**GEOMETRY
**

This course
will provide the students with experience that deepens the understanding of two
and three-dimensional objects and their properties.
Deductive and inductive reasoning, as well as investigative strategies in
drawing conclusions are stressed. Properties
and relationships of geometric figures will include the study of points, lines,
angles, planes, polygons with a special emphasis on quadrilaterals, triangles
and right triangles, circles, polyhedra, and other solids.
An understanding of proof and logic will be developed.
Use of graphing calculators and computer drawing programs will be used.

Notes:

(1)
Prerequisite:
Completion of Algebra I with a “C” or better

(2)
A Core 40 and AHD
course with standards defined

**Recommended
Textbook: Glencoe Mathematics: Geometry
by Glencoe/McGraw-Hill
**

**prepared Spring 2004
**

**By
**

**OAK HILL
UNITED SCHOOL CORPORATION
**

**Course**: **Geometry
**

**Unit:** 1

**Textbook
References: Chapters 1 - 3
**

**Suggested
Time Frame:**
9 weeks

**Content
Outline:
**

Ø
Points

Ø
Lines

Ø
Planes

Ø
Angles

Ø
Distance

Ø
Midpoint

Ø
Logic

Ø
Reasoning

Ø
Proofs

Ø
Conditional
Statements

Ø
Parallel Lines

Ø
Perpendicular Lines

Ø
Transversal

Ø
Slope

Ø
Equations of Lines

**Standards
to be Emphasized:
**

Standard
1

Points, Lines, Angles, and Planes

Students find
lengths and midpoints of lines. 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.

Example: Find the length and midpoint of the line joining the points *A*
(3, 8) and *B* (9, 0).

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.

Example: Construct the perpendicular bisector of a given line segment,
justifying each step of the process.

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

Example: In the diagram, the lines *k*
and *l* are parallel. What is the
measure of angle *x*? Explain your
answer.

G.1.4
Use coordinate geometry to find slopes, parallel lines, perpendicular
lines, and equations of lines.

Example: Find an equation of a line perpendicular to *y* = 4*x* – 2.

**Standard
8
Mathematical Reasoning and Problem Solving
**

*Students use a
variety of strategies to solve problems.
*

G.8.1
Use a variety of problem-solving strategies, such as drawing a diagram,
making a chart, guess-and-check, solving a simpler problem, writing an equation,
and working backwards.

Example: How far does the tip of the minute hand of a clock move in 20 minutes
if the tip is 4 inches from the center of the clock?

G.8.2
Decide
whether a solution is reasonable in the context of the original situation.

Example: John says the answer to the problem in the first example is
“12 inches.” Is his

answer reasonable? Why or why not?

*Students develop
and evaluate mathematical arguments and proofs.
*

G.8.3
Make conjectures about geometric ideas. Distinguish between information
that supports a conjecture and the proof of a conjecture.

Example: Calculate the ratios of side lengths in several different-sized
triangles with angles of 90°, 50°, and 40°. What do you notice about the
ratios? How might you prove that your observation is true (or show that it is
false)?

G.8.4
Write and interpret statements of the form “if – then” and “if
and only if.”

Example: Decide whether this statement is true: “If today is Sunday, then we
have school tomorrow.”

G.8.5
State, use, and examine the validity of the converse, inverse, and
contrapositive of “if – then” statements.

Example: In the last example, write the converse of the statement.

Example: Do you prove axioms from theorems or theorems from axioms?

G.8.7
Construct logical arguments, judge their validity, and give
counterexamples to disprove statements.

Example: Find an example to show that triangles with two sides and one angle
equal are not necessarily congruent.

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 charts, paragraphs, and two-column and indirect
proofs.

Example: In triangle *LMN*, *LM*
= *LN*. Prove that Ð*LMN*
@
Ð*LNM*.

G.8.9
Perform
basic constructions, describing and justifying the procedures used.

Distinguish between constructing and drawing geometric figures.

Example: Construct a line parallel to a given line through a given point
not

on the line, explaining and justifying each step.

