Download ACADEMIC GEOMETRY - Kingsway Regional School District

Kingsway Regional School District
Committed to Excellence
Course Name: Academic Geometry
Department: Mathematics
BOE Adoption Date: September 22, 2014
Grade Level(s): 9th and 10th
Credits: 5
Revision Date(s): October 22, 2015
ABSTRACT
The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students
explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments.
Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are
emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high
school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students
experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. There are six
critical areas in which to focus on for Geometry:
Critical Area 1: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid
motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this
unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a
familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles,
quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Critical Area 2: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They
identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle
trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order
to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They
are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles.
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KRSD Office of Curriculum and Instruction
Critical Area 3: 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.
Critical Area 4: 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, which relates
back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.
Critical Area 5: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and
theorems about chords, secants, and tangents dealing with segment lengths and angle measures. 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, which relates back to work done in the first course, to determine intersections between lines and circles or
parabolas and between two circles.
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KRSD Office of Curriculum and Instruction
TABLE OF CONTENTS
Mission Statement
Page 4
Curriculum and Instruction Goals
Page 4
Philosophy of Shared Curriculum Service with South Harrison Township Elementary
Page 4
How to Read this Document
Page 4
Terms to Know
Page 5
Pacing Guide
Page 7-12
Curriculum Units
Page 13-63
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KRSD Office of Curriculum and Instruction
Mission Statement
The Kingsway Regional School District believes that this school district is responsible for developing and maintaining a comprehensive educational program that
will foster the academic, social, and personal growth of all students. The Kingsway Regional School District provides a secure, supportive environment. It also
provides high quality resources to challenge and empower each individual to pursue his/her potential, to develop a passion for learning in a diverse and
challenging world, to encourage active citizenship, and to reach a high standard of achievement at all grade levels as defined by the New Jersey Core Curriculum
Content Standards (NJCCCS) & Common Core State Standards (CCSS).
Curriculum & Instruction Goals
Goal(s):
1. To ensure students are college and career ready upon graduation
2. To vertically and horizontally align curriculum K-12 to ensure successful transition of students at each grade level
3. To identify individual student strengths and weaknesses utilizing various assessment measures (formative, summative, alternative, etc.) so as to
differentiate instruction while meeting the rigor of the applicable content standards
4. To improve student achievement as assessed through multiple measures including, but not limited to, state testing, local assessments, and intermediate
benchmarking
Philosophy of the Shared Curriculum Service with South Harrison Township Elementary
The ultimate goal of the newly established shared curriculum service with South Harrison Township Elementary is to provide clearly coherent curriculum for
grades K-12 to enhance student growth and achievement and provide learning experiences that assist in providing an inherent love of learning. With true
vertical and horizontal curricular alignment all students will be effectively prepared for their arrival onto the campus of Kingsway Regional Middle School.
Through this shared vision, both school districts are able to work earlier and more productively with students to ensure they are properly equipped with the
knowledge and skills necessary to be successful in college and career upon graduation from high school. The alignment of curriculum K-12 safeguards countless
benefits for our children; it is the very foundation for the improved teaching and learning that is our goal as educators, parents, and community members. Most
notably, an aligned curriculum K-12 creates a common ownership and understanding of what must be taught and learned at each grade level for each subject
area. No matter where a student attends, the curriculum requirements are the same across buildings, grade levels and teachers. Additionally, an aligned
curriculum serves to provide valuable information to parents who will know what each child is expected to learn while in the classroom.
How to Read this Document
This curricular document contains both pacing guides and curriculum units. The pacing guides serve to communicate an estimated timeframe as to when skills
and topics will be taught throughout the year. The pacing, however, may differ slightly depending upon the unique needs of each learner. The curriculum units
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KRSD Office of Curriculum and Instruction
contain more detailed information as to the content, goals, and objectives of the course well as how students will be assessed. The terms and definitions below
will assist the reader to better understand the sections and components of this curriculum document.
Terms to Know
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Accommodation(s): The term "accommodation" may be used to describe an alteration of environment, curriculum format, or equipment that allows an
individual with a disability to gain access to content and/or complete assigned tasks. They allow students with disabilities to pursue a regular course of
study. The term accommodation is often used interchangeable with the term modification. However, it is important to remember that modifications
change or modify the intended learning goal while accommodations result in the same learning goal being expected but with added assistance in that
achievement. Since accommodations do not alter what is being taught, instructors should be able to implement the same grading scale for students
with disabilities as they do for students without disabilities.
Differentiated Instruction: Differentiation of instruction relies on the idea that instructional approaches should be tailored to each individual student’s
learning needs. It provides students an array of options during the learning process that allows them make sense of ideas as it relates to them. The integration
of differentiated instructional techniques is a curriculum design approach to increase flexibility in teaching and decrease the barriers that frequently limit
student access to materials and learning in classrooms.
Enduring Understanding: Enduring understandings (aka big ideas) are statements of understanding that articulate deep conceptual understandings at
the heart of each content area. Enduring understandings are noted in the alongside essential questions within each unit in this document.
Essential Question: These are questions whose purpose is to stimulate thought, to provoke inquiry, and to spark more questions. They extend beyond a
single lesson or unit. Essential questions are noted in the beginning of each unit in this document.
Formative Assessments: Formative assessments monitor student learning to provide ongoing feedback that can be used by (1) instructors to improve
teaching and (2) by students to improve their learning. Formative assessments help identify students’ strengths and weaknesses and address problems
immediately.
Learning Activity(s): Learning activities are those activities that take place in the classroom for which the teacher facilitates and the students participate
in to ensure active engagement in the learning process. (Robert J. Marzano, The Art and Science of Teaching)
Learning Assignment(s): Learning assignments are those activities that take place independently by the student inside the classroom or outside the
classroom (i.e. homework) to extend concepts and skills within a lesson.
Learning Goal(s): Learning goals are broad statements that note what students “should know” and/or “be able to do” as they progress through a unit.
Learning goals correlate specifically to the NJCCCS and CCSS are noted within each unit.
Learning Objective(s): Learning objectives are more specific skills and concepts that students must achieve as they progress towards the broader
learning goal. These are included within each unit and are assessed frequently by the teacher to ensure students are progressing appropriately.
Model Assessment: Within the model curriculum, model assessments are provided that included assessments that allow for measuring student
proficiency of those target skills as the year of instruction progresses.
Model Curriculum: The model curriculum has been provided by the state of New Jersey to provide a “model” for which districts can properly implement
the Common Core State Standards and New Jersey Core Curriculum Content Standards by providing an example from which to work and/or a product
for implementation.
Modification(s): The term "modification" may be used to describe a change in the curriculum. Modifications are typically made for students with
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KRSD Office of Curriculum and Instruction
13.
14.
15.
16.
disabilities who are unable to comprehend all of the content an instructor is teaching. The term modification is often used interchangeable with the term
accommodations. However, it is important to remember that modifications change or modify the intended learning goal while accommodations result
in the same learning goal being expected but with assistance in that achievement.
Performance Assessments: (aka alternative or authentic assessments) Performance assessments are a form of assessment that requires students to
perform tasks that generate a more authentic evaluation of a student’s knowledge, skills, and abilities. Performance assessments stress the application
of knowledge and extend beyond traditional assessments (i.e. multiple-choice question, matching, true & false, etc.).
