Download Geometry Narrative 08.2011Final UPDATED OCTOBER 8 2013

Geometry – Grades 9 and 10
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Course Philosophy
“The universe cannot be read until we have learnt the language and become familiar with
the characters in which it is written. It is written in mathematical language, and the letters
are triangles, circles and other geometrical figures, without... it is humanly impossible to
comprehend a single word.” Galilei, Galileo (1564-1642) Opere II Saggiatore p.171.
All students will develop an understanding of the attributes and relationships of geometric
objects which can be applied in diverse contexts – interpreting a schematic drawing,
estimating the amount of wood needed to frame a sloping roof, rendering computer
graphics, or designing a sewing pattern for the most efficient use of materials. (CCSS,
Common Core State Standards)
Course Goals
The goals of geometry (as aligned with the CCSS)
Spatial sense is an intuitive feel for shape and space. Geometry and measurement both
involve describing the shapes we see all around us in art, nature, and the things we make.
Students will use spatial sense, geometric modeling, and measurement to describe and
interpret their physical environment and to solve problems.
Students will understand and use geometric properties to identify, describe and classify
standard geometric objects, describe and compare properties of geometric objects, making
conjectures concerning them, and use reasoning and proof to verify or refute conjectures
and theorem. Also included are such concepts as symmetry, congruence, and similarity.
Students will analyze how various transformations affect geometric objects to enhance their
spatial sense. This includes combining shapes to form new ones and decomposing complex
shapes into simpler ones. It includes the standard geometric transformations of translation,
refection, rotation, and dilation. It also includes using tessellations and fractals to create
geometric patterns.
Students will use coordinate geometry to understand the connection between geometry and
algebra. It facilitates the visualization of algebraic relationships, as well as an analytical
understanding of geometry.
Measurement helps describe our world using numbers. Students will recognize that how we
attach numbers to real-world phenomena, familiarity with common measurement units (e.g.
inches, liters, and miles per hour), and a practical knowledge of measurement tools and
techniques are critical for understanding the world around them.
Students will apply the knowledge and understands of units of measurement in order to
actually perform measurement. While students will eventually apply formulas, it is
important that students develop and apply strategies that derive from their understanding
of the attributes of geometric objects. In addition to measuring objects directly, students
will apply indirect measurement skills, using for example, similar triangles and
trigonometry.
Through these experiences, students will develop appropriate skills as outlined in the New
Jersey Core Curriculum Content Standards for Content Area and Technology.
New Brunswick Public Schools
1 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Course Description
Geometry is a branch of mathematics that deals with the measurement, properties, and
relationships of points, lines, angles, surfaces and solids. Students need to understand that
the study of these areas helps them to describe and interpret our physical environment.
This course follows a structured scope and sequence, introducing, reinforcing and extending
topics identified by the Common Core State Standards and the National Council of
Mathematics Teachers. This course may include the use of labs, group explorations,
calculators, and computer software to study real world problem solving techniques and
presentation strategies. Student will model and solve geometric problems that involve the
study of transformational geometry, congruence and similarity of closed, plan figures,
parallel and perpendicular lines and related angles, circles, coordinate geometry, right
triangle trigonometry, fractals, matrices, and sequences. A portion of weekly lessons will be
dedicated to reinforcing those mathematical cluster skills that have been identified on the
High School Proficiency Assessment.
Student proficiency will be measured in the following way:
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60 % Major Assessments
o Course Benchmark Assessments
o Unit Tests/Quizzes
o Projects
30 % Class work and Class Participation
o In-Class Assignments
o Class Discussions
o Group Work
10 % Homework
o Reflection, exploration, and evaluation of material outside of class
o Application of skills and content introduced in class
o Preparation for upcoming, in-class, work
The student must earn a cumulative grade of “D” or better in order to receive
credit for this course.
