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 111.48 - Algebraic Reasoning, Adopted 2015 (One Credit)
Universal Citation: 19 TX Admin Code ยง 111.48
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 Algebraic
Reasoning, students will build on the knowledge and skills for mathematics in
Kindergarten-Grade 8 and Algebra I, continue with the development of
mathematical reasoning related to algebraic understandings and processes, and
deepen a foundation for studies in subsequent mathematics courses. Students
will broaden their knowledge of functions and relationships, including linear,
quadratic, square root, rational, cubic, cube root, exponential, absolute
value, and logarithmic functions. Students will study these functions through
analysis and application that includes explorations of patterns and structure,
number and algebraic methods, and modeling from data using tools that build to
workforce and college readiness such as probes, measurement tools, and software
tools, including spreadsheets.
(4)
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, or justify mathematical
ideas and arguments using precise mathematical language in written or oral
communication.
(2)
Patterns and structure. The student applies mathematical processes to connect
finite differences or common ratios to attributes of functions. The student is
expected to:
(A) determine the patterns that
identify the relationship between a function and its common ratio or related
finite differences as appropriate, including linear, quadratic, cubic, and
exponential functions;
(B) classify
a function as linear, quadratic, cubic, and exponential when a function is
represented tabularly using finite differences or common ratios as
appropriate;
(C) determine the
function that models a given table of related values using finite differences
and its restricted domain and range; and
(D) determine a function that models
real-world data and mathematical contexts using finite differences such as the
age of a tree and its circumference, figurative numbers, average velocity, and
average acceleration.
(3) Patterns and structure. The student
applies mathematical processes to understand the connections among
representations of functions and combinations of functions, including the
constant function, f(x) = x,
f(x) =
x2,
f(x) = [Square Root]x, f(x)
=
1/x,
f(x) =
x3,
f(x) =
3[Square Root]x,
f(x) =
bx,
f(x) = |x|, and f(x) =
logb
(x) where b is 10
or e; functions and their inverses; and key attributes of
these functions. The student is expected to:
(A) compare and contrast the key attributes,
including domain, range, maxima, minima, and intercepts, of a set of functions
such as a set comprised of a linear, a quadratic, and an exponential function
or a set comprised of an absolute value, a quadratic, and a square root
function tabularly, graphically, and symbolically;
(B) compare and contrast the key attributes
of a function and its inverse when it exists, including domain, range, maxima,
minima, and intercepts, tabularly, graphically, and symbolically;
(C) verify that two functions are inverses of
each other tabularly and graphically such as situations involving compound
interest and interest rate, velocity and braking distance, and
Fahrenheit-Celsius conversions;
(D)
represent a resulting function tabularly, graphically, and symbolically when
functions are combined or separated using arithmetic operations such as
combining a 20% discount and a 6% sales tax on a sale to determine
h(x), the total sale, f(x) =
0.8x,
g(x) = 0.06(0.8x), and
h(x) = f(x) +
g(x);
(E) model a
situation using function notation when the output of one function is the input
of a second function such as determining a function h(x), =
g(f(x)) = 1.06(0.8x) for the final purchase
price, h(x), of an item with price x dollars
representing a 20% discount, f(x) = 0.8x
followed by a 6% sales tax, g(x) = 1.06x;
and
(F) compare and contrast a
function and possible functions that can be used to build it tabularly,
graphically, and symbolically such as a quadratic function that results from
multiplying two linear functions.
(4) Number and algebraic methods. The student
applies mathematical processes to simplify and perform operations on functions
represented in a variety of ways, including real-world situations. The student
is expected to:
(A) connect tabular
representations to symbolic representations when adding, subtracting, and
multiplying polynomial functions arising from mathematical and real-world
situations such as applications involving surface area and volume;
(B) compare and contrast the results when
adding two linear functions and multiplying two linear functions that are
represented tabularly, graphically, and symbolically;
(C) determine the quotient of a polynomial
function of degree three and of degree four when divided by a polynomial
function of degree one and of degree two when represented tabularly and
symbolically; and
(D) determine the
linear factors of a polynomial function of degree two and of degree three when
represented symbolically and tabularly and graphically where
appropriate.
(5) Number
and algebraic methods. The student applies mathematical processes to represent,
simplify, and perform operations on matrices and to solve systems of equations
using matrices. The student is expected to:
(A) add and subtract matrices;
(B) multiply matrices;
(C) multiply matrices by a scalar;
(D) represent and solve systems of two linear
equations arising from mathematical and real-world situations using matrices;
and
(E) represent and solve systems
of three linear equations arising from mathematical and real-world situations
using matrices and technology.
(6) Number and algebraic methods. The student
applies mathematical processes to estimate and determine solutions to equations
resulting from functions and real-world applications with fluency. The student
is expected to:
(A) estimate a reasonable
input value that results in a given output value for a given function,
including quadratic, rational, and exponential functions;
(B) solve equations arising from questions
asked about functions that model real-world applications, including linear and
quadratic functions, tabularly, graphically, and symbolically; and
(C) approximate solutions to equations
arising from questions asked about exponential, logarithmic, square root, and
cubic functions that model real-world applications tabularly and
graphically.
(7)
Modeling from data. The student applies mathematical processes to analyze and
model data based on real-world situations with corresponding functions. The
student is expected to:
(A) represent domain
and range of a function using interval notation, inequalities, and set
(builder) notation;
(B) compare and
contrast between the mathematical and reasonable domain and range of functions
modeling real-world situations, including linear, quadratic, exponential, and
rational functions;
(C) determine
the accuracy of a prediction from a function that models a set of data compared
to the actual data using comparisons between average rates of change and finite
differences such as gathering data from an emptying tank and comparing the
average rate of change of the volume or the second differences in the volume to
key attributes of the given model;
(D) determine an appropriate function model,
including linear, quadratic, and exponential functions, for a set of data
arising from real-world situations using finite differences and average rates
of change; and
(E) determine if a
given linear function is a reasonable model for a set of data arising from a
real-world situation.
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