Download Geometry - Morris School District

Morris School District
Morristown, NJ
2012
Geometry
Superintendent
Dr. Thomas J. Ficarr
1
Table of Content
Part I
Rationale and Philosophy
3
Goals and Objectives
5
Part II
Units of Study
9
Mastery Objectives
Teaching and Learning Activities
Assessment and Testing strategies
Text and Materials
Procedures for use of Supplemental Materials
Part III
Curriculum Map
19
Part IV: Appendix
References
32
Do Now Problems
Unit of Study Summative Problems
Rubrics (forthcoming)
Internet Web Sites
2
ale and Philosophy
In order to prepare for global competition and high expectations for all, Morris School District students must have increased
opportunities for mathematical experiences that extend critical thinking and reasoning. Specifically, access to higher mathematics is
essential. Geometry is the second or third course in a traditional mathematics sequence that builds mathematical reasoning through
mathematical proof, and improves algebraic thinking by embedding algebra in geometry problems.
Key considerations:
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State and National Expectations
There has been great activity on the state and national level in terms of expectations for the skills, knowledge and expertise
students should master in mathematics to succeed in work and life in the 21 st century. NJ is currently moving forward with the
“Framework for 21st Century Learning,” a partnership with business, education, and government to develop a collective vision
to strengthen American education (www.21stcenturyskills.org). The six components include: Core Content – Mathematics, 21st
Century Content, Learning and Thinking skills, Information and Communications Technology (ICT) Literacy, Life Skills – Real
World Applications, and 21st Century Assessments. Students will be assessed on their depth of knowledge of Geometry on
national examinations beginning in 2010, with additional high school mathematics courses to follow.
●
Equity and Access to Higher Mathematics
The belief that all students, not just a select few, have access to mathematical learning environments that enable them to
meet world-class standards for both college and the world of work continues to be an essential goal of the Morris School
District. The National Council of Teachers of Mathematics initially reflected this important consideration in The Equity Principle
whereby, “Excellence in mathematics education requires equity – high expectations and strong support for all students (NCTM
2000).” Subsequent research has revealed that “It is important that high schools do everything to promote success among all
students – encouraging enrollment by students from all demographics in advanced math courses.” NCTM’s recent release of
the scientifically research-based, Focus in High School Mathematics: Reasoning and Sense Making (2009), stressed the role of
educators to help students with a wide range of backgrounds develop connections between applications of new learning and
their existing knowledge, increasing their likelihood of understanding and thereby allowing increased options and entry into
advanced mathematics.
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Building on Existing Partnerships
The Geometry course fills a critical need for extensive study towards the development of abstract thinking and mathematics.
This goal is aligned with the district’s vision of providing rich opportunities for all students to move forward on Blooms’
Knowledge Taxonomy continuum from Knowledge and Awareness to Comprehension, Application, Analysis, Synthesis, and
Evaluation. Specifically, scaffolding on a foundation of geometry skills, this course moves students from the acquisition and
assimilation of concepts towards increased application and adaptation. Adaptation occurs when students have the
competence to think in complex ways and to apply their knowledge and skills. Building on the district’s existing partnerships,
this course would provide a springboard for entrance into higher level mathematics courses for a greater number of Morris
School District students. [Rigor and Relevance Framework: International Center for Leadership in Education, Bill Daggett and
Ray McNulty, www.leadered.com]
and Objectives (outcomes):
Geometry is the study of congruence through formal mathematical reasoning and the application of reasoning to problem solve. In
teaching and learning Geometry, it is important for teachers and students to comprehend the following Big Ideas and Enduring
Understandings and to establish connections and applications of the individual skills and concepts to these broad principles as the
critical goals and objectives of the course:
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Congruence
The concepts of congruence, similarity, and symmetry can be understood fromthe perspective of geometric transformation.
Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular
type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an
isosceles triangle assures that its base angles are congruent.
●
Similarity, Right Triangles and Trigonometry
Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define
congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades. These
transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent. The
definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean
Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to nonright
triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases
where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions
in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion.
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Extending to Three Dimensions
Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of
circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional
shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.
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Connecting Algebra and Geometry Through Coordinates
Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate
system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes
of parallel and perpendicular lines. Students continue their study of quadratics by connecting the geometric and
algebraic definitions of the parabola.