New Brunswick Public Schools
2 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
New Jersey Core Curricular Content Standards – Technology
The NJCCCS for Technology are not mastered in any one class. They are practiced and
assessed in New Brunswick in a myriad of ways at all times of the year. Therefore, no one
class contains all technology standards for the given grade level. Specific references of
technology infusion can be found on the curriculum maps, aligned with exemplar
activities/strategies.
4.5.F. Technology-Mathematics
Standard 4.5.F requires students use calculators and computers along with other
mathematical tools in both instructional and assessment activities. These tools should be
used, not to replace mental math and paper-and-pencil computational skills, but to enhance
understanding of mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should also become
proficient in using technology as it is used by adults.
8.1 Technological Literacy
Standard 8.1 requires students to use technology effectively. Geometry provides students
with numerous and varied activities that require the creative use of technology. Students
use technology to investigate geometric concepts, create geometric figures and prepare
presentations. Students relate their use of technology to the professional world, making
judgments between similar applications and platforms as to which is best suited for a task.
As well, students are forced not only to select the best application for a task, but to use
multiple applications in conjunction with each other in order to complete one task
effectively. In Geometry students are forced to use technology as a means for research, to
evaluate any material these searches produce, and to solve problems and make decisions
using technology.
Standards: 8.1.A.1, .5-.9; 8.1.B.1-.2, .4, .6-.7, .9, .11-.12
8.2 Technology Literacy
Standard 8.2 requires students to develop and apply an understanding of how technology
impacts individuals, societies, and the environment.
Standards: 8.2.A.3; 8.2.C.3
New Brunswick Public Schools
3 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Common Core State Standards
Congruence (G-CO. 1 – 12)
 Experiment with transformations in the plane
 Understand congruence in terms of rigid motions
 Prove geometric theorems
 Make geometric constructions
Similarity, Right Triangles, and Trigonometry (G-SRT. 1 – 11)
 Understand similarity in terms of similarity transformations
 Prove theorems involving similarity
 Define trigonometric ratios and solve problems involving right triangles
 Apply trigonometry to general triangles
Circles (G-C.1 – 5)
 Understand and apply theorems about circles
 Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations (G-GPE. 1 – 7)
 Translate between the geometric description and the equation for a conic section
 Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension (G-GMD. 1 – 4)
 Explain volume formulas and use them to solve problems
 Visualize relationships between two-dimensional and three-dimensional objects
Modeling with Geometry (G-MG. 1 – 3)
 Apply geometric concepts in modeling situations
New Brunswick Public Schools
4 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Enduring Understandings
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Geometric relationships provide a means to make sense of a variety of phenomena
Measurements can be used to describe, compare, and make sense of phenomena
The development of mathematical reasoning allows you to understand why
mathematics makes sense
Mathematical reasoning is used to evaluate situations, select problem-solving
strategies, and draw logical conclusions
Geometric properties can be used to construct geometric figures
Everyday objects have a variety of attributes, each of which can be measured in
many ways
Coordinate geometry can be used to represent and verify geometric/algebraic
relationships
What we measure affects how we measure it
Shape and area can be conserved during mathematical transformations
Essential Questions
Open-ended and content directed, the Essential Questions for Geometry to challenge
students to become avid explorers of our country’s past and creative observers of and
critical thinkers in the world in which they live.
Unit 1
How can you communicate ideas using the language of geometry?
How can concepts such as equality and congruence be used in real life situations?
How can geometric figures be used to describe and represent real world situations?
What alternatives are available to prove a statement when it is not possible to do it directly?
How do you reach valid conclusions in Geometry and in life?
How can logical reasoning be used to analyze situations and solve problems?
How are concepts, such as parallelism and perpendicularity, used in real life situations?
Unit 2
How can the application of the properties of triangles be used to underst6and and solve real
life problems?
How much information is needed to tell whether two figures are congruent?
How can you use triangle relationships to find and compare angle measures and distances?
How are congruent triangles used to calculate inaccessible distances?
How can you use similarity in indirect measurement to find distances?
When a figure is reduced, how are the corresponding angles related: How are the
corresponding lengths related?
Unit 3
What relationship exists among the sides of a right triangle?
