Download Grade 10 - Practical Geometry

PRACTICAL GEOMETRY
Course #240
Course of Study
Findlay City Schools
2013
TABLE OF CONTENTS
1. Findlay City Schools’ Mission Statement and Beliefs
2. Practical Geometry Course of Study
3. Practical Geometry Pacing Guide
Course Description: The student will describe two and three-dimensional figures; recognize properties of figures; draw, construct and make
models of figures. Much emphasis will be placed upon measurement, including the use of different units of measures and working with various
devices to measure segment lengths and angles. Students will use coordinate and transformational approaches to solving problems. Real life
applications of geometric concepts will be explored. Calculators will be employed to solve problems. This course is necessary to help prepare for
the OGT test and other standardized tests.
PRACTICAL GEOMETRY
Course of Study
Writing Team
Ryan Headley
Ellen Laube
Karen Ouwenga
Carrie Soellner
Text: Geometry Common Core, 2012 edition; Pearson (publisher); ISBN: 978-0-13-318582-9
Mission Statement
The mission of the Findlay City Schools, a community partnership committed to educational
excellence, is to instill in each student the knowledge, skills and virtues necessary to be lifelong
learners who recognize their unique talents and purpose and use them in pursuit of their dreams
and for service to a global society.
This is accomplished through a passion for knowledge, discovery and vision shared by students,
families, staff and community.
Beliefs
Our beliefs form the ethical foundation of the Findlay City Schools.
We believe….
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every person has worth
every individual can learn
family is the most important influence on the development of personal values.
attitude is a choice and always affects performance
motivation and effort are necessary to achieve full potential
honesty and integrity are essential for building trust.
people are responsible for the choices they make.
performance is directly related to expectations.
educated citizens are essential for the survival of the democratic process.
personal fulfillment requires the nurturing of mind, body and spirit.
every individual has a moral and ethical obligation to contribute to the well-being of society.
education is a responsibility shared by students, family, staff and community.
the entire community benefits by investing its time, resources and effort in educational excellence.
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a consistent practice of shared morals and ethics is essential for our community to thrive.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
1.1
Practical Geometry
10th grade
Chapter 1: Tools of Geometry
22 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
3D objects can be represented with a 2D figure using special drawing techniques
 Prepares for G.CO.1
1.2
Geometry is a mathematical system built on accepted facts, basic terms, and definitions
 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.
1.3
Number operations can be used to find and compare the lengths of segments and the measures of angles
 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.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
1.4
Number operations can be used to find and compare the lengths of segments and the measures of angles
 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.
1.5
Special angle pairs can used to identify geometric relationships and to find angle measures
 Prepares for G.CO.1
1.6
Geometric figures can be constructed using straightedge and compass
 G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an
angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
 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.
1.7
Formulas can be used to find the midpoint and length of any segment in the coordinate plane
 Prepares for G.PE.4
 Prepares for G.PE.7
 G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
1.8
Perimeter and area are two different ways of measuring the size of geometric figures
 N.Q.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and
interpret units consistently in formulas…
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Know (G.CO.1)
Find (G.GPE.6)
Make (G.CO.12)
“Unwrapped Concepts”
(Students need to know)
Definitions of
 Angle
 Circle
 Perpendicular line
 Parallel line
 Line segment
Locations of points using segment addition
Formal geometric constructions
Bloom’s
Taxonomy Levels
Understand
Understand
Create
Use (N.Q.1)
Choose (N.Q.1)
Appropriate units for solutions
Appropriate units and scales:
 Formulas
 Graphs
 Data Displays
Appropriate units and scales:
 Formulas
 Graphs
 Data Displays
Interpret (N.Q.1)
Vocabulary
1.1
Net, Isometric drawing, orthographic drawing
1.2
Point, Line, Plane, Collinear Points, Coplanar, Space,
Segment, Ray, Opposites Rays, Postulate, Axiom,
Intersection
1.3
Coordinate, Distance, Congruent Segments, Midpoint,
Segment Bisector
1.4
Angle, Side of Angle, Vertex of an Angle, Measure of an
Angle, Acute Angle, Right Angle, Obtuse Angle, Straight
Angle, Congruent Angles
1.5
Adjacent Angles, Vertical Angles, Complementary Angles,
Supplementary Angles, Linear Pair, Angle Bisector
Understand
Apply
Analyze
Resources
Textbook with Supplementals
1.6
Straightedge, Compass, Construction, Perpendicular Lines,
Perpendicular Bisector
1.8
Perimeter, Area
Essential Questions
1. How can you represent a 3D figure with a 2D
drawing?
2. What are the building blocks of geometry?
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3. How can you describe the attributes of a segment or
angle?
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Understanding/Corresponding Big Ideas
Students will make nets for solid figures.
Isometric drawings and orthographic drawings will be
used to show attributes of figures.
Students will define basic geometric figures.
Undefined terms such as point, line, plane will be shown
with visual representations.
Postulates, which lead to proofs later in the text, will be
presented.
Segments will be measured with and without a
coordinate grid.
Students will use the midpoint and distance formulas.
Protractors will be used to measure angles.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 2: Reasoning and Proof
7 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
2.1 Patterns in some number sequences and some sequences of geometric figures can be used to discover relationships
 Prepares for G.CO.9
 Prepares for G.CO.10
 Prepares for G.CO.11
2.2
