(a) General requirements. Students shall be awarded
one credit for successful completion of this course. Prerequisite:
Algebra I.
(b) Introduction.
(1) The desire to achieve educational excellence is
the driving force behind the Texas essential knowledge and skills
for mathematics, guided by the college and career readiness standards.
By embedding statistics, probability, and finance, while focusing
on fluency and solid understanding, Texas will lead the way in mathematics
education and prepare all Texas students for the challenges they will
face in the 21st century.
(2) The process standards describe ways in which students
are expected to engage in the content. The placement of the process
standards at the beginning of the knowledge and skills listed for
each grade and course is intentional. The process standards weave
the other knowledge and skills together so that students may be successful
problem solvers and use mathematics efficiently and effectively in
daily life. The process standards are integrated at every grade level
and course. When possible, students will apply mathematics to problems
arising in everyday life, society, and the workplace. Students will
use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying
the solution, and evaluating the problem-solving process and the reasonableness
of the solution. Students will select appropriate tools such as real
objects, manipulatives, paper and pencil, and technology and techniques
such as mental math, estimation, and number sense to solve problems.
Students will effectively communicate mathematical ideas, reasoning,
and their implications using multiple representations such as symbols,
diagrams, graphs, and language. Students will use mathematical relationships
to generate solutions and make connections and predictions. Students
will analyze mathematical relationships to connect and communicate
mathematical ideas. Students will display, explain, or justify mathematical
ideas and arguments using precise mathematical language in written
or oral communication.
(3) In Geometry, students will build on the knowledge
and skills for mathematics in Kindergarten-Grade 8 and Algebra I to
strengthen their mathematical reasoning skills in geometric contexts.
Within the course, students will begin to focus on more precise terminology,
symbolic representations, and the development of proofs. Students
will explore concepts covering coordinate and transformational geometry;
logical argument and constructions; proof and congruence; similarity,
proof, and trigonometry; two- and three-dimensional figures; circles;
and probability. Students will connect previous knowledge from Algebra
I to Geometry through the coordinate and transformational geometry
strand. In the logical arguments and constructions strand, students
are expected to create formal constructions using a straight edge
and compass. Though this course is primarily Euclidean geometry, students
should complete the course with an understanding that non-Euclidean
geometries exist. In proof and congruence, students will use deductive
reasoning to justify, prove and apply theorems about geometric figures.
Throughout the standards, the term "prove" means a formal proof to
be shown in a paragraph, a flow chart, or two-column formats. Proportionality
is the unifying component of the similarity, proof, and trigonometry
strand. Students will use their proportional reasoning skills to prove
and apply theorems and solve problems in this strand. The two- and
three-dimensional figure strand focuses on the application of formulas
in multi-step situations since students have developed background
knowledge in two- and three-dimensional figures. Using patterns to
identify geometric properties, students will apply theorems about
circles to determine relationships between special segments and angles
in circles. Due to the emphasis of probability and statistics in the
college and career readiness standards, standards dealing with probability
have been added to the geometry curriculum to ensure students have
proper exposure to these topics before pursuing their post-secondary
education.
(4) These standards are meant to provide clarity and
specificity in regards to the content covered in the high school geometry
course. These standards are not meant to limit the methodologies used
to convey this knowledge to students. Though the standards are written
in a particular order, they are not necessarily meant to be taught
in the given order. In the standards, the phrase "to solve problems"
includes both contextual and non-contextual problems unless specifically
stated.
(5) Statements that contain the word "including" reference
content that must be mastered, while those containing the phrase "such
as" are intended as possible illustrative examples.
(c) Knowledge and skills.
(1) Mathematical process standards. The student uses
mathematical processes to acquire and demonstrate mathematical understanding.
The student is expected to:
(A) apply mathematics to problems arising in everyday
life, society, and the workplace;
(B) use a problem-solving model that incorporates analyzing
given information, formulating a plan or strategy, determining a solution,
justifying the solution, and evaluating the problem-solving process
and the reasonableness of the solution;
(C) select tools, including real objects, manipulatives,
paper and pencil, and technology as appropriate, and techniques, including
mental math, estimation, and number sense as appropriate, to solve
problems;
(D) communicate mathematical ideas, reasoning, and
their implications using multiple representations, including symbols,
diagrams, graphs, and language as appropriate;
(E) create and use representations to organize, record,
and communicate mathematical ideas;
(F) analyze mathematical relationships to connect and
communicate mathematical ideas; and
(G) display, explain, and justify mathematical ideas
and arguments using precise mathematical language in written or oral
communication.
(2) Coordinate and transformational geometry. The student
uses the process skills to understand the connections between algebra
and geometry and uses the one- and two-dimensional coordinate systems
to verify geometric conjectures. The student is expected to:
(A) determine the coordinates of a point that is a
given fractional distance less than one from one end of a line segment
to the other in one- and two-dimensional coordinate systems, including
finding the midpoint;
(B) derive and use the distance, slope, and midpoint
formulas to verify geometric relationships, including congruence of
segments and parallelism or perpendicularity of pairs of lines; and
(C) determine an equation of a line parallel or perpendicular
to a given line that passes through a given point.
