Chapter 111. Texas Essential Knowledge and
Skills for Mathematics
Subchapter C. High School
Statutory Authority: The provisions of this Subchapter C
issued under the Texas Education Code, §28.002, unless
otherwise noted.
The provisions of this subchapter shall be implemented
beginning September 1, 1998, and at that time, shall
supersede §75.63(e)-(g) of this title (relating to
Mathematics).
Source: The provisions of this §111.31 adopted to
be effective September 1, 1996, 21 TexReg 7371.
(a) Basic understandings.
(1) Foundation concepts for high school mathematics.
As presented in Grades K-8, the basic
understandings of number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry; measurement; and
probability and statistics are essential
foundations for all work in high school
mathematics. Students will continue to build on
this foundation as they expand their understanding
through other mathematical experiences.
(2) Algebraic thinking and symbolic reasoning.
Symbolic reasoning plays a critical role in
algebra; symbols provide powerful ways to
represent mathematical situations and to express
generalizations. Students use symbols in a variety
of ways to study relationships among quantities.
(3) Function concepts. Functions represent the
systematic dependence of one quantity on another.
Students use functions to represent and model
problem situations and to analyze and interpret
relationships.
(4) Relationship between equations and functions.
Equations arise as a way of asking and answering
questions involving functional relationships.
Students work in many situations to set up
equations and use a variety of methods to solve
these equations.
(5) Tools for algebraic thinking. Techniques for
working with functions and equations are essential
in understanding underlying relationships.
Students use a variety of representations
(concrete, numerical, algorithmic, graphical),
tools, and technology, including, but not limited
to, powerful and accessible hand-held calculators
and computers with graphing capabilities and model
mathematical situations to solve meaningful
problems.
(6) Underlying mathematical processes. Many processes
underlie all content areas in mathematics. As they
do mathematics, students continually use problem-
solving, computation in problem-solving contexts,
language and communication, connections within and
outside mathematics, and reasoning, as well as
multiple representations, applications and
modeling, and justification and proof.
(b) Foundations for functions: knowledge and skills and
performance descriptions.
(1) The student understands that a function represents
a dependence of one quantity on another and can be
described in a variety of ways.
Following are performance descriptions.
(A) The student describes independent and
dependent quantities in functional
relationships.
(B) The student gathers and records data, or uses
data sets, to determine functional
(systematic) relationships between quantities.
(C) The student describes functional relationships
for given problem situations and writes
equations or inequalities to answer questions
arising from the situations.
(D) The student represents relationships among
quantities using concrete models, tables,
graphs, diagrams, verbal descriptions,
equations, and inequalities.
(E) The student interprets and makes inferences
from functional relationships.
(2) The student uses the properties and attributes of
functions.
Following are performance descriptions.
(A) The student identifies and sketches the general
forms of linear (y = x) and quadratic (y = x2)
parent functions.
(B) For a variety of situations, the student
identifies the mathematical domains and ranges
and determines reasonable domain and range
values for given situations.
(C) The student interprets situations in terms of
given graphs or creates situations that fit
given graphs.
(D) In solving problems, the student collects and
organizes data, makes and interprets
scatterplots, and models, predicts, and makes
decisions and critical judgments.
(3) The student understands how algebra can be used to
express generalizations and recognizes and uses
the power of symbols to represent situations.
Following are performance descriptions.
(A) The student uses symbols to represent unknowns
and variables.
(B) Given situations, the student looks for
patterns and represents generalizations
algebraically.
(4) The student understands the importance of the
skills required to manipulate symbols in order to
solve problems and uses the necessary algebraic
skills required to simplify algebraic expressions
and solve equations and inequalities in problem
situations.
Following are performance descriptions.
(A) The student finds specific function values,
simplifies polynomial expressions, transforms
and solves equations, and factors as necessary
in problem situations.
(B) The student uses the commutative, associative,
and distributive properties to simplify
algebraic expressions.
(c) Linear functions: knowledge and skills and performance
descriptions.