Some mathematical relationships can be described using a variety of if-then statements
 Prepares for G.CO.9
 Prepares for G.CO.10
 Prepares for G.CO.11
2.5
Algebraic properties of equality are used in geometry to solve problems and justify reasoning
 Prepares for G.CO.9
 Prepares for G.CO.10
 Prepares for G.CO.11
2.6
Given information, definitions, properties, postulates, and previously proven theorems, can be used as a reason in a
proof
 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.
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Prove (G.CO.9)
“Unwrapped Concepts”
(Students need to know)
Theorems about lines and angles
Vocabulary
2.1
Inductive Reasoning, Conjecture, Counterexample
Bloom’s
Taxonomy Levels
Evaluate
Resources
Textbook with Supplementals
2.2
Conditional, Hypothesis, Conclusion, Negation, Converse,
Inverse, Contrapositive, Equivalent Statements
2.5
Reflexive Property, Symmetric Property, Transitive
Property, Proof, Two-Column Proof
2.6
Theorem, Paragraph Proof
Essential Questions
1. How can you make a conjecture and prove that it is
true?
Understanding/Corresponding Big Ideas
 Students will observe patterns leading to making
conjectures.
 Students will solve equations giving their reasons for
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each step and connect this to simple proofs.
Students will prove geometric relationships using
given information, definitions, properties, postulates,
and theorems.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 3: Parallel and Perpendicular Lines
18 Days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
3.1 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.
 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.
 Prepares for G.CO.9
3.2
The special angle pairs formed by parallel lines and a transversal are either congruent or supplementary.
 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.
3.3
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Certain angle pairs can be used to decide whether two lines are parallel.
Extends G.CO.9
3.4
The relationships of two lines to a third line can be used to decide whether two lines are parallel or perpendicular to
each other.
 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).*
3.5
The sum of the angle measures of a triangle is always the same.
 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.
3.6
Construct parallel and perpendicular lines using a compass and a straightedge.
 G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of
a line segment; and constructing a line parallel to a given line through a point not on the line.
 G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
3.7
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.
 Prepares for G.GPE.5
3.8
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Comparing the slopes of two lines can show whether the lines are parallel or perpendicular.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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).
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Know (G.CO.1)
“Unwrapped Concepts”
(Students need to know)
Definitions of
 Angle
 Circle
 Perpendicular line
 Parallel line
 Line segment
Bloom’s
Taxonomy Levels
Understand
Prove (G.CO.9)
Prove (G.CO.11)
Make (G.CO.12)
Construct (G.CO.13)
Theorems about lines and angles
Theorems about parallelograms
Formal geometric constructions
Construct
 Equilateral triangle
 Square
 Regular hexagon inscribed in a circle
Geometric methods to solve design problems
Apply (G.MG.3)
Vocabulary
3.1
Skew Lines, Parallel Lines, Parallel Planes, Transversal,
Alternate Interior Angles, Same Side Interior Angles,
Corresponding Angles, Alternate Interior Angles
Evaluate
Evaluate
Create
Create
Apply
Resources
Textbook with Resources
3.3
Flow Proof
3.5
Auxiliary Line, Exterior Angle of a Polygon, Remote Interior
Angles
3.7
Slope, Slope Intercept Form, Point Slope Form
Essential Questions
1. How to prove that two lines are parallel and
perpendicular?
2. What is the sum of the measures of the angles of a
triangle?
3. How do you write an equation of a line in the
coordinate plane?
Understanding/Corresponding Big Ideas
 Students will use postulates and theorems to explore
lines in a plane.
 Students will use coordinate geometry to examine
the slopes of parallel and perpendicular lines.
 Students will use the triangle angle sum theorem.
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Students will write equations using slope intercept
form.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 4: Congruent Triangles
16 Days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
4.1 Comparing the two corresponding parts of two figures can show whether the figures are congruent
 Prepares for G.SRT.5
4.2
Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent.
Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
4.3
Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent.
Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
4.4
If two triangles are congruent, then every pair of their corresponding parts is also congruent.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
 G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of
a line segment; and constructing a line parallel to a given line through a point not on the line.
4.5
The angles and the sides of isosceles and equilateral triangles have special relationships.
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.
 G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
4.6
Two triangles can be proven to be congruent without having to show that all corresponding parts are congruent.
Triangles can be proven to be congruent by using SSS, SAS, ASA, AAS or HL.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
4.7
Congruent corresponding parts of one pair of congruent triangles can sometimes be used to prove another pair of
triangles congruent. This often involves overlapping triangles.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Use (G.SRT.5)
Make (G.CO.12)
Prove (G.CO.10)
Construct (G.CO.13)
“Unwrapped Concepts”
(Students need to know)
Congruence and similarity criteria to solve
problems
Formal geometric constructions
Theorems about triangles
Construct
 Equilateral triangle
Bloom’s
Taxonomy Levels
Apply
Create
Evaluate
Create
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Square
Regular hexagon inscribed in a circle
Vocabulary
4.1
Congruent Polygons
Resources
Textbook with Supplementals
4.5
Legs of an Isosceles Triangle, Base of an Isosceles Triangle,
Vertex Angle of an Isosceles Triangle, Base Angles of an