(3) Coordinate and transformational geometry. The student
uses the process skills to generate and describe rigid transformations
(translation, reflection, and rotation) and non-rigid transformations
(dilations that preserve similarity and reductions and enlargements
that do not preserve similarity). The student is expected to:
(A) describe and perform transformations of figures
in a plane using coordinate notation;
(B) determine the image or pre-image of a given two-dimensional
figure under a composition of rigid transformations, a composition
of non-rigid transformations, and a composition of both, including
dilations where the center can be any point in the plane;
(C) identify the sequence of transformations that will
carry a given pre-image onto an image on and off the coordinate plane;
and
(D) identify and distinguish between reflectional and
rotational symmetry in a plane figure.
(4) Logical argument and constructions. The student
uses the process skills with deductive reasoning to understand geometric
relationships. The student is expected to:
(A) distinguish between undefined terms, definitions,
postulates, conjectures, and theorems;
(B) identify and determine the validity of the converse,
inverse, and contrapositive of a conditional statement and recognize
the connection between a biconditional statement and a true conditional
statement with a true converse;
(C) verify that a conjecture is false using a counterexample;
and
(D) compare geometric relationships between Euclidean
and spherical geometries, including parallel lines and the sum of
the angles in a triangle.
(5) Logical argument and constructions. The student
uses constructions to validate conjectures about geometric figures.
The student is expected to:
(A) investigate patterns to make conjectures about
geometric relationships, including angles formed by parallel lines
cut by a transversal, criteria required for triangle congruence, special
segments of triangles, diagonals of quadrilaterals, interior and exterior
angles of polygons, and special segments and angles of circles choosing
from a variety of tools;
(B) construct congruent segments, congruent angles,
a segment bisector, an angle bisector, perpendicular lines, the perpendicular
bisector of a line segment, and a line parallel to a given line through
a point not on a line using a compass and a straightedge;
(C) use the constructions of congruent segments, congruent
angles, angle bisectors, and perpendicular bisectors to make conjectures
about geometric relationships; and
(D) verify the Triangle Inequality theorem using constructions
and apply the theorem to solve problems.
(6) Proof and congruence. The student uses the process
skills with deductive reasoning to prove and apply theorems by using
a variety of methods such as coordinate, transformational, and axiomatic
and formats such as two-column, paragraph, and flow chart. The student
is expected to:
(A) verify theorems about angles formed by the intersection
of lines and line segments, including vertical angles, and angles
formed by parallel lines cut by a transversal and prove equidistance
between the endpoints of a segment and points on its perpendicular
bisector and apply these relationships to solve problems;
(B) prove two triangles are congruent by applying the
Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side,
and Hypotenuse-Leg congruence conditions;
(C) apply the definition of congruence, in terms of
rigid transformations, to identify congruent figures and their corresponding
sides and angles;
(D) verify theorems about the relationships in triangles,
including proof of the Pythagorean Theorem, the sum of interior angles,
base angles of isosceles triangles, midsegments, and medians, and
apply these relationships to solve problems; and
(E) prove a quadrilateral is a parallelogram, rectangle,
square, or rhombus using opposite sides, opposite angles, or diagonals
and apply these relationships to solve problems.
(7) Similarity, proof, and trigonometry. The student
uses the process skills in applying similarity to solve problems.
The student is expected to:
(A) apply the definition of similarity in terms of
a dilation to identify similar figures and their proportional sides
and the congruent corresponding angles; and
(B) apply the Angle-Angle criterion to verify similar
triangles and apply the proportionality of the corresponding sides
to solve problems.
(8) Similarity, proof, and trigonometry. The student
uses the process skills with deductive reasoning to prove and apply
theorems by using a variety of methods such as coordinate, transformational,
and axiomatic and formats such as two-column, paragraph, and flow
chart. The student is expected to:
(A) prove theorems about similar triangles, including
the Triangle Proportionality theorem, and apply these theorems to
solve problems; and
(B) identify and apply the relationships that exist
when an altitude is drawn to the hypotenuse of a right triangle, including
the geometric mean, to solve problems.
(9) Similarity, proof, and trigonometry. The student
uses the process skills to understand and apply relationships in right
triangles. The student is expected to:
(A) determine the lengths of sides and measures of
angles in a right triangle by applying the trigonometric ratios sine,
cosine, and tangent to solve problems; and
(B) apply the relationships in special right triangles
30°-60°-90° and 45°-45°-90° and the Pythagorean
theorem, including Pythagorean triples, to solve problems.
(10) Two-dimensional and three-dimensional figures.
The student uses the process skills to recognize characteristics and
dimensional changes of two- and three-dimensional figures. The student
is expected to:
(A) identify the shapes of two-dimensional cross-sections
of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional
objects generated by rotations of two-dimensional shapes; and
(B) determine and describe how changes in the linear
dimensions of a shape affect its perimeter, area, surface area, or
volume, including proportional and non-proportional dimensional change.
(11) Two-dimensional and three-dimensional figures.
The student uses the process skills in the application of formulas
to determine measures of two- and three-dimensional figures. The student
is expected to:
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