ESSENTIALS OF GEOMETRY
FULL YEAR COURSE
IN
MATHEMATICS
RUTHERFORD HIGH SCHOOL
JUNE 2005
RUTHERFORD HIGH SCHOOL
Rutherford, New Jersey
COURSE OUTLINE
ESSENTIALS OF GEOMETRY
I.
INTRODUCTION
Essentials of Geometry is the third year in the essentials track sequence. It is
designed for junior students who have completed the Essentials of Algebra 1-A
and B program. The regular Geometry content is taught at a level more suitable
to the needs of these students.
Students who take this course will develop the ability to think creatively about
mathematics and reason logically. They will focus on problem solving and
communicating mathematically. They will gain skills that are needed to succeed
in the workplace of the twenty-first century.
Students in this course will use computers on a regular basis. Through discovery
exercises and laboratory explorations with computer software they will discover
many of the concepts for themselves. They also take an active part in using
various manipulatives in cooperative learning situations, thus applying teamwork
to the learning process.
Students in this course will appreciate the application of mathematics to the real
world by applying geometric concepts to everyday life.
Note: References to the New Jersey Core Curriculum Content Standards appear
as a numeral in parentheses.
II.
OBJECTIVES
A. SKILLS
The student will be able to:
1. Develop logical reasoning skills using inductive and deductive
processes. (4.2)
2. Apply Euclidian definitions and principles. (4.2)
3. Make connections between Geometry with Algebra. (4.5)
4. Understand transformations and their properties. (4.2)
5. Explore the nature of proof. (4.2)
6. Construct figures, then analyze, interpret, and verbalize the
relationships between them. (4.2)
B. CONTENT
The student will be able to:
1. Employ various reasoning methods by:
(4.2.12A)
a. Defining deductive and inductive reasoning.
b. Using inductive reasoning to draw conclusions about
numerical and geometric patterns.
c. Using deductive reasoning and algebraic postulates to
prove simple relationships.
d. Creating and translating conditional statements into
corresponding inverse, converse, and contrapositive modes.
e. Constructing a truth table from conditional statements.
f. Solving logic problems through the Venn diagrams and
other strategies.
2. Learn basic definitions by:
(4.2.12A)
a. Expanding on the meanings of undefined terms in
geometry.
b. Measuring angles and working with vertical angles.
c. Relating angles formed by parallel lines and transversals.
d. Proving angle and line relationships using deductive
reasoning.
e. Identifying types of angles.
f. Classifying the types of triangles by their angles and sides.
g. Applying properties of interior and exterior angles of
polygons.
3. Determine congruence and inequality of geometric figures by:
(4.2.12A)
a. Identifying congruent triangles.
b. Proving triangles congruent.
c. Proving parts of triangles congruent.
d. Determining the existence of triangles.
e. Discovering the relationship between sides and angles in a
single triangle and in two triangles.
4. Investigate quadrilaterals by:
(4.4.2.12A)
a. Identifying special types of quadrilaterals.
b. Discovering properties of parallelograms.
c. Discovering properties of special parallelograms.
d. Developing a hierarchal relationship among the types of
quadrilaterals.
e. Proving that quadrilaterals are parallelograms.
f. Performing addition and subtraction of vectors, using
parallelograms.
5. Calculate area and perimeter by: (4.2.12,D, E)
a. Developing and applying formulas for finding area and
perimeter.
b. Finding the maximum area of a rectangle given a fixed
perimeter.
c. Exploring Pick’s and Heron’s formulas.
6. Investigate the Pythagorean Theorem and right triangles by:
(4.1.12E)
a. Discovering the Pythagorean theorem.
b. Using the Pythagorean Theorem to find parts of right
triangles.
c. Identifying Pythagorean triplets.
d. Applying the converse of the Pythagorean Theorem.
e. Using algebraic properties to simplify radicals.
f. Discovering and applying relationships between the sides
of special right triangles.
g. Exploring right triangle trigonometric relationships.
7. Determine similarity of figures by: (4.2.12E)
a. Solving problems using ratio and proportion.
b. Finding missing parts of similar figures.
c. Exploring the Golden ratio as it relates to similar figures.
d. Discovering the postulates and theorems of similar figures.
8. Investigate circles by:
(4.2.12A)
a. Identifying important parts of circles and related segments.
b. Defining angles related to circles.
c. Discovering the arc/angle relationships.
d. Discovering the relationships between tangents, secants,
and chords.
e. Solving problems involving common internal and external
tangents of circles.
f. Solving problems dealing with area and circumference of a
circle.
g. Solving problems involving sector area and arc length in a
circle.