Standards: Academic standards, from which the curriculum is built, are statements that of what students “should know” or “be able to do” upon
completion of a grade-level or course of study. Educational standards help teachers ensure their students have the skills and knowledge they need to be
successful by providing clear goals for student learning.
o State: The New Jersey Core Curriculum Content Standards (NJCCCS) include Preschool Teaching and Learning Standards as well as K-12 standards
for: Visual and Performing Arts; Comprehensive Health and Physical Education; Science; Social Studies; World Languages; Technology; and 21stCentury Life and Careers.
o Common Core: Standards for mathematics and English-language arts literacy are part of the Common Core State Standards (CCSS) which are
nationally adopted standards coordinated by the Council of Chief States School Officers (CCSSO) and the National Governor’s Association (NGA)
in partnership with other national organizations.
Summative Assessments: Summative assessments evaluate student learning at the end of an instructional time period by comparing it against some
standard or benchmark. Information from summative assessments can be used formatively when students or faculty use it to guide their efforts and
activities in subsequent courses.
21st Century Skills: These skills emphasis the growing need to focus on those skills that prepare students successfully by focusing on core subjects and
21st century themes; learning and innovation skills; information, media and technology skills; and life and career skills. These concepts are embedded in
each unit of the curriculum.
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KRSD Office of Curriculum and Instruction
Proficiencies and Pacing:
Course Title: Academic Geometry
Prerequisite(s): Algebra I
Unit Title:
Duration/
Month(s)
Unit 1: Points, Lines, &
Planes
15 Days
Related Standards:
Learning Goals:
Topics and Skills:
Subject Area:
Students will understand the precise
definition of line segment and circle,
based on the undefined terms (point,
line and plane). Students will be able to
calculate distance along a line and
around a circular arc. Students will also
be able to use coordinates to partition
a segment, compute perimeter of
polygon and area of triangles and
rectangles as well as prove simple
geometric theorems.
1. Define basic geometry terms
(point, line, plan, and distance)
and will be able to describe and
name simple geometric
structures.
2. Use algebra and geometry to
determine the length of a line
segment, explore congruence,
and construct congruent line
segments.
3. Use the Pythagorean Theorem to
solve problems.
4. Use the midpoint formula to find
the midpoint or the endpoint of a
segment on the coordinate plane
and construct a segment
bisector.
5. Find the point on a directed line
segment between two given
points that partitions the
segment in a given ratio and
divide a segment into a given
number of congruent parts using
constructions.
6. Use the Distance Formula to
determine the length of
segments, and use coordinates to
compute perimeters of polygons
G.GPE.6
G.GPE.7
G.CO.1
G.CO.4
Interdisciplinary:
9.1.12.A.1
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KRSD Office of Curriculum and Instruction
Unit Title:
Duration/
Month(s)
Related Standards:
Subject Area:
Unit 2: Angle Relationships
& Proofs
13 Days
G.CO.9
G.CO.1
Interdisciplinary:
9.1.12.A.1
Learning Goals:
Students will understand the precise
definition of an angle and investigate
relationships between angles. Students
will be able to prove that vertical
angles are congruent.
Topics and Skills:
1.
2.
3.
4.
5.
6.
7.
Subject Area:
Unit 3:Parallel and
Perpendicular Lines
14 Days
G.CO.9
G.GPE.5
G.CO.10
Interdisciplinary:
9.1.12.A.1
Students will understand the precise
definitions of parallel and
perpendicular lines and investigate the
relationship between their slopes.
Students will be able to compare slopes
of lines and prove theorems involving
corresponding and alternate interior
angles.
1.
2.
3.
4.
5.
6.
7.
Subject Area:
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KRSD Office of Curriculum and Instruction
Students will understand the
1.
and areas of triangles and
rectangles.
Measure angles and use angle
pair relationships to solve
problems.
Use the angle addition postulate
to solve problems.
Use vertical angles to solve
problems.
Identify and use basic postulates
about points, lines, and planes to
develop paragraph and T-chart
proofs.
Write proofs using algebra
equations.
Prove the Vertical Angle
Theorem.
Use angle bisectors to solve
problems.
Identify relationships between
figures in space.
Identify angles formed by two
lines and a transversal.
Use properties of parallel lines to
find angle measures.
Solve problems involving
perpendicular lines.
Find slopes of lines.
Compare slopes to determine if
lines are parallel or
perpendicular.
Prove theorems involving parallel
and perpendicular lines.
Draw translated images.
Unit Title:
Duration/
Month(s)
Unit 4: Transformations
17 Days
Related Standards:
Learning Goals:
Topics and Skills:
G.SRT.1
G.CO.2
G.CO.3
G.CO.4
G.CO.5
definitions and properties of
reflections, translations, rotations and
dilations. Students will be able to draw
the image figure, given a
transformation rule. Students will also
be able to write the transformation
rule when given two figures.
2. Write the rule for a translated
figure.
3. Draw reflected images.
4. Write the reflection rule when
given two figures.
5. Recognize and draw lines of
symmetry.
6. Specify a reflection that will carry
a given figure onto itself.
7. Understand the properties of
rotations.
8. Draw rotated images.
9. Write the rule for a rotated
figure.
10. Determine if a figure has
rotational symmetry.
11. Draw dilated images.
12. Write the rule for a dilated figure.
13. Locate the images of figures by a
composite of transformations.
1. Use congruence properties to
show that triangles are
congruent.
2. Use SSS, SAS, and HL congruence
theorems to prove that triangles
are congruent.
3. Use the ASA and AAS theorems
to test for triangle congruence.
4. Prove that triangles are
congruent by SSS, SAS, ASA, AAS,
or HL.
1. Classify triangles by sides and
angle measures.
2. Solve problems involving exterior
Interdisciplinary:
9.1.12.A.1
Subject Area:
Unit 5: Congruent Triangles
11 Days
G.CO.6
G.CO.7
G.CO.8
Subject Area:
Unit 6: Triangles
9
10 Days
G-CO.9
KRSD Office of Curriculum and Instruction
Students will understand the definition
of congruence in terms of rigid motions
(reflections, translations, and
rotations). Students will be able to
prove triangles are congruent and
explain the criteria for triangle
congruence (ASA, SAS, and SSS) in
terms of rigid motions.
Students will understand the
relationships of lines and angles in
triangles (exterior angles,
Unit Title:
Duration/
Month(s)
Related Standards:
Learning Goals:
Topics and Skills:
G-CO.10
perpendicular bisectors, medians,
altitudes). Students will be able to
show that the medians of a triangle
meet at a point. Students will also be
able to prove two theorems about
triangles. (Isosceles Triangle Theorem
and the Perpendicular Bisector
Theorem).
angles.
3. Use the properties of isosceles
triangles
4. Prove the isosceles triangle
theorem.
5. Prove the perpendicular bisector
theorem and solve problems
involving perpendicular bisectors
of triangles.
6. Use the medians and altitudes of
a triangle to solve problems.
1. Find the measures of interior and
exterior angles of a polygon.
2. Identify polygons.
3. Recognize and apply properties
of parallelograms.
4. Prove theorems about
parallelograms
5. Recognize and apply properties
of rhombi, rectangles, and
squares.
6. Prove theorems about
rectangles.
7. Recognize and apply properties
of trapezoids and kites.
8. Compare the different types of
quadrilaterals.
9. Prove a figure is a rectangle given
4 points on a coordinate plane.
1. Write ratios and solve
proportions.
2. Identify and apply similar
polygons.
3. Use the AA Similarity Postulate
Interdisciplinary:
9.1.12.A.1
Subject Area:
Unit 7: Quadrilaterals
12 Days
G-CO.11
G.GPE.4
Interdisciplinary:
9.1.12.A.1
Subject Area:
Unit 8: Similar Figures
10
14 Days
G.SRT.1
G.SRT.2
G.SRT.3
KRSD Office of Curriculum and Instruction
Students will understand the properties
of each type of quadrilateral. Students
will also be able to prove theorems
involving parallelograms and prove a
figure is a rectangle given 4 points on a
coordinate plane.
