Download Unit Overview - Pompton Lakes School District

POMPTON LAKES SCHOOL DISTRICT
GEOMETRY FOUNDATIONS
COURSE OF STUDY
(January 2012)
Submitted By
The Mathematic Department
Dr. Paul Amoroso, Superintendent
Mr. Vincent Przybylinski, Principal
Mr. Anthony Mattera, Vice Principal
Frances J. Macdonald, District Mathematics Supervisor
BOARD MEMBERS
Mr. Jose A. Arroyo, Mrs. Catherine Brolsma, Mr. Shawn Dougherty,
Mrs. Nancy Lohse-Schwartz, Mr. Garry Luciani, Mr. Carl Padula,
Mr. Tom Salus, Mrs. Stephanie Shaw, Mr. Timothy Troast, Jr.
Unit Overview
Content Area:
MATH UNIT 1
Unit Title:
LINES AND ANGLES
Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10
Unit Summary:
Students learn about points, lines and planes, the building blocks of Geometry. Line segments,
rays, angles, polygons, parallel lines and perpendicular lines are also introduced in this unit.
Students explore congruent segments and angles and learn to construct them with a compass and
straightedge. Students expand on their knowledge of the Pythagorean theorem to master the
distance formula and use the midpoint formula to find the midpoint of a segment. Students also
compute the perimeter of a given polygon. Students identify the special angle relationships that
result when a transversal intersects parallel lines. Students solve problems by writing linear
equations and use slope to determine whether two lines are parallel, perpendicular or neither.
Primary interdisciplinary connections: Science, Business, Economics, History, Art
21st century themes: Mathematical Literacy
Unit Rationale:
The content and skills acquired in the unit are the tools necessary to study polygons. The terms
and notation will help students progress throughout the different geometric topics.
Learning Targets
Standards:
4.2 (Geometry and Measurement)
All students will develop spatial sense and the ability to use geometric properties, relationships
and measurement to model, describe, and analyze phenomena.
Content Statements:
Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve
the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric
modeling and measurement can help us describe and interpret our physical environment and to
solve problems.
CPI #
4.1.12C.1
Cumulative Progress Indicator (CPI)
Recognize the limitations of estimation, assess the amount of error resulting from
estimation, and determine whether the error is within acceptable tolerance limits.
4.2.12A.5
Perform basic geometric constructions using a variety of methods (e.g. straightedge
and compass, patty/tracing paper, or technology).
4.5B.1
Use communication to organize and clarify their mathematical thinking.
4.5B.2
Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others both orally and in writing.
4.5B.3
Analyze and evaluate the mathematical thinking and strategies of others.
4.5B.4
Use the language of mathematics to express mathematical ideas precisely.
4.5C.1
Recognize recurring themes across mathematical domains (e.g. patterns in number,
algebra, and geometry.)
4.5C.2
Use connections among mathematical ideas to explain concepts (e.g. two linear
equations have a unique solution because the lines they represent intersect at a
single point).
4.5D.1
Recognize that mathematical facts, procedures, and claims must be justified.
4.5D.2
Use reasoning to support their mathematical conclusions and problem solutions.
4.5D.5
Make and investigate mathematical conjectures.
Unit Essential Questions:
 What is the meaning and representation for
geometric terms?
 What is the relationship between slope and
rate of change?
Unit Enduring Understandings:
 Students will learn all the definition and how
to identify and label all geometric terms.
Students will be able to calculate slope and
express its meaning and relationship to a rate of
change of a quantity.
Unit Learning Targets:
Students will ...
 Solve problems by making models of points, lines, planes, and angles.
 Find the slope of various lines and interpret its meaning in terms of a rate of change.
 Calculate the midpoint and distance of a segment.
Explain the relationship between different angle pairs.
Evidence of Learning
Summative Assessment:
Students will work collaboratively to complete the task of solving real life problems involving
angles and lines. This task consists of group or individual questioning, class discussions, teacher
developing standardized tests/quizzes. Also, students will complete a written assignment to
explain their findings.