**Key
Vocabulary:
**

point
conclusion

line
parallel

plane
perpendicular

ray
proof

angle
theorem

acute angle
postulate

obtuse angle
midpoint

right angle
bisector

straight angle
compass

vertical angle
transversal

linear pair
alternate interior angle

complementary
angle
alternate exterior angle

supplementary
angle
corresponding angle

collinear
consecutive interior angle

coplanar
skew lines

congruent
slope

concave
converse

convex
contrapositive

hypothesis
inductive and deductive reasoning

**Suggestions
for Incorporating Technology (for students):
**

Geometer's
Sketchpad

5 Minute
Checks on Overhead

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

**Suggestions
for Incorporating Technology (for teachers):**

Geometer's
Sketchpad

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

TestCheck

Worksheet
Builder

**Suggestions for
Differentiated Instruction:
**

·
Illustrate points,
lines, planes, and angles in nature

·
Use meterstick to
find distance between points or objects

·
List angle
relationships

·
Choose a traffic
sign and describe the polygon

·
Brainstorm
conjectures and counterexamples in nature

·
Use index cards to
explain converse, inverse, and contrapositve statements

·
Use index cards for
proofs

·
Illustrate
alternate interior, alternate exterior, corresponding, and consecutive
interior angles by marking a transversal and two parallel lines on the floor

·
Compare slopes of
parallel and perpendicular lines using note cards

**Notes
. . .
**

**Course:**
**GEOMETRY **

**Unit:** 2

**Textbook
References: Chapters 4 - 6
**

**Suggested
Time Frame:**
9 weeks

**Content
Outline:
**

Ø
Triangle
Relationships

Ø
Congruent Triangles

Ø
Proportions and
Similarities

Ø
Right Triangles

Ø
SSS

Ø
SAS

Ø
ASA

Ø
AAS

Ø
HL

Ø
Bisectors

Ø
Medians

Ø
Altitudes

Ø
Geometric Mean

Ø
Pythagorean Theorem

Ø
Trigonometry

**Standards
to be Emphasized:
**

**Standard
4
Triangles
**

Students identify
and describe types of triangles. They identify and draw altitudes, medians, and
angle bisectors. They use congruence and similarity. 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.

Example: Use a drawing program to create examples of right, acute, obtuse,
scalene, isosceles, equilateral, and equiangular triangles. Identify and
describe the attributes of each triangle.

Example: Draw several triangles. Construct their angle bisectors. What do you
notice?

G.4.3
Construct triangles congruent to given triangles.

Example: Construct a triangle given the lengths of two sides and the measure of
the angle between the two sides.

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

Example: Of two similar triangles, the second has sides half the length of the
first. The area of the first triangle is 20 cm^{2}. What is the area of
the second?

Example: In triangle *ABC*,
is parallel to
. What is the length of
?

Example: In the last example, prove that triangles *ABC* and *APQ* are similar.

G.4.7
Find and use measures of sides, perimeters, and areas of triangles.
Relate these measures to each other using formulas.

Example: The gable end of a house is a triangle 20 feet long and 13 feet high.
Find its area.

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

Example: Can you draw a triangle with
sides of length 7 cm, 4 cm, and 15 cm?

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

Example: Draw a triangle with vertices at (1, 3), (2, 5), and (6, 1).
Draw another triangle with vertices at (-3, -1), (-2, 1), and (2, -3). Are
these triangles the same shape and size?

**Standard
5
Right Triangles**

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.

Example: On each side of a right triangle, draw a square with that side of the
triangle as one side of the square. Find the areas of the three squares. What
relationship is there between the areas?

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

Example: In triangle *ABC* with right
angle at *C*, draw the altitude
from *C*
to
. Name all similar triangles in the diagram. Use these similar triangles to
prove the Pythagorean Theorem.

G.5.3
Use special right triangles (30° - 60° and 45° - 45°) to solve
problems.

Example: An isosceles right triangle has one short side of 6 cm. Find the
lengths of the other two sides.

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

Example: In triangle *ABC*, tan *A*
=
1/5
. Find sin *A* and cot *A*.

G.5.5
Know and use the relationship sin^{2} *x*
+ cos^{2} *x* = 1.

Example: Show that, in a right triangle, sin^{2} *x* + cos^{2} *x* = 1
is an example of the Pythagorean Theorem.

G.5.6
Solve word problems involving right triangles.

Example: The force of gravity pulling an object down a hill is its weight
multiplied by the sine of the angle of elevation of the hill. What is the force
on a 3,000-pound car on a hill with a 1 in 5 grade? (A grade of 1 in 5 means
that the hill rises one unit for every five horizontal units.)

