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Geometry Scope and Sequence
General Information
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Suggested Pacing is 1 lesson per day.
Each lesson has an exit ticket that may be used as a formative assessment.
As of now, Engageny only has Module 1 completed, the rest of the modules will be added once they are complete. The modules listed are
suggested topics and lessons. If the remaining modules are unavailable when needed, please refer to the Topics/Standards below and
use materials from 2013-2014.
Suggestions
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Teachers may create additional assessments as they feel necessary.
Modules (student materials) may be printed and bound for students to use as a workbook.
Common Core belief is to provide students with answer keys to practice correctly.
The tests online have answer keys that are available to the public. It is suggested to use them for either reviews or correctives.
Suggested Materials
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Compass and straightedge
Geometer’s Sketchpad or Geogebra Software
Patty paper
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Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Summary of Year (from engageny)
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.
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Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Quarter 1
Module 1: Congruence, Proof, and Constructions
Module 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 strategically applied a rigid motion to informally show that two triangles are congruent. In this module,
students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They build upon this familiar foundation of triangle
congruence to develop formal proof techniques. Students make conjectures and construct viable arguments to prove theorems—using a variety of formats—
and solve problems about triangles, quadrilaterals, and other polygons. They construct figures by manipulating appropriate geometric tools (compass, ruler,
protractor, etc.) and justify why their written instructions produce the desired figure.
Topic A: Basic Constructions (G-CO.1)
Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role
transformations play in defining congruence.
Students begin this module with Topic A, Constructions. Major constructions include an equilateral triangle, an angle bisector, and a perpendicular bisector.
Students synthesize their knowledge of geometric terms with the use of new tools and simultaneously practice precise use of language and efficient
communication when they write the steps that accompany each construction (G-CO.1).
Lesson
Description
Mathematical
Practices
MP.5
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Students learn to construct an equilateral triangle.
Students communicate mathematic ideas effectively and efficiently.
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Students apply the equilateral triangle construction to more challenging problems.
Students communicate mathematical concepts clearly and concisely.
MP.5
3: Copy and Bisect an Angle
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
MP.5
MP.6
4: Construct a Perpendicular
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Students learn how to bisect an angle as well as how to copy an angle.
Note: These more advanced constructions require much more consideration in the communication of
the student’s steps.
Students learn to construct a perpendicular bisector and about the relationship between symmetry
with respect to a line and a perpendicular bisector.
Students become familiar with vocabulary regarding two points of concurrencies and understand why
the points are concurrent.
MP.5
1: Construct an Equilateral
Triangle
2: Construct an Equilateral
Triangle II
Bisector
5: Points of Concurrencies

MP.5
MP.6
3
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic B: Unknown Angles (G-CO.9)
Constructions segue into Topic B, Unknown Angles, which consists of unknown angle problems and proofs. These exercises consolidate students’ prior body of
geometric facts and prime students’ reasoning abilities as they begin to justify each step for a solution to a problem. Students began the proof writing process
in Grade 8 when they developed informal arguments to establish select geometric facts (8.G.5).
Lesson
Description
Mathematical
Practices
MP.6
MP.7
6:
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Students review formerly learned geometry facts and practice citing the geometric justifications in
anticipation of unknown angle proofs.
7: Solve for Unknown
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Students review formerly learned geometry facts and practice citing the geometric justifications in
anticipation of unknown angle proofs.
MP.7
8: Solve for Unknown
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MP.7
Angles—Angles in a Triangle
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Students review formerly learned geometry facts and practice citing the geometric justifications
regarding angles in a triangle in anticipation of unknown angle proofs.
Students recognize the relationship between angle measures and side lengths
Students write unknown angle proofs, which does not require any new geometric facts. Rather, writing
proofs requires students to string together facts they already know to reveal more information.
Note: Deduction is a mental disciplining by which we get more from our thinking.
Students write unknown angle proofs involving auxiliary lines.
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Students write unknown angle proofs involving known facts.
MP.7
Solve for Unknown
Angles—Angles and Lines at
a Point
Angles—Transversals
9: Unknown Angle Proofs—
Writing Proofs
10: Unknown Angle Proofs—
MP.7
MP.8
MP.7
Proofs with Constructions
11:
Unknown Angle Proofs—
Proofs of Known Facts
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Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic C: Transformations/Rigid Motions (G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7, G-CO.12)
Topic C, Transformations, builds on students’ intuitive understanding developed in Grade 8. With the help of manipulatives, students observed how reflections,
translations, and rotations behave individually and in sequence (8.G.1, 8.G.2). In Grade 10, this experience is formalized by clear definitions (G.CO.4) and more
in-depth exploration (G.CO.3, G.CO.5). The concrete establishment of rigid motions also allows proofs of facts formerly accepted to be true (G.CO.9). Similarly,
students’ Grade 8 concept of congruence transitions from a hands-on understanding (8.G.2) to a precise, formally notated understanding of congruence
(G.CO.6). With a solid understanding of how transformations form the basis of congruence, students next examine triangle congruence criteria. Part of this
examination includes the use of rigid motions to prove how triangle congruence criteria such as SAS actually work (G.CO.7, G.CO.8).