Equipment needed: graphing calculator, Smart Board, computer access, protractors, rulers
Teacher Resources: Math websites, textbooks, and resource books
Formative Assessments:
 Discussions
 Journal Entries
 Smart board presentation
 Evaluation Questions
 Tests/Quizzes
 Create/draw geometric models
Lesson Plans
Lesson
Timeframe
Lesson 1
Points, Lines, and Planes
Lesson 2
Linear Measure and Precision
Lesson 3
Distance and Midpoints
Lesson 4
Angle Measure
Lesson 5
Angle Relationships
Lesson 6
Polygons
Lesson 7
Parallel Lines and Transversals
Lesson 8
Angles and Parallel Lines
Lesson 9
Slopes of Lines
Lesson 10
Equations of Lines
Lesson 11
Proving Lines Parallel
Lesson 12
Perpendiculars and Distance
1 Day
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
2 Days
Teacher Notes:
Students can seek input from their peers and teachers throughout collaborative assignments and
activities.
Curriculum Development Resources
The completed Curriculum Design Template shows how this unit is situated within this district’s
Math Course.
Unit Overview
Content Area:
MATH UNIT 2
Unit Title:
TRIANGLES
Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10
Unit Summary:
In this unit, students prove triangles congruent and similar using various methods. Students
classify triangles angles according to their angles or sides and apply the angle sum theorem and
the exterior angle theorem. Special segments of triangles including bisectors, medians, and
altitudes are identified and explored. Students apply properties of inequalities relating to the
measures of angles and sides of a triangle and then extend those properties to two triangles.
Students learn to solve right triangles via various methods including Pythagorean Theorem,
trigonometric ratios, and geometric mean. Students also use the Law of Sines and Cosines to
solve non-right triangles.
Primary interdisciplinary connections: Wood shop, Science, Art,
21st century themes: Mathematical Literacy
Unit Rationale:
Students’ knowledge of properties of triangles in essential in a discovery of properties of other
polygons including quadrilaterals. This knowledge will help them in higher level mathematics
courses including pre-calculus and calculus. Architects, surveyors, and civil engineers use
trigonometric ratios in their work.
Learning Targets
Standards:
4.2 (Geometry and Measurement)
All students will develop spatial sense and the ability to use geometric properties, relationships
and measurement to model, describe, and analyze phenomena.
Content Statements:
Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve
the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric
modeling and measurement can help us describe and interpret our physical environment and to
solve problems.
CPI #
Cumulative Progress Indicator (CPI)
Use geometric models to represent real-world situations and objects and to solve
4.2.12A.1
problems using those models (e.g. use Pythagorean theorem to determine if an
object can fit through a door).
4.2.12A.3
Apply the properties of geometric shapes.
4.2.12A.5
Perform basic geometric constructions using a variety of methods.
4.2.12E.1
Use techniques of indirect measurement to represent and solve problems.
4.5B.1
Use communication to organize and clarify their mathematical thinking.
4.5B.2
Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others both orally and in writing.
4.5B.3
Analyze and evaluate the mathematical thinking and strategies of others.
4.5B.4
Use the language of mathematics to express mathematical ideas precisely.
4.5C.1
Recognize recurring themes across mathematical domains (e.g. patterns in number,
algebra, and geometry.)
4.5C.2
Use connections among mathematical ideas to explain concepts (e.g. two linear
equations have a unique solution because the lines they represent intersect at a
single point).
4.5D.1
Recognize that mathematical facts, procedures, and claims must be justified.
4.5D.2
Use reasoning to support their mathematical conclusions and problem solutions.
4.5D.5
Make and investigate mathematical conjectures.
Unit Essential Questions:
 What is the difference between congruent
triangles and similar triangle?
 What method(s) is most appropriate for
solving a right triangle or non-right triangle?
Unit Enduring Understandings:
 Students will compare the sides and angles of
triangles to determine whether they are
congruent or similar.