(1) The student understands that linear functions can
be represented in different ways and translates
among their various representations.
Following are performance descriptions.
(A) The student determines whether or not given
situations can be represented by linear
functions.
(B) The student determines the domain and range
values for which linear functions make sense
for given situations.
(C) The student translates among and uses
algebraic, tabular, graphical, or verbal
descriptions of linear functions.
(2) The student understands the meaning of the slope
and intercepts of linear functions and interprets
and describes the effects of changes in parameters
of linear functions in real-world and mathematical
situations.
Following are performance descriptions.
(A) The student develops the concept of slope as
rate of change and determines slopes from
graphs, tables, and algebraic representations.
(B) The student interprets the meaning of slope
and intercepts in situations using data,
symbolic representations, or graphs.
(C) The student investigates, describes, and
predicts the effects of changes in m and b on
the graph of y = mx + b.
(D) The student graphs and writes equations of
lines given characteristics such as two
points, a point and a slope, or a slope and
y-intercept.
(E) The student determines the intercepts of
linear functions from graphs, tables, and
algebraic representations.
(F) The student interprets and predicts the
effects of changing slope and y-intercept in
applied situations.
(G) The student relates direct variation to linear
functions and solves problems involving
proportional change.
(3) The student formulates equations and inequalities
based on linear functions, uses a variety of
methods to solve them, and analyzes the solutions
in terms of the situation.
Following are performance descriptions.
(A) The student analyzes situations involving
linear functions and formulates linear
equations or inequalities to solve problems.
(B) The student investigates methods for solving
linear equations and inequalities using
concrete models, graphs, and the properties of
equality, selects a method, and solves the
equations and inequalities.
(C) For given contexts, the student interprets and
determines the reasonableness of solutions to
linear equations and inequalities.
(4) The student formulates systems of linear equations
from problem situations, uses a variety of methods
to solve them, and analyzes the solutions in terms
of the situation.
Following are performance descriptions.
(A) The student analyzes situations and formulates
systems of linear equations to solve problems.
(B) The student solves systems of linear equations
using concrete models, graphs, tables, and
algebraic methods.
(C) For given contexts, the student interprets and
determines the reasonableness of solutions to
systems of linear equations.
(d) Quadratic and other nonlinear functions: knowledge and
skills and performance descriptions.
(1) The student understands that the graphs of
quadratic functions are affected by the parameters
of the function and can interpret and describe the
effects of changes in the parameters of quadratic
functions.
Following are performance descriptions.
(A) The student determines the domain and range
values for which quadratic functions make
sense for given situations.
(B) The student investigates, describes, and
predicts the effects of changes in a on the
graph of y = ax2.
(C) The student investigates, describes, and
predicts the effects of changes in c on the
graph of y = x2 + c.
(D) For problem situations, the student analyzes
graphs of quadratic functions and draws
conclusions.
(2) The student understands there is more than one way
to solve a quadratic equation and solves them
using appropriate methods.
Following are performance descriptions.
(A) The student solves quadratic equations using
concrete models, tables, graphs, and algebraic
methods.
(B) The student relates the solutions of quadratic
equations to the roots of their functions.
(3) The student understands there are situations
modeled by functions that are neither linear nor
quadratic and models the situations.
Following are performance descriptions.
(A) The student uses patterns to generate the laws
of exponents and applies them in problem-
solving situations.
(B) The student analyzes data and represents
situations involving inverse variation using
concrete models, tables, graphs, or algebraic
methods.
(C) The student analyzes data and represents
situations involving exponential growth and
decay using concrete models, tables, graphs,
or algebraic methods.
Source: The provisions of this §111.32 adopted to
be effective September 1, 1996, 21 TexReg 7371.
(a) Basic understandings.
(1) Foundation concepts for high school mathematics.
As presented in Grades K-8, the basic
understandings of number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry; measurement; and
probability and statistics are essential
foundations for all work in high school
mathematics. Students continue to build on this
foundation as they expand their understanding
through other mathematical experiences.
(2) Algebraic thinking and symbolic reasoning.