Students will understand the definition
of similarity in terms of similarity
transformations. Students will also be
able to decide if two figures are similar
using congruence and similarity criteria
Unit Title:
Duration/
Month(s)
Related Standards:
Learning Goals:
G.SRT.4
G.SRT.5
G.CO.10
(AA Similarity Postulate) as well as
proving the Mid-Segment Theorem.
Interdisciplinary:
9.1.12.A.1
Subject Area:
Unit 9: Trigonometry
14 Days
G.SRT.6
G.SRT.7
G.SRT.8
Topics and Skills:
4.
5.
6.
Students will understand the
relationship between sine and cosine of
complementary angles. Students will
be able to use trigonometric ratios to
solve right triangles.
Interdisciplinary:
9.1.12.A.1
7.
1.
2.
3.
4.
5.
6.
7.
Subject Area:
Unit 10: Surface Area
12 Days
G.MG.1
G.MG.2
G.MG.3
Interdisciplinary:
9.1.12.A.1
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KRSD Office of Curriculum and Instruction
Students will understand the concepts
of surface area and be able to use
geometric shapes, their measures, and
their properties to describe objects.
Students will be able to apply
geometric methods to solve design
problems.
1.
2.
3.
4.
and the SAS and SSS Similarity
Theorems.
Use similarity to find indirect
measurements.
Find and use relationships in
similar right triangles.
Use the Side-Splitter Theorem
and the Triangle-Angle Bisector
Theorem.
Prove the Midsegment Theorem.
Apply the Pythagorean Theorem
to classify a triangle as obtuse,
right, or acute.
Apply the properties of 45-45-90
triangles.
Apply the properties of 30-60-90
triangles.
Find trigonometric ratios using
right triangles.
Solve problems using
trigonometric ratios.
Solve triangles using
trigonometric ratios and inverse
ratios.
Solve problems involving angles
of depression and elevation.
Find the areas and perimeters of
various polygons.
Calculate lateral and surface
areas of Prisms.
Calculate the lateral area and
surface area of cones, cylinders,
and spheres.
Find the lateral and surface area
Unit Title:
Duration/
Month(s)
Related Standards:
Subject Area:
Unit 11: Volume
10 Days
G.MG.2
G.GMD.1
G.GMD.2
G.GMD.3
G.GMD.4
Interdisciplinary:
9.1.12.A.1
Subject Area:
Unit 12: Circles
12 days
G.C.1
G.C.2
G.C.3
G.C.5
Interdisciplinary:
9.1.12.A.1
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KRSD Office of Curriculum and Instruction
Learning Goals:
Students will understand how to use
volume formulas for cylinders,
pyramids, cones, and spheres to solve
problems. Students will be able to
apply concepts of density based on
area and volume in real life situations.
Students will identify and describe
relationships among radii, chords,
inscribed angles, arcs, central angles
and inscribed angles. Students will be
able to write the equation of a circle.
Topics and Skills:
of a pyramid.
5. Use surface areas to solve design
problems.
1. Find the volume of a prism and
the volume of a cylinder.
2. Visualize cross sections of space
figures and determine equivalent
volumes of solids using Cavalieri’s
principle.
3. Find the volumes of pyramids,
cones, and spheres.
4. Describe the ratio of mass to area
or volume of real life objects.
1. Find the center and radius of a
circle.
2. Use congruent chords, arcs, and
central angles.
3. Use perpendicular bisectors to
chords.
4. Find the measure of an inscribed
angle.
5. Use properties of a tangent to a
circle.
6. Find the measure of an angle
formed by a tangent and a chord.
7. Find the measures of angles
formed by chords, secants, and
tangents.
8. Find the lengths of segments
associated with circles.
9. Write the equation of a circle.
Unit 1: Points, Lines, and Planes
Recommended Duration: 14 days
Unit Description: Students learn about points, lines, and planes, and the building blocks of geometry. Line segments, rays, and polygons are also introduced in
this unit. Students explore congruent segments and angles and learn to construct them with a compass and straight-edge. Students expand on their knowledge
of the Pythagorean Theorem to master the Distance Formula and Midpoint Formula to find the midpoint of a segment. Students will also compute the
perimeter of a given polygon.
Essential Questions:
•
•
•
•
What are the building blocks of Geometry?
How can you describe the attributes of a segment?
Have you ever noticed that a four-legged chair sometimes wobbles,
but a three-legged stool never wobbles?
How do points, lines, and planes exist in nature?
Enduring Understandings:
•
•
•
•
•
•
•
•
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KRSD Office of Curriculum and Instruction
Geometry is a mathematical system built on accepted facts, basic
terms, and definitions.
Number operations can be used to find and compare the length of
segments.
The Ruler and Segment Addition Postulates can be used in reasoning
about lengths.
Formulas can be used to find the midpoint and length of any segment
in the coordinate plane.
Perimeter, circumference, and area are different ways of measuring
the size of geometric figures.
The area of a region is the sum of the areas of its non-overlapping
parts.
Special geometric tools can be used to make a figure that is congruent
to an original figure without measuring.
Construction with a straightedge and compass is more accurate than
sketching and drawing.
Relevant Standards:
Content Standards:
Primary(Power):
G.GPE.6 - Find the point on a directed line
segment between two given points that
partitions the segment in a given ratio.
G.GPE.7 - Use coordinates to compute
perimeters of polygons and areas of triangles
and rectangles, e.g., using the distance formula.
Secondary(Supportive):
G.CO.1 - Know precise definitions of angle,
circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of
point, line, distance along a line, and distance
around a circular arc.
G.PE.4 - Use coordinates to prove simple
geometric theorems algebraically. For example,
prove or disprove that a figure defined by four
given points in the coordinate plane is a
rectangle; prove or disprove that the point (1,
√3) lies on the circle centered at the origin and
containing the point (0, 2).
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
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KRSD Office of Curriculum and Instruction
Learning Goals:
Students will understand the precise definition of
line segment and circle, based on the undefined
terms (point, line and plane). Students will be
able to calculate distance along a line and around
a circular arc. Students will also be able to use
coordinates to partition a segment, compute
perimeter of polygon and area of triangles and
rectangles as well as prove simple geometric
theorems.
Learning Objectives:
•
•
•
•
•
•
The students will be able to define basic
geometry terms (point, line, plan, and distance)
and will be able to describe and name simple
geometric structures.
The students will use algebra and geometry to
determine the length of a line segment, explore
congruence, and construct congruent line
segments.
The students will be able to use the
Pythagorean Theorem to solve problems.
The students will be able to use the midpoint
formula to find the midpoint or the endpoint of
a segment on the coordinate plane and
construct a segment bisector.
The student will be able to find the point on a
directed line segment between two given
points that partitions the segment in a given
ratio and divide a segment into a given number
of congruent parts using constructions.
The students will be able to use the Distance
Formula to determine the length of segments,
and use coordinates to compute perimeters of
polygons and areas of triangles and rectangles.
Relevant Standards:
Learning Goals:
Learning Objectives:
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Pre-Assessment
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: Postulate, point, line, plane, collinear, ray, opposite rays, coplanar, line segment, congruent, parallel, Pythagorean Theorem, Distance Formula, area,
midpoint, directed line segment, ratio
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KRSD Office of Curriculum and Instruction
Non-Essential:
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
Students will be able to analyze
videos on how to do
constructions.