Lesson
Description
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Students discover the gaps in specificity regarding their understanding of transformations.
Students identify the parameters they need to complete any rigid motion.
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Students manipulate rotations by each parameter—center of rotation, angle of rotation, and a point
under the rotation.
14: Reflections
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Students learn the precise definition of a reflection.
Students construct the line of reflection of a figure and its reflected image. Students construct the
image of a figure when provided the line of reflection.
15: Rotations, Reflections, and
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Students learn the relationship between a reflection and a rotation.
Students examine rotational symmetry within an individual figure.
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Students learn the precise definition of a translation and perform a translation by construction.
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Students understand that any point on a line of reflection is equidistant from any pair of pre-image and
image points in a reflection.
18:
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19:
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Students learn to construct a line parallel to a given line through a point not on that line using a
rotation by 180˚. They learn how to prove the alternate interior angles theorem using the parallel
postulate and the construction.
Students begin developing the capacity to speak and write articulately using the concept of
congruence. This involves being able to repeat the definition of congruence and use it in an accurate
and effective way.
12:
Transformations—The
Next Level
13:
Rotations
Symmetry
16: Translations
17: Characterize Points on a
Perpendicular Bisector
Looking More Carefully at
Parallel Lines
Construct and Apply a
Sequence of Rigid Motions
Mathematical
Practices
MP.7
MP.8
MP.5
MP.6
MP.7
MP.5
MP.6
MP.5
MP.6
MP.7
MP.7
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Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
20:
Applications of
Congruence in Terms of Rigid
Motions
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Students will understand that a congruence between figures gives rise to a correspondence between
parts such that corresponding parts are congruent, and they will be able to state the correspondence
that arises from a given congruence.
Students will recognize that correspondences may be set up even in cases where no congruence is
present, they will know how to describe and notate all the possible correspondences between two
triangles or two quadrilaterals and they will know how to state a correspondence between two
polygons.
Students practice applying a sequence of rigid motions from one figure onto another figure in order to
demonstrate that the figures are congruent.
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Correspondence and
Transformations
Mid-Module Assessment Topics A through C (assessment 1 day, return 1 day, remediation or further applications 2 days)
21:
Topic D: Congruence (G-CO.7, G-CO.8)
In Topic D, Proving Properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties—those that have been
accepted as true and those that are new—of parallelograms and triangles (G.CO.10, G.CO.11). The module closes with a return to constructions in Topic E
(G.CO.13), followed by a review that of the module that highlights how geometric assumptions underpin the facts established thereafter (Topic F).
Lesson
22: Congruence Criteria for
Mathematical
Practices
Description
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Students learn why any two triangles that satisfy the SAS congruence criterion must be congruent.
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Students examine two different proof techniques via a familiar theorem.
Students complete proofs involving properties of an isosceles triangle.
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Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be
congruent.
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Students learn why any two triangles that satisfy the SAA or HL congruence criteria must be congruent.
Students learn why any two triangles that meet the AAA or SSA criteria are not necessarily congruent.
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Students complete proofs requiring a synthesis of the skills learned in the last four lessons.
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Students complete proofs requiring a synthesis of the skills learned in the last four lessons.
Triangles—SAS
23: Base Angles of Isosceles
Triangles
24: Congruence Criteria for
Triangles—ASA and SSS
25: Congruence Criteria for
Triangles—SAA and HL
26: Triangle Congruency
Proofs—Part 1
27: Triangle Congruency
Proofs—Part 2
6
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic E: Proving Properties of Geometric Figures (G-CO.9, G-CO.10, G-CO.11)
In Topic E, students extend their work on rigid motions and proof to establish properties of triangles and parallelograms. In Lesson 28, students apply their
recent experience with triangle congruence to prove problems involving parallelograms. In Lessons 29 and 30, students examine special lines in triangles,
namely mid-segments and medians. Students prove why a mid-segment is parallel to and half the length of the side of the triangle it is opposite from. In Lesson
30, students prove why the medians are concurrent.