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Circles With and Without Coordinates
In this unit, students prove basic theorems about circles, with particular attention to perpendicularity and inscribed
angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships
among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate
system, students use the distance formula to write the equation of a circle when given the radius and the coordinates
of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving
quadratic equations to determine intersections between lines and circles or parabolas and between two circles.
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Connections Between Geometry & Probability
Geometry provides techniques for analyzing situations that involve chance and uncertainty, including area of region and
segment length to find the probability of events. Building on probability concepts that began in the middle grades, students
use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for
compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should
make use of geometric probability models wherever possible. They use probability to make informed decisions.
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Units of Study:
(Note: emphasis on problem solving, applications, and modeling)
The Common Core Mathematical Practices
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop
in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It
Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive
disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own
efficacy). The Common Core Mathematical Practice Standards are:
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its
solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution
and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special
cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and
change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change
the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships,
graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize
and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually
ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify
correspondences between different approaches.
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2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two
complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract
a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order
to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing
and flexibly using different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical progression of statements to explore the
truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about
data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are
also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such
as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or
made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can
listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
Engineers will often minimize volume of material in order to reduce cost. In addition concepts of volume can be applied to real world situations like the
consumption of a volume of material per unit time.
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5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil
and paper, concrete models, a ruler, a protractor, a calculator, or dynamic geometry software. Proficient students are sufficiently
familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful,
recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze
graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation
and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various
grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to
pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others
and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and
appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a
problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem
context. Initially, students give carefully formulated explanations to each other. By the time they reach high school they have learned to
examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well remembered 7
× 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, algebra students can see the 14 as
2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an
auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such
as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5
minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. By paying
attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, algebra
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students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x
– 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to
solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
The Common Core Standards in Geometry
Congruence
• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions
• Prove geometric theorems
• Make geometric constructions
Similarity, Right Triangles, and Trigonometry
• 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
• Understand and apply theorems about circles
• Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations
• Translate between the geometric description and the equation for a conic section
• Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension
• Explain volume formulas and use them to solve problems
• Visualize relationships between two-dimensional and three-dimensional objects
Modeling with Geometry
• Apply geometric concepts in modeling situations
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Mastery Objectives:
MASTERY OBJECTIVES
(NJCCCS)
All course of study must include the following, which replace the Workplace readiness standards:
Career Education and Consumer, Family, and Life Skills
Career and Technical Education: All students will develop career awareness and planning, employability skills, and foundational
knowledge necessary for success in the workplace.
Consumer, Family, and Life Skills: All students will demonstrate critical life skills in order to be functional members of society.
Scans Workplace Competencies
Effective workers can productively use:
Resources: They know how to allocate time, money, materials, space and staff.
Interpersonal Skills: They can work on teams, teach others, serve customers, lead, negotiate, and work well with people from culturally
diverse backgrounds.
Information: They can acquire and evaluate data, organize and maintain files, interpret and communicate, and use computers to process
information.
Systems: They understand social, organizational, and technological systems; they can monitor and correct performance; and they can
design or improve systems.
Technology: They can select equipment and tools, apply technology to specific tasks, and maintain and troubleshoot equipment.
SCANS Foundations Skills
Competent workers in the high-performance workplace need:
Basic Skills: reading, writing, arithmetic, and mathematics, speaking and listening.
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Thinking Skills – the ability to learn, reason, think creatively, make decisions, and solve problems.
Personal Qualities – individual responsibility self-esteem and self-management, sociability, integrity, and honesty.
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Teaching/Learning Activities
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Assessment and Testing Strategies
Sound and productive classroom assessments are built on a foundation of the following five key dimensions (Stiggins et al, 2006):
Key 1: Assessment serves a clear and appropriate purpose.
Did the teacher specify users and uses, and are these appropriate?
Key 2: Assessment reflects valued achievement targets.
Has the teacher clearly specified the achievement targets to be reflected in the exercises? Do these represent important learning
outcomes?
Key 3: Design.
Does the selection of the method make sense given the goals and purposes? Is there anything in the assessment that might lead
to misleading results?
Key 4: Communication.
Is it clear how this assessment helps communication with others
about student achievement?
Key 5: Student Involvement.