Symbolic reasoning plays a critical role in
algebra; symbols provide powerful ways to
represent mathematical situations and to express
generalizations. Students study algebraic concepts
and the relationships among them to better
understand the structure of algebra.
(3) Functions, equations, and their relationship. The
study of functions, equations, and their
relationship is central to all of mathematics.
Students perceive functions and equations as means
for analyzing and understanding a broad variety of
relationships and as a useful tool for expressing
generalizations.
(4) Relationship between algebra and geometry.
Equations and functions are algebraic tools that
can be used to represent geometric curves and
figures; similarly, geometric figures can
illustrate algebraic relationships. Students
perceive the connections between algebra and
geometry and use the tools of one to help solve
problems in the other.
(5) Tools for algebraic thinking. Techniques for
working with functions and equations are essential
in understanding underlying relationships.
Students use a variety of representations
(concrete, numerical, algorithmic, graphical),
tools, and technology, including, but not limited
to, powerful and accessible hand-held calculators
and computers with graphing capabilities and model
mathematical situations to solve meaningful
problems.
(6) Underlying mathematical processes. Many processes
underlie all content areas in mathematics. As they
do mathematics, students continually use problem-
solving, computation in problem-solving contexts,
language and communication, connections within and
outside mathematics, and reasoning, as well as
multiple representations, applications and
modeling, and justification and proof.
(b) Foundations for functions: knowledge and skills and
performance descriptions.
(1) The student uses properties and attributes of
functions and applies functions to problem
situations.
Following are performance descriptions.
(A) For a variety of situations, the student
identifies the mathematical domains and ranges
and determines reasonable domain and range
values for given situations.
(B) In solving problems, the student collects data
and records results, organizes the data, makes
scatterplots, fits the curves to the
appropriate parent function, interprets the
results, and proceeds to model, predict, and
make decisions and critical judgments.
(2) The student understands the importance of the
skills required to manipulate symbols in order to
solve problems and uses the necessary algebraic
skills required to simplify algebraic expressions
and solve equations and inequalities in problem
situations.
Following are performance descriptions.
(A) The student uses tools including matrices,
factoring, and properties of exponents to
simplify expressions and transform and solve
equations.
(B) The student uses complex numbers to describe
the solutions of quadratic equations.
(C) The student connects the function notation of
y = and ƒ(x) =.
(3) The student formulates systems of equations and
inequalities from problem situations, uses a
variety of methods to solve them, and analyzes the
solutions in terms of the situations.
Following are performance descriptions.
(A) The student analyzes situations and formulates
systems of equations or inequalities in two or
more unknowns to solve problems.
(B) The student uses algebraic methods, graphs,
tables, or matrices, to solve systems of
equations or inequalities.
(C) For given contexts, the student interprets and
determines the reasonableness of solutions to
systems of equations or inequalities.
(c) Algebra and geometry: knowledge and skills and
performance descriptions.
(1) The student connects algebraic and geometric
representations of functions.
Following are performance descriptions.
(A) The student identifies and sketches graphs of
parent functions, including linear (y = x),
quadratic (y = x2), square root (y = Öx),
inverse (y = 1/x), exponential (y = ax), and logarithmic
(y = logax) functions.
(B) The student extends parent functions with
parameters such as m in y = mx and describes
parameter changes on the graph of parent
functions.
(C) The student recognizes inverse relationships
between various functions.
(2) The student knows the relationship between the
geometric and algebraic descriptions of conic
sections.
Following are performance descriptions.
(A) The student describes a conic section as the
intersection of a plane and a cone.
(B) In order to sketch graphs of conic sections,
the student relates simple parameter changes
in the equation to corresponding changes in
the graph.
(C) The student identifies symmetries from graphs
of conic sections.
(D) The student identifies the conic section from
a given equation.
(E) The student uses the method of completing the
square.
(d) Quadratic and square root functions: knowledge and
skills and performance descriptions.
(1) The student understands that quadratic functions
can be represented in different ways and
translates among their various representations.
Following are performance descriptions.