Lesson
Mathematical
Practices
Description
28:
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Students complete proofs that incorporate properties of parallelograms.
29: Special Lines in Triangles
30: Special Lines in Triangles
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Students examine the relationships created by special lines in triangles, namely mid-segments.
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Students examine the relationships created by special lines in triangles, namely medians.
Properties of
Parallelograms
End-of-Module Assessment Topics A through G (F and G are in Honors only. Assessment 1 day, return 1 day, remediation or further applications 3 days)
Focus Standards for Mathematical Practice for Module 1
MP.3 Construct viable arguments and critique the reasoning of others. Students articulate steps needed to construct geometric figures, using relevant
vocabulary. Students develop and justify conclusions about unknown angles and defend their arguments with geometric reasons.
MP.4 Model with mathematics. Students apply geometric constructions and knowledge of rigid motions to solve problems arising with issues of design or
location of facilities.
MP.5 Use appropriate tools strategically. Students consider and select from a variety of tools in constructing geometric diagrams, including (but not limited to)
technological tools.
MP.6 Attend to precision. Students precisely define the various rigid motions. Students demonstrate polygon congruence, parallel status, and perpendicular
status via formal and informal proofs. In addition, students will clearly and precisely articulate steps in proofs and constructions throughout the module.
7
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Module 2: Similarity, Proof, and Trigonometry
Module 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, make sense of and persevere in solving similarity problems, and apply similarity to right triangles to prove the Pythagorean Theorem.
Students attend to precision in showing that trigonometric ratios are well defined, and apply trigonometric ratios to find missing measures of general (not
necessarily right) triangles. Students model and make sense out of indirect measurement problems and geometry problems that involve ratios or rates.
Topic A: Similarity
Standards: G-SRT.1, 2, 3
Lesson
1: Ratios and Proportions
2: Similar Polygons
3: Similar Triangles
4: Dilations
Mathematical
Practices
Description
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Students learn to construct an equilateral triangle.
Students communicate mathematic ideas effectively and efficiently.
Students apply the equilateral triangle construction to more challenging problems.
Students communicate mathematical concepts clearly and concisely.
Establish AA
AA, SAS, SSS
Students learn to construct a perpendicular bisector and about the relationship between symmetry
with respect to a line and a perpendicular bisector.
Topic B: Proofs
Standards: G-SRT.1, 2, 3
Lesson
Mathematical
Practices
Description
5:
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Triangle Proportionality
o A line parallel to one side of a triangle divides the other two proportionally
6: Prove Pythagorean
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Prove Pythagorean Theorem through similarity
o Not application of Pythagorean theorem, but just deriving the formula from similar triangles
Triangle Proportionality
Theorem
Theorem
8
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic C: Trigonometry
Standards: G-SRT.6, 7, 8
Lesson
7:
8:
Pythagorean Theorem
Pythagorean Theorem
Converse
9: Trig Ratios
10: Solve for a missing side
11: Solve for a missing angle
12: Special Right Triangles
Mathematical
Practices
Description
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Solve problems using the Pythagorean Theorem
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Apply the converse of the Pythagorean Theorem to classify triangles
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Set up trig ratios given side lengths
Complementary angle relationship with trig ratios
Use trig ratios to solve for a missing side given an angle and a side length
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Use trig ratios to solve for a missing angle given two side lengths
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Identify special right triangles in relation to trig.
o 30-60-90
o 45-45-90
Topic D: Modeling
Standards: G-MG.1, 2, 3
Lesson
13: Angles of Elevation and
Mathematical
Practices
Description
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Apply trigonometric ratios to solve real-world problems.
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Apply similarity and trigonometric ratios to solve real-world problems.
Depression
14: Further Applications of
Similarity and Trig
9
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Module 3: Extending to Three Dimensions
Module 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. They reason abstractly and quantitatively to model problems using volume formulas.