Is it clear how students are involved in the assessment as a way to help them understand achievement targets, practice hitting
those targets, see themselves growing in their achievement, and communicate with others about their success as learners?
The Algebra I course will include a variety of assessment tools for the effective teaching and learning of mathematics. In addition to
classroom and district assessments, students will demonstrate proficiency of algebraic reasoning and skills on the New Jersey Algebra I
State-wide Assessment as required for graduation.
Indicators of Sound Classroom Assessment Practice will consist of both formative and summative assessments that may include, but are
not limited to:
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Observation
Interviews
Portfolios (Project, Growth, Achievement, Competence, Celebration)
Paper-and-pencil tests/quizzes
Performance Tasks
Journals/Self-Reflection
d Materials
Student Text:
○ TBD
Teacher Materials and Resources:
○ Blueprints for Success Gold Seal Lessons: Successful Practice Network. www.leadered.com/spn.html
○ Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston,
VA: National Council of Teachers of Mathematics.
○ National Council of Teachers of Mathematics. (2001). Navigating Through Algebra in Grades 6-8. Reston, VA: NCTM.
○ National Council of Teachers of Mathematics. (2001). Navigating Through Algebra in Grades 9-12. Reston, VA: NCTM.
○ National Council of Teachers of Mathematics. (2006). Navigating Through Mathematical Connections in Grades 9-12. Reston,
VA: NCTM.
○ National Council of Teachers of Mathematics. (2004). Navigating Through Probability in Grades 9-12. Reston, VA: NCTM.
○ New Jersey Common Core Curriculum Standards 2010. www.njcccs.org.
Technology/Computer Software
○ Geometer’s Sketchpad: Key Curriculum Press
○ Fathom: Key Curriculum Press
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Procedures for Use of Supplemental Instructional Materials
Instructional materials not approved by the Board of Education must be brought to the attention of the building principal or viceprincipal before use in any instructional area. Materials that are approved include all textbooks, videos, and other supplemental
material acquired through purchase orders, and/or other school funds. Resources from the County Education Media and Technology
Center are also acceptable, with age appropriateness reviewed.
All instructional materials not explicitly Board approved as outlined in above, which are intended for use in any instructional setting
must be approved by the building principal or vice- principal at least 5 schools days prior to use. The principal or vice-principal may
request to review a copy of the materials, video, etc, prior to use in the classroom.
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Unit 1 - Congruence, Proofs, and Constructions
Part a - Points, Lines, Planes, and Angles (1, no section 1.7)
Part b - Transformations (7)
Part c - Reasoning and Proof (2 no section 2.3)
Part d - Perpendicular and Parallel Lines (3, 5.1)
Part e - Congruent Triangles (4, 5.5)
Part f - Polygons (Quadrilaterals) (6, section 11.1)
Content/Objective
Points, Lines, Planes, and Angles
The Learner will
● define geometric terms such
as point, line, plane.
● measure with and without the
coordinate grid.
● quantitatively add and
subtract line segments and
angles.
● derive the midpoint and
distance formula.
● measure angles using
protractors.
● bisect segments and angles.
● classify angles.
● identify vertical,
complementary,
supplementary angles, and
linear pairs.
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Essential Questions/
Enduring Understandings
Suggested Activity/
Appropriate Materials-Equipment
Suggested
Evaluation/Assessment
-How do you name and draw Instructional Focus:
the basic elements of geometry
as a perspective drawing?
Sample Assessments:
- How can you describe the
attributes of a segment or
Instructional Strategies:
angle?
- Make formal geometric constructions with a variety of tools
and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.) such
as copying a segment and angle and bisecting a segment and
angle. (G.CO.12)
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- Special angle pairs help to
- Students will find vertical, complementary, supplementary
identify geometric
angles in the real-world.
relationships and use them to
PreAssessment,
find angle measures.