Isosceles Triangle, Corollary
4.6
Hypotenuse, Legs of a Right Triangle
Essential Questions
1. How do you identify corresponding parts of
congruent triangles?
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2. How do you show that two triangles are congruent?
3. How can you tell whether a triangle is isosceles or
equilateral?
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Understanding/Corresponding Big Ideas
Students will visualize the triangles placed on top of
each other.
Students will use tick marks and angle marks to label
corresponding sides and corresponding angles.
Students will use SSS, SAS, ASA, AAS and HL.
Students will use the definitions and look at the
number of congruent sides and angles.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
PRACTICAL Geometry
Grade / Course
10th Grade
Unit of Study
Chapter 5: Relationships within Triangles
Pacing
10 Days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
5.1 The midsegment of a triangle can be used to uncover relationships within a triangle
 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.
 G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of
a line segment; and constructing a line parallel to a given line through a point not on the line.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
5.2
Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. Geometric figures
such as angle bisectors and perpendicular bisectors can be used to cut the measure of an angle or segment in half.
 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.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
5.3
There are special parts of a triangle that are always concurrent. A triangle’s three perpendicular bisectors are always
concurrent, as are a triangle’s three angle bisectors, it’s three medians, and it’s three altitudes. Angle bisectors and
segment bisectors can be used in triangles to determine various angle and segment measures.
 G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
5.4
There are special parts of a triangle that are always concurrent. A triangle’s three perpendicular bisectors are always
concurrent, as are a triangle’s three angle bisectors, it’s three medians, and it’s three altitudes. The length of
medians and altitudes in a triangle can be determined given the measures of other triangle segments.
 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 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.
5.6
The measures of the angles of a triangle are related to the lengths of the opposites sides.
 Extends G.CO.10
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Prove (G.CO.10)
Make (G.CO.12)
Use (G.SRT.5)
Prove (G.SRT.5)
Prove (G.CO.9)
Construct (G.C.3)
Prove (G.C.3)
“Unwrapped Concepts”
(Students need to know)
Theorems about triangles
Formal geometric constructions
Congruence and similarity criteria to solve
problems
Relationships in geometric figures
Theorems about lines and angles
Inscribed and circumscribed circles of a triangle
Properties of a cyclic quadrilateral
Bloom’s
Taxonomy Levels
Evaluate
Create
Apply
Evaluating
Evaluate
Create
Evaluate
Vocabulary
5.1
Midsegment of a Triangle
Resources
Textbook with Supplementals
5.2
Equidistant, Distance from a point to a line
5.3
Concurrent, Point of Concurrency, Circumcenter,
Circumscribe, Incenter of a Triangle, Inscribed in
5.4
Median of a Triangle, Centroid of a Triangle, Altitude of a
Triangle, Orthocenter of a Triangle
Essential Questions
1. How do you use coordinate geometry to find
relationships within triangles?
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2. How do you solve problems that involve
measurements of triangles?
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Understanding/Corresponding Big Ideas
Students will use the midpoint formula to find
midsegments of triangles.
Students will use the distance formula to examine
relationships in triangles.
Students will examine inequalities in one triangle.
Students will examine inequalities in two triangles.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
PRACTICAL Geometry
Grade / Course
10th Grade
Unit of Study
Chapter 6: Polygons and Quadrilaterals
Pacing
19 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
6.1 The sum of the angle measures of a polygon depends on the number of sides the polygon has.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.2
Parallelograms have special properties regarding their sides, angles, and diagonals.
 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.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.3
If a quadrilateral’s sides, angles, and diagonals have certain properties, it can be shown that the quadrilateral is a
parallelogram.
 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.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.4
The special parallelograms, rhombus, rectangle and square, have basic properties of their sides, angles, and
diagonals that help to identify them.
 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.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.5
The special parallelograms, rhombus, rectangle and square, have basic properties of their sides, angles, and
diagonals that help to identify them.
 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.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.6
The angles, sides, and diagonals of a trapezoid have certain properties.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
6.7
The formulas for slope, distance and midpoint can be used to classify and to prove geometric relationships for
figures in the coordinate plane. Using variables to name the coordinates of a figure allow relationships to be shown
to be true for a general case.
 G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.★
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Prove (G.SRT.5)
Prove (G.CO.11)
Use (G.GPE.7)
“Unwrapped Concepts”
Bloom’s
(Students need to know)
Taxonomy Levels
Relationships in geometric figures
Evaluating
Theorems about parallelograms
Evaluate
Coordinate geometry to compute perimeters and
Apply
areas of polygons
Vocabulary
Resources
6.1
Textbook with Supplementals
Equilateral Polygon, Equiangular Polygon, Regular Polygon
6.2
Parallelogram, Opposite Sides, Opposite Angles,
Consecutive Angles
6.4
Rhombus, Rectangle, Square
6.6
Trapezoid, Base, Leg, Base Angle, Isosceles Trapezoid,
Midsegment of a Trapezoid, Kite
Essential Questions
1. How can you find the sum of the measures of
polygon angles?
2. How can you classify quadrilaterals?