9. Calculate surface area and volume by:
(4.2.12A, E)
a. Constructing and determining the relationship between
parts of various polyhedra.
b. Constructing and investigating the five platonic solids.
c. Developing and applying the formulas for surface area and
volume of various solids.
d. Understanding the relationship between surface areas and
volumes of pyramids and cones, prisms and cylinders.
e. Exploring the problem of creating a box of maximum
volume from a rectangle of fixed dimensions.
f. Drawing perspective views of three-dimensional objects.
10. Use coordinate geometry by:
(4.2.12C)
a. Determining the midpoint and distance between points on a
coordinate plane.
b. Solving problems involving parallel, perpendicular and
intersecting lines.
c. Using slope and distance to classify figures in the plane.
d. Solving vertex-edge graph problems and algorithms.
11. Explore transformational geometry by:
(4.2.12B)
a. Defining mappings, transformations and isometries and
developing the hierarchal relationship among them.
b. Constructing and solving problems dealing with the basic
transformations of translations, reflections, rotations, and
dilations.
c. Identifying types of symmetry.
d. Sketching points and lines of symmetry and drawing
symmetric figures.
e. Creating figures that tessellate the plane.
12. Explore fractals and chaos by:
(4.2.12B)
a. Examining the iterative process of fractal formation.
b. Identifying fractals in nature and other settings.
c. Solving problems involving numerical and geometric
patterns in fractals.
III.
PROFICIENCY LEVELS
The course in Essentials of Geometry is designed for the special needs student
who is preparing for college.
IV.
METHODS OF ASSESSMENT
Students will be evaluated by a variety of assessment tools and strategies, which
include teacher-made tests and quizzes, homework, notebooks, portfolios, computer
labs, projects, presentations, and a final exam.
Students will also be encouraged to assess their own work in order to strive for
the highest level of achievement they can attain. Through perseverance, a strong
work ethic, and regular participation, students can gain self-confidence in their ability
to do mathematics and often improve their overall marking period grade.
It should be noted that in accordance with the district homework policy,
homework is twenty percent of the student’s overall grade.
The teacher will provide the subject area supervisor with suggestions for changes
to the curriculum or assessment procedures.
V.
GROUPING
Students in Essentials of Geometry have successfully completed Essentials of
Algebra 1-A and Essentials of Algebra 1-B. These two courses complete their study
of first year Algebra so as to enable them to see the connections between numerical
and algebraic operations and geometry.
VI.
ARTICULATION/SCOPE & SEQUENCE
Essentials of Geometry is intended to be a full year course. It provides the
framework necessary for students to either complete their three-year mathematics
requirement or to continue their mathematics coursework in Algebra 2.
VII.
RESOURCES
A. TEXT
Geometry, Holt, Rinehart and Winston, 2001.
B. REFERENCES
Geometry, Prentice Hall, 1993.
Discovering Geometry, Key Curriculum Press, 1997.
Geometry, Houghton Mifflin Company, 1983.
Kaplan SAT Math Workbook, 1998-99 Edition.
C. SOFTWARE
Geometer’s Sketchpad
Geometric SuperSupposer
Geometric Golfer
Appleworks 6.0
MacBestGrapher
Green Globs
Tesselmania
D. MANIPULATIVES
Tangrams
Geoboards
Tesselation Tiles, Polyhedra Models
Compass, Protractors, Rulers
Wax Paper
Scissors, Straws, String, Tape, Paper
Toothpicks, Gumdrops
VIII. METHODOLOGIES
Students in this course will use technology frequently in the form of the TI-34
scientific calculator and various computer software programs. Through discovery
exercises and laboratory explorations they will discover many of the concepts for
themselves. They will take an active part in using various forms of manipulatives
in cooperative learning situations, thus applying teamwork to the learning process.
IX.
SUGGESTED ACTIVITIES
A. Collaborative projects with appropriate level science course.
B. Portfolio work
C. Oral Presentations
D. Use of appropriate software programs to reinforce concepts
X.
INTERDISCIPLINARY CONNECTIONS
Connections are made to science, particularly environmental science, by means
of collaborative projects coordinating topics in the two subject areas. Writing
assignments and portfolios strengthen the connection between mathematics and
language arts literacy and the fine arts.
XI.
PROFESSIONAL DEVELOPMENT
As per the PIP/100 hour statement: the teacher will continue to improve
through participation in a variety of professional development opportunities.