■
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Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- How can you change a
-Represent transformations in the plane using, e.g.,
PreStudents will be able to
identify and use reflections, figure’s position without
transparencies and geometry software; describe
Assessment,
translations, and rotations. changing its size and shape? transformations as functions that take points in the plane as
Checkpoint
- How can transformations be
(part b)
inputs and give other points as outputs. Compare
exercises,
described mathematically?
transformations
that
preserve
distance
and
angle
to
those
that
DoNows,
- How can you represent a
portfolios, oral
transformation in a coordinate do not. (G.CO.2)
-Given a rectangle, parallelogram, trapezoid, or regular
questioning,
plane?
polygon, describe the rotations and reflections that carry it onto closure, selfitself. (G.CO.3)
reflection
-Given a geometric figure and a rotation, reflection, or
journals,
translation, draw the transformed figure using, e.g., graph
projects, tests,
paper, tracing paper, or geometry software. Specify a sequence technologyof transformations that will carry a given figure onto another. based
(G.CO.5)
assessments, teacher
- Use geometric descriptions of rigid motions to transform
constructed quizzes
figures and to predict the effect of a given rigid motion on a
and tests
given figure; given two figures, use the definition of congruence
in terms of rigid motions to decide if they are congruent.
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■
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(G.CO.6)
-Transformations will be conducted both on and off a
coordinate plane. Students will determine the new coordinates
of a polygon after any given transformation.
- Analyze figures in terms of their symmetries using the
concepts of reflection, rotation, and translation, and
combinations of these.
How
can
mathematical
-Observe patterns leading in order to make conjectures.
Students will be able to
reasoning help you make
- Write definitions as biconditionals.
recognize and analyze
- Solve logic puzzles.
conditional statements and generalizations?
- How do you know when you -Use the process of elimination and assume a statement is true,
biconditional statements
have proved something?
unless you can determine a counterexample.
(part c)
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
Students will be able to use -How can you justify the steps -Solve equations giving their reasons for each step and connect Preof an algebraic equation?
this to simple proofs.
properties from algebra
Assessment,
-Prove geometric relationships using given information,
and prove statements about - Properties of congruence
Checkpoint
definitions, properties, postulates, and theorems to verify
segments and angles. (part allow you to justify segment
exercises,
and angle relationship in real statements about segments and angles.
c)
DoNows,
life.
-Prove theorems about segments and angles. (G.CO.9)
portfolios, oral
- Explain and illustrate the role of definitions, conjectures,
questioning,
theorems, proofs, and counterexamples in mathematical
reasoning, using geometric examples to illustrate these concepts. closure, selfreflection
journals,
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■
■
■
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projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- How do you prove that two - Develop definitions of perpendicular and parallel lines.
PreStudents will be able to
lines are parallel or
(G.CO.4)
recognize and apply the
Assessment,
-Use postulates and theorems to explore lines in plane.
properties of parallel lines, perpendicular?
Checkpoint
- Any point on the
-Use coordinate geometry to examine the slopes of parallel and
perpendicular lines, and
exercises,
perpendicular lines. (G.GPE.5)
perpendicular bisectors of perpendicular bisector is
equidistant from the endpoints - Make formal geometric constructions with a variety of tools DoNows,
segment in geometric
portfolios, oral
of the segment.
figures. (part d)
and methods (compass and straightedge, string, reflective
questioning,
devices, paper folding, dynamic geometric software, etc.).
Constructing perpendicular lines, including the perpendicular closure, selfbisector of a line segment and constructing a line parallel to a reflection
journals,
given line through a point not on the line. (G.CO.12)
projects, tests,
- Prove theorems about lines and angles. Theorems include:
technologywhen a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; based
points on a perpendicular bisector of a line segment are exactly assessments, teacher
constructed quizzes
those equidistant from the segment’s endpoints. (G.CO.9)
and tests
Students will be able to
identify and classify
triangles.
Students will be able to
prove triangles are
congruent.
Students will develop and
use the triangle-sum and
angle-measure theorems
and the Triangle Inequality
Theorem. (part e)
- How can you tell whether a
triangle is scalene, isosceles, or
equilateral?
-How can you identify
corresponding parts of
congruent triangles?
-How do you show that two
triangles are congruent?
- Prove theorems about triangles. Theorems include: measures
of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining midpoints
of two sides of
a triangle is parallel to the third side and half the length; the
medians of
a triangle meet at a point. (G.CO.10)
- Use the definition of congruence in terms of rigid motions to
show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
are congruent. (G.CO.7)
- Explain how the criteria for triangle congruence (ASA, SAS,
and SSS) follow from the definition of congruence in terms of
rigid motions. (G.CO.8)
- Apply the Triangle Inequality Theorem.
●
●
●
Students will be able to
identify, name, and
describe polygons and
regular polygons.