3. How can you use coordinate geometry to prove
general relationships?


Understanding/Corresponding Big Ideas
The formula for angle measures of a polygon will be
derived using diagonals.
Students will use the properties of parallel and
perpendicular lines and diagonals to classify
quadrilaterals.
Students will use coordinate geometry to classify
special parallelograms.
Students will examine slope and segment length in
the coordinate plane.
Students will use the distance formula in the
coordinate plane.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 7 – Similarity
17 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
7.1 – An equation can be written stating that two ratios are equal, and if the equation contains a variable, it can be solved
to find the value of the variable.
 Prepares for G.SRT.5
7.2 – Ratios and proportions can be used to decide whether two polygons are similar and to find unknown side lengths of
similar figures.
7.2 – Ratios and proportions can be used to prove whether two polygons are similar and to find unknown side lengths.
Triangles can be shown to be similar based on the relationship of two or three pairs of corresponding parts.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
7.3 – Ratios and proportions can be used to prove whether two polygons are similar and to find unknown side lengths.
Triangles can be shown to be similar based on the relationship of two or three pairs of corresponding parts.
7.3 – Two triangles can be shown to be similar. Drawing in the altititude to the hypotenuse of a right triangle forms three
pairs of similar right triangles.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
 G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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).
7.4 – Drawing in the altitude to the hypotenuse of a right triangle forms the three pairs of similar right triangles.
7.4 – It can be proven that the three pairs of right triangles formed by drawing in the altitude to the hypotenuse are
similar.
7.4 – Two triangles can be shown to be similar. Drawing in the altitude to the hypotenuse of a right triangle forms three
pairs of similar right triangles.
 G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in

geometric figures.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses 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).
7.5 – When two or more parallel lines intersect other lines, proportional segments are formed.
 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.
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Use (G.SRT.5)
Prove (G.SRT.5)
Prove (G.GPE.5)
Use (G.GPE.5)
Prove (G.SRT.4)
“Unwrapped Concepts”
(Students need to know)
Congruence and similarity criteria to solve
problems
Relationships in geometric figures
Relationship of the slopes of parallel and
perpendicular lines
Slopes to solve geometric problems
Theorems about triangles
Bloom’s
Taxonomy Levels
Apply
Evaluating
Evaluate
Apply
Evaluate
Vocabulary
7.1
Ratio, extended ratio, proportion, extremes, means, cross
products properties
Resources
Textbook with Supplementals
7.2
Similar figures, similar polygons, extended proportions,
scale factor, scale drawing, scale
7.3
Indirect measurement
7.4
Geometric mean
Essential Questions
How do you use proportions to find side lengths in
similar polygons?
2. How do you show two triangles are similar?
1.




3. How do you identify corresponding parts of similar
triangles?

Understanding/Corresponding Big Ideas
Student will form proportions based on known
lengths of corresponding sides.
Students will use the Angle-Angle Similarity
Postulate.
Students will use the Side-Angle-Side Similarity
Theorem.
Students will use the Side-Side-Side Similarity
Theorem.
A key to understanding corresponding parts of
similar triangle is to show the triangle in like
orientations.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 8 – Right Triangles and Trigonometry
14
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
8.1 – If the lengths of any two sides of a right triangle are known, the length of the third side can be found by using the
Pythagorean Theorem.
 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.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
8.2 – Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean
Theorem.
 G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
8.3 – If certain combinations of side lengths and angle measure of a right triangle are known, ratios can be used to find
other side lengths and angle measures.
8.3 – Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side
lengths and angle measures are known.
 G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
 G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
 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).*
8.4 – The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a
vertical height.
 G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Prove (G.SRT.4)
Use (G.SRT.8)
Understand (G.SRT.6)
Explain (G.SRT.7)
Use (G.SRT.7)
Use (G.MG.1)
“Unwrapped Concepts”
(Students need to know)
Theorems about triangles
Trig ratios and Pythagorean theorem to solve
application problems
That right triangle similarity leads to trig ratios
Relationship between the sine and cosine of
complementary angles
Relationship between the sine and cosine of
complementary angles
Geometric shapes, measures and properties to
describe objects
Vocabulary
8.1
Pythagorean triple
8.3
Trigonometric ratios, sine, cosine, tangent
8.4
Angle of elevation, angle of depression
Bloom’s
Taxonomy Levels
Evaluate
Apply
Understand
Understand
Apply
Apply
Resources
Textbook with Supplementals
Essential Questions
1. How do you find a side length or angle measure in a
right triangle?
2. How do trig ratios relate to similar right triangles?






