Download Westerville City School District COURSE OF STUDY Math Lab for Geometry

Westerville City School District
COURSE OF STUDY
Math Lab for Geometry
MA 102
Recommended Grade Level:
10, 11
Course Length:
Semester
Credits:
.25/semester mathematics elective credit
Course Weighting:
1.0
Course Fee:
None
Course Description
Math Lab is designed to help students be successful in their Geometry course. Math lab is a .25
elective credit per semester. In order to receive credit, students must earn a grade of “satisfactory.”
Math Lab is designed to give additional opportunities to learn and apply algebraic and geometric
concepts. The class will be activity based and students will need to fully participate. Some homework
help and review for Geometry assessments will also be incorporated into the course.
Course Rationale
Students may enter Geometry having not been successful in previous math courses or not having the
prerequisite knowledge and skills necessary to be successful in Geometry. The goal of Math Lab is
to support each student to be successful in the Geometry classroom by providing the opportunity to
acquire the background understandings necessary as well as to front load the Geometry content.
Math Lab will provide opportunities for students to fill gaps and improve on their foundational
mathematical knowledge and skills. Some students may be ready to exit this course during or at the
end of the year based on the grades in their Geometry class.
Math Lab does not have a scope and sequence in the traditional sense. As each student will have an
individual path through the school year, the units below are meant as general guidelines; the work in
the classroom must necessarily be responsive to the gaps and needs of the particular students in
each individual section.
Math Lab
Scope and Sequence
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Topics of Study
Evaluating Expressions and the Order of Operations
Shapes and Transformations
Angle Relationships and Measurement
Solving and Writing Linear Equations
Similarity, Right Triangles, and Trigonometry
Solving Linear Systems of Equations
Probability
Congruence and Proof
Review for Semester Exam
Factoring & Multiplying Polynomials
Quadrilaterals and Coordinate Geometry
Polygons and Circles
Applications of Probability
Writing Exponential Equations
Three Dimensions
Writing Equations of Circles
Laws of Exponents
Review for Semester Exam
Estimated Time (in weeks)
1 week
3 weeks
3 weeks
2 weeks
4 weeks
1 week
2 weeks
2 weeks
1 week
1 week
4 weeks
5 weeks
1 week
1 week
2 weeks
1 week
1 week
1 week
Primary Material Recommendation
Text:
● CPM Texbooks - Course 3, Algebra 1, and Geometry
Other Resources:
● ALEKS for diagnostic and gap filling; assessment tool
● braingenie - online individualized practice; assessment tool
● kahoot - assessment tool
● quizizz - assessment tool
● desmos - online graphing utility
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Math Lab
Topics of Study
Topic #1: Transformations and Congruence
Content Standards G-CO.2 Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in
the plane as inputs and give other points as outputs. Compare transformations
that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon,
describe the rotations and reflections that carry it onto itself.
G-CO.4 Develop definitions of rotations, reflections, and translations in terms
of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw
the transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
G-CO.6 Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if they are
congruent.
G-CO.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-GMD.4 Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects generated by
rotations of two-dimensional objects.
Essential
1. Which transformations preserve the integrity of the shape?
Questions
2. What is symmetry?
3. What characteristics can a shape have?
Enduring
Rigid transformations preserve angle measure and distance relationships and
Understandings
are at the foundation of the concept of congruence.
Key Concepts/
reflection, rotation, translation, rigid transformation, symmetry, right angle,
Vocabulary
circular angle, acute, obtuse, straight angle, protractor, regular, line of
reflection, prime notation, perpendicular, isosceles, line segment, midpoint,
polygon, triangle, quadrilaterals, pentagons, hexagon, octagon, decagon,
parallel, perpendicular, scalene, venn diagram, image, equilateral
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Math Lab
Content
Elaborations
Build on student experience with rigid transformations from earlier grades.
Students will also need fluency in these checkpoints:
Course #1
Checkpoint #6: Locating Points on a Coordinate Graph
Course #3
Checkpoint #8: Transformations
Algebra
Checkpoint #1 & #4: Solving Linear Equations
Checkpoint #5B: Writing the Equation of a Line
Learning Targets
Assessments
Instructional
Strategies and
Materials
Considerations for
Intervention and
Acceleration
I can:
● use my spatial visualization skills to investigate reflection, rotation, and
translation.
● use prime notation for corresponding parts of transformations.
● show that objects and their images are equidistant from the line of
reflection, and that the line segment connecting a point with its
reflected image is perpendicular to the line of reflection.
● recognize that the slopes of perpendicular lines are opposite
reciprocals.
● use symmetry to discover relationships within basic geometric shapes
such as rhombus, square, parallelogram, isosceles triangle, right
triangle, kite, and dart.
● classify shapes by their attributes using Venn diagrams.