Students will be able to
find the sum of the
interior and exterior
angles of a polygon.
Students will be able to
identify and classify
special quadrilaterals.
(part a)
- What shapes are polygons?
- Where do polygons occur
in nature?
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- Sample Activity: Explore the definition of polygons through
weather symbols and programming symbols.
- The formula for angle measures of a polygon will be derived
using diagonals.
- Students will use the properties of parallel and
perpendicular lines and diagonals to classify quadrilaterals.
- Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent
diagonals. (G.CO.11)
Unit 2 - Similarity, Proofs, and Trigonometry
Part a - Similarity (8)
Part b - Right Triangles and Trigonometry (9 no section 9.7)
Content/Objective
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Students will be able to
Essential Questions/
Enduring Understandings
- What makes shapes alike
Suggested Activity/
Appropriate Materials-Equipment
- Given two figures, use the definition of similarity in terms
Suggested
Evaluation/Assessment
Pre-
●
apply ratios and
proportions to identify
similar polygons.
Students will be able to
prove theorems. (part a)
and different?
- How are similarity,
congruence, and symmetry
related?
- How do you show two
polygons are similar?
-How do you use proportions
to find side lengths in similar
polygons?
Enduring Understanding:
- Congruent shapes are also
similar, but similar shapes
may not be congruent.
-An origin-centered dilation
with scale factor r maps
every point (x,y) in the
coordinate plane to the point
(rx,ry).
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●
●
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Students will be able to
solve right triangle
problems using the
Pythagorean Theorem.
Students will be able to
solve special right
triangles.
Students will be to use
trigonometry ratios to
-How do you find a side
length or angle measure in a
right triangle?
-How do trigonometric ratios
relate to similar right
triangles?
of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of
similarity for triangles as the equality of all corresponding
pairs of angles and the proportionality of all corresponding
pairs of sides. (G.SRT.2)
- Apply the definition and characteristics of similarity to
verify basic properties of angles and triangles and to
perform constructions using geometric software.
- Prove theorems including a line parallel to one side of a
triangle divides the other two proportionally, and
conversely. (G.SRT.4)
- Use the properties of similarity transformations to
establish the AA, SSS, and SAS criterion for two triangles to
be similar. (G.SRT.3)
- Use congruence and similarity criteria for triangles to
solve
problems and to prove relationships in geometric figures.
(G.SRT.5)
- Properties of iterative geometric patterns can be analyzed
and described mathematically.
- Verify experimentally the properties of dilation given by a
center and a scale factor. (G.SRT.1)
- Represent dilation using function notation f(x,y) = (rx,ry).
Assessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- Prove theorems about triangles. Theorems include: the
Pythagorean Theorem proved using triangle similarity.
(G.SRT.4)
- Understand that by similarity, side ratios in right triangles
are properties of the angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
(G.SRT.6)
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
solve right triangles in
applied
problems.(G.SRT.8) (part
b)
●
Enrichment Extension:
Students will be able to
generalize the
Pythagorean Theorem to
non-right triangles by the
Law of Cosines and Sines.
(part b)
- The Laws of Sines and
Cosines embody the triangle
congruence criteria for the
cases where three pieces of
information suffice to
completely solve a triangle.
These laws yield two possible
solutions in the ambiguous
case, illustrating the SideSide-Angles is not
congruence criterion.
Unit 3 - Extending to Three Dimensions
Part a - Area of Polygons and Circles (11 no 11.1, 11.6)
Part b- Surface Area and Volume (12)
25
- Explain and use the relationship between the sine and
cosine of complementary angles. (G.SRT.7)
- Use trigonometric ratios to calculate the length of sides
and measure of angles.
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
- Prove the Laws of Sines and Cosines and use them to solve
problems. (G.SRT.10)
- Understand and apply the Law of Sines and the Law of
Cosines to find unknown measurements in right and nonright triangles. (G.SRT.11)
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
Content/Objective
●
●
●
26
Essential Questions/
Enduring Understandings
Suggested Activity/
Appropriate Materials-Equipment
Suggested
Evaluation/Assessment
Students will be able to
find the area of a
polygon.
Students will be able to
find the circumference
and area of a circle. (part
a)
- How do perimeters and
areas of similar polygons
compare?