(A) For given contexts, the student determines the
reasonable domain and range values of
quadratic functions, as well as interprets and
determines the reasonableness of solutions to
quadratic equations and inequalities.
(B) The student relates representations of
quadratic functions, such as algebraic,
tabular, graphical, and verbal descriptions.
(C) The student determines a quadratic function
from its roots or a graph.
(2) The student interprets and describes the effects
of changes in the parameters of quadratic
functions in applied and mathematical situations.
Following are performance descriptions.
(A) The student uses characteristics of the quadratic
parent function to sketch the related graphs and
connects between the y = ax2 + bx + c and the y =
a(x - h)2 + k symbolic representations of
quadratic functions.
(B) The student uses the parent function to
investigate, describe, and predict the effects of
changes in a, h, and k on the graphs of y = a(x -
h)2 + k form of a function in applied and purely
mathematical situations.
(3) The student formulates equations and inequalities
based on quadratic functions, uses a variety of
methods to solve them, and analyzes the solutions
in terms of the situation.
Following are performance descriptions.
(A) The student analyzes situations involving
quadratic functions and formulates quadratic
equations or inequalities to solve problems.
(B) The student analyzes and interprets the
solutions of quadratic equations using
discriminants and solves quadratic equations
using the quadratic formula.
(C) The student compares and translates between
algebraic and graphical solutions of quadratic
equations.
(D) The student solves quadratic equations and
inequalities.
(4) The student formulates equations and inequalities
based on square root functions, uses a variety of
methods to solve them, and analyzes the solutions
in terms of the situation.
Following are performance descriptions.
(A) The student uses the parent function to
investigate, describe, and predict the effects
of parameter changes on the graphs of square
root functions and describes limitations on
the domains and ranges.
(B) The student relates representations of square
root functions, such as algebraic, tabular,
graphical, and verbal descriptions.
(C) For given contexts, the student determines the
reasonable domain and range values of square
root functions, as well as interprets and
determines the reasonableness of solutions to
square root equations and inequalities.
(D) The student solves square root equations and
inequalities using graphs, tables, and
algebraic methods.
(E) The student analyzes situations modeled by
square root functions, formulates equations or
inequalities, selects a method, and solves
problems.
(F) The student expresses inverses of quadratic
functions using square root functions.
(e) Rational functions: knowledge and skills and
performance descriptions. The student formulates
equations and inequalities based on rational functions,
uses a variety of methods to solve them, and analyzes
the solutions in terms of the situation.
Following are performance descriptions.
(1) The student uses quotients to describe the
graphs of rational functions, describes
limitations on the domains and ranges, and
examines asymptotic behavior.
(2) The student analyzes various representations
of rational functions with respect to problem
situations.
(3) For given contexts, the student determines
the reasonable domain and range values of
rational functions, as well as interprets and
determines the reasonableness of solutions to
rational equations and inequalities.
(4) The student solves rational equations and
inequalities using graphs, tables, and
algebraic methods.
(5) The student analyzes a situation modeled by a
rational function, formulates an equation or
inequality composed of a linear or quadratic
function, and solves the problem.
(6) The student uses direct and inverse variation
functions as models to make predictions in
problem situations.
(f) Exponential and logarithmic functions: knowledge
and skills and performance descriptions. The student
formulates equations and inequalities based on
exponential and logarithmic functions, uses a variety
of methods to solve them, and analyzes the solutions in
terms of the situation.
Following are performance descriptions.
(1) The student develops the definition of
logarithms by exploring and describing the
relationship between exponential functions
and their inverses.
(2) The student uses the parent functions to
investigate, describe, and predict the
effects of parameter changes on the graphs of
exponential and logarithmic functions,
describes limitations on the domains and
ranges, and examines asymptotic behavior.
(3) For given contexts, the student determines
the reasonable domain and range values of
exponential and logarithmic functions, as
well as interprets and determines the
reasonableness of solutions to exponential
and logarithmic equations and inequalities.
