,
important theorems and results are used to make logical arguments, find
measures of missing angles and sides, and solve problems in a real world
context. The Pythagorean theorem is then revisited in the coor
dinate plane, and
lengths of missing sides are found. The converse of the Pythagorean theorem is
used to develop conditions for when a triangle is acute, right, or obtuse. Finally,
the area of a triangle is explored, and formulas for the area of a triang
le

are derived and used to find the area of triangles in the coordinate plane.

Transformations in the plane are then used on triangles.



High Priority Standards:

Geometry: 2.0, 4.0, 10.0, 12.0, 13.0,




Supporting Standards:

Geometry: 1.0, 3.0, 5.0,

6.0, 8.0, 15.0, 22.0




Unit 4: Quadrilaterals


This unit begins in a similar fashion as the previous unit, by proving the
Quadrilateral Angle Sum theorem. Parallelograms are then defined and
important results are proven regarding the sides and angles

of a parallelogram.
These theorems are used to find the measure of missing sides and angles.
Other theorems are introduced in order to prove that certain quadrilaterals are
parallelograms. Special parallelograms are then defined, including the rhombus
,
rectangle, and square. Conditions are developed that describe when a
parallelogram is a rhombus, rectangle, or square. Special quadrilaterals are

defined, including a trapezoid and kite. Conditions for an isosceles trapezoid are
developed. The midseg
ment theorem for trapezoids is proven. Formulas for the
area of various quadrilaterals are developed, and used in a real world context.
The Pythagorean theorem is used again in order to find the lengths of the
diagonals of a rhombus and a kite. Area is

revisited in terms of quadrilaterals.
Formulas for the areas of parallelograms and trapezoids are developed and
used. Finally, quadrilaterals are plotted on the coordinate plane, and translated,
reflected, or rotated. Each of these transformations is
identified as an isometry,
in that they preserve length.



High Priority Standards:

Geometry: 2.0, 7.0, 10.0, 12.0, 13.0, 22.0




Supporting Standards:

Geometry: 1.0, 3.0, 4.0, 8.0, 15.0


Unit 5: Polygons


Polygons are introduced by developing the pol
ygon angle sum theorem. The

differences between convex and concave polygons are explored, and sets of
polygons are classified. A regular polygon is defined, and the formula for the
measure of the interior angle of a regular polygon is derived. This for
mula is also
used to find the number of sides of a polygon given the angle measure. The
polygon exterior angle theorem is proven. The concept of area is applied to
polygons. Areas of both regular and irregular polygons are found. Throughout
this unit,
important theorems and results are used to find measures of missing
angles and sides, and solve problems in a real world context. Isometries in the
coordinate plane are further discussed in terms of polygons.



High Priority Standards:

Geometry: 2.0, 12.
0, 13.0, 22.0




Supporting Standards:

Geometry: 1.0, 3.0, 4.0, 7.0, 8.0, 10.0

Unit 6: Similar Polygons


Ratios and proportions are used to find missing lengths in geometric figures.
Similarity is then introduced in terms of a polygon. Corresponding par
ts of similar
polygons are identified, and results are proven showing the proportional
relationship of the perimeter of a polygon and the lengths of its sides. Ratios are
used to relate corresponding sides of similar polygons. Areas of similar polygons
a
re investigated. Triangles then become the focus. The Angle
-
Angle similarity
postulate is introduced and used to prove two triangles are similar. The Side
-
Side
-
Side and the Side
-
Angle
-
Side similarity theorems are proven and also used
to prove two trian
gles similar. The Triangle Proportionality Theorem and its
converse are also proven and used to solve problems in a real world

context. Pythagorean triples and similarity are discussed, including an
introduction to the special right triangles (30
-
60
-
90

and 45
-
45
-
90). Similarity is
discussed in the coordinate plane by using similar triangles to illustrate the
concept of the slope of a line. Throughout this unit, important theorems and
results are used to find measures of missing angles and sides, and s
olve
problems in a real world context.



