| (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 4 are expected to perform their work without the use of calculators.
(4) The primary focal areas in Grade 4 are use of operations,
fractions, and decimals and describing and analyzing geometry and
measurement. 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 represent points on a number line that correspond
to a given fraction or terminating decimal. In algebraic reasoning,
students will represent and solve multi-step problems involving the
four operations with whole numbers with expressions and equations
and generate and analyze patterns. In geometry and measurement, students
will classify two-dimensional figures, measure angles, and convert
units of measure. 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 whole numbers and
decimals and understand relationships related to place value. The
student is expected to:
(A) interpret the value of each place-value position
as 10 times the position to the right and as one-tenth of the value
of the place to its left;
(B) represent the value of the digit in whole numbers
through 1,000,000,000 and decimals to the hundredths using expanded
notation and numerals;
(C) compare and order whole numbers to 1,000,000,000
and represent comparisons using the symbols >, <, or =;
(D) round whole numbers to a given place value through
the hundred thousands place;
(E) represent decimals, including tenths and hundredths,
using concrete and visual models and money;
(F) compare and order decimals using concrete and visual
models to the hundredths;
(G) relate decimals to fractions that name tenths and
hundredths; and
(H) determine the corresponding decimal to the tenths
or hundredths place of a specified point on a number line.
(3) Number and operations. The student applies mathematical
process standards to represent and generate fractions to solve problems.
The student is expected to:
(A) represent a fraction a/b as
a sum of fractions 1/b, where a and b are
whole numbers and b > 0, including
when a > b;
(B) decompose a fraction in more than one way into
a sum of fractions with the same denominator using concrete and pictorial
models and recording results with symbolic representations;
(C) determine if two given fractions are equivalent
using a variety of methods;
(D) compare two fractions with different numerators
and different denominators and represent the comparison using the
symbols >, =, or <;
(E) represent and solve addition and subtraction of
fractions with equal denominators using objects and pictorial models
that