(4) The student solves exponential and
logarithmic equations and inequalities using
graphs, tables, and algebraic methods.
(5) The student analyzes a situation modeled by
an exponential function, formulates an
equation or inequality, and solves the
problem.
Source: The provisions of this §111.33 adopted to
be effective September 1, 1996, 21 TexReg 7371.
(a) Basic understandings.
(1) Foundation concepts for high school mathematics.
As presented in Grades K-8, the basic
understandings of number, operation, and
quantitative reasoning; patterns, relationships,
and algebraic thinking; geometry; measurement; and
probability and statistics are essential
foundations for all work in high school
mathematics. Students continue to build on this
foundation as they expand their understanding
through other mathematical experiences.
(2) Geometric thinking and spatial reasoning. Spatial
reasoning plays a critical role in geometry;
shapes and figures provide powerful ways to
represent mathematical situations and to express
generalizations about space and spatial
relationships. Students use geometric thinking to
understand mathematical concepts and the
relationships among them.
(3) Geometric figures and their properties. Geometry
consists of the study of geometric figures of
zero, one, two, and three dimensions and the
relationships among them. Students study
properties and relationships having to do with
size, shape, location, direction, and orientation
of these figures.
(4) The relationship between geometry, other
mathematics, and other disciplines. Geometry can
be used to model and represent many mathematical
and real-world situations. Students perceive the
connection between geometry and the real and
mathematical worlds and use geometric ideas,
relationships, and properties to solve problems.
(5) Tools for geometric thinking. Techniques for
working with spatial figures and their properties
are essential in understanding underlying
relationships. Students use a variety of
representations (concrete, pictorial, algebraic,
and coordinate), tools, and technology, including,
but not limited to, powerful and accessible hand-
held calculators and computers with graphing
capabilities to solve meaningful problems by
representing figures, transforming figures,
analyzing relationships, and proving things about
them.
(6) Underlying mathematical processes. Many processes
underlie all content areas in mathematics. As they
do mathematics, students continually use problem-
solving, computation in problem-solving contexts,
language and communication, connections within and
outside mathematics, and reasoning, as well as
multiple representations, applications and
modeling, and justification and proof.
(b) Geometric structure: knowledge and skills and
performance descriptions.
(1) The student understands the structure of, and
relationships within, an axiomatic system.
Following are performance descriptions.
(A) The student develops an awareness of the
structure of a mathematical system, connecting
definitions, postulates, logical reasoning,
and theorems.
(B) Through the historical development of
geometric systems, the student recognizes that
mathematics is developed for a variety of
purposes.
(C) The student compares and contrasts the
structures and implications of Euclidean and
non-Euclidean geometries.
(2) The student analyzes geometric relationships in
order to make and verify conjectures.
Following are performance descriptions.
(A) The student uses constructions to explore
attributes of geometric figures and to make
conjectures about geometric relationships.
(B) The student makes and verifies conjectures
about angles, lines, polygons, circles, and
three-dimensional figures, choosing from a
variety of approaches such as coordinate,
transformational, or axiomatic.
(3) The student understands the importance of logical
reasoning, justification, and proof in
mathematics.
Following are performance descriptions.
(A) The student determines if the converse of a
conditional statement is true or false.
(B) The student constructs and justifies
statements about geometric figures and their
properties.
(C) The student demonstrates what it means to
prove mathematically that statements are true.
(D) The student uses inductive reasoning to
formulate a conjecture.
(E) The student uses deductive reasoning to prove
a statement.
(4) The student uses a variety of representations to
describe geometric relationships and solve
problems.
Following is a performance description. The student
selects an appropriate representation (concrete,
pictorial, graphical, verbal, or symbolic) in order to
solve problems.
(c) Geometric patterns: knowledge and skills and
performance descriptions.
The student identifies, analyzes, and describes
patterns that emerge from two- and three-dimensional
geometric figures.
Following are performance descriptions.
(1) The student uses numeric and geometric
patterns to make generalizations about
geometric properties, including properties of
polygons, ratios in similar figures and
solids, and angle relationships in polygons
and circles.
