GRADE 9 MATH SYLLABUS: GEOMETRY
Course Description
The Grade 9 Geometry course has been developed taking into consideration the five strands of
Mathematics defined by NCTM. The course develops student understanding and skills in Geometry,
including numbers, operations, algebra, measurement, data analysis, probability and problem solving.
The Geometry course will focus on segments, angles, patterns, similarity, ratio, 3Dvisualization and
2D proof. In addition students learn different ways to represent math, to communicate their math
understanding and to make connections to real world situations and other subject areas. Students
are assessed in a variety of ways continuously throughout the program of study to ensure they are
understanding content and are able to use and apply the knowledge and skills they are developing.
Curricular Content:
APPENDIX A contains the NCTM mathematical standards. APPENDIX B contains the performance
indicators for each standard for High School, APPENDIX C contains the Mathematics Focal Points for
Grade 9. Students use the Holt McDougal – Geometry textbook.
Quarter 1
Unit 0: Numeracy
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
Theory behind number
Operations with Integers, Fractions, and Decimals
Unit 1: Foundations for Geometry
 Collinear and coplanar points; basic terms and notation; points, lines, planes, and space
(1.1)
 Find length and mid-points of segments (1.2)
 Angle definitions, notation, and properties; classify angles as acute, right, or obtuse;
estimate and construct angles (1.3)
 Construct angle bisectors (1.3)
 Linear pairs, vertically opposite, complementary & supplementary angles (1.4)
 Distance in a coordinate plane using the distance formula and pythagoras’ theorem (1.6)
 Midpoints in the coordinate plane (1.6)
Unit 2: Parallel and Perpendicular lines
 Identify parallel, perpendicular and skew lines (3.1)
 Identify angles formed by 2 lines and a transversal; corresponding, alternate interior,
alternate exterior and same-side interior (3.1)
 Identify the angles formed by parallel lines and a transversal (3.2)
 Use the angles formed by a transversal to prove two lines are parallel (3.3)
 Construct the perpendicular bisector of a line segment (3.4)
 Find the slope of a line (3.5)
 Prove lines are parallel or perpendicular using slope (3.5)
 Graph lines and write equation in slope-intercept form and point-slope form (3.6)
Unit 3: Triangle Congruence (Part 1)
 Name parts of triangles (4.1)
 Classify triangles as; scalene, isosceles, equilateral / acute, obtuse, right (4.1)
 Find the measures or interior and exterior angles in triangles (4.2)

Apply interior and exterior angle theorem (4.2)
Quarter 2
Unit 3: Triangle Congruence (Part 2)
 Use properties of congruent triangles (4.3)
 Apply SSS and SAS to construct triangle and solve problems. Prove triangles are congruent
using SSS, SAS (4.4)
 Apply ASA , AAS & HL to construct triangle and solve problems. Prove triangles are
congruent using ASA, AAS & HL (4.5)
 Prove theorems about isosceles and equilateral triangles, apply properties of isosceles and
equilateral triangles (4.8)
 Angle measures in isosceles triangles (4.8)
Unit 4: Properties and Attributes of Triangles
 Solve problems using perpendicular and angle bisectors of a triangle (5.1)
 Construct the circumcenter of a triangle, know and use its properties (5.2)
 Construct the incenter of a triangle, know and use its properties (5.2)
 Apply properties of medians and altitudes in triangles, know how to find the centroid and
orthocenter of a triangle (5.3)
 Prove and use properties of triangle mid-segments (5.4)
 Apply inequalities in a triangle (5.5)
Unit 5: Pythagoras’ Theorem and Right angled triangles
 Know and use Pythagoras’ Theorem to find unknowns sides (5.7)
 Use Pythagorean inequalities to classify triangles and know Pythagorean triples (5.7)
 Apply the 450, 450, 900 special triangle (5.8)
 Apply the 300, 600, 900 special triangle (5.8)
 Apply similarity in right triangles to solve problems (8.1)
 Find the sine, cosine and tangent of an acute angle (8.2)
 Use the trigonometric ratios to find side lengths in right triangles and to solve real world
problems (8.2)
 Use the trigonometric ratios to find angles in right triangles and to solve real world
problems (8.3)
Unit 6: Polygons and Quadrilaterals
 Identify polygons by number of sides and angles (6.1)
 Find and use the measures of interior and exterior angles of polygons (6.1)
 Prove and apply properties of parallelograms, use properties to solve problems (6.2)
 Prove that a give quadrilateral is a parallelogram (6.3)
 Prove and apply properties of rectangles, squares and rhombuses (6.4)
 Use the properties of rectangles, squares and rhombuses to solve problems (6.4)
Semester Review for End of Semester Exam
Quarter 3
Unit 7: Similarity

