| (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) For students to become fluent in mathematics, students
must develop a robust sense of number. The National Research Council's
report, "Adding It Up," defines procedural fluency as "skill in carrying
out procedures flexibly, accurately, efficiently, and appropriately."
As students develop procedural fluency, they must also realize that
true problem solving may take time, effort, and perseverance. Students
in Grade 5 are expected to perform their work without the use of calculators.
(4) The primary focal areas in Grade 5 are solving
problems involving all four operations with positive rational numbers,
determining and generating formulas and solutions to expressions,
and extending measurement to area and volume. These focal areas are
supported throughout the mathematical strands of number and operations,
algebraic reasoning, geometry and measurement, and data analysis.
In Grades 3-5, the number set is limited to positive rational numbers.
In number and operations, students will apply place value and identify
part-to-whole relationships and equivalence. In algebraic reasoning,
students will represent and solve problems with expressions and equations,
build foundations of functions through patterning, identify prime
and composite numbers, and use the order of operations. In geometry
and measurement, students will classify two-dimensional figures, connect
geometric attributes to the measures of three-dimensional figures,
use units of measure, and represent location using a coordinate plane.
In data analysis, students will represent and interpret data.
(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.
(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, compare, and order positive rational
numbers and understand relationships as related to place value. The
student is expected to:
(A) represent the value of the digit in decimals through
the thousandths using expanded notation and numerals;
(B) compare and order two decimals to thousandths and
represent comparisons using the symbols >, <, or =; and
(C) round decimals to tenths or hundredths.
(3) Number and operations. The student applies mathematical
process standards to develop and use strategies and methods for positive
rational number computations in order to solve problems with efficiency
and accuracy. The student is expected to:
(A) estimate to determine solutions to mathematical
and real-world problems involving addition, subtraction, multiplication,
or division;
(B) multiply with fluency a three-digit number by a
two-digit number using the standard algorithm;
(C) solve with proficiency for quotients of up to a
four-digit dividend by a two-digit divisor using strategies and the
standard algorithm;
(D) represent multiplication of decimals with products
to the hundredths using objects and pictorial models, including area
models;
(E) solve for products of decimals to the hundredths,
including situations involving money, using strategies based on place-value
understandings, properties of operations, and the relationship to
the multiplication of whole numbers;
(F) represent quotients of decimals to the hundredths,
up to four-digit dividends and two-digit whole number divisors, using
objects and pictorial models, including area models;
(G) solve for quotients of decimals to the hundredths,
up to four-digit dividends and two-digit whole number divisors, using
strategies and algorithms, including the standard algorithm;
(H) represent and solve addition and subtraction of
fractions with unequal denominators referring to the same whole using
objects and pictorial models and properties of operations;
(I) represent and solve multiplication of a whole number
and a fraction that refers to the same whole using objects and pictorial
models, including area models;
(J) represent division of a unit fraction by a whole
number and the division of a whole number by a unit fraction such
as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models,
including area models;
(K) add and subtract positive rational numbers fluently;
and
(L) divide whole numbers by unit fractions and unit
fractions by whole numbers.
(4) Algebraic reasoning. The student applies mathematical
process standards to develop concepts of expressions and equations.
The student is expected to:
(A) identify prime and composite numbers;
(B) represent and solve multi-step problems involving
the four operations with whole numbers using equations with a letter
standing for the unknown quantity;
(C) generate a numerical pattern when given a rule
in the form y = ax or y = x + a and graph;
(D) recognize the difference between additive and multiplicative
numerical patterns given in a table or graph;
(E) describe the meaning of parentheses and brackets
in a numeric expression;
(F) simplify numerical expressions that do not involve
exponents, including up to two levels of grouping;
(G) use concrete objects and pictorial models to develop
the formulas for the volume of a rectangular prism, including the
special form for a cube (V = l x w x h, V = s x s x s, and V = Bh); and
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