- Students will use formulas to find areas of parallelograms,
triangles, trapezoids, rhombuses, and kites.
- Students will explore area concepts related to regular
polygons.
- Students will use trigonometry to find areas.
- Students will find circumferences and areas of circles.
- Students will examine ratios of perimeters, circumferences,
and areas among similar figures.
- Given a figure and its area and perimeter, students will be
able to find the area and perimeter of a figure similar to the
original figure.
- Give an informal argument for the formulas for the
circumference of a circle and area of a circle (G.GMD.1)
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
Students will be able to
find the surface area and
volume of solid figures.
(part b)
- How are 1-, 2-, and 3dimensional shapes related?
- How can 3-dimensional
objects be represented in 2
dimensions?
- How can 3-dimensional
objects be measured?
- How can you determine the
intersection of a solid and a
plane?
- How do the surface areas
- Give an informal argument for the formulas for the volume
of a cylinder, pyramid, and cone. (G.GMD.1)
- Use surface area and volume formulas for cylinders,
pyramids, cones, and spheres to solve problems. (G.GMD.2)
- Identify the shapes of two-dimensional cross-sections of
three- dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
(G.GMD.4)
- Use geometric shapes, their measures, and their properties
to describe objects. (G.MG.1)
- Students will examine ratios among similar solids.
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
and volumes of similar solids
compare?
- Given a figure and its surface area, students will be able to
find the surface area and volume of a solid similar to the
original solid.
- Students will construct their own representation of 3dimensional objects, nets, projective view (tops, front, side),
and prospectives views using isometric dot paper.
technologybased
assessments, teacher
constructed quizzes
and tests
Unit 4 - Connecting Algebra and Geometry through Coordinates
Part a - Coordinate Geometry
***G.GPE.2 must be added - directrix and focus of a parabola
Content/Objective
●
27
Students will develop
and test conjectures
using coordinate
geometry. (part a)
Essential Questions/
Enduring Understandings
- How do you use coordinate
geometry to prove simple
geometric theorems
algebraically?
Suggested Activity/
Appropriate Materials-Equipment
- Prove the slope criteria for parallel and perpendicular lines
and
uses them to solve geometric problems. (G.GPE.5)
- Prove that the diagonals of a rhombus are perpendicular
using coordinate geometry.
- Prove that a triangle is right using the slope formula.
- Investigate the properties of the quadrilateral formed by
joining the midpoint of the sides of a parallelogram using
computer software.
- Investigate properties of quadrilaterals by using coordinate
geometry. (G.GPE.4)
- Prove a quadrilateral is a parallelogram, rhombus,
rectangle, or square by using slope, midpoint, and distance
formulas.
Suggested
Evaluation/Assessment
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
- Use coordinates to compute perimeters of polygons and
areas of triangles and rectangles, e.g., using the distance
formula. (G.GPE.7)
- Students will use coordinate geometry to classify special
parallelograms.
- Students will examine slope and segment length in
coordinate plane.
- Students will use the Distance Formula in the coordinate
plane.
- Find the point on a directed line segment between two
given points that partitions the segment in a given ratio.
(G.GPE.6)
constructed quizzes
and tests
Unit 5 - Circles With and Without Coordinates
Part a - Circles (10, no section 10.7)
Content/Objective
●
●
28
Students will be able to
understand and apply
theorems about circles.
(part a)
Essential Questions/
Enduring Understandings
Suggested Activity/
Appropriate Materials-Equipment
- How can you prove
relationships between angles
and arcs in a circle?
- When lines intersect a circle,
or within a circle, how do you
find the measure of resulting
angles, arcs, and segments?
- How do you find the equation
of a circle in the coordinate
- Prove that all circles are similar. (G.C.1)
- Identify and describe relationships among inscribed
angles, radii, and chords. Include the relationship between
central, inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the radius of a circle
is perpendicular to the tangent where the radius intersects
the circle. (G.C.2)
- Construct the inscribed and circumscribed circles of a
triangle, and prove properties of angles for a quadrilateral
Suggested
Evaluation/Assessment
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
plane?
- What geometric relationships
can be found in circles?