____ Civic Literacy
____ Media Literacy
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Visual and Performing Arts:
____ Information & Communication
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
____ Information Literacy
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): Pre-Assessment
Major Activities (required): Station Activities, Scavenger Hunt
16
KRSD Office of Curriculum and Instruction
Unit 2: Angle Relationships and Proofs
Recommended Duration: 13 days
Unit Description: In this unit, students are introduced to angles and angle measures. Students will explore angle relationships and congruent angles. They will
learn to construct angles with a straightedge and compass. Students will also have an introduction to proofs and investigate proofs involving angles such as
Vertical Angles.
Essential Questions:
•
•
•
•
•
•
•
Enduring Understandings:
•
How can you describe the attributes of an angle?
How can you prove a statement is true?
What kinds of angles are formed when streets intersect?
How big is a degree?
How is mathematical evidence similar to evidence in law?
How can segment relationships be used for travel?
How do scissors illustrate supplementary angles?
•
•
•
•
•
•
•
•
17
KRSD Office of Curriculum and Instruction
Number operations can be used to find and compare measures of
angles.
The Protractor and Angle Addition Postulates can be used in reasoning
about angle measures.
Special angle pairs, such as adjacent, vertical, complementary, and
supplementary angles, can be used to identify geometric relationships
and to find angle measures.
Special geometric tools can be used to make a figure that is congruent
to an original figure without measuring.
Construction with a straightedge and compass is more accurate than
sketching and drawing.
A postulate is an accepted statement of fact.
Logical reasoning from one step to another is essential in building a
proof.
Reasons in a proof include given information, definitions, properties,
postulates, and previously proven theorems.
Given information, definitions, properties, postulates, and previously
proven theorems can be used as reasons in a proof.
Relevant Standards:
Content Standards:
Primary(Power):
G.CO.9 - Prove theorems about lines and angles.
Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are
exactly those equidistant from the segment’s
endpoints.
Secondary(Supportive):
G.CO.1 - Know precise definitions of angle,
circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of
point, line, distance along a line, and distance
around a circular arc.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
18
KRSD Office of Curriculum and Instruction
Learning Goals:
Students will understand the precise definition of
an angle and investigate relationships between
angles. Students will be able to prove that vertical
angles are congruent.
Learning Objectives:
•
•
•
•
•
•
•
The students will be able to measure angles
and use angle pair relationships to solve
problems.
The students will be able to use the angle
addition postulate to solve problems.
The students will be able to use vertical angles
to solve problems.
The students will be able to identify and use
basic postulates about points, lines, and planes
to develop paragraph and T-chart proofs.
The students will write proofs using algebra
equations.
The students will be able to prove the Vertical
Angle Theorem.
The students will be able to use angle bisectors
to solve problems.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: Angle, vertex, exterior, interior, acute, obtuse, right, straight, adjacent angles, supplementary, complementary, linear pair, vertical angles, postulate,
theorem, congruent, angle bisector
Non-Essential:
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
Students will be able to analyze
videos on how to do
constructions.
____ Civic Literacy
____ Media Literacy
19
KRSD Office of Curriculum and Instruction
Interdisciplinary Connections
(Applicable Standards):
Visual and Performing Arts:
Integration of Technology:
21st Century Themes:
21st Century Skills:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
____ Information & Communication
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
____ Information Literacy
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): Station Activities, Scavenger Hunt
20
KRSD Office of Curriculum and Instruction
Unit 3: Parallel and Perpendicular Lines
Recommended Duration: 14 days
Unit Description: Students identify the special angle relationships that result when a transversal intersects parallel lines. Slope and forms for equations of a line
are reviewed. Students solve problems by writing linear equations and use slope to determine whether two lines are parallel, perpendicular, or neither.
Students will explore the proofs of the Corresponding Angles Theorem and Alternate Interior Angles Theorem. Students will also construct parallel lines,
perpendicular lines, and perpendicular bisectors.
Essential Questions:
•
•
•
•
•
•
•
•
•
How do you prove that two lines are parallel or perpendicular?
What is the sum of the measures of the angles of a triangle?
How do you write the equation of a line in the coordinate plane?
How are parallel lines and planes used in architecture?
How can angles and lines be used in art?
How is slope used in transportation?
How can the equation of a line describe the cost of a cellphone
service?
How do you know that the sides of a parking space are parallel?
How does the distance between parallel lines relate to hanging new
shelves?
Enduring Understandings:
•
•
•
•
•
•
•
•
•
•
•
•
21
KRSD Office of Curriculum and Instruction
Not all lines and not all planes intersect.
When a line intersects two or more lines, the angles formed at the
intersection points create special angle pairs.
The special angle pairs formed by parallel lines and a transversal are
either congruent or supplementary.
Geometric postulates and theorems can be combined with Algebra to
find some angle measures.
Certain angle pairs can be used to decide whether two lines are parallel.
Paragraph and two-column proofs are two forms of proofs.
The relationships of two lines to a third line can be used to decide
whether two lines are parallel or perpendicular to each other.
The sum of the angle measures of a triangle is always the same.
Parallel and perpendicular lines can be constructed with a straightedge
and compass.
A line can be graphed and its equation written when certain facts about
the line, such as its slope and a point on the line, are known.
The equation of a line can be written in various forms comparing the
slopes of two lines can show whether the lines are parallel or
perpendicular.
The relationships between parallel or perpendicular lines can sometimes
be used to write the equation of its line.
Relevant Standards:
Content Standards:
Primary(Power):
G.CO.9 - Prove theorems about lines and angles.
Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are
exactly those equidistant from the segment’s
endpoints.
G.GPE.5 - Prove the slope criteria for parallel
and perpendicular lines and use them to solve
geometric problems (e.g., find the equation of a
line parallel or perpendicular to a given line that
passes through a given point).
Secondary(Supportive):
G.CO.1 - Know precise definitions of angle,
circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of
point, line, distance along a line, and distance
around a circular arc.
G.CO.10 - 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.
22
KRSD Office of Curriculum and Instruction
Learning Goals:
Students will understand the precise definitions of
parallel and perpendicular lines and investigate
the relationship between their slopes. Students
will be able to compare slopes of lines and prove
theorems involving corresponding and alternate
interior angles.
Learning Objectives:
•
•
•
•
•
•
•
Students will be able to identify relationships
between figures in space.
Students will be able to identify angles formed
by two lines and a transversal.
Students will be able to use properties of
parallel lines to find angle measures.
Students will be able to solve problems
involving perpendicular lines.
Students will be able to find slopes of lines.
Students will be able to compare slopes to
determine if lines are parallel or perpendicular.
Students will be able to prove theorems
involving parallel and perpendicular lines.
Relevant Standards:
Learning Goals:
Learning Objectives:
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Summative Assessment #1
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
23
KRSD Office of Curriculum and Instruction
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: parallel, skew, transversal, alternate interior, alternate exterior, corresponding, same-side interior, perpendicular, slope, slope-intercept form, pointslope form
Non-Essential: rise, run
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
24
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): Summative Assessment #1, Performance-based Activity, Choice of Assessment (PARCC vs. Teacher-generated),
Major Activities (required): Scavenger Hunt, Jigsaw, Kahoot
25
KRSD Office of Curriculum and Instruction
Unit 4: Transformations
Recommended Duration: (18 days)
Unit Description: Students explore the different types of transformations: reflections, translations, rotations, and dilations. The students will be able to identify
the transformation given two images and draw an image given a transformation rule.
Essential Questions:
•
•
•
•
•
•
•
Enduring Understandings:
•
How can you change a figure’s position without changing its size and
shape?
How can you change a figure’s size without changing its shape?
How can you represent a transformation in the coordinate plane?
How are reflections found in nature?
