Download Geometry - Pompton Lakes School District

POMPTON LAKES SCHOOL DISTRICT
GEOMETRY
COURSE OF STUDY
June 2012
Submitted by
The Math 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 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 #
Cumulative Progress Indicator (CPI)
Recognize the limitations of estimation, assess the amount of error resulting from
4.1.12C.1
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 written assignments 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
1 Day
Lesson 2
Linear Measure and Precision
2 Days
Lesson 3
Distance and Midpoints
2 Days
Lesson 4
2 Days
Angle Measure
Lesson 5
2 Days
Angle Relationships
Lesson 6
2 Days
Polygons
Lesson 7
2 Days
Parallel Lines and Transversals
Lesson 8
2 Days
Angles and Parallel Lines
Lesson 9
2 Days
Slopes of Lines
Lesson 10
2 Days
Equations of Lines
Lesson 11
2 Days
Proving Lines Parallel
Lesson 12
2 Days
Perpendiculars and Distance
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 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 problems
4.2.12A.1
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
Classify Triangles
1 Day
Lesson 2
Angles of Triangles
2 Days
Lesson 3
Congruent Triangles
2 Days
Lesson 4
Proving Congruence-SSS, SAS
2 Days
Lesson 5
Proving Congruence-ASA, AAS
2 Days
Lesson 6
Isosceles Triangles
2 Days
Lesson 7
Bisectors, Medians, and Altitudes
3 Days
Lesson 8
Inequalities and Triangles
2 Days
Lesson 9
The Triangle Inequality
2 Days
Lesson 10
Proportions
1 Day
Lesson 11
Similar Polygons
1 Day
Lesson 12
Similar Triangles
2 Days
Lesson 13
Parallel Lines and Proportional Parts
2 Days
Lesson 14
Parts of Similar Triangles
2 Days
Lesson 15
Geometric Mean
1 Day
Lesson 16
The Pythagorean Theorem and Its
2 Days
Converse
Lesson 17
Special Right Triangles
2 Days
Lesson 18
Trigonometry
2 Days
Lesson 19
Angle of Elevation and Depression
1 Day
Lesson 20
The Law of Sines
2 Days
Lesson 21
The Law of Cosines
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 3
Unit Title:
QUADRILATERALS AND CIRCLES
Target Course/Grade Level: GEOMETRY 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
Angles of Polygons
1 Day
Lesson 2
Parallelograms
2 Days
Lesson 3
Tests for Parallelograms
2 Days
Lesson 4
Rectangles
1 Day
Lesson 5
Rhombi and Squares
2 Days
Lesson 6
Trapezoids
2 Days
Lesson 7
Reflections
2 Days
Lesson 8
Translations
1 Day
Lesson 9
Rotations
2 Days
Lesson 10
Dilations
2 Days
Lesson 11
Circles and Circumference
1 Day
Lesson 12
Angles and Arcs
2 Days
Lesson 13
Arcs and Chords
2 Days
Lesson 14
Inscribed Angles
2 Days
Lesson 15
Tangents
2 Days
Lesson 16
Secants, Tangents, and Angle Measures
2 Days
Lesson 17
Special Segments in a Circle
2 Days
Lesson 18
Equations of Circles
1 Day
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 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 #
4.2.12A.1
Cumulative Progress Indicator (CPI)
Use geometric models to represent real-world situations and objects and to solve 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
Lesson 1
Areas of Parallelograms
Lesson 2
Areas of Triangles, Trapezoids and
Rhombi.
Lesson 3
Areas of regular polygons and circles
Lesson 4
Areas of irregular figures.
Lesson 5
Three dimensional figures
Timeframe
1 Day
2 Days
2 Days
2 Days
1 Day
Lesson 6
Surface Areas of prisms, cylinders,
2 Days
pyramids
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.