Understanding/Corresponding Big Ideas
Students will use the Pythagorean Theorem.
Students will use concepts of 30-60-90 and 45-45-90
triangles.
Students will use trig ratios to form proportions.
Student will examine the sine ratio.
Student will examine the cosine ratio.
Student will examine the tangent ratio.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 9– Transformations
5
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
9.1 – The distance between any two points and the angles in a geometric figure stay the same when (1) its location and
orientation changes, (2) it is flipped across a line, or (3) it is turned about a point.
 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.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.
 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.
9.2. – The distance between any two points and the angles in a geometric figure stay the same when (1) its location and
orientation changes, (2) it is flipped across a line, or (3) it is turned about a point.
 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.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.
 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.
9.3 – The distance between any two points and the angles in a geometric figure stay the same when (1) its location and
orientation changes, (2) it is flipped across a line, or (3) it is turned about a point.
 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.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.
 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.
9.4 – One of two congruent figures in a plane can be mapped onto the other by a single reflection, translation, oration, or
glide reflection.
 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.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.
 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.
9.6 – A scale factor can be used to make a larger or smaller copy of a figure that is similar to the original figure.
9.6 – You can use coordinate geometry to prove triangle congruence and verify properties of similarity.
 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).
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Represent (G.CO.2)
Describe (G.CO.2)
Compare (G.CO.2)
Develop (G.CO.4)
Bloom’s
Taxonomy Levels
Understand
Understand
Analyze
Creating
Draw (G.CO.5)
Identify (G.CO.5)
Use (G.CO.6)
“Unwrapped Concepts”
(Students need to know)
Transformations in the plane
Transformations as functions
Congruent and non-congruent transformations
Definitions of rotations, reflections and translations
in terms of
 Angles
 Circles
 Perpendicular lines
 Parallel lines
 Line segments
A transformed geometric figure
A composition of transformations
Geometric descriptions to transform figures
Predict (G.CO.6)
The effect of a given rigid motion
Analyze
Create
Understand
Apply
Vocabulary
9.1
Transformation, pre-image, image, rigid motion,
translation, composition of transformations
Resources
Textbook with Supplementals
9.2
Reflection, Line of Reflection
9.3
Rotation, center of rotation, angle of rotation
9.4
Glide reflection, isometry
9.6
Dilation, center of dilation, scale factor of dilation,
enlargement, reduction
Essential Questions
1. How can you change a figure’s position without
changing it size and shape? How can you change a
figure’s size without changing its shape?
2. How can you represent a transformation in the
coordinate plane?




Understanding/Corresponding Big Ideas
Students will explore translations, reflections, and
rotations.
Students will explore dilations.
Transformations will be conducted both on and off a
coordinate plane.
Students will determine the new coordinates of a
polygon after any given transformation.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 10-Area
12 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
10.1 – The area of a parallelogram or a triangle can be found when the length of its base and its height are known.
 G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.★
 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).*
10.2 – 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.
 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).*
10.3 – The area of a regular polygon is a function of the distance from the center to a side and the perimeter.
 G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
 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).*
10.6 – The length of part of a circle’s circumference can be found by relating it to an angle in the circle.
 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.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.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.
10.7 – The area of parts of a circle formed by radii and arcs can be found when the circle’s radius is known.
 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 Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Use (G.GPE.7)
Use (G.MG.1)
Construct (G.CO.13)
Know (G.CO.1)
“Unwrapped Concepts”
(Students need to know)
Coordinate geometry to compute perimeters and
areas of polygons
Geometric shapes, measures and properties to
describe objects
Construct
 Equilateral triangle
 Square
 Regular hexagon inscribed in a circle
Definitions of
 Angle
 Circle
 Perpendicular line
 Parallel line
Bloom’s
Taxonomy Levels
Apply
Apply
Create
Understand
 Line segment
Circles are similar
Relationships among inscribed angles, radii and
chords
Relationships among inscribed angles, radii and
chords
Arc length
Formula for the area of a sector
Prove (G.C.1)
Identify (G.C.2)
Describe (G.C.2)
Derive (G.C.5)
Derive (G.C.5)
Vocabulary
10.1
Base of a parallelogram, altitude of a parallelogram, height
of a parallelogram, base of a triangle, height of a triangle
Evaluate
Understand
Understand
Create
Create
Resources
Textbook with Supplementals
10.2
Height of a trapezoid
10.3
Radius of a regular polygon, apothem
10.6
Circle, center, diameter, radius, congruent circles, central
angle, semicircle, minor arc, major arc, adjacent arcs,
circumference, pi, concentric circles, arc length
10.7
Sector of a circle, segment of a circle
Essential Questions
1. How do you find the area of a polygon or find the
circumference and area of a circle?
Understanding/Corresponding Big Ideas
 Students will use formulas to find areas of
parallelograms, triangles, trapezoids, rhombuses,
and kites.
 Students will explore area concepts related to regular