How are translations used in a marching band show?
How do amusement rides illustrate rotations?
How do you use dilations when you use a computer?
•
•
•
•
•
•
Relevant Standards:
Content Standards:
Primary(Power):
G.SRT.1 – Verify experimentally the properties
of dilations given by a center and a scale factor:
a. A dilation takes a line not passing
through the center of the dilation to a
parallel line, and leaves a line passing
through the center unchanged.
b. The dilation of a line segment is longer
26
KRSD Office of Curriculum and Instruction
The location and orientation of a geometric figure can be changed
while preserving distance and angle measures.
The distance between any two points, angle measures, and orientation
of a geometric figure remain the same when the figure is translated in
one direction.
When you reflect a figure across a line, each point of the figure goes to
another point the same distance from the line, but on the other side.
Distances, angle measures, and orientation of a geometric figure stay
the same when a figure is rotated about a center of rotation.
If two figures can be mapped to each other by a sequence of rigid
motions, then the figures are congruent.
A scale factor can be used to make a larger or smaller copy of a figure
that is also similar to the original figure.
If there is an isometry that maps a figure to another, then you can map
one onto the other by using a composition of reflections.
Learning Goals:
Students will understand the definitions and
properties of reflections, translations, rotations
and dilations. Students will be able to draw the
image figure, given a transformation rule.
Students will also be able to write the
transformation rule when given two figures.
Learning Objectives:
•
•
•
•
•
•
Student will be able to draw translated images.
Students will be able to write the rule for a
translated figure.
Student will be able to draw reflected images.
Students will also be able to write the reflection
rule when given two figures.
Students will be able to recognize and draw
lines of symmetry.
Students will be able to specify a reflection that
Relevant Standards:
or shorter in the ratio given by the scale
factor.
Secondary(Supportive):
G.CO.2 - Represent transformations in the plane
using, e.g., transparencies and geometry
software; describe transformations as functions
that take points in the plane as inputs and give
other points as outputs. Compare
transformations that preserve distance and
angle to those that do not (e.g., translation
versus horizontal stretch).
G.CO.3 - Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
G.CO.4 - Develop definitions of rotations,
reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and
line segments.
G.CO.5 - Given a geometric figure and a
rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper,
tracing paper, or geometry software. Specify a
sequence of transformations that will carry a
given figure onto another.
Mathematical Practice (Supportive):
27
KRSD Office of Curriculum and Instruction
Learning Goals:
Learning Objectives:
•
•
•
•
•
•
•
will carry a given figure onto itself.
Students will understand the properties of
rotations.
Student will be able to draw rotated images.
Students will be able to write the rule for a
rotated figure.
Student will be able to determine if a figure has
rotational symmetry.
Student will be able to draw dilated images.
Students will be able to write the rule for a
dilated figure.
Student will be able to locate the images of
figures by a composite of transformations.
Relevant Standards:
Learning Goals:
Learning Objectives:
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major
Activities/
(required):
Assignments
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
28
KRSD Office of Curriculum and Instruction
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: transformation, reflection, translation, rotation, dilation, rigid motion, image, preimage, prime notation, line of symmetry, rotational symmetry, scale
factor, isometry
Non-Essential: counterclockwise, clockwise
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
29
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum, Rulers,
TI 30 Calculators
Major Assignments (required): None
Major Activities (required): Kahoot
30
KRSD Office of Curriculum and Instruction
Unit 5: Congruent Triangles
Recommended Duration: (11 days)
Unit Description: The unit builds the students’ understanding and skills related to angles and triangles. Students will explore how to prove triangles congruent
by using SSS, SAS, ASA, AAS, and HL. Students will also use transformations to show that two triangles are congruent by moving one triangle unto the other.
Essential Questions:
•
•
•
•
•
•
•
Enduring Understandings:
•
How do you identify corresponding parts of congruent triangles?
How do you show that two triangles are congruent?
Why are triangles important in construction?
How are angles in triangles used to make kites?
Why are triangles used in bridges?
How do land surveyors use congruent triangles?
How are congruent triangles used in construction?
•
•
•
•
Relevant Standards:
Content Standards:
Primary(Power):
G.CO.6 - Use geometric descriptions of rigid
motions to transform figures and to predict the
effect of a given rigid motion on a given figure;
given two figures, use the definition of
congruence in terms of rigid motions to decide if
they are congruent.
G.CO.7 - 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 are
31
KRSD Office of Curriculum and Instruction
Comparing the corresponding parts of two figures can show whether
the figures are congruent.
Two triangles can be proven to be congruent without having to show
that all corresponding parts are congruent.
Two ways triangles can be proven to be congruent are by using three
pairs of corresponding sides or by using two pairs of corresponding
sides and one pair of corresponding angles.
Another way triangles can be proven to be congruent is by using one
pair of corresponding sides and two pairs of corresponding angles.
If two triangles are congruent, then every pair of their corresponding
part is also congruent.
Learning Goals:
Students will understand the definition of
congruence in terms of rigid motions (reflections,
translations, and rotations). Students will be able
to prove triangles are congruent and explain the
criteria for triangle congruence (ASA, SAS, and SSS)
in terms of rigid motions.
Learning Objectives:
•
•
•
•
The student will be able to use congruence
properties to show that triangles are
congruent.
The students will be able to use SSS, SAS, and
HL congruence theorems to prove that
triangles are congruent.
Students will be able to use the ASA and AAS
theorems to test for triangle congruence.
The students will be able to prove that triangles
are congruent by SSS, SAS, ASA, AAS, or HL.
Relevant Standards:
Learning Goals:
Learning Objectives:
congruent.
G.CO.8 - Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid
motions.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
Assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
32
KRSD Office of Curriculum and Instruction
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: congruence statement, SSS, SAS, ASA, AAS, HL, included angle, included side, CPCTC (corresponding parts of congruent triangles are congruent)
Non-Essential:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
33
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Interdisciplinary Connections
(Applicable Standards):
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum, TI 30
Calculators
Major Assignments (required): None
Major Activities (required): None
34
KRSD Office of Curriculum and Instruction
Unit 6: Triangles
Recommended Duration: (10 days)
Unit Description: In this unit students expand their knowledge of triangles and their properties. Students will learn how to classify triangles by their side
measures and their angle measures. Bisectors, medians, and altitudes are identified and explored.
Essential Questions:
•
•
Enduring Understandings:
•
What different types of triangles are used in construction?
How are the angles of triangles used to make kites?
•
•
•
•
•
•
There are two special relationships between a midsegment of a
triangle and the third side of the triangle.
There is a special relationship between the points on the perpendicular
bisector of a segment and the endpoints of the segment.
A triangle has three medians that meet at the centroid.
A triangle has three perpendicular bisectors that meet at the
circumcenter.
A triangle has three angle bisectors that meet at the incenter.
A triangle has three altitudes that meet at the orthocenter.
The angles and sides of isosceles and equilateral triangles have special
relationships.
Relevant Standards:
Learning Goals:
Content Standards:
Primary(Power):
G-CO.9 - Prove theorems about lines and angles.
Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are
exactly those equidistant from the segment's
endpoints.
Students will understand the classifications of
triangles (equilateral, isosceles, scalene, acute,
equiangular, right, and obtuse) and the
relationships of angles and lines in triangles
(exterior angles, perpendicular bisectors, medians,
altitudes). Students will be able to prove the
Isosceles Triangle Theorem and the Perpendicular
Bisector Theorem.
G-CO.10 - Prove theorems about triangles.
Theorems include: measures of interior angles of
35
KRSD Office of Curriculum and Instruction
Learning Objectives:
•
•
•
•
•
•
Students will be able to classify triangles by
sides and angle measures.