(2) The student uses properties of
transformations and their compositions to
make connections between mathematics and the
real world in applications such as
tessellations or fractals.
(3) The student identifies and applies patterns
from right triangles to solve problems,
including special right triangles (45-45-90
and 30-60-90) and triangles whose sides are
Pythagorean triples.
(d) Dimensionality and the geometry of location: knowledge
and skills and performance descriptions.
(1) The student analyzes the relationship between
three-dimensional objects and related two-
dimensional representations and uses these
representations to solve problems.
Following are performance descriptions.
(A) The student describes, and draws cross
sections and other slices of three-dimensional
objects.
(B) The student uses nets to represent and
construct three-dimensional objects.
(C) The student uses top, front, side, and corner
views of three-dimensional objects to create
accurate and complete representations and
solve problems.
(2) The student understands that coordinate systems
provide convenient and efficient ways of
representing geometric figures and uses them
accordingly.
Following are performance descriptions.
(A) The student uses one- and two-dimensional
coordinate systems to represent points, lines,
line segments, and figures.
(B) The student uses slopes and equations of lines
to investigate geometric relationships,
including parallel lines, perpendicular lines,
and special segments of triangles and other
polygons.
(C) The student develops and uses formulas
including distance and midpoint.
(e) Congruence and the geometry of size: knowledge and
skills and performance descriptions.
(1) The student extends measurement concepts to find
area, perimeter, and volume in problem situations.
Following are performance descriptions.
(A) The student finds areas of regular polygons
and composite figures.
(B) The student finds areas of sectors and arc
lengths of circles using proportional
reasoning.
(C) The student develops, extends, and uses the
Pythagorean Theorem.
(D) The student finds surface areas and volumes of
prisms, pyramids, spheres, cones, and
cylinders in problem situations.
(2) The student analyzes properties and describes
relationships in geometric figures.
Following are performance descriptions.
(A) Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties of parallel
and perpendicular lines.
(B) Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties and
attributes of polygons and their component
parts.
(C) Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties and
attributes of circles and the lines that
intersect them.
(D) The student analyzes the characteristics of
three-dimensional figures and their component
parts.
(3) The student applies the concept of congruence to
justify properties of figures and solve problems.
Following are performance descriptions.
(A) The student uses congruence transformations to
make conjectures and justify properties of
geometric figures.
(B) The student justifies and applies triangle
congruence relationships.
(f) Similarity and the geometry of shape: knowledge
and skills and performance descriptions. The student
applies the concepts of similarity to justify
properties of figures and solve problems.
Following are performance descriptions.
(1) The student uses similarity properties and
transformations to explore and justify
conjectures about geometric figures.
(2) The student uses ratios to solve problems
involving similar figures.
(3) In a variety of ways, the student develops,
applies, and justifies triangle similarity
relationships, such as right triangle ratios,
trigonometric ratios, and Pythagorean
triples.
(4) The student describes the effect on
perimeter, area, and volume when length,
width, or height of a three-dimensional solid
is changed and applies this idea in solving
problems.
Source: The provisions of this §111.34 adopted to
be effective September 1, 1996, 21 TexReg 7371.
(a) General requirements. The provisions of this section
shall be implemented beginning September 1, 1998, and
at that time shall supersede §75.63(bb) of this title
(relating to Mathematics). Students can be awarded one-
half to one credit for successful completion of this
course. Recommended prerequisites: Algebra II,
Geometry.
(b) Introduction.
(1) In Precalculus, students continue to build on the
K-8, Algebra I, Algebra II, and Geometry
foundations as they expand their understanding
through other mathematical experiences. Students
use symbolic reasoning and analytical methods to
represent mathematical situations, to express
generalizations, and to study mathematical
concepts and the relationships among them.
Students use functions, equations, and limits as
useful tools for expressing generalizations and as
means for analyzing and understanding a broad
variety of mathematical relationships. Students
also use functions as well as symbolic reasoning
to represent and connect ideas in geometry,
probability, statistics, trigonometry, and
calculus and to model physical situations.