How are the definitions and properties of special quadrilaterals used in designs and
creations in the real world?
Unit 4
How can the application of area and perimeter be used to solve real world problems in
situations such as manufacturing, architecture, sports and landscaping?
How can you apply the properties of surface area and volume of geometric solid objects to
solve real life problems?
New Brunswick Public Schools
5 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Unit 5
How do you describe the types of motions we encounter in real life situations?
How are dilations used in scale drawings, maps, photographs, and blueprints?
Where do you see tessellations in nature, architecture, art and in other real life situations?
What relationship exists among the sides of right triangles?
How can properties of trigonometric ratios of right triangles be used to indirectly solve
problems?
What are the relationships among lengths, arc measures, and angles formed when lines
intersect circles in one of two points?
How can circles be used to model natural phenomena?
Objectives
Unit 1 Points, Lines and Planes
1.1 Foundations of Geometry (Review)
 Identify, label, and describe the basic properties of points, lines, line segments, rays,
angles, and planes
 Use segment postulates to identify congruent segments
 Find lengths of segments in the coordinate plane
 Use the Midpoint Formula and the Distance Formula
 Measure and classify angles
 Describe angle pair relationships
 Use special angle relationships to find angle measures
 Identify and classify polygons
 Find the perimeter, circumference and area of polygons and circles
 Use constructions to copy and bisect segments and angles
 Solve real world problems
1.2 Logical Reasoning and Proof
 Analyze, describe, and extend geometric patterns
 Construct conjectures from a group of related examples using inductive reasoning
 Analyze conditional statements
 Use postulates and diagrams
 Reason using properties from algebra
 Prove statements about segments and angles
 Prove angle pair relationships
1.3 Parallel and Perpendicular Lines
 Identify pairs of lines and angles
 Identify angle pairs formed by three intersecting lines
 Use angles formed by parallel lines and transversals
 Use angle relationships to prove that lines are parallel
 Find and compare slopes of lines
 Writhe and graph equations of lines
 Prove theorems about perpendicular lines
 Find the distance between a point and a line
 Use constructions to draw parallel and perpendicular lines
 Solve real world problems
New Brunswick Public Schools
6 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Unit 2 Triangles
2.1 Congruent Triangles
 Classify triangles and find the measures of their angles
 Identify congruent figures
 Prove triangles congruent by SSS, SAS, HL, ASA, and AAS
 Use congruent triangles in proofs
 Apply properties and theorems of isosceles and equilateral triangles
 Use constructions to copy triangles
 Solve real world problems
2.2 Triangle Relationships
 Use properties of Mid-Segments of Triangles
 Use perpendicular bisectors to solve problems
 Use angle bisectors to find distance relationships
 Explore the use of medians and altitudes of triangles
 Investigate points of concurrency
 Use inequalities involving triangle side lengths and angle measures to solve problems
2.3 Similarity
 Solve problems using rations, proportions, and the geometric mean
 Use proportions to solve geometry problems
 Use proportions to identify similar polygons
 Prove triangles are similar by using the AA, SSS, and SAS similarity postulates and
theorems
 Use the Proportionality theorems to solve problems with triangles or parallel lines
 Explore Fractals
Unit 3 Plane Figures
3.1 Right Triangles
 Apply the Pythagorean Theorem to find side lengths in right triangles
 Use the converse of the Pythagorean Theorem to determine if a triangle is a right
triangle
 Use properties of the altitude of a right triangle
 Use the relationships among the sides in special right triangles to solve problems
3.2 Quadrilaterals
 Find angle measures in polygons
 Find the relationships among angles, sides, and diagonals of parallelograms
 Find the characteristics of quadrilaterals that indicate that they are parallelograms
 Determine and apply the properties of rectangles, rhombuses, and squares
 Determine and apply the properties of trapezoids and kites
 Classify quadrilaterals by their properties
Unit 4 Measuring Plane Figures and Geometric Solids
New Brunswick Public Schools
7 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
4.1 Understand Perimeter and Area of Polygons
 Find the areas of triangles and parallelograms
 Find the areas of quadrilaterals