Students will be able to solve problems
involving exterior angles.
Students will be able to use the properties of
isosceles triangles
Students will be able to prove the isosceles
triangle theorem.
The students will be able to prove the
perpendicular bisector theorem and solve
problems involving perpendicular bisectors of
triangles.
Students will be able to use the medians and
Relevant Standards:
Learning Goals:
Learning Objectives:
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.
altitudes of a triangle to solve problems.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Summative Assessment #2
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
36
KRSD Office of Curriculum and Instruction
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: acute, right, obtuse, equiangular, scalene, isosceles, equilateral, perpendicular bisector, circumcenter, incenter, median, centroid, altitude,
orthocenter
Non-Essential:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
37
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Interdisciplinary Connections
(Applicable Standards):
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Integration of Technology:
21st Century Themes:
21st Century Skills:
Library:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): Summative Assessment #2
Major Activities (required): None
38
KRSD Office of Curriculum and Instruction
Unit 7 : Quadrilaterals
Recommended Duration: (12 days)
Unit Description: In this chapter, students explore quadrilaterals. They begin by investigating the interior and exterior angles of polygons. Then students learn to
recognize and apply the properties of parallelograms. Students’ knowledge of parallelograms is extended as they explore rectangles, rhombi, and squares and
their special properties. Trapezoids and kites are also explored. Finally, students position quadrilaterals on the coordinate plane for use in coordinate proofs.
Essential Questions:
•
•
•
•
•
•
•
Enduring Understandings:
How does a scallop shell illustrate the angles of polygons?
How are parallelograms used to represent data?
How are parallelograms used in architecture?
How are rectangles used in sports?
How can you ride a bicycle with square wheels?
How are trapezoids used in careers in construction?
How can you use a coordinate plane to prove theorems about
quadrilaterals?
•
•
•
•
•
•
•
•
The sum of the interior angle measures of a polygon.
By dividing a polygon with n sides into (n-2) triangles, you can show
that the sum of the interior angle measures of any polygon is multiple
of 180.
Parallelograms have special properties regarding their sides, angles,
and diagonals.
You can decide whether a quadrilateral is a parallelogram if its sides,
angles, and diagonals have certain properties.
Rhombi, rectangles, and squares have basic properties about their
sides and angles that help identify them. The diagonals of these
parallelograms also have certain properties.
You can determine whether a parallelogram is a rhombus or a
rectangle based on the properties of its diagonals.
The angles, sides, and diagonals of a trapezoid have certain properties.
You can classify figures in the coordinate plane using the formulas for
slope, distance, and midpoint.
Relevant Standards:
Learning Goals:
Content Standards:
Primary(Power):
G-CO.11 - 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.
Students will understand the properties of each
type of quadrilateral. Students will also be able to
prove theorems involving parallelograms and
prove a figure is a rectangle given 4 points on a
coordinate plane.
39
KRSD Office of Curriculum and Instruction
Learning Objectives:
•
•
•
•
•
The students will be able to find the measures
of interior and exterior angles of a polygon.
The students will be able to identify polygons.
The students will be able to recognize and
apply properties of parallelograms.
The students will be able to prove theorems
about parallelograms
The students will be able to recognize and
Relevant Standards:
Learning Goals:
Learning Objectives:
G.GPE.4 - Use coordinates to prove simple
geometric theorems algebraically. For example,
prove or disprove that a figure defined by four
given points in the coordinate plane is a
rectangle; prove or disprove that the point (1,
√3) lies on the circle centered at the origin and
containing the point (0, 2).
•
•
•
•
Mathematical Practice (Supportive):
apply properties of rhombi, rectangles, and
squares.
The students will be able to prove theorems
about rectangles.
The students will be able to recognize and
apply properties of trapezoids and kites.
The students will be able to compare the
different types of quadrilaterals.
The students will be able to prove a figure is a
rectangle given 4 points on a coordinate plane.
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major
Activities/
(required):
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
40
KRSD Office of Curriculum and Instruction
Assignments
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: parallelogram, rectangle, rhombus, square, kite, trapezoid, diagonal, isosceles trapezoid, median
Non-Essential: slope, distance, midpoint
21st Century Themes:
21st Century Skills:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary
(Applicable Standards):
Mathematics:
Connections Integration of Technology:
Technology:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
41
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Interdisciplinary
Connections Integration of Technology:
(Applicable Standards):
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
21st Century Themes:
21st Century Skills:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): None
42
KRSD Office of Curriculum and Instruction
Unit 8 : Similar Figures
Recommended Duration: 3 weeks (14 days)
Unit Description: Students extend their knowledge of ratios and proportions to similar figures. Students will compare scale factors by cross multiplying the
proportions. They also learn to recognize and use proportional parts and relationships to solve problems involving similar triangles and parallel lines.
Essential Questions:
•
•
•
Enduring Understandings:
How do you use proportions to find side lengths in similar polygons?
How do you show two triangles are similar?
How do you identify corresponding parts of similar triangles?
•
•
•
•
•
•
•
•
•
•
•
Relevant Standards:
Content Standards:
Primary(Power):
G.SRT.1 – Verify experimentally the properties
43
KRSD Office of Curriculum and Instruction
A ratio can be written to compare two quantities.
An equation can be written stating that two ratios are equal.
If the equation contains a variable, it can be solved to find the value of
the variable.
Ratios are proportions can be used to decide whether two polygons
are similar and to find unknown side lengths of similar figures.
All lengths in a scale drawing are proportional to their corresponding
actual lengths.
Triangles can be shown to be similar based on the relationship of two
or three pairs of corresponding parts.
Similar triangles can be used to find unknown measurements.
Drawing in the altitude to the hypotenuse of a right triangle forms
three pairs of similar right triangles.
The altitude of the hypotenuse of a right triangle, the segments
formed by the altitude, and the sides of the right triangle has lengths
that are related using geometric means.
When two or more parallel lines intersect other lines, proportional
segments are formed.
The bisector of an angle of a triangle divides the opposite side into two
segments with lengths proportional to the sides of the triangle that
form the angle.
Learning Goals:
Students will understand the definition of
similarity in terms of similarity transformations.
Students will also be able to decide if two figures
Learning Objectives:
•
•
Students will be able to write ratios and solve
proportions.
Students will be able to identify and apply
similar polygons.
Relevant Standards:
Learning Goals:
of dilations given by a center and a scale factor:
c. A dilation takes a line not passing
through the center of the dilation to a
parallel line, and leaves a line passing
through the center unchanged.
d. The dilation of a line segment is longer
or shorter in the ratio given by the scale
factor.
are similar using congruence and similarity criteria
(AA Similarity Postulate) as well as proving the
Midsegment Theorem.
G.SRT.2 - Given two figures, use the definition of
similarity in terms 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.3 - Use the properties of similarity
transformations to establish the AA criterion for
two triangles to be similar.
G.SRT.4 - Prove theorems about triangles.
Theorems include: a line parallel to one side of a
triangle divides the other two proportionally,
and conversely; the Pythagorean Theorem
proved using triangle similarity.
G.SRT.5 - Use congruence and similarity criteria
for triangles to solve problems and to prove
relationships in geometric figures.
G.CO.10 - Prove theorems about triangles.
Theorems include: measures of interior angles of
44
KRSD Office of Curriculum and Instruction
Learning Objectives:
•
•
•
•
•
Students will be able to use the AA Similarity
Postulate and the SAS and SSS Similarity
Theorems.
Students will also be able to use similarity to
find indirect measurements.
Students will be able to find and use
relationships in similar right triangles.