Unit 6 - Applications of Probability
29
inscribed in a circle. (G.C.3)
- Construct a tangent line from a point outside a given
circle to
the circle. (G.C.4)
- Derive using similarity the fact that the length of the arc
intercepted by an angle is proportional to the radius, and
define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
(G.C.5)
- Derive the equation of a circle of given center and radius
using
the Pythagorean Theorem; complete the square to find the
center and radius of a circle given by an equation.
(G.GPE.1)
- Students will examine angles formed by lines that
intersect inside and outside a circle.
Students will relate arcs and angles.
- Students will use properties of tangent lines.
- Students will use the relationships among chords, arcs,
and central angles.
- Students will solve problems with angles formed by
secants and tangents.
- The center and radius of a circle in a coordinate plane can
be used to find the equation of a circle.
- Determine the length of line segments and arcs, the
measure of angles, and the areas of shapes that they define
in complex geometric drawings.
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
Part a - in NJ Standards and Core - In Stats text and Section 11.6
Content/Objective
●
●
30
Students will use
geometric
representations to
solve probability
problems.
Students will apply
what they know about
probability to
situations where
probability is
calculated in regards
to length and area.
Essential Questions/
Enduring Understandings
- How can geometry help
determine probability?
- Geometric probability is the
study of probabilities involved in
geometric problems under stated
conditions, such as length, area
and volume of geometric objects.
Suggested Activity/
Appropriate Materials-Equipment
- Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of
other events (“or,” “and,” “not”). (S.CP.1)
- Understand that two events A and B are independent if
the
probability of A and B occurring together is the product
of their probabilities, and use this characterization to
determine if they are independent. (S.CP.2)
- Understand the conditional probability of A given B as
P(A
and B)/P(B), and interpret independence of A and B as
saying that the conditional probability of A given B is the
same as the probability of A, and the conditional
probability of B given A is the same as the probability of
B. (S.CP.3)
- Construct and interpret two-way frequency tables of
data when two categories are associated with each object
being classified. Use the two-way table as a sample space
to decide if events are independent and to approximate
conditional probabilities. (S.CP.4)
- Recognize and explain the concepts of conditional
probability
and independence in everyday language and everyday
situations. (S.CP.5)
- Find the conditional probability of A given B as the
Suggested
Evaluation/Assessment
PreAssessment,
Checkpoint
exercises,
DoNows,
portfolios, oral
questioning,
closure, selfreflection
journals,
projects, tests,
technologybased
assessments, teacher
constructed quizzes
and tests
fraction of B’s outcomes that also belong to A, and
interpret the answer in terms of the model. (S.CP.6)
- Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A
and B), and interpret the answer in terms of the model.
(S.CP.7)
- Students will use composite figures to find areas of
shaded regions and calculate the probability of a point
falling in that region.
31
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Completing the Right Courses (www.sreb.org).
Daggett, Bill and McNulty, Ray. Rigor and Relevance Framework: International Center for Leadership in Education
(www.leadered.com).
Danielson, Charlotte (2007). Enhancing Professional Practice: A Framework for Teaching, 2nd Edition. Alexandria, VA: Association for
Supervision and Curriculum Development.
Killion, Joellen P. (2008). Collaborative Professional Learning in School and Beyond: A Toolkit for New Jersey Educators. Trenton, NJ:
New Jersey Department of Education, the New Jersey Professional Teaching Standards Board, and the National Development Council.
National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA:
National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: National Council
of Teachers of Mathematics.
National Research Council (1999). A Guide for Using Mathematics and Science Education Standards. Washington, D.C.: National
Academy Press.
New Jersey Department of Education (2010). New Jersey Common Core Curriculum Standards 2010. www.njcccs.org.
Romagnano, Lew (2006). Mathematics Assessment Literacy: Concepts and Terms in Large Scale Assessment. Reston, VA: National
Council of Teachers of Mathematics.
32
Stiggins, Rick, Arter, Judith, Chappuis, Jan, and Chappuis, Steve (2006). Classroom Assessment for Student Learning—
SUPPLEMENTARY MATERIAL. Portland, OR: Educational Testing Service.
The Secretary’s Commission on Achieving Necessary Skills. (1992). Learning a Living: A Blueprint for High Performance. A SCANS
Report for America 2000. Executive Summary. Washington, DC: U.S. Department of Labor.
Webb, Norman. Depth-of-Knowledge Levels. Wisconsin Center for Educational Research (www.facstaff.wcer.wisc.educ/normw).
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