**Key
Vocabulary:
**

right triangle
centroid

acute triangle
circumcenter

obtuse
triangle
incenter

Scalene
triangle
orthocenter

Isosceles
triangle
angle bisector

equilateral
triangle
perpendicular bisector

equiangular
triangle
ratio

base angle
proportion

included side
similar

included angle
cross products

remote
interior angle
midsegment

altitude
geometric mean

median
Pythagorean triple

Trigonometric
functions – sine, cosine, tangent

**Suggestions
for Incorporating Technology (for students):
**

Geometer's
Sketchpad

5 Minute
Checks on Overhead

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

Scientific
Calculator

**Suggestions
for Incorporating Technology (for teachers):
**

Geometer's
Sketchpad

Scientific
Calculator

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

TestCheck

Worksheet
Builder

**Suggestions for
Differentiated Instruction:
**

·
Illustrate
different types of triangles in nature

·
Show that the
angles of a triangle total 180 degrees by cutting the angles and arranging
them in a straight line

·
Students create
notecards illustrating the different ways to prove triangles are congruent

·
Explore
circumcenter, centroid, orthocenter, and incenter by folding paper triangles

·
Summarize a proof

·
Compare and
contrast theorems

·
Students can find
the height of a wall by using a shadow and proportions

·
Explore the lengths
of sides in special right triangles by paper folding

**Notes
. . .
**

**Course:**
**Geometry
**

**Unit**: 3

**Textbook
References: Chapters 7 - 9
**

**Suggested
Time Frame:** 9 weeks

**Content
Outline:
**

Ø
Quadrilaterals

Ø
Parallelograms

Ø
Rectangle

Ø
Square

Ø
Rhombus

Ø
Trapezoid

Ø
Kite

Ø
Polygons

Ø
Transformations

Ø
Rotation

Ø
Reflection

Ø
Translation

Ø
Vector

Ø
Circles

Ø
Arcs

Ø
Chords

Ø
Secant

Ø
Tangent

**Standards
to be Emphasized:
**

**Standard
2
Polygons
**

Students identify
and describe polygons and measure interior and exterior angles. They use
congruence, similarity, symmetry, tessellations, and transformations. They find
measures of sides, perimeters, and areas.

G.2.1
Identify and describe convex, concave, and regular polygons.

Example: Draw a regular hexagon. Is it convex or concave? Explain your answer.

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

Example: Calculate the measure of one interior angle of a regular
octagon.

Explain your method.

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

Example: Divide a regular hexagon into triangles by joining the center to
each vertex. Show that these triangles are all the same size and shape
and find the sizes of the interior angles of the hexagon.

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

Example: Use a drawing program to create regular hexagons, regular octagons, and
regular pentagons. Under the drawings, describe which of the polygons would be
best for tiling a rectangular floor. Explain your reasoning.

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

Example: Do these four points form a square: (2, 1), (6, 2), (5, 6), (1,
5)?

**Standard
3
Quadrilaterals
**

Students identify
and describe 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 quadrilateral square, rectangle, rhombus, parallelogram, trapezoid, and kite.

Example: Use a drawing program to create a square, rectangle, rhombus, parallelogram, trapezoid, and kite. Judge which of the quadrilaterals has
perpendicular diagonals and draw those diagonals in the figures. Give a
convincing argument that your judgment is correct.

G.3.4
Use coordinate
geometry to prove properties of quadrilaterals, such as

Example: Is rectangle *ABCD* with
vertices at (0, 0), (4, 0), (4, 2), (0, 2) congruent
to rectangle *PQRS* with vertices at
(-2, -1), (2, -1), (2, 1), (-2, 1)?

**Standard
6
Circles
**

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
circles.

G.6.1
Find the center of a given circle. Construct the circle that passes
through three given points (not in a straight line).

Example: Given a circle, find its center by drawing the perpendicular bisectors
of two chords.

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

Example: What is the angle between a tangent to a circle and the radius at the
point where the tangent meets the circle?

G.6.3
Prove theorems related to circles.

Example: Prove that an inscribed angle in a circle is half the measure of the
central angle with the same arc.

G.6.4
Construct tangents to circles and circumscribe and inscribe circles.

Example: Draw an acute triangle and construct the circumscribed circle.

G.6.5
Define, find, and use measures of arcs and related angles (central,
inscribed, and intersections of secants and tangents).

Example: Find the measure of angle *ABC*
in the diagram below.

Example: Are circles with the same center always the same shape? Are they always
the same size?

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

Example: Which will give you more: three 6-inch pizzas or two 8-inch pizzas?
Explain your answer.