Students will be able to use the Side-Splitter
Theorem and the Triangle-Angle Bisector
Theorem.
Students will be able to prove the Midsegment
Theorem.
Relevant Standards:
Learning Goals:
Learning Objectives:
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.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
45
KRSD Office of Curriculum and Instruction
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: ratio, proportion, extremes, means, cross product, similar figures, extended proportion, scale factor, AA similarity, SAS similarity, SSS similarity,
proportional, altitude, geometric mean, side-splitter, triangle-angle-bisector, midsegment, CASTC, CSSTP
Non-Essential:
21st Century Themes:
21st Century Skills:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary
(Applicable Standards):
Mathematics:
Connections Integration of Technology:
Technology:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
46
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Interdisciplinary
(Applicable Standards):
Library:
Connections Integration of Technology:
21st Century Themes:
21st Century Skills:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): None
47
KRSD Office of Curriculum and Instruction
Unit 9 : Trigonometry
Recommended Duration: 3 weeks (14 days)
Unit Description: Provides students with an introduction to trigonometry. They solve problems using the Pythagorean theorem and its converse. Trigonometric
ratios are defined and then used to solve right triangle problems.
Essential Questions:
•
•
•
•
•
•
Enduring Understandings:
•
How do you find a side length or angle measure of a right triangle?
How do trigonometric ratios relate to similar right triangles?
How are right triangles used to build suspension bridges?
How is triangle tiling used in wallpaper design?
How can surveyors determine angle measures?
How do airline pilots use angles of elevation and depression?
•
•
•
•
If the lengths of any two sides of a right triangle are known, the length
of the third side can be found using the Pythagorean Theorem.
If the lengths of all sides of a triangle are known it can be determined
whether the triangle is acute, right, or obtuse.
Certain right triangles have properties that allow their side lengths to
be determined without using Pythagorean Theorem.
If certain combinations of side lengths and angle measures of a right
triangle are known, ratios can be used to find other side lengths and
angle measures.
The angles of elevation and depression are the acute angles of right
triangles formed by horizontal distance and a vertical height.
Relevant Standards:
Learning Goals:
Content Standards:
Primary(Power):
Students will understand the relationship between
sine and cosine of complementary angles.
Students will be able to use trigonometric ratios to
solve right triangles.
G.SRT.6 – Understand that by similarity, side
ratios in right triangles are properties of the
angles in a triangle, leading to definitions of
trigonometric ratios for acute angles.
Learning Objectives:
•
•
•
G.SRT.7 - Explain and use the relationship
between the sine and cosine of complementary
angles.
•
G.SRT.8 - Use trigonometric ratios and the
•
48
KRSD Office of Curriculum and Instruction
•
Students will be able to apply the Pythagorean
Theorem to classify a triangle as obtuse, right,
or acute.
Students will be able to apply the properties of
45-45-90 triangles.
The students will be able to apply the
properties of 30-60-90 triangles.
Students will be able to find trigonometric
ratios using right triangles.
Students will be able to solve problems using
trigonometric ratios.
Students will be able to solve triangles using
Relevant Standards:
Learning Goals:
Learning Objectives:
Pythagorean Theorem to solve right triangles in
applied problems.
•
Mathematical Practice (Supportive):
trigonometric ratios and inverse ratios.
Students will be able to solve problems
involving angles of depression and elevation.
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
49
KRSD Office of Curriculum and Instruction
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: Pythagorean Theorem, Pythagorean Triple, trigonometry, trigonometric ratio, sine, cosine, tangent, angle of elevation, angle of depression
Non-Essential:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
50
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): None
51
KRSD Office of Curriculum and Instruction
Unit 10 : Area and Surface Area
Recommended Duration: (12 days)
Unit Description: In this chapter, students find areas of parallelograms, rhombi, trapezoids, and triangles. They also begin their exploration of solids. The basic
types of geometric figures are described and their characteristics are discussed. Students find the lateral areas of prisms, cylinders, pyramids, and cones. They
also find the surface areas of these figures. Students identify the parts of a sphere and find the surface areas of sphere.
Essential Questions:
•
•
•
•
•
•
•
•
•
•
Enduring Understandings:
•
Why are drawings of three-dimensional structures valuable to
archaeologists?
How is area related to garden design?
How is the area of a triangle related to beach umbrellas?
How do windsurfers use area?
Why is surface area important to car manufacturers?
How do brick masons know how many bricks to order for a project?
How are cylinders used in extreme sports?
How are pyramids used in architecture?
How is the lateral area of a cone used to cover a tepee?
How do manufacturers of sports equipment use the surface areas of
spheres?
•
•
•
•
•
•
The area of a parallelogram or a triangle can be found when the length
of its base and its height are known.
The area of a trapezoid can be found when the height and the lengths
of its bases are known.
The area of a rhombus or a kite can be found when the lengths of its
diagonals are known.
Ratios can be used to compare the perimeters and areas of similar
figures.
The area of a circle can be found when the circles radius is known.
The area of a three-dimensional figure is equal to the sum of the areas
of each surface of the figure.
The surface area of a sphere can be found when its radius is known.
Relevant Standards:
Learning Goals:
Content Standards:
Primary(Power):
G.MG.1- Use geometric shapes, their measures,
and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a
cylinder).
Students will understand the concepts of surface
area and be able to use geometric shapes, their
measures, and their properties to describe
objects. Students will be able to apply geometric
methods to solve design problems.
G.MG.2 - Apply concepts of density based on
area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot).
52
KRSD Office of Curriculum and Instruction
Learning Objectives:
•
•
•
•
•
Students will be able to find the areas and
perimeters of various polygons.
Students will be able to calculate lateral and
surface areas of Prisms.
Students will be able to calculate the lateral
area and surface area of cones, cylinders, and
spheres.
The students will be able to find the lateral and
surface area of a pyramid.
Students will use surface areas to solve design
problems.
Relevant Standards:
Learning Goals:
Learning Objectives:
G.MG.3 - Apply geometric methods to solve
design problems (e.g., designing an object or
structure to satisfy physical constraints or
minimize cost; working with typographic grid
systems based on ratios).
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major
Activities/
(required):
Assignments
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
53
KRSD Office of Curriculum and Instruction
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: radius, diameter, perimeter, area, circumference, polyhedron, face, edge, vertex, prism, lateral faces, right prism, oblique prism, surface area,
cylinder, cone, sphere, slant height, height, lateral area, pyramid
Non-Essential:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
54
____ Information & Communication
KRSD Office of Curriculum and Instruction
____ Information Literacy
Interdisciplinary Connections
(Applicable Standards):
Library:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): None
55
KRSD Office of Curriculum and Instruction
Unit 11: Volume
Recommended Duration: (10 days)
Unit Description: In this chapter, students explore volume. They find the volume of prisms, cylinders, pyramids, cones, and spheres. Students investigate density
and how it relates to mass and volume.
Essential Questions:
•
•
•
•
•
•
•
•
Enduring Understandings:
•
How do architects use volume?
How can you find the volume of the Earth?
How are similar solids applied to miniature collectables?
What is the difference between surface area and volume?
What is population density?
How would you define density?
What is the difference between weight and mass?
How is the cross section of an object related to its volume?
Relevant Standards:
Content Standards:
Primary(Power):
G.MG.2 - Apply concepts of density based on
area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot).
Secondary(Supportive):
G.GMD.1 - Give an informal argument for the
formulas for the circumference of a circle, area
of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri's
principle, and informal limit arguments.
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KRSD Office of Curriculum and Instruction
•
•
•
•
•
The volume of a prism and a cylinder can be found when its height and
the area of its base are known.