 Student will use their knowledge of various
methods learning included Pythagorean
Theorem, geometric mean, trigonometry, Law
of Sines and Cosines to solve a triangle.
Unit Learning Targets:
Students will ...
 Classify triangles according to their angles and sides.
 Determine whether triangles are congruent or similar.
 Prove triangles are congruent using various postulates and theorems for congruence.
 Identify special segments in triangles including median, angle bisector, perpendicular bisector,
and altitude.
 Use triangle inequality theorem to determine if numbers can be lengths of a side of triangle.
 Determine relationships between angles and sides of a triangle.
 Use proportions to solve problems.
 Use relationships between proportional parts of triangles.
 Use appropriate methods to solve right triangles including geometric mean, Pythagorean
Theorem, special right triangles, and trigonometry.
 Solve non-right triangles using the Law of Sines and Law of Cosines.
 Use Converse of Pythagorean Theorem to determine if a triangle is a right triangle.
Evidence of Learning
Summative Assessment:
Students will work collaboratively to complete the task of solving real life problems involving
triangles. This task consists of group or individual questioning, class discussions, teacher
developing standardized tests/quizzes. Also, students will complete a written assignment to
explain their findings.
Equipment needed: Graphing Calculator, Smart Board, Computer Access, Protractors, Rulers
Teacher Resources: Math Websites, textbooks, and resource books
Formative Assessments:
 Discussions
 Journal Entries
 Smart board presentation
 Evaluation Questions
 Tests/Quizzes
 Create/draw congruent and similar triangles
Lesson Plans
Lesson
Timeframe
Lesson 1
1 Day
Classify Triangles
Lesson 2
2 Days
Angles of Triangles
Lesson 3
2 Days
Congruent Triangles
Lesson 4
2 Days
Proving Congruence-SSS, SAS
Lesson 5
2 Days
Proving Congruence-ASA, AAS
Lesson 6
2 Days
Isosceles Triangles
Lesson 7
3 Days
Bisectors, Medians, and Altitudes
Lesson 8
2 Days
Inequalities and Triangles
Lesson 9
2 Days
The Triangle Inequality
Lesson 10
1 Day
Proportions
Lesson 11
1 Day
Similar Polygons
Lesson 12
2 Days
Similar Triangles
Lesson 13
2 Days
Parallel Lines and Proportional Parts
Lesson 14
2 Days
Parts of Similar Triangles
Lesson 15
1 Day
Geometric Mean
Lesson 16
2 Days
The Pythagorean Theorem and Its Converse
Lesson 17
2 Days
Special Right Triangles
Lesson 18
2 Days
Trigonometry
Lesson 19
1 Day
Angle of Elevation and Depression
Lesson 20
2 Days
The Law of Sines
Lesson 21
2 Days
The Law of Cosines
Teacher Notes: Students can seek input from their peers and teachers throughout collaborative
assignments and activities.
Curriculum Development Resources: The completed Curriculum Design Template shows how
this unit is situated within this district’s Math Course.
Unit Overview
Content Area:
MATH UNIT 3
Unit Title:
QUADRILATERALS AND CIRCLES
Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10
Unit Summary:
In this unit, students explore polygons by investigating the exterior and interior angles of
polygons. Students learn to recognize and apply the properties of parallelograms, rectangles,
rhombi, squares, and trapezoids. Students explore the different types of transformations:
reflections, translations, rotations, and dilations. They learn to identify, draw, and recognize
figures that have been transformed. Students identify the parts of a circle and solve problems
involving circumference and area. Arc and angle measures and the measures of segments within
a circle are explored. Equations of circles are derived and applied.
Primary interdisciplinary connections: Art, Science
21st century themes: Mathematical Literacy
Unit Rationale:
The area of quadrilaterals is necessary for calculating the surface area of prisms. Understanding
the properties is essential to success in engineering, architecture, and design. Circles are needed
to understanding spheres.
Learning Targets
Standards:
4.2 (Geometry and Measurement)
All students will develop spatial sense and the ability to use geometric properties, relationships
and measurement to model, describe, and analyze phenomena.