2. How do perimeters and areas of similar polygons
compare?


polygons.
Students will use trigonometry to find areas.
Students will find circumferences and areas of
circles.
Students will examine ratios among similar figures.
Given a figure and its area, students will be able to
find the area of a figure similar to the original figure.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 11 Surface Area and Volume
10 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
11.1 – A three-dimensional figure can be analyzed by describing the relationships among its vertices, edges, and faces.
 G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects.
11.2 – The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the figure.
 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).*
11.3 - The surface area of a three-dimensional figure is equal to the sum of the areas of each surface of the figure.
 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).*
11.4 – The volume of a prism and a cylinder can be found when its height and the area of its base are known.
 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.
 G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
 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).*
11.5 – The volume of a pyramid is related to the volume of a prism with the same base and height.
 G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
 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).*
11.6 – The surface area and the volume of a sphere can be found when its radius is known.


G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
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).*
11.7 – Ratios can be used to compare the areas and volumes of similar solids.
 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).*
 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).*
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Identify (G.GMD.4)
Identify (G.GMD.4)
Use (G.MG.1)
Give (G.GMD.1)
“Unwrapped Concepts”
(Students need to know)
The shapes of cross sections of 3D figures
3D objects generated by rotations of 2D objects
Geometric shapes, measures and properties to
describe objects
An informal argument for the formulas of
 Circumference
 Area of a circle
 Volume of a cylinder
 Volume of a pyramid
Bloom’s
Taxonomy Levels
Understand
Understand
Apply
Evaluate
Use (G.GMD.3)
Use (G.MG.1)
Apply (G.MG.2)
 Volume of a cone
Volume formulas to solve problems for
 Cylinders
 Pyramids
 Cones
 spheres
Geometric shapes, measures and properties to
describe objects
Concepts of density based on area and volume in
modeling situations
Vocabulary
11.1
Polyhedron, face, edge, vertex, cross section
11.2
Prism(base, lateral face, altitude, height, lateral area,
surface area), right prism, oblique prism, cylinder (base,
altitude, height, lateral area, surface area), right cylinder,
oblique cylinder
11.3
Pyramid (base, lateral face, vertex, altitude, height, slant
height, lateral area, surface area), regular pyramid, cone
(base, altitude, vertex, height, slant height, lateral area,
surface area), right cone
11.4
Volume, composite space figure
11.5
Volumes of Pyramids and Cones
Apply
Apply
Apply
Resources
Textbook with Supplementals
11.6
Sphere, center of a sphere, radius of a sphere, diameter of a
sphere, circumference of a sphere, great circle, hemisphere
11.7
Similar solids
Essential Questions
1. How can you determine the intersection of a solid
and a plane?
2. How do you find the surface area and volume of a
solid?
Understanding/Corresponding Big Ideas
 Students will examine cross sections.



3. How do the surface areas and volumes of similar
solids compare?



Students will use formulas to find surface areas and
volumes of prisms and cylinders.
Students will use formulas to find surface areas and
volumes of pyramids and cones.
Students will use formulas to find surface areas and
volumes of spheres.
Students will examine ratios among similar solids.
Given a figure and its surface area, students will be
able to find the surface area of a solid similar to the
original solid.
Given a figure and its volume, students will be able to
find the volume of a solid similar to the original
solid.
FINDLAY CITY SCHOOLS
Curriculum Design – Grades 6 – 12
Subject(s)
Grade / Course
Unit of Study
Pacing
Practical Geometry
10th Grade
Chapter 12 Circles
14 days
ESSENTIAL UNDERSTANDINGS AND SUPPORTING STANDARDS
12.1 – A radius of a circle and the tangent that intersects the endpoint of the radius on the circle have a special
relationship.
 Prepares for G.C.2
12.2 – Information about congruent parts of a circle (or congruent circles) can be used to find information about other
parts of the circle (or circles).
 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.
12.3 – Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. This
includes (1) arcs formed by chords that inscribe angles, (2) angles and arcs formed by lines intersecting either
within a circle of outside a circle, and (3) intersecting chords, intersecting secants, or a secant that intersects a
tangent.
 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.
12.4 - Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. This
includes (1) arcs formed by chords that inscribe angles, (2) angles and arcs formed by lines intersecting either within
a circle of outside a circle, and (3) intersecting chords, intersecting secants, or a secant that intersects a tangent.
 Extends G.C.2
12.5 - 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.
 G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the
square to find the center and radius of a circle given by an equation.
Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
“Unwrapped Skills”
(Students need to be able to do)
Identify (G.C.2)
Describe (G.C.2)
Construct (G.C.3)
Prove (G.C.3)
Derive (G.GPE.1)
Vocabulary
12.1
Tangent to a circle, point of tangency
12.2
Chord
“Unwrapped Concepts”
(Students need to know)
Relationships among inscribed angles, radii and
chords
Relationships among inscribed angles, radii and
chords
Inscribed and circumscribed circles of a triangle
Properties of a cyclic quadrilateral
The equation of a circle
Bloom’s
Taxonomy Levels
Understand
Understand
Create
Evaluate
Create
Resources
Textbook with Supplementals
12.3
Inscribed angle, intersected arc
12.4
Secant
12.5
Standard form of an equation of a circle
Essential Questions
1. How can you prove relationships between angles and
arcs in a circle?
2. When lines intersect a circle, or within a circle, how
do you find the measures of resulting angles, arcs,
and segments?