Ratio and Proportion, write and simplify ratios. Use proportions to solve problems. (7.1)
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Identify similar polygons, apply properties of similar problems to solve problems. (7.2)
Prove certain triangles are similar by using AAA, SSS and SAS, use these properties to solve
problems. (7.3)
Use properties of similar triangles to find segment lengths. (7.4)
Use ratios to make indirect measurements. (7.4)
Use scale drawings to solve problems. (7.5)
Apply similarity properties in the co-ordinate plane. (7.6)
Unit 8: Right Triangles and Trigonometry
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Use the geometric mean to find segment lengths in right triangles. (8.1)
Apply similarity in right triangles to solve problems (8.1)
Find the sine, cosine and tangent of an acute angle (8.2)
Us e the trigonometric ratios to find side lengths in right triangles and to solve real world problems
(8.2)
Us e the trigonometric ratios to find angles in right triangles and to solve real world problems (8.3)
Solve problems involving angles of elevation and angles of depression. (8.4)
Use the Law of Sines and the Law of Cosines to solve problems. (8.5)
Find the magnitude and direction of a vector. (8.6)
Unit 9: Extending Perimeter, Circumference and Area
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Develop and apply the formulas for the areas of triangles and special quadrilaterals. (9.1)
Solve problems involving perimeters and areas of triangles and special triangles. (9.2)
Develop and apply the formulas for the area and circumference of a circle. (9.2)
Develop and apply the formula for the area of a regular polygon. (9.2)
Find the area of composite figures. (9.3)
Describe the effect on Perimeter and area when one or more dimensions of a figure are changed.
(9.5)
Apply the relationship between perimeter and area in problem solving. (9.5)
Calculate geometric probabilities. (9.6)
Quarter 4
Unit 10: Circles
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Identify tangents, secants and chords. (11.1)
Use the properties of tangents to solve problems. (11.1)
Apply the properties of arcs and chords. (11.2)
Find the areas of sectors, find arc lengths. (11.3)
Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems.
(11.4)
Prove the theorems for inscribed angles. (See attachment on calendar)
Find the measures of angles formed by lines that intersect circles. (11.5)
Find the length of segments formed by lines that intersect circles. (11.6)
Unit 11: Spatial Reasoning
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Classify 3-D figures according to their properties. (10.1)
Use nets and cross-sections to analyze 3-D figures. (10.1)
Draw representations of 3-D figures, recognize a 3-D figure from a given representation. (10.2)
Apply Euler’s formula to find the number of vertices, edges and faces of a polyhedron. (10.3)
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Apply the distance and mid-point formulas in 3 dimensions. (10.3)
Learn and apply the formula for the surface area of a prism and cylinder. (10.4)
Learn and apply the formula for the surface area of a pyramid and cone. (10.5)
Learn and apply the formula for the volume of a prism and cylinder. (10.6)
Learn and apply the formula for the volume of a pyramid and cone. (10.7)
Learn and apply the formula for the volume and surface area of a sphere. (10.8)
Review for end of semester exams
Grading Policy:
Mathematical literacy and communication of understanding and thinking are at the center of class
activity therefore student grades are viewed in this context. The teacher continuously assesses
student performance and progress, as evidenced by in-class task commitment, finished classroom
assignments, on-demand demonstration of learning, homework, tests and quizzes, homework, class
notes, and daily preparation. Appendix F contains additional information on the ISS Assessment
Policy.
Evaluation System:
 Summative: Tests and projects
 Formative: Quizzes and class work
 Effort: Participation, Homework, etc.
40%
45%
15%
Semester grades are developed based upon the following formula:
 Quarterly Grades
(average)
80%
 Exam Grade or Semester Project 20%
APPENDIX A: Mathematics Standards
1. Number and Operations
Instructional programs from prekindergarten through grade 12 should enable all
students to-


understand numbers, ways of representing numbers, relationships among
numbers, and number systems;
understand meanings of operations and how they relate to one another;
compute fluently and make reasonable estimates
2. Algebra
Instructional programs from prekindergarten through grade 12 should enable all
students to-



understand patterns, relations, and functions;
represent and analyze mathematical situations and structures using algebraic
symbols;
use mathematical models to represent and understand quantitative relationships;
analyze change in various contexts
3. Geometry
Instructional programs from prekindergarten through grade 12 should enable all
students to-