Content Statements:
Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve the
shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric
modeling and measurement can help us describe and interpret our physical environment and to
solve problems.
CPI #
4.2.12B.1
Cumulative Progress Indicator (CPI)
Determine, describe, and draw the effect of a transformation, or a sequence of
transformations, on a geometric or algebraic representation, and conversely,
determine whether and how one representation can be transformed to another by a
transformation or sequence of transformations.
4.2.12B.4
Generate and analyze iterative geometric patterns.
4.2.12C.3
Find an equation of circle given its center and radius and, given an equation of a
circle in standard form, find its center and radius.
4.2.5.D.1
Select and use appropriate units to measure angles and areas.
4.2.5.D.4
Use measurements and estimates to describe and compare phenomena.
4.5B.1
Use communication to organize and clarify their mathematical thinking.
4.5B.2
Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others both orally and in writing.
4.5B.3
Analyze and evaluate the mathematical thinking and strategies of others.
4.5B.4
Use the language of mathematics to express mathematical ideas precisely.
4.5C.1
Recognize recurring themes across mathematical domains (e.g. patterns in number,
algebra, and geometry.)
4.5C.2
Use connections among mathematical ideas to explain concepts (e.g. two linear
equations have a unique solution because the lines they represent intersect at a
single point).
4.5D.1
Recognize that mathematical facts, procedures, and claims must be justified.
4.5D.2
Use reasoning to support their mathematical conclusions and problem solutions.
4.5D.5
Make and investigate mathematical conjectures.
Unit Essential Questions:
 What are the similarities/differences of
special quadrilaterals?
 What are the parts of a circle?
 What are the relationships betweens lines
and circles?
Unit Enduring Understandings:
 Trapezoid has only pair of parallel sides;
squares and rectangles have congruent
diagonals and four right angles; squares and
rhombi have four congruent sides.
 Students will be able to identify and label
diameter, radius, chord, secant, tangent, and
center.
 Students will use various theorems involving
chords, secants, and tangents to solve
problems.
Unit Learning Targets:
Students will ...
 Find the interior and exterior angle sum for different convex polygons.
 Classify and compare different quadrilaterals based on their properties.
 Use the different properties of quadrilaterals to solve problems.
 Use slope, distance, and midpoint formulas to justify what type of quadrilateral it is.
 Create different transformations of figures.
 Identify parts of a circle.
 Determine relationships between segments and lines and circles.
 Identify the relationship of different angles to a circle.
 Write equations of circles and use to graph circles in coordinate plane.
Evidence of Learning
Summative Assessment:
Students will work collaboratively to complete the task of solving real life problems involving
quadrilaterals, circles, and transformations. This task consists of group or individual questioning,
class discussions, teacher developing standardized tests/quizzes. Also, students will complete a
written assignment to explain their findings.
Equipment needed: Graphing calculator, Smart Board, Computer Access, Protractors, Rulers
Teacher Resources: Math Web Sites, textbooks, and resource books
Formative Assessments:
 Discussions
 Journal Entries
 Smart board presentation
 Evaluation Questions
 Tests/Quizzes
 Create/draw transformations
Lesson Plans
Lesson
Timeframe
Lesson 1
1 Day
Angles of Polygons
Lesson 2
2 Days
Parallelograms
Lesson 3
2 Days
Tests for Parallelograms
Lesson 4
1 Day
Rectangles
Lesson 5
2 Days
Rhombi and Squares
Lesson 6
2 Days
Trapezoids
Lesson 7
2 Days
Reflections
Lesson 8
1 Day
Translations
Lesson 9
2 Days
Rotations
Lesson 10
2 Days
Dilations
Lesson 11
1 Day
Circles and Circumference
Lesson 12
2 Days
Angles and Arcs
Lesson 13
2 Days
Arcs and Chords
Lesson 14
2 Days
Inscribed Angles
Lesson 15
2 Days
Tangents
Lesson 16
2 Days
Secants, Tangents, and Angle Measures
Lesson 17
2 Days
Special Segments in a Circle
Lesson 18
1 Day
Equations of Circles
Teacher Notes: Students can seek input from their peers and teachers throughout collaborative
assignments and activities.