 Find the perimeter and area of similar figures
 Find the circumference and are lengths of circles
 Find the area of circles and sectors
 Find the perimeters and areas of regular polygons
4.2 Surface Areas and volumes of Solid Figures
 Explore and identify solid figures
 Recognize nets of various space figures
 Determine the surface areas of prisms and cylinders
 Determine the surface areas of pyramids and cones
 Determine the volume of prisms and cylinders
 Determine the volume of pyramids and cones
 Determine the surface areas and volumes of spheres
 Explore similar solids
Unit 5 Extension – Plane Figures
5.1 Transformations: Reflections, Translations, Rotations and Compositions of Reflections
 Translate figures and use vectors
 Identify isometries
 Use properties of matrices to perform translations
 Reflect figures over a given line
 Rotate figures about a point
 Apply compositions of transformations
 Identify and describe glide reflections
 Explore tessellations
 Identify line and rotational symmetries of a figure
 Identify and perform dilations
5.2 Rights Triangles and Trigonometry
 Identify the tangent ration in right triangles and solve problems using the tangent
ratio
 Identify and apply the sine and cosine ratios and solve problems using the sine or
cosine ratios
 Solve problems involving angles of elevations and angles of depression
5.3 Properties of Circles
 Identify special segments and lines that intersect circles
 Use properties of a tangent to a circle to solve problems
 Use angle measures to find arc measures
 Use relationships of arcs and chords in a circle to solve problems
 Use inscribed angles to circles to solve problems
 Apply other angle relationships in circles to solve problems
 Find segment lengths in circles
 Explore writing equations of circles given their graph
New Brunswick Public Schools
8 of 9
2011/2012
Reviewed: October 8, 2013
Geometry – Grades 9 and 10
____________________________________________________________
Suggested Activities/Strategies
The curriculum includes suggested activities/strategies for each course objective. These
suggestions are found on the Geometry curriculum map. These activities reflect the
standard of both best practice and rigor that the District expects in each classroom. Each of
the provided activities/strategies is aligned with specific focus levels of Bloom’s Taxonomy.
This is not to say that the given activity does not require additional levels within the
taxonomy, it merely highlights those levels of instructional focus.
Interdisciplinary Study
The curriculum maps for Geometry indicate connections for interdisciplinary study. These
connections are specific to the suggested activities/strategies, and as such are, again, both
for direct use and to provide teachers with a model for designing instruction to include
connections across the content areas through use of reading texts representative of the
various content areas referenced on the Geometry map.
Assessments
Geometry includes two uniform benchmark assessments:
Final Examination.
Midterm Examination and the
As well, during each unit student will demonstrate core proficiencies and content knowledge
on teacher-design assessments including: exams and quizzes; written projects; homework
and class participation; and individual and group performance-based assessments.
Suggested Resources
The Suggested Resources for Geometry is a starting point for teachers to expand their
instruction beyond a core text.
United Streaming is a standards-based, internet resource that New Brunswick Public Schools
uses to stretch instruction beyond the classroom. Aligned to state and national standards,
United Streaming allows teachers access to planning and instructional tools that bring math
alive for students. The listed resources in this category include real life applications of
mathematical concepts.
Internet Resources provide teachers with exemplar tools on the internet that expand and
challenge student thinking about Mathematics. Whether through a web quest, an online
instructional site, the internet resources mark the quality, variety, and substance of internet
resources that New Brunswick classrooms utilize.
Teacher’s Reflections
At the end of each of the four units teachers are required to complete the district form:
Teacher’s Reflections. By following the guiding questions provided by the district, teachers
reflect on the unit that students have just completed. The focus is instruction, including
planning, delivery, assessment, and student performance. The dual purpose of this process
is to improve individual teacher performance and to provide data that will inform continual
curriculum revision.
New Brunswick Public Schools
9 of 9
2011/2012
Reviewed: October 8, 2013