3. How do you find the equation of a circle in the
coordinate plane?

Understanding/Corresponding Big Ideas
Students will examine angles formed by lines that
intersect inside and outside a circle.
Students will relate arcs and angles.
Students will use properties of tangent lines.
Students will use the relationships among chords,
arcs, and central angles.
Students will solve problems with angles formed by
secants and tangents.
The center and radius of a circle in a coordinate
plane can be used to find the equation of a circle.
Chapter 1
Section
CP Geometry
Review how to solve
equations
Pacing
1
Students take SLO test
and do more algebra
review
Also plan on getting
books this day
1
Students take SLO test and
do more algebra review
1
1
Also plan on getting books
this day
1
Teach 1.2 and 1.3
together. Spend one day
on each, then a 3rd day
combining them
See above
1½
Teach 1.2 and 1.3 together.
Spend one day on each,
then a 3rd day combining
them
See above
1½
1
1.4
Not a full period…begin
1.4
After quiz, do measuring
and classifying activity
1.5
On day 3, pull 1.4 and
1.5 together
3
On day 3, pull 1.4 and 1.5
together
3
Only half period quiz
Then practice with
compass and concept
byte
Day 1: #1 and #3
Day 2: #2 and #4
1
Only half period quiz
Then practice with
compass and concept byte
1
2
Day 1: #1 and #3
Day 2: #2 and #4
2
2
Also do page 57 Review
2
First Day of
School
Second Day
of School
1.1
1.2
1.3
QUIZ
QUIZ
1.6
Also do page 57 Review
1½
1½
Practical Geometry
Review how to solve
equations
Not a full period…begin
1.4
After quiz, do measuring
and classifying activity
Pacing
1
1½
1
1½
1.7
Concept
1.8
Performance
Task
Review
Test
Total Days
No performance task on
this test
Concept
1
2
1
1
1
2
1
20
No performance task on
this test
1
22
Chapter 2
Section
2.1
2.2
2.3
2.4
CP Geometry
Teach 2.1 and 2.2
together…only do #1
and #2
See above
½
Practical Geometry
Teach 2.1 and 2.2
together…only do #1 and
#2
See above
SKIP
0
SKIP
0
SKIP
0
SKIP
0
2.5
2.6
Performance
Task
Review
Test
Total Days
SKIP!!!
Pacing
½
Pacing
½
½
2
2
2
2
0
SKIP!!!
0
1
1
1
1
7
7
Chapter 3
Section
3.1
3.2
CP Geometry
Combine 3.1 and 3.2
Day 1: naming and
looking at relationships
Day 2: Proofs
Day 3: Start angle project
Day 4: Workday on
angle project
Only ½ day quiz, then
start 3.4
Day 1: Teach 3.5
Day 2: Teach 5.6
Day 3: Practice problems
finding missing angles
NO point slope form
3
1
Do #2 and #3
1
2
½
1½
Day 1: Teach 3.5
Day 2: Teach 5.6
Day 3: Practice problems
finding missing angles
3
1
NO point slope form
2
3.8
Performance
Task
Only ½ day quiz, then
start 3.4
1
3.6
3.7
½
Pacing
2
2
1½
3.4
3.5
2
Practical Geometry
Combine 3.1 and 3.2
Day 1: naming and looking
at relationships
Day 2: Proofs
Day 3: Start angle project
Day 4: Workday on angle
project **easier angle
project than CP**
1
3.3
QUIZ
Pacing
2
1
2
Do #2 and #3
1
1
Review
Test
Total Days
Include a performance
task question
1
16
2
Include a performance task
question
1
18
Chapter 4
Section
CP Geometry
4.1
Concept
Byte
4.2
4.3
Page 225
1
Combine 4.2 and 4.3
Day 1: Identify with pics
Day 2: Proving
See above
1
Combine 4.2 and 4.3
Day 1: Identify with pics
Day 2: Proving
See above
1
Total Days
1
2
Send home graded HW
problem after day 2
2
Day 1: all overlapping
Day 1: all proofs
Day 2: Find angles and
performance tasks
1
2
Send home graded HW
problem after day 2
1
Performance See below
Task
Do performance task #1
Review
with review
Test
Pacing
1
1
4.6
4.7
Practical Geometry
Page 225
4.4
4.5
Pacing
1
2
1
2
Day 1: all overlapping
2
0
See below
0
2
Do performance task #1