Students use a variety of representations
(concrete, numerical, algorithmic, graphical),
tools, and technology to model functions and
equations and solve real-life problems.
(2) As students do mathematics, they continually use
problem-solving, language and communication,
connections within and outside mathematics, and
reasoning. Students also use multiple
representations, applications and modeling,
justification and proof, and computation in
problem-solving contexts.
(c) Knowledge and skills.
(1) The student defines functions, describes
characteristics of functions, and translates among
verbal, numerical, graphical, and symbolic
representations of functions, including polynomial,
rational, radical, exponential, logarithmic,
trigonometric, and piecewise-defined functions. The
student is expected to:
(A) describe parent functions symbolically and
graphically, including y = xn, y = ln x,
y =loga x, y = 1/x, y = ex,
y = ax, y = sin x, etc.;
(B) determine the domain and range of functions
using graphs, tables, and symbols;
(C) describe symmetry of graphs of even and odd
functions;
(D) recognize and use connections among
significant points of a function (roots,
maximum points, and minimum points), the graph
of a function, and the symbolic representation
of a function; and
(E) investigate continuity, end behavior, vertical
and horizontal asymptotes, and limits and
connect these characteristics to the graph of
a function.
(2) The student interprets the meaning of the symbolic
representations of functions and operations on
functions within a context. The student is expected
to:
(A) apply basic transformations, including
a•ƒ(x), ƒ(x) + d, ƒ(x - c), ƒ(b•x), |ƒ(x)|,
ƒ(|x|), to the parent functions;
(B) perform operations including composition on
functions, find inverses, and describe these
procedures and results verbally, numerically,
symbolically, and graphically; and
(C) investigate identities graphically and verify
them symbolically, including logarithmic
properties, trigonometric identities, and
exponential properties.
(3) The student uses functions and their properties to
model and solve real-life problems. The student is
expected to:
(A) use functions such as logarithmic,
exponential, trigonometric, polynomial, etc.
to model real-life data;
(B) use regression to determine a function to
model real-life data;
(C) use properties of functions to analyze and
solve problems and make predictions; and
(D) solve problems from physical situations using
trigonometry, including the use of Law of
Sines, Law of Cosines, and area formulas.
(4) The student uses sequences and series to represent,
analyze, and solve real-life problems. The student
is expected to:
(A) represent patterns using arithmetic and
geometric sequences and series;
(B) use arithmetic, geometric, and other sequences
and series to solve real-life problems;
(C) describe limits of sequences and apply their
properties to investigate convergent and
divergent series; and
(D) apply sequences and series to solve problems
including sums and binomial expansion.
(5) The student uses conic sections, their properties,
and parametric representations to model physical
situations. The student is expected to:
(A) use conic sections to model motion, such as
the graph of velocity vs. position of a
pendulum and motions of planets;
(B) use properties of conic sections to describe
physical phenomena such as the reflective
properties of light and sound;
(C) convert between parametric and rectangular
forms of functions and equations to graph
them; and
(D) use parametric functions to simulate problems
involving motion.
(6) The student uses vectors to model physical
situations. The student is expected to:
(A) use the concept of vectors to model situations
defined by magnitude and direction; and
(B) analyze and solve vector problems generated by
real-life situations.
Source: The provisions of this §111.35 adopted to
be effective September 1, 1998, 22 TexReg 7623.
(a) General requirements. The provisions of this section
shall be implemented beginning September 1, 1998.
Students can be awarded one-half to one credit for
successful completion of this course. Recommended
prerequisite: Algebra I.
(b) Introduction.
(1) In Mathematical Models with Applications, students
continue to build on the K-8 and Algebra I
foundations as they expand their understanding
through other mathematical experiences. Students
use algebraic, graphical, and geometric reasoning
to recognize patterns and structure, to model
information, and to solve problems from various
disciplines. Students use mathematical methods to
model and solve real-life applied problems
involving money, data, chance, patterns, music,
design, and science. Students use mathematical
models from algebra, geometry, probability, and
statistics and connections among these to solve
problems from a wide variety of advanced
applications in both mathematical and
nonmathematical situations. Students use a variety
of representations (concrete, numerical,
algorithmic, graphical), tools, and technology to
link modeling techniques and purely mathematical
concepts and to solve applied problems.