The volume of a composite space figure is the sum of the volumes of
the figures that are combined.
The volume of a pyramid is related to the volume of a prism with the
same base and height.
The volume of a cone is related to the volume of a cylinder with the
same base and height.
The volume of a sphere can be found when its radius is known.
A cross section is the intersection of a three-dimensional figure and a
plane.
Learning Goals:
Students will understand how to use volume
formulas for cylinders, pyramids, cones, and
spheres to solve problems. Students will be able to
apply concepts of density based on area and
volume in real life situations.
Learning Objectives:
•
•
•
•
Students will be able to find the volume of a
prism and the volume of a cylinder.
Students will be able to visualize cross sections
of space figures and determine equivalent
volumes of solids using Cavalieri’s principle.
Students will be able to find the volumes of
pyramids, cones, and spheres.
Students will be able to describe the ratio of
mass to area or volume of real life objects.
Relevant Standards:
Learning Goals:
Learning Objectives:
G.GMD.2 - Give an informal argument using
Cavalieri's principle for the formulas for the
volume of a sphere and other solid figures.
G.GMD.3 - Use volume formulas for cylinders,
pyramids, cones, and spheres to solve problems.
G.GMD.4 - Identify the shapes of twodimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects
generated by rotations of two-dimensional
objects.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
57
KRSD Office of Curriculum and Instruction
Performance Assessments:
Major
Activities/
(required):
Assignments
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: volume, composite figure, cross section, Cavalieri’s principle, density, population density
Non-Essential: ratio, cone, sphere, prism, base, height, cylinder, pyramid, area
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
Technologies Literacy
Social Studies:
58
____ Information & Communication
KRSD Office of Curriculum and Instruction
Interdisciplinary Connections
(Applicable Standards):
Technology:
Integration of Technology:
21st Century Life and Careers:
21st Century Themes:
21st Century Skills:
X Communication & Collaboration
____ Information Literacy
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): None
Major Activities (required): None
59
KRSD Office of Curriculum and Instruction
Unit 12 : Circles
Recommended Duration: (12 days)
Unit Description: This chapter focuses exclusively on circles and their special properties. A circle is a unique geometric shape in which the angles, arcs, and
segments intersecting that circle have special relationships. In this chapter, students identify the parts of a circle and solve problems involving circumference.
They find arc and angle measures and the measures of segments in a circle. In addition, students write the equation of a circle and graph circles in the
coordinate.
Essential Questions:
•
•
•
•
•
•
•
How far does a carousel animal travel in one rotation?
What kinds of angles do the hands on a clock form?
How do the grooves in a Belgian waffle iron model segments in a
circle?
How is a socket like an inscribed polygon?
How are tangents related to track and field events?
How is a rainbow formed by segments of a circle?
How are lengths of intersecting chords related?
Enduring Understandings:
•
•
•
•
•
•
•
•
•
•
•
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KRSD Office of Curriculum and Instruction
The length of part of a circle’s circumference can be found by relating
it to the central angle in the circle.
A radius of a circle and the tangent that intersects the endpoint of the
radius on the circle has a special relationship.
A circle has a special relationship to a triangle whose sides are tangent
to the circle.
Information about congruent parts of a circle (or congruent circles) can
be used to find information about other parts of the circle (or circles).
Angles formed by intersecting lines have special relationship to the
arcs the intersecting lines intercept.
Specifically, arcs intercepted by chords that form inscribed angles are
related to the inscribed angles.
Angles formed by intersecting lines have a special relationship to the
arcs the intersecting lines intercept.
Arcs formed by lines intersecting either within a circle or outside a
circle are related to the angles formed by the lines.
There are special relationships between intersecting chords,
intersecting secants, or a secant and tangent that intersect.
The information in the equation of a circle allows the circle to be
graphed.
The equation of a circle can be written if its center and radius are
known.
Relevant Standards:
Content Standards:
Primary(Power):
N/A
Secondary(Supportive):
G.C.1 - Prove that all circles are similar.
G.C.2 - 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.3 - Construct the inscribed and
circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed
in a circle.
G.C.5 - 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.
Mathematical Practice (Supportive):
MP1 Make sense of problems and persevere in
solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique
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KRSD Office of Curriculum and Instruction
Learning Goals:
Students will identify and describe relationships
among radii, chords, inscribed angles, arcs, central
angles and inscribed angles. Students will be able
to write the equation of a circle.
Learning Objectives:
•
•
•
•
•
•
•
•
•
Students will be able to find the center and
radius of a circle.
Students will be able to use congruent chords,
arcs, and central angles.
Students will be able to use perpendicular
bisectors to chords.
Students will be able to find the measure of an
inscribed angle.
Students will be able to use properties of a
tangent to a circle.
Students will be able to find the measure of an
angle formed by a tangent and a chord.
Students will be able to find the measures of
angles formed by chords, secants, and
tangents.
Students will be able to find the lengths of
segments associated with circles.
Students will be able to write the equation of a
circle.
Relevant Standards:
Learning Goals:
Learning Objectives:
the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated
reasoning.
Formative Assessments
Summative Assessments:
Teacher Observation, Class
Participation, Warm ups, Homework,
Exit Slips, Learning Goals and Scales,
I-clicker responses, Group work
Unit Tests, Quizzes, Graded
Homework, open-ended
questions, benchmark
assessments
Performance Assessments:
Major Activities/ Assignments
(required):
Post Assessment
Possible Assessment Modifications /Accommodations:
Calculators, bold key words, reduced answer choices, increased time on tests/quizzes, answers recorded, reword questions, prior notice of tests
Instructional Strategies (refer to Robert Marzano’s 41 Elements):
Providing clear learning goals, celebrating student success, establish classroom routines, identifying critical information, preview new content, chunk content
into “digestible bites”, reflect on learning, organize students to practice and deepen knowledge, using homework, examine errors in reasoning, practice skills
and strategies, revising knowledge, provide resource and guidance, organizing/engaging students for cognitively complex tasks involving generating hypothesis
and testing
Possible Instructional Modifications /Accommodations/Differentiation:
Copy of teacher’s notes or outline of notes, preferential seating, pair with partner, cueing strategies to keep students focused, additional or online textbook
Unit Vocabulary:
Essential: circle, center, chord, radius, diameter, circumference, pi, central arc, minor arc, major arc, semicircle, inscribed, circumscribed, intercepted, tangent,
point of tangency, secant
62
KRSD Office of Curriculum and Instruction
Non-Essential:
Integration of Technology:
21st Century Themes:
21st Century Skills:
Technology:
____ Global Awareness
____ Creativity & Innovation
Science:
____ Civic Literacy
____ Media Literacy
Visual and Performing Arts:
____ Financial, Economic, Business,
X Critical Thinking and Problem Solving
Health/PE:
& Entrepreneurial Literacy
X Life and Career Skills
World Languages:
____ Health Literacy
Interdisciplinary Connections
(Applicable Standards):
Mathematics:
____ Information & Communication
Technologies Literacy
Social Studies:
X Communication & Collaboration
Technology:
21st Century Life and Careers:
____ Information Literacy
9.1.12.A.1 - Apply critical thinking and
problem-solving strategies during
structured learning experiences.
Library:
Resources:
Texts/Materials:
Geometry Textbook by Glencoe/McGraw Hill copyright 2005, Pearson Geometry Common Core textbook copyright 2012, PMI, PARCC, Model Curriculum,
Protractors, Compasses, Rulers, TI 30 Calculators
Major Assignments (required): Final Benchmark
Major Activities (required): None
63
KRSD Office of Curriculum and Instruction