with review
2
2
Day 1: all proofs
Day 2: Find angles and
performance tasks
2
16
16
Chapter 5
Section
CP Geometry
5.1
1
5.2
5.7
Test
Total Days
2
2
Day 3: Review 5.3-5.4
3
SKIP
0
SKIP
0
TEACH THIS AFTER
SECTION 3.5
0
TEACH THIS AFTER
SECTION 3.5
0
SKIP
0
SKIP
0
1
Only #1
Use more from resources
1
Only #1
Performance Use more from resources
Task
Review
Day 1: Teach 5.2
Day 2: Review 5.1-5.2
Pacing
1
2
5.4
5.6
Practical Geometry
2
5.3
5.5
Pacing
1
1
1
1
1
9
10
Chapter 6
Section
6.1
6.2
CP Geometry
Also talk about the
Quadrilateral Family
Tree
Day 1: #1 and #2
Day 2: #3 and #4
6.3
Teach 6.4 and 6.5
together
REVIEW
Day 1: #1 and #2
Day 2: #3 and #4
2
2
SKIP PROOFS
1
½
1
Teach 6.4 and 6.5 together
1
1
Day 1: Trapezoids
Day 2: Kites
2
Day 1: Trapezoids
Day 2: Kites
2
Review 6.1 – 6.6
1
Review 6.1 – 6.6
2
QUIZ
6.7
6.8
Pacing
1
2
½
6.5
6.6
Practical Geometry
Also talk about the
Quadrilateral Family Tree
1
QUIZ
6.4
Pacing
1
Day 1: Instruction
Day 2: Group work
1
1
2
3
2
SKIP
0
6.9
Performance
Task
Day 1: Instruction
Day 2: Group Work
2
SKIP
0
All 3
1
Do only #1 and #3
1
2
Review
Test
Total Days
Day 1: 6.1 – 6.6
Day 2: 6.7 – 6.9
2
22
2
Only 6.1 – 6.7
1
19
Chapter 7
Section
PROJECT
CP Geometry
Start the chapter with the
Similarity Cartoon/Comic
project (see Ellen, Karen
or Carrie for help)
7.1
7.2
7.3
QUIZ
7.4
7.5
Performance
Task
Review
Test
Total Days
Pacing
1
Practical Geometry
Start the chapter with the
Similarity Cartoon/Comic
project (see Ellen, Karen or
Carrie for help)
Pacing
1
1
1
1
1
Days 1-2: Inside
Day 3: Go outside!
3
Days 1-3: Inside
Day 4: Go outside!
4
Only half the period
Review how to simplify
radicals the other half
1
Day 1: Quiz
Day 2: Review how to
simplify radicals
Skip #4
2
2
2
2
2
1
1
1
2
1
1
14
17
Chapter 8
Section
CP Geometry
Pacing
Practical Geometry
Pacing
8.1
1
1
8.2
2
3
Quiz
½
½
8.3
8.4
½ Trig ratios
1 day solving angles and
sides
½ day solving angles and
sides
½ day application
1 day of application
1 day outside (p 515)
8.5
8.6
8.5 – 8.6
2
2.5
1
Teach only one formula
Pull it all together
2
1
½ Trig ratios
2 days of solving for sides
and angles
2 days of application
1 day outside (p 515)
2.5
3
SKIP
SKIP
SKIP
Performance
Task
1
1
Review
2
2
Test
1
1
Total Days
16
14
Chapter 9
Section
9.1
9.2
CP Geometry
Combine 9.1 and 9.2
Pacing
1
See above
Practical Geometry
Combine 9.1 and 9.2
Pacing
1
See above
9.3
1
1
9.4
1
1
9.5
9.6
9.7
Performance
Task
Review
SKIP
Talk about compositions
SKIP
1
Talk about compositions
SKIP
SKIP
SKIP
SKIP
1
½
Test
½
Total Days
5
Use a project instead of a
test
1
5
Chapter 10
Section
10.1
10.2
CP Geometry
Combine 10.1 and 10.2
1
See above
10.3
10.4
Pacing
Practical Geometry
Combine 10.1 and 10.2
Pacing
1
See above
2
SKIP
2
SKIP
SKIP
10.5
1
Quiz
1
1
10.6
2
2
10.7
2
2
10.8
Performance
Task
Review
SKIP
SKIP
Only problems 1 and 2
Only problems 1 and 2
1
1
1
2
Test
1
1
Total Days
12
12
Chapter 11
Section
11.1
CP Geometry
Include rotations of 2D
objects
11.2
Combine 11.2 and 11.3
*give formulas
11.3
11.4
11.5
Pacing
1
Practical Geometry
Include rotations of 2D
objects
1
Combine 11.2 and 11.3
*give formulas
See above
See above
Combine 11.4 and 11.5
Combine 11.4 and 11.5
1
See above
Pacing
1
2
1
See above
11.6
1
1
11.7
2
2
Performance
Task
1
1
Review
1
1
Test
1
1
Total Days
9
10
Chapter 12
Section
CP Geometry
Pacing
Practical Geometry
Pacing
12.1
1
1
12.2
2
2
12.3
2
2
Quiz
1
1
1 day – angles
1 day – lengths
2
1
Pull it all together
1
2
1
1
12.4
12.1 – 12.4
12.5
12.6
SKIP
SKIP
Performance
Task
1
1
Review
1
1
Test
1
1
Total Days
13
14