High Priority Standards:

Geometry: 2.0, 4.0, 5.0, 6.0




Supporting Standards:

Geometry: 1.0, 3.0, 8.0, 10.0, 17.0, 20.0



Unit 7: Right Triangle Trigonometry

This unit begins with an exploration of several
different proofs of the Pythagorean

theorem. The three trigonometric ratios (sine, cosine, tangent) are defined in
terms of a right triangle. The trigonometric ratios are found for given angles in a
triangle, including the special right triangles. Valu
es of the trigonometric ratios
for 30
o
, 45
o
, 60
o
, and 90
o

angles are found. The trigonometric ratios are used in
a real world context by solving problems using indirect measurement. The
inverse trigonometric relationships are introduced, and used to solv
e triangles for
missing sides and angles in a right triangle. The trigonometric ratios are defined
in the coordinate plane using a right triangle placed at the origin along the

x
-
axis.
Using these definitions, basic trigonometric identities are developed
, including
the fundamental trigonometric identities.



High Priority Standards:

Geometry: 2.0, 14.0, 18.0, 19.0, 20.0, 15.0




Supporting Standards:

Geometry: 1.0, 3.0


Unit 8: Circles

This unit begins with the geometric definition of a circle, and

definitions of
important vocabulary, including center, radius, diameter, chord, secant, and
tangent. The concept of tangency is further explored, and the perpendicular
relationship between a circles tangent and its radius, is proven. The chord of a
circ
le is used in the construction of the circle’s center, and results about
congruent chords are proven. The focus of the unit then shifts to angles inside of
a circle. Arcs and inscribed angles are defined, and results relating theses are
proven. Theorems

about the congruence of inscribed angles are proven. This
leads to a discussion of inscribed and circumscribed polygons. Conditions for
when a certain polygon can be inscribed in a circle are developed and

proven. Throughout this unit, important resul
ts about circles are used to find the

measure of missing angles or side lengths. Circles are then studied in the
coordinate plane. The equation of a circle centered at the origin is developed,
and knowledge of transformations of the plane is used to wri
te the standard form
of a circle centered at any point in the plane.



High Priority Standards:

Geometry: 2.0, 4.0, 7.0, 17.0, 21.0, 22.0




Supporting Standards:

Geometry: 1.0, 3.0, 8.0, 16.0


Unit 9: Solid Geometry


The concept of three
-
dimensional
geometry is introduced, and the definition of a

polyhedron is given, and connected with the definition of a polygon. Important

vocabulary is defined, including the faces, edges, and vertices of polyhedra.
Polyhedra are classified, and regular polyhedra

are defined, and compared to
regular polygons. The concept of convex and concave is revisited in terms of the
polyhedra. Euler’s theorem is introduced and used to relate the number of faces,
vertices, and sides of polyhedra. The concept of a net is int
roduced to help
define the surface area of prisms, cylinders, pyramids and cones. The concept
of volume is reviewed in terms of the volume of a cube. Formulas for the
volumes of prisms, cylinders, pyramids and cones are developed and used. The
concept o
f a sphere is defined, and formulas for the surface area and volume of
a sphere are found. Formulas for area and volume of geometric objects are
written as functions, and graphed on a coordinate plane. Throughout this unit
formulas and theorems relating
to three
-
dimensional solids are used to solve
problems in a real world context.



High Priority Standards:

Geometry: 2.0, 8.0, 9.0, 11.0


Supporting Standards:

Geometry: 1.0, 3.0, 10.0


Unit 10: CST Review Unit

In this final unit, the connections that
have been formed from unit to unit are
solidified. The relationships between all the skill sets and concepts that have
been learned throughout the course are communicated. As a way of
differentiating instruction, understanding of skills and concepts from

the course is
self
-
assessed, and an individualized plan is developed to address any
deficiencies.