analyze characteristics and properties of two- and three-dimensional geometric
shapes and develop mathematical arguments about geometric relationships;
specify locations and describe spatial relationships using coordinate geometry
and other representational systems;
apply transformations and use symmetry to analyze mathematical situations;
use visualization, spatial reasoning, and geometric modeling to solve problems.
4. Measurement
Instructional programs from prekindergarten through grade 12 should enable all
students to-

understand measurable attributes of objects and the units, systems, and
processes of measurement;
apply appropriate techniques, tools, and formulas to determine measurements.
5. Data Analysis and Probability
Instructional programs from prekindergarten through grade 12 should enable all
students to-



formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them;
select and use appropriate statistical methods to analyze data;
develop and evaluate inferences and predictions that are based on data;
understand and apply basic concepts of probability
6. Problem Solving
Instructional programs from prekindergarten through grade 12 should enable all
students to-



build new mathematical knowledge through problem solving;
solve problems that arise in mathematics and in other contexts;
apply and adapt a variety of appropriate strategies to solve problems;
monitor and reflect on the process of mathematical problem solving.
7. Reasoning and Proof
Instructional programs from prekindergarten through grade 12 should enable all
students to-
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

recognize reasoning and proof as fundamental aspects of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and proofs;
select and use various types of reasoning and methods of proof.
8. Communication
Instructional programs from prekindergarten through grade 12 should enable all
students to-
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

organize and consolidate their mathematical thinking through communication;
communicate their mathematical thinking coherently and clearly to peers,
teachers, and others;
analyze and evaluate the mathematical thinking and strategies of others;
use the language of mathematics to express mathematical ideas precisely.
9. Connections
Instructional programs from prekindergarten through grade 12 should enable all
students to-


recognize and use connections among mathematical ideas;
understand how mathematical ideas interconnect and build on one another to
produce a coherent whole;
recognize and apply mathematics in contexts outside of mathematics.
10. Representation
Instructional programs from prekindergarten through grade 12 should enable all
students to-


create and use representations to organize, record, and communicate
mathematical ideas;
select, apply, and translate among mathematical representations to solve
problems;
use representations to model and interpret physical, social, and mathematical
phenomena.
APPENDIX B: Performance Indicators - Grades 9-12
1. Numbers and Operations Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to
1.1 Understand
numbers, ways of
representing numbers,
relationships among
numbers, and number
systems
1.2 Understand
meanings of
operations and how
they relate to one
another
1.3 Compute fluently
and make reasonable
estimates
In grades 9 through 12 all students should—
 develop a deeper understanding of very large and very small
numbers and of various representations of them;
 compare and contrast the properties of numbers and number
systems, including the rational and real numbers, and understand
complex numbers as solutions to quadratic equations that do not
have real solutions;
 understand vectors and matrices as systems that have some of the
properties of the real-number system;
 use number-theory arguments to justify relationships involving
whole numbers.
 judge the effects of such operations as multiplication, division, and
computing powers and roots on the magnitudes of quantities;
 develop an understanding of properties of, and representations for,
the addition and multiplication of vectors and matrices;
 develop an understanding of permutations and combinations as
counting techniques.
 develop fluency in operations with real numbers, vectors, and
matrices, using mental computation or paper-and-pencil
calculations for simple cases and technology for more-complicated
cases.
 judge the reasonableness of numerical computations and their
results.
2. Algebra Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
2.1 Understand
patterns, relations, and
functions
In grades 9 through 12 all students should—

generalize patterns using explicitly defined and recursively defined
functions;
 understand relations and functions and select, convert flexibly
among, and use various representations for them;
 analyze functions of one variable by investigating rates of change,
intercepts, zeros, asymptotes, and local and global behavior;
 understand and perform transformations such as arithmetically
combining, composing, and inverting commonly used functions,
using technology to perform such operations on more-complicated
symbolic expressions;

