| (a) 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 computational thinking, mathematical 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, algorithms, paper and pencil, and technology
and techniques such as mental math, estimation, number sense, and
generalization and abstraction to solve problems. Students will effectively
communicate mathematical ideas, reasoning, and their implications
using multiple representations such as symbols, diagrams, graphs,
computer programs, 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) The primary focal areas in Grade 6 are number and
operations; proportionality; expressions, equations, and relationships;
and measurement and data. Students use concepts, algorithms, and properties
of rational numbers to explore mathematical relationships and to describe
increasingly complex situations. Students use concepts of proportionality
to explore, develop, and communicate mathematical relationships. Students
use algebraic thinking to describe how a change in one quantity in
a relationship results in a change in the other. Students connect
verbal, numeric, graphic, and symbolic representations of relationships,
including equations and inequalities. Students use geometric properties
and relationships, as well as spatial reasoning, to model and analyze
situations and solve problems. Students communicate information about
geometric figures or situations by quantifying attributes, generalize
procedures from measurement experiences, and use the procedures to
solve problems. Students use appropriate statistics, representations
of data, and reasoning to draw conclusions, evaluate arguments, and
make recommendations. While the use of all types of technology is
important, the emphasis on algebra readiness skills necessitates the
implementation of graphing technology.
(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.
(b) 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) Number and operations. The student applies mathematical
process standards to represent and use rational numbers in a variety
of forms. The student is expected to:
(A) classify whole numbers, integers, and rational
numbers using a visual representation such as a Venn diagram to describe
relationships between sets of numbers;
(B) identify a number, its opposite, and its absolute
value;
(C) locate, compare, and order integers and rational
numbers using a number line;
(D) order a set of rational numbers arising from mathematical
and real-world contexts; and
(E) extend representations for division to include
fraction notation such as a/b represents
the same number as a ÷ b where b ≠
0.
(3) Number and operations. The student applies mathematical
process standards to represent addition, subtraction, multiplication,
and division while solving problems and justifying solutions. The
student is expected to:
(A) recognize that dividing by a rational number and
multiplying by its reciprocal result in equivalent values;
(B) determine, with and without computation, whether
a quantity is increased or decreased when multiplied by a fraction,
including values greater than or less than one;
(C) represent integer operations with concrete models
and connect the actions with the models to standardized algorithms;
(D) add, subtract, multiply, and divide integers fluently;
and
(E) multiply and divide positive rational numbers fluently.
(4) Proportionality. The student applies mathematical
process standards to develop an understanding of proportional relationships
in problem situations. The student is expected to:
(A) compare two rules verbally, numerically, graphically,
and symbolically in the form of y = ax or y = x + a in order to differentiate between
additive and multiplicative relationships;
(B) apply qualitative and quantitative reasoning to
solve prediction and comparison of real-world problems involving ratios
and rates;
(C) give examples of ratios as multiplicative comparisons
of two quantities describing the same attribute;
(D) give examples of rates as the comparison by division
of two quantities having different attributes, including rates as
quotients;
(E) represent ratios and percents with concrete models,
fractions, and decimals;
(F) represent benchmark fractions and percents such
as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by
10 grids, strip diagrams, number lines, and numbers;
(G) generate equivalent forms of fractions, decimals,
and percents using real-world problems, including problems that involve
money; and
(H) convert units within a measurement system, including
the use of proportions and unit rates.
(5) Proportionality. The student applies mathematical
process standards to solve problems involving proportional relationships.
The student is expected to:
(A) represent mathematical and real-world problems
involving ratios and rates using scale factors, tables, graphs, and
proportions;
(B) solve real-world problems to find the whole given
a part and the percent, to find the part given the whole and the percent,
and to find the percent given the part and the whole, including the
use of concrete and pictorial models; and
(C) use equivalent fractions, decimals, and percents
to show equal parts of the same whole.
(6) Expressions, equations, and relationships. The
student applies mathematical process standards to use multiple representations
to describe algebraic relationships. The student is expected to:
(A) identify independent and dependent quantities from
tables and graphs;
(B) write an equation that represents the relationship
between independent and dependent quantities from a table; and
(C) represent a given situation using verbal descriptions,
tables, graphs, and equations in the form y
= kx or y = x + b.
(7) Expressions, equations, and relationships. The
student applies mathematical process standards to develop concepts
of expressions and equations. The student is expected to:
(A) generate equivalent numerical expressions using
order of operations, including whole number exponents and prime factorization;
(B) distinguish between expressions and equations verbally,
numerically, and algebraically;
(C) determine if two expressions are equivalent using
concrete models, pictorial models, and algebraic representations;
and
(D) generate equivalent expressions using the properties
of operations: inverse, identity, commutative, associative, and distributive
properties.
(8) Expressions, equations, and relationships. The
student applies mathematical process standards to use geometry to
represent relationships and solve problems. The student is expected
to:
(A) extend previous knowledge of triangles and their
properties to include the sum of angles of a triangle, the relationship
between the lengths of sides and measures of angles in a triangle,
and determining when three lengths form a triangle;
(B) model area formulas for parallelograms, trapezoids,
and triangles by decomposing and rearranging parts of these shapes;
(C) write equations that represent problems related
to the area of rectangles, parallelograms, trapezoids, and triangles
and volume of right rectangular prisms where dimensions are positive
rational numbers; and
(D) determine solutions for problems involving the
area of rectangles, parallelograms, trapezoids, and triangles and
volume of right rectangular prisms where dimensions are positive rational
numbers.
(9) Expressions, equations, and relationships. The
student applies mathematical process standards to use equations and
inequalities to represent situations. The student is expected to:
(A) write one-variable, one-step equations and inequalities
to represent constraints or conditions within problems;
(B) represent solutions for one-variable, one-step
equations and inequalities on number lines; and
(C) write corresponding real-world problems given one-variable,
one-step equations or inequalities.
(10) Expressions, equations, and relationships. The
student applies mathematical process standards to use equations and
inequalities to solve problems. The student is expected to:
(A) model and solve one-variable, one-step equations
and inequalities that represent problems, including geometric concepts;
and
(B) determine if the given value(s) make(s) one-variable,
one-step equations or inequalities true.
(11) Measurement and data. The student applies mathematical
process standards to use coordinate geometry to identify locations
on a plane. The student is expected to graph points in all four quadrants
using ordered pairs of rational numbers.
(12) Measurement and data. The student applies mathematical
process standards to use numerical or graphical representations to
analyze problems. The student is expected to:
(A) represent numeric data graphically, including dot
plots, stem-and-leaf plots, histograms, and box plots;
(B) use the graphical representation of numeric data
to describe the center, spread, and shape of the data distribution;
(C) summarize numeric data with numerical summaries,
including the mean and median (measures of center) and the range and
interquartile range (IQR) (measures of spread), and use these summaries
to describe the center, spread, and shape of the data distribution;
and
(D) summarize categorical data with numerical and graphical
summaries, including the mode, the percent of values in each category
(relative frequency table), and the percent bar graph, and use these
summaries to describe the data distribution.
(13) Measurement and data. The student applies mathematical
process standards to use numerical or graphical representations to
solve problems. The student is expected to:
(A) interpret numeric data summarized in dot plots,
stem-and-leaf plots, histograms, and box plots; and
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