Topic A: Area and Surface Area
Lesson
1: Review of Area Formulas
2: Area of Regular Polygons
3: Area of Regular Polygons
Continued
4: Surface Area of Prisms
Mathematical
Practices
Description
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Triangles, Parallelograms, Trapezoids, Rhombi, Kites, Circles
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Given a side length and an apothem
HONORS: find apothem or side
Given a side length and an apothem
HONORS: find apothem or side
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Make sure to include regular polygon bases
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Make sure to include regular polygon bases
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Find the SA of a Sphere
and Cylinders
5: Surface Area of Pyramids
and Cones
6: Surface Area of Spheres
10
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic B: Volume
Standards: G-GMD.1, 3
Lesson
7:
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Find the volume of prisms and cylinders
8: Volume of Pyramids and
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Find the volume of pyramids and cones
Volume of Prisms and
Cylinders
Mathematical
Practices
Description
Cones
 Find the volume of spheres
9: Volume of Spheres
Topic C: Relationships and Modeling
Standards: G-GMD.4; G-MG.1
Lesson
Mathematical
Practices
Description
10:
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Students will solve problems involving relationships between 3-D figures
11:
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Students will model situations involving 3-D figures
Relationships between
3-D figures
Modeling with 3-D
figures
11
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Module 4: Coordinate Geometry
Module 4: Building on their work with the Pythagorean theorem in 8th grade to find distances, students analyze geometric relationships in the context of a
rectangular coordinate system, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, relating back to work
done in the first module. Students attend to precision as they connect the geometric and algebraic definitions of parabola. They solve design problems by
representing figures in the coordinate plane, and in doing so, they leverage their knowledge from synthetic geometry by combining it with the solving power of
algebra inherent in analytic geometry.
Topic A: Coordinate Formulas
Standards: G-GPE.5, 6
Lesson
Mathematical
Practices
Description
1: Midpoint Formula
2: Distance Formula
3: Distance Ratio
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Given two points, find the midpoint
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Given two points, find the distance
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4: Slope Formula
5: Comparing Slopes
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Find the point on a directed line segment between two given points that partitions the segment in a
given ratio (example, 2/3)
Given two points, find the slope
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Given two equations or graphs determine the relationship between the lines
o Parallel
o Perpendicular
o Coincident
o Intersecting
Topic B: Coordinate Proofs
Standards: G-GPE.4
Lesson
6:
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Prove parallelism using slope
Use distance to prove congruence
7: Prove 4 Points Make a
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Continued from above.
Prove 4 points make a
specific quadrilateral
Mathematical
Practices
Description
Specific Quadrilateral
Continued
12
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic C: Areas and Perimeters in Coordinate Plane
Standards: G-GPE.7
Lesson
8:
9:
Area in Coordinate Plane
Perimeter in Coordinate
Plane
10:
Solve Problems involving
the Coordinate Plane
Mathematical
Practices
Description
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Given 3-4 points, calculate the area of the polygon on a coordinate plane
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Given 3-4 points, calculate the perimeter of a polygon
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Students will solve problems that can be placed a coordinate plane.
13
Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Module 5: Circles
Module 5: In this module students prove and apply basic theorems about circles, such as: a tangent line is perpendicular to a radius theorem, the 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 explain the correspondence between
the definition of a circle and the equation of a circle written in terms of the distance formula, its radius, and 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.
Topic A: Circle Basics
Standards: G-C.1, 2, 3
Lesson
1: Definitions
2: Measure of Angles and
Mathematical
Practices
Description
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Center, Radius, Diameter
Chord, Secant, Tangent
Central angle, Inscribed angle, Arc
Find measure of angles and arcs given a circle with many radii and values
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Find measure of arcs and chords given angle or arc measures
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Calculate the value of inscribed angles
An inscribed angle with endpoints on a diameter equal 90 degrees
Identify and solve problems given tangents
Solve problems given a tangent and a radius (right angle)
Find measure of arcs or angles given tangents, secants and angle measures
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Solve problems using:
o Perpendicular chords
o Secant lengths
o Chord lengths
o Tangent lengths
Arcs
3: Arcs and Chords
4: Inscribed Angles
5: Tangents
6: Secants, Tangents, and
Angle Measures
7: Segments
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Dysart USD – engageny
2014 ~ 2015
Geometry Scope and Sequence
Topic B: Arc Length and Area of Sector
Standards: G-C.5
Lesson
Mathematical
Practices
Description
 Find the length of a circular arc
8: Arc Length
 Find the area of a sector
9: Area of a Sector
Topic C: Coordinates of Circles & Proof
Standards: G-GPE.1, 4
Lesson
10:
Derive the Equation of
a Circle
11:
Equations of Circles
Mathematical
Practices
Description

Use the Pythagorean Theorem to derive the equation of a circle
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Complete the Square to write the equation
Write the equation of a circle given:
o Center and radius
o Center and point
o Radius and point
Topic D: Modeling
Standards: G-MG.1
Lesson
13: Modeling involving Circle
Mathematical
Practices
Description

Students will solve problems involving circle parts

Students will solve problems involving area and sector area
Parts
14: Modeling involving Area
and Sector Area
15
Dysart USD – engageny
2014 ~ 2015