Texas Administrative Code
Title 19 - EDUCATION
Part 2 - TEXAS EDUCATION AGENCY
Chapter 111 - TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS
Subchapter C - HIGH SCHOOL
Section 111.41 - Geometry, Adopted 2012 (One Credit)
Universal Citation: 19 TX Admin Code ยง 111.41
Current through Reg. 49, No. 38; September 20, 2024
(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:
(A) apply the formula for the area of regular
polygons to solve problems using appropriate units of measure;
(B) determine the area of composite
two-dimensional figures comprised of a combination of triangles,
parallelograms, trapezoids, kites, regular polygons, or sectors of circles to
solve problems using appropriate units of measure;
(C) apply the formulas for the total and
lateral surface area of three-dimensional figures, including prisms, pyramids,
cones, cylinders, spheres, and composite figures, to solve problems using
appropriate units of measure; and
(D) apply the formulas for the volume of
three-dimensional figures, including prisms, pyramids, cones, cylinders,
spheres, and composite figures, to solve problems using appropriate units of
measure.
(12) Circles.
The student uses the process skills to understand geometric relationships and
apply theorems and equations about circles. The student is expected to:
(A) apply theorems about circles, including
relationships among angles, radii, chords, tangents, and secants, to solve
non-contextual problems;
(B) apply
the proportional relationship between the measure of an arc length of a circle
and the circumference of the circle to solve problems;
(C) apply the proportional relationship
between the measure of the area of a sector of a circle and the area of the
circle to solve problems;
(D)
describe radian measure of an angle as the ratio of the length of an arc
intercepted by a central angle and the radius of the circle; and
(E) show that the equation of a circle with
center at the origin and radius r is
x2 +
y2 =
r2 and determine the equation for
the graph of a circle with radius r and center (h,
k),(x - h)2 + (y -
k)2 =
r2.
(13) Probability. The student uses the
process skills to understand probability in real-world situations and how to
apply independence and dependence of events. The student is expected to:
(A) develop strategies to use permutations
and combinations to solve contextual problems;
(B) determine probabilities based on area to
solve contextual problems;
(C)
identify whether two events are independent and compute the probability of the
two events occurring together with or without replacement;
(D) apply conditional probability in
contextual problems; and
(E) apply
independence in contextual problems.
Disclaimer: These regulations may not be the most recent version. Texas may have more current or accurate information. We make no warranties or guarantees about the accuracy, completeness, or adequacy of the information contained on this site or the information linked to on the state site. Please check official sources.
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