(2) As students do mathematics, they continually use
problem-solving, language and communication,
connections within and outside mathematics, and
reasoning. Students also use multiple
representations, applications and modeling,
justification and proof, and computation in
problem-solving contexts.
(c) Knowledge and skills.
(1) The student uses a variety of strategies and
approaches to solve both routine and non-routine
problems. The student is expected to:
(A) compare and analyze various methods for
solving a real-life problem;
(B) use multiple approaches (algebraic, graphical,
and geometric methods) to solve problems from
a variety of disciplines; and
(C) select a method to solve a problem, defend the
method, and justify the reasonableness of the
results.
(2) The student uses graphical and numerical techniques
to study patterns and analyze data. The student is
expected to:
(A) interpret information from various graphs,
including line graphs, bar graphs, circle
graphs, histograms, and scatterplots to draw
conclusions from the data;
(B) analyze numerical data using measures of
central tendency, variability, and correlation
in order to make inferences;
(C) analyze graphs from journals, newspapers, and
other sources to determine the validity of
stated arguments; and
(D) use regression methods available through
technology to describe various models for data
such as linear, quadratic, exponential, etc.,
select the most appropriate model, and use the
model to interpret information.
(3) The student develops and implements a plan for
collecting and analyzing data in order to make
decisions. The student is expected to:
(A) formulate a meaningful question, determine the
data needed to answer the question, gather the
appropriate data, analyze the data, and draw
reasonable conclusions;
(B) communicate methods used, analysis conducted,
and conclusions drawn for a data-analysis
project by written report, visual display,
oral report, or multi-media presentation; and
(C) determine the appropriateness of a model for
making predictions from a given set of data.
(4) The student uses probability models to describe
everyday situations involving chance. The student
is expected to:
(A) compare theoretical and empirical probability;
and
(B) use experiments to determine the
reasonableness of a theoretical model such as
binomial, geometric, etc.
(5) The student uses functional relationships to solve
problems related to personal income. The student is
expected to:
(A) use rates, linear functions, and direct
variation to solve problems involving personal
finance and budgeting, including compensations
and deductions;
(B) solve problems involving personal taxes; and
(C) analyze data to make decisions about banking.
(6) The student uses algebraic formulas, graphs, and
amortization models to solve problems involving
credit. The student is expected to:
(A) analyze methods of payment available in retail
purchasing and compare relative advantages and
disadvantages of each option;
(B) use amortization models to investigate home
financing and compare buying and renting a
home; and
(C) use amortization models to investigate
automobile financing and compare buying and
leasing a vehicle.
(7) The student uses algebraic formulas, numerical
techniques, and graphs to solve problems related to
financial planning. The student is expected to:
(A) analyze types of savings options involving
simple and compound interest and compare
relative advantages of these options;
(B) analyze and compare coverage options and rates
in insurance; and
(C) investigate and compare investment options
including stocks, bonds, annuities, and
retirement plans.
(8) The student uses algebraic and geometric models to
describe situations and solve problems. The student
is expected to:
(A) use geometric models available through
technology to model growth and decay in areas
such as population, biology, and ecology;
(B) use trigonometric ratios and functions
available through technology to calculate
distances and model periodic motion; and
(C) use direct and inverse variation to describe
physical laws such as Hook's, Newton's, and
Boyle's laws.
(9) The student uses algebraic and geometric models to
represent patterns and structures. The student is
expected to:
(A) use geometric transformations, symmetry, and
perspective drawings to describe mathematical
patterns and structure in art and
architecture; and
(B) use geometric transformations, proportions,
and periodic motion to describe mathematical
patterns and structure in music.
Source: The provisions of this §111.36 adopted to
be effective September 1, 1998, 22 TexReg 7623.