Curriculum Development Resources: The completed Curriculum Design Template shows how
this unit is situated within this district’s Math Course.
Unit Overview
Content Area:
MATH UNIT 4
Unit Title:
AREA AND VOLUME
Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10
Unit Summary: Area and volume can be used to analyze real-world situations. In this unit, you
will learn about formulas used to find the areas of two-dimensional figures and the surface and
the surface areas and volumes of three-dimensional figures.
Primary interdisciplinary connections: Science, Business, Art
21st century themes: Mathematical Literacy
Unit Rationale: The Knowledge about Area and Volume that students gain while studying this
unit will be important to them in the future mathematics courses, in physics, and in many careers
that they might choose.
Learning Targets
Standards:
4.2 (Geometry and Measurement)
All students will develop spatial sense and the ability to use geometric properties, relationships
and measurement to model, describe, and analyze phenomena.
Content Statements: Spatial sense is an intuitive feel for shape and space. Geometry and
measurement both involve the shapes we see all around us in art, nature, and the things we make.
Spatial sense, geometric modeling and measurement can help us describe and interpret our
physical environment and to solve problems.
CPI #
Cumulative Progress Indicator (CPI)
Use geometric models to represent real-world situations and objects and to solve
4.2.12A.1
problems using those models.
4.2.12A.2
Draw perspective views of 3D objects on isometric dot paper, given 2D
representations (e.g., nets or projective views.)
4.2.12E.2
Develop and apply strategies and formulas for finding perimeter and area of
squares rectangles.
Unit Essential Questions:
 What is the difference between Area and
Volume?
 What are the units of measure of Area and
Volume?
 Is the Area used to find the Volume of a
solid figure?
Unit Enduring Understandings:
 Area measures the surface of a two
dimensional figure and the volume shows how
much a sold figure can hold.
 The unit of measure for Area is square units
and Volume is cubic units.
 Yes because the volume of any solid figure is
based on the area of its base and height.
Unit Learning Targets:
Students will ...
 Solve problems by making a model.
 Find the area of various 2-dimensional figures such as parallelograms, rectangles, trapezoid,
rhombus, squares, triangles, circles, and regular polygons.
 Find the lateral areas and surface areas and volumes of various solid figures such as rectangular
prisms, cubes, pyramids, cones, and spheres
Evidence of Learning
Summative Assessment:
Students will work collaboratively to complete the task of solving real life problems involving
area and volume. This task consists of group or individual questioning, class discussions, teacher
developing standardized tests/quizzes. Also, students will complete a written assignment to
explain their findings.
Equipment needed: Graphing calculator, Smart board, and computer access.
Teacher Resources: Math web sites, textbooks and resource books.
Formative Assessments:
 Discussions
 Journal Entries
 Smart board presentation
 Evaluation Questions
 Tests/Quizzes
 Create/draw/fold 3-dimensional figures.
Lesson Plans
Lesson
Timeframe
Lesson 1
Areas of Parallelograms
1 Day
Lesson 2
Areas of Triangles, Trapezoids and Rhombi.
2 Days
Lesson 3
Areas of regular polygons and circles.
2 Days
Lesson 4
Areas of irregular figures.
2 Days
Lesson 5
Three dimensional figures
1 Day
Lesson 6
Surface Areas of prisms, cylinders, pyramids
2 Days
Lesson 7
Volumes of Cones and Spheres
2 Days
Lesson 8
Volumes of prisms and cylinders.
2 Days
Lesson 9
Volumes of pyramids and cones.
2 Days
Lesson 10
Volumes of Spheres
1 Day
Teacher Notes:
Students can seek input from their peers and/or teachers before developing their projects.
Curriculum Development Resources
The completed Curriculum Design Template shows how this unit is situated within this district’s
Math Course.