Performance on Geometry assessments in their regular Geometry course and
the online benchmark and progress monitoring tools determined for the
course.
● CPM Geometry textbook
● CPM Checkpoints from previous courses
● CPM Parent Guides
● ALEKS dynamic computer software for tracking skill mastery
● Study team strategies
● Teacher created additional practice
This is an intervention class. The teacher will modify, as needed, on an
individual basis.
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Math Lab
Topic #2: Proving Theorems
Content Standards 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.CO.10 Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180º; 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.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve
right triangles in applied problems.
Essential
Questions
Enduring
Understandings
Key Concepts/
Vocabulary
Content
Elaborations
Learning Targets
Assessments
1. What are the conditions necessary for angle relationships to hold?
2. What are the preconditions to use the Pythagorean Theorem?
Students will understand the necessary connections between parallel lines
and the angle relationships.
Students will be able to use the Pythagorean theorem to solve problems.
theorem, linear pair, vertical angles, alternate interior angles, same-side
interior angles, corresponding angles, perpendicular bisector,
supplementary angles, complimentary angles, equidistant,, midpoint, midsegment, angle bisector, altitude, isosceles triangle,bisector, congruence
properties, Pythagorean Theorem
Fluency in the following checkpoints from Course 1, 2, and 3 needed to be
successful in this course:
Course 3
Checkpoint #5 Solving Equations
Algebra 1
Checkpoint #10B Factoring Polynomials
I can:
● use the angle relationships to solve problems.
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for the
course.
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Math Lab
Instructional
Strategies and
Materials
Considerations for
Intervention and
Acceleration
● Students will be given checkpoint materials from College Preparatory
Mathematics Course 1,2, and 3
● Students will use ALEKS dynamic computer software for tracking
skill mastery.
● CPM checkpoint materials
● CPM parent guide
This is an intervention class. The teacher will modify, as needed, on an
individual basis.
Topic #3: Similarity And Trigonometry
Content Standards
G.SRT.1 Verify experimentally the properties of dilations given by a
center and a scale factor.
1. A dilation takes a line not passing through the center of the dilation
to a parallel line, and leaves a line passing through the center
unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by
the scale factor.
G.SRT.2 Given two figures, use the definition of similarity in terms of
similarity transformations to decide if they are similar; explain using
similarity transformation the meaning of similarity for triangles a s the
equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
G.SRT.3 Use the properties of similarity transformations to establish the
AA criterion for two triangles to be similar.
G.SRT.4 Prove theorems about triangles. Theorems include: a line
parallel to one side of a triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve
problems and to prove relationships in geometric figures.
G.SRT.6 Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
G.SRT.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).
G.MG.2 Apply concepts of density based on area and volume in
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Math Lab
modeling situations (e.g., persons per square mile, BTUs per cubic foot).
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).
Essential Questions
Enduring
Understandings
Key Concepts/
Vocabulary
Content Elaborations
Learning Targets
Assessments
1. What are the conditions for similarity?
2. What are trigonometric ratios? Where and when can they be used?
3. How can trigonometric ratios be used to solve problems involving
right triangles?
Students will understand that Trigonometry is a special case of similarity.
Students will understand the importance of Pythagorean Theorem and its
applicability to many real world contexts.
transformation, dilation, scale factor, similarity, AA criterion, ratio,
corresponding angles, corresponding sides, proportionality, SSS, SAS,
vertex, Pythagorean Theorem, congruence, hypotenuse, leg,
Pythagorean triples, Trigonometric ratios, sine, cosine, tangent, opposite
side, adjacent side, right triangle
Fluency in the following checkpoints from Course 1, 2, and 3 needed to
be successful in this course:
Course 1
Checkpoint #8A Rewriting and Evaluating Variable Expressions
Course 2
Checkpoint #6 Writing and Evaluating Variable Expressions
Checkpoint #8 Solving Multi-Step Equations
Course 3
Checkpoint #2 Unit Rates and Proportions
Checkpoint #5 Solving Equations
Checkpoint #7 Solving Equations with Fractions
Algebra 1
Checkpoint #6A Rewriting Equations with More than One Variable
I can:
● use trigonometric ratios to solve real life problems.
● use trigonometric ratios to solve problems involving right triangles.
● use similarity to define the relationships between sides and angles
of right triangles.
● model real life situations using Trigonometry.
● use the Law of Cosines and Sines to solve problems.
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for
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Math Lab
Instructional
Strategies and
Materials
Considerations for
Intervention and
Acceleration
the course.
● ALEKS dynamic computer software for tracking skill mastery
● CPM Parent Guides
● Checkpoints from previous courses
● Study Team Strategies
● Teacher created additional practice
Please see attached documents addressing the challenges and response
to exception students of all levels.
Topic #4: Circles
Content Standards
G.C.1 Prove that all circles are similar.
G.C.2 Identify and describe relationships among inscribed angles, radii,
and chords. Include the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where the radius
intersects the circle.
G.C.3 Construct the inscribed and circumscribed circles of a triangle, and
prove properties of angles for a quadrilateral inscribed in a circle.
G.C.5 Derive using similarity the fact that the length of the arc intercepted
by an angle is proportional to the radius, and define the radian measure of
the angle as the constant of proportionality; derive the formula for the area
of a sector.
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
G.GPE.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that
the point (1, ) lies on the circle centered at the origin and containing the
point (0, 2).
G.MG.1 Use geometric shapes, their measures, and their properties to
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Math Lab
describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder.
Essential Questions
Enduring
Understandings
Key Concepts/
Vocabulary
Content Elaborations
Learning Targets
Assessments
Instructional
Strategies and
Materials
Considerations for
Intervention and
1. Are all circles similar?
2. What is a sector and segment of a circle and how does one find it’s
area?
3. What is a radian?
4. What is the difference between arc measure and arc length?
Students can use the properties of circles to solve problems.
diameter, radii, chords, arcs, arc measure , arc length, inscribed angle,
inscribed shape, circumscribed shape, sector, circumference, radian,
degree, segment of circle, central angle
Fluency in the following checkpoints from Course 1, 2, and 3 needed to
be successful in this course:
Course 1
Checkpoint #8A Rewriting and Evaluating Variable Expressions
Course 2
Checkpoint # 8 Solving Multi-Step Equations
Course 3
Checkpoint #4 Area and Perimeter of Circles and Composite Figures
Algebra 1
Checkpoint #11 The Quadratic Web
Completing the Square
I can:
● decide if a point lies outside, inside, or on a given circle.
● find an equation of the line tangent to the circle at that point given
the equation of a circle and a point on it.
● complete the square in circle problems
● derive the equation of a circle in a variety of ways
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for
the course.
● ALEKS dynamic computer software for tracking skill mastery
● CPM Parent Guides
● Checkpoints from previous courses
● Study Team Strategies
● Teacher created additional practice
Please see attached documents addressing the challenges and response
to exception students of all levels.
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Math Lab
Acceleration
Topic #5: Connecting Algebra and Geometry Through Coordinates
Content Standards
G.GPE.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that
the point (1, ) lies on the circle centered at the origin and containing the
point (0, 2).
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.)
G.GPE.6 Find the point on a directed line segment between two given
points that partitions the segment in a given ratio.
G.GPE.7 Use coordinates to compute perimeters of polygons and areas
of triangles and rectangles, e.g., using the distance formula
Essential Questions
1. How does Algebra support Geometry?
2. Can the tools of Algebra the student in their investigation of
Geometry?
Students will come to understand that Algebraic Geometry and Synthetic
Geometry are mutually supportive. Algebra and Geometry are working
towards the same goals,addressing the regularities found in our world,
from different prospectives.
Enduring
Understandings
Key Concepts/
Vocabulary
Content Elaborations
slope, distance formula, vertex, midpoint, equations of parallel,
perpendicular, or intersecting lines, slope-intercept form, point-slope
form, quadrilaterals: parallelogram, rectangle, rhombus, square, tangent
Fluency in the following checkpoints from Course 1, 2, and 3 needed to
be successful in this course:
Course 1
Checkpoint #6
Course 2
Checkpoints #1, 6, & 7A
Course 3
Checkpoints #4, 5, & 6
Algebra 1
Checkpoint #5B & 6B
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Math Lab
Learning Targets
Assessments
Instructional
Strategies and
Materials
Considerations for
Intervention and
Acceleration
I can:
● draw conclusions about figures in the coordinate plane using linear
relationships.
● use the Pythagorean theorem, slope, and the properties of
coordinate geometry to model real world situations.
● use special right triangles in a myriad of ways.
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for
the course.
● ALEKS dynamic computer software for tracking skill mastery
● CPM Parent Guides
● Checkpoints from previous courses
● Study Team Strategies
● Teacher created additional practice
Please see attached documents addressing the challenges and response
to exception students of all levels.
Topic #6: Extending to Three Dimensions
Content Standards
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.GMD.4 Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects generated by
rotations of two-dimensional objects.
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Math Lab
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).
Essential Questions
Enduring
Understandings
Key Concepts/
Vocabulary
Content Elaborations
Learning Targets
Assessments
Instructional
Strategies and
Materials
Considerations for
1. What situations in real life can be modeled with geometric
concepts?
Many real world problems can be solved using area, surface area, and
volume to model the problem situation.
Many everyday objects can be formed by rotating geometric shapes.
circle, circumference, perimeter, diameter, area, polygon, radius, base,
prism, cylinder, pyramid, cone, sphere, volume, height, volume
Fluency in the following checkpoints from Course 1, 2, and 3 needed to
be successful in this course:
Course 2
Checkpoint #1 Area and Perimeter of Polygons
Checkpoint #6 Writing and Evaluating Algebraic Expressions
Checkpoint #7A Simplifying Expressions
Course 3
Checkpoint #4 Area and Perimeter of Circles and Composite Figures
Checkpoint #5 Solving Equations
Algebra 1
Checkpoint #4 Solving Linear Equations with Fractional Coefficients
Checkpoint #5B Writing the Equation of a line
Checkpoint #7A Solving Problems by Writing Equations
I can:
● use my prior knowledge develop formulas and equations.
● solve many real world problems using area, surface area, and
volume.
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for
the course.
● ALEKS dynamic computer software for tracking skill mastery
● CPM Parent Guides
● Checkpoints from previous courses
● Study Team Strategies
● Teacher created additional practice
Please see attached documents addressing the challenges and response
to exception students of all levels.
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Math Lab
Intervention and
Acceleration
Topic #7: Probability
Content Standards
Essential Questions
S.CP.1 Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events (“or”, “and”, “not”).
S.CP.2 Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are
independent.
S.CP.3 Understand the conditional probability of A given B as P(A and
B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A, and
the conditional probability of B given A is the same as the probability of B.
S.CP.4 Construct and interpret two-way frequency tables of data when
two categories are associated with each object being classified. Use the
two-way table as a sample space to decide if events are independent and
to approximate conditional probabilities. For example, collect data from a
random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student
is in tenth grade. Do the same for other subjects and compare the results.
S.CP.5 Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For
example, compare the chance of having lung cancer if your are a smoker
with the chance of being a smoker if you have lung cancer.
S.CP.6 Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms of the
model.
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B),
and interpret the answer in terms of the model.
S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability
model,
P(A and B) = P(A) P(B|A) = P(B)P(A|B), and interpret the answer in terms
of the model.
S.CP.9 (+) Use permutations and combinations to compute probabilities
of compound events and solve problems.
1. How can probability be used to predict outcomes?
2. What are independent events?
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Math Lab
Enduring
Understandings
Key Concepts/
Vocabulary
Content Elaborations
Learning Targets
3. What methods can be used to organize data?
4. What real life situations are influenced by probabilities?
5. What are the rules of probability ? How and when can they be
applied?
6. How does the order of events affect whether or not it is considered
a different outcome?
7. How can we determine whether a decision is fair?
8. How can probablility be utilized to analyze the impact of a possible
decsion?
Probability is a real life tool for predicting outcomes.
Organizing data into tables is helpful for analysis of independence and
estimating probabilities.
Interpreting probability can be useful in real life situations
Probability can be used to make fair decisions in real life situations.
Probability can be used analyze and assist with decision making..
expected probability, fair decision, probability, events, outcomes, subsets,
sample space, unions, intersections, complements, independent,
dependent, conditional probability, frequency table, random, data
compound events, mutually exclusive, addition rule, multiplication rule,
permutation, combination, venn diagram, tree diagram
Fluency in the following checkpoints from Course 1, 2, and 3 needed to
be successful in this course:
Course 1
Checkpoint #7A Multiplication of Fractions and Decimals
Checkpoint #8B Division of Fractions and Decimals
Checkpoint #9A Displays of Data
Course 2
Checkpoint #3 Multiplying Fractions and Decimals
Checkpoint #7B Displays of Data
Checkpoint #9 Unit Rates and Proportions
Course 3
Checkpoint #3 Unit Rates and Proportions
Checkpoint #9 Satterplots and Association
Algebra 1
Checkpoint #7A Solving Problems by Writing Equations
Checkpoint #8 Interpreting Associations
I can:
● I know that probability can be represented as a fraction or decimal
on the interval
.
● decide if two events are independent or dependent.
● organize data into tables is helpful for analysis of independence
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Math Lab
Assessments
Instructional
Strategies and
Materials
Considerations for
Intervention and
Acceleration
and estimate probabilities.
● interpret probability in real life situations
● explain the rules than govern probability relationships.
● find expected value and use it as a tool to determine a situation is
fair for all participants.
Performance on Geometry assessments in their regular Geometry course
and the online benchmark and progress monitoring tools determined for
the course.
● ALEKS dynamic computer software for tracking skill mastery
● CPM Parent Guides
● Checkpoints from previous courses
● Study Team Strategies
● Teacher created additional practice
Please see attached documents addressing the challenges and response
to exception students of all levels.
Acknowledgements
It is through the hard work and dedication of high school Mathematics team that the Westerville City Schools’
High School Geometry Math Lab Course of Study is presented to the Board of Education. Sincere
appreciation is extended to the following individuals for their assistance and expertise.
Central HS
North HS
South Hs
District
Jennifer Horn
Jessica Martin
Kasandra Sliney
Michael Huler
Brittany Barnhart
Richard Gary
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