2.2 Represent and
analyze mathematical
situations and
structures using
algebraic symbols
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2.3 Use mathematical
models to represent
and understand
quantitative
relationships
2.4 Analyze change in
various contexts
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understand and compare the properties of classes of functions,
including exponential, polynomial, rational, logarithmic, and
periodic functions;
interpret representations of functions of two variables
understand the meaning of equivalent forms of expressions,
equations, inequalities, and relations;
write equivalent forms of equations, inequalities, and systems of
equations and solve them with fluency—mentally or with paper and
pencil in simple cases and using technology in all cases;
use symbolic algebra to represent and explain mathematical
relationships;
use a variety of symbolic representations, including recursive and
parametric equations, for functions and relations;
judge the meaning, utility, and reasonableness of the results of
symbol manipulations, including those carried out by technology.
identify essential quantitative relationships in a situation and
determine the class or classes of functions that might model the
relationships;
use symbolic expressions, including iterative and recursive forms,
to represent relationships arising from various contexts;
draw reasonable conclusions about a situation being modeled.
approximate and interpret rates of change from graphical and
numerical data.
3. Geometry Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
3.1 Analyze
characteristics and
properties of two- and
three-dimensional
geometric shapes and
develop mathematical
arguments about
geometric
relationships
3.2 Specify locations
and describe spatial
relationships using
coordinate geometry
and other
representational
systems
3.3 Apply
transformations and
use symmetry to
In grades 9 through 12 all students should—
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analyze properties and determine attributes of two- and threedimensional objects;
explore relationships (including congruence and similarity) among
classes of two- and three-dimensional geometric objects, make and
test conjectures about them, and solve problems involving them;
establish the validity of geometric conjectures using deduction,
prove theorems, and critique arguments made by others;
use trigonometric relationships to determine lengths and angle
measures.
use Cartesian coordinates and other coordinate systems, such as
navigational, polar, or spherical systems, to analyze geometric
situations;
investigate conjectures and solve problems involving two- and
three-dimensional objects represented with Cartesian coordinates.
understand and represent translations, reflections, rotations, and
dilations of objects in the plane by using sketches, coordinates,
vectors, function notation, and matrices;
analyze mathematical
situations
3.4 Use visualization,
spatial reasoning,
and geometric
modeling to solve
problems
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use various representations to help understand the effects of simple
transformations and their compositions.
draw and construct representations of two- and three-dimensional
geometric objects using a variety of tools;
visualize three-dimensional objects and spaces from different
perspectives and analyze their cross sections;
use vertex-edge graphs to model and solve problems;
use geometric models to gain insights into, and answer questions in,
other areas of mathematics;
use geometric ideas to solve problems in, and gain insights into,
other disciplines and other areas of interest such as art and
architecture.
4. Measurement Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
4.1 Understand
measurable attributes
of objects and the
units, systems, and
processes of
measurement
4.2 Apply appropriate
techniques, tools, and
formulas to determine
measurements
In grades 9 though 12 all students should—

make decisions about units and scales that are appropriate for
problem situations involving measurement.

analyze precision, accuracy, and approximate error in measurement
situations;
 understand and use formulas for the area, surface area, and volume
of geometric figures, including cones, spheres, and cylinders;
 apply informal concepts of successive approximation, upper and
lower bounds, and limit in measurement situations;
 use unit analysis to check measurement computations
5. Data Analysis and Probability Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
5.1 Formulate
questions that can be
addressed with data
and collect, organize,
and display relevant
data to answer them
In grades 9 through 12 all students should—

understand the differences among various kinds of studies and
which types of inferences can legitimately be drawn from each;
 know the characteristics of well-designed studies, including the role
of randomization in surveys and experiments;
 understand the meaning of measurement data and categorical data,
of univariate and bivariate data, and of the term variable;
 understand histograms, parallel box plots, and scatterplots and use
them to display data;

5.2 Select and use
appropriate statistical
methods to analyze
data
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5.3 Develop and
evaluate inferences
and predictions that
are based on data
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5.4 Understand and
apply basic concepts
of probability
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compute basic statistics and understand the distinction between a
statistic and a parameter.
for univariate measurement data, be able to display the distribution,
describe its shape, and select and calculate summary statistics;
for bivariate measurement data, be able to display a scatterplot,
describe its shape, and determine regression coefficients, regression
equations, and correlation coefficients using technological tools;
display and discuss bivariate data where at least one variable is
categorical;
recognize how linear transformations of univariate data affect
shape, center, and spread;
identify trends in bivariate data and find functions that model the
data or transform the data so that they can be modeled.
use simulations to explore the variability of sample statistics from a
known population and to construct sampling distributions;
understand how sample statistics reflect the values of population
parameters and use sampling distributions as the basis for informal
inference;
evaluate published reports that are based on data by examining the
design of the study, the appropriateness of the data analysis, and the
validity of conclusions;
understand how basic statistical techniques are used to monitor
process characteristics in the workplace.
understand the concepts of sample space and probability distribution
and construct sample spaces and distributions in simple cases
use simulations to construct empirical probability distributions;
compute and interpret the expected value of random variables in
simple cases;
understand the concepts of conditional probability and independent
events;
understand how to compute the probability of a compound event.
APPENDIX C: Focal Points - Grade 9 Geometry

Students demonstrate understanding by identifying and giving examples of undefined
terms, axioms, theorems, and inductive and deductive reasoning.