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

Example: Find the equation of the circle with radius 10 and center (6,
-3).

**Key
Vocabulary:
**

diagonal
circle
isometry

trapezoid
center
major arc

Isosceles
trapezoid
circumference
minor arc

rectangle
chord

square
circumscribed

rhombus
inscribed

kite
radius

rotation
secant

translation
tangent

reflection
diameter

transformation
semicircle

vector
central angle

glide
reflection
tessellation

**Suggestions
for Incorporating Technology (for students):
**

Geometer's
Sketchpad

5 Minute
Checks on Overhead

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

**Suggestions
for Incorporating Technology (for teachers):
**

Geometer’s
Sketchpad

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

TestCheck

Worksheet
Builder

**Suggestions for
Differentiated Instruction:
**

·
Illustrate angle
bisectors and diagonals with paper polygons

·
Compose a picture
using right triangles

·
List similarities
between a rectangle, square, and rhombus

·
Use string to show
congruent diagonals intersect at their midpoints

·
Name examples of
trapezoids in the real world

·
Name examples of
reflections in nature

·
Explore the idea of
tessellations with regular and irregular polygons

·
Apply knowledge of
circles to a 16 inch pizza

**Notes
. . .
**

**Course:**
**Geometry
**

**Unit:** 4

**Textbook
References: Chapters 10 - 13
**

**Suggested
Time Frame**: 9 weeks

**Content
Outline:
**

Ø
Areas of Polygons

Ø
Areas of Circles

Ø
Surface Area

Ø
Prisms

Ø
Cylinders

Ø
Pyramids

Ø
Cones

Ø
Spheres

Ø
Volume

Ø
Congruent Solids

Ø
Similar Solids

**Standards
to be Emphasized:
**

G.2.5
Find
and use measures of sides, perimeters, and areas of polygons. Relate
these measures to each other using formulas.

Example: A rectangle of area 360 square yards is ten times as long as it
is wide. Find its length and width.

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

Example: Of two similar rectangles, the second has sides three times the length
of the first. How many times larger in area is the second rectangle?

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

Example: A section of roof is a trapezoid with length 4 m at the ridge
and 6 m at the gutter. The shortest distance from ridge to gutter is 3 m.
Construct a model using a drawing program, showing how to find the area of this
section of roof.

**Polyhedra
and Other Solids
**

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

G.7.1
Describe and make regular and nonregular polyhedra.

Example: Is a cube a regular polyhedron? Explain why or why not.

G.7.2
Describe the polyhedron that can be made from a given net (or pattern).
Describe the net for a given polyhedron.

Example: Make a net for a tetrahedron out of poster board and fold it up to make
the tetrahedron.

G.7.3
Describe relationships between the faces, edges, and vertices of
polyhedra.

Example: Count the sides, edges, and corners of a square-based pyramid. How are
these numbers related?

G.7.4
Describe symmetries of geometric solids.

Example: Describe the rotation and reflection symmetries of a square-based
pyramid.

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

Example: On Earth, is the equator a great circle?

G.7.6
Identify and know properties of congruent and similar solids.

Example: Explain how the surface area and volume of similar cylinders are
related.

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

Example: An ice cube is dropped into a glass that is roughly a right cylinder
with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ice
cube?

**Key
Vocabulary:
**

polyhedra
great circle

apothem
hemisphere

sector
oblique

face
slant height

edge
sphere

vertices
surface area

cone
volume

prism
congruent solids

pyramid
similar solids

cylinder

**Suggestions
for Incorporating Technology (for students):
**

Geometer's
Sketchpad

5 Minute
Checks on Overhead

GeomPASS:
Tutorial Plus

**Suggestions
for Incorporating Technology (for teachers):
**

Geometer's
Sketchpad

GeomPASS:
Tutorial Plus

MindJogger
Videoquizzes

TestCheck

Worksheet
Builder

**Suggestions for
Differentiated Instruction:**

·
Draw tangents using
a CD and a ruler

·
Work with partners
to the apothem by drawing a polygon and using the radius of the circumscribed
circle

·
Illustrate
irregular shapes by taking pictures of real life objects

·
Explain nets by
cutting out shapes and folding them together

·
Find the surface
area of a paper towel roll or toilet paper roll

·
Illustrate an
oblique cylinder using a stack poker chips

·
Find examples of
cylinders, cones, pyramids, and spheres

**Notes
. . .
**