Students write geometric proofs, including proofs by contradiction.

Students construct and judge the validity of a logical argument and give counterexamples to
disprove a statement.

Students prove basic theorems involving congruence and similarity.

Students prove that triangles are congruent or similar, and they are able to use the concept
of corresponding parts of congruent triangles.

Students know and are able to use the triangle inequality theorem.

Students prove and use theorems involving the properties of parallel lines cut by a
transversal, the properties of quadrilaterals, and the properties of circles.

Students know, derive, and solve problems involving the perimeter, circumference, area,
volume, lateral area, and surface area of common geometric figures.

Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and
spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.

Students compute areas of polygons, including rectangles, scalene triangles, equilateral
triangles, rhombi, parallelograms, and trapezoids.

Students determine how changes in dimensions affect the perimeter, area, and volume of
common geometric figures and solids

Students find and use measures of sides and of interior and exterior angles of triangles and
polygons to classify figures and solve problems.

Students prove relationships between angles in polygons by using properties of
complementary, supplementary, vertical, and exterior angles.

Students prove the Pythagorean theorem.

Students use the Pythagorean theorem to determine distance and find missing lengths of
sides of right triangles.

Students perform basic constructions with a straightedge and compass, such as angle
bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the
line.

Students prove theorems by using coordinate geometry, including the midpoint of a line
segment, the distance formula, and various forms of equations of lines and circles.

Students know the definitions of the basic trigonometric functions defined by the angles of a
right triangle. They also know and are able to use elementary relationships between them.
2
2
For example, tan(x) = sin(x)/cos(x), (sin(x)) + (cos(x)) = 1.

Students use trigonometric functions to solve for an unknown length of a side of a right
triangle, given an angle and a length of a side.

Students know and are able to use angle and side relationships in problems with special
right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.

Students prove and solve problems regarding relationships among chords, secants,
tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

Students know the effect of rigid motions on figures in the coordinate plane and space,
including rotations, translations, and reflections.
APPENDIX D: Secondary Assessment Policy
Assessment monitors the progress of student learning and produces feedback for students,
teachers, parents and external institutions. The following policy outlines the general assessment
procedures for the school. Teachers are responsible for communicating their individual
assessment policies to the students and parents at the beginning of the school year.
Teachers are expected to communicate assessment expectations and criteria, including major
assignments and projects clearly to students prior to a chunk of learning. Assessment should take
into account the ISS diverse group of learners and learning styles. Feedback on assignments should
be positive, constructive and prompt. Teachers should provide a wide variety of different
assessment opportunities which are relevant and motivational to students. Formative assessments
assist student in building understanding, knowledge and skills and summative assessments assess
students’ acquired understanding, knowledge and skills.
External
External
assessments are
assessments
which are
designed and
marked externally
Primary Purpose To measure
growth and
progress, to
inform teaching,
to identify needs,
to collect data, to
determine level of
understanding, to
determine
reading or math
levels against
national norms,
assessing student
learning,
providing a
qualification for
university or
college entry.
Policies
Some external
assessments are
taken twice a
year, some are
Definition
Summative
Summative assessments
are those assessments
given within a class at
the end of a chunk of
learning (such as a unit).
To inform teaching, to
identify needs, to
determine level of
understanding, to
measure progress, to
communicate with
parents
Formative
Formative assessments
are those given
regularly and
continuously
throughout the school
year.
To determine prior
knowledge, to
determine student
interest, to modify
teacher practice,
measure
understanding,
ensuring short-term
knowledge and
understanding
objectives and targets
are being met, to
ensure students are
progressing
Assessments are aligned
to curriculum, teachers
model in advance,
authentic assessments,
Assessments are
aligned to curriculum,
differentiated if
necessary.
Practices
once and some
are on-going.
STAR Math,
NWEA,
Accelerated Math,
PSAT, SAT, AP
differentiated if
necessary.
Essays, projects, test,
RAFTS, portfolio,
investigations, realworld examples, exams,
oral presentation,
reports, reflections, midtrimester reports, midquarter reports
Observation, journal,
quiz, exit cards, peer
assessment or selfassessment (not
graded on Gradequick),
role play, conferencing,
small group discussion,
debate, create/present,
note-taking, reflection,
homework, classwork,
effort, behavior,
participation,
Gradequick reporting,
Teachers will be asked to implement IEP's/ILP's in their classroom should it contain students
receiving necessary support. Teachers will be provided with the document, as well as support in
how to effectively implement the modifications in order to ensure student success. We strongly
suggest that teachers consult with the learning specialist or principal during the design and
implementation of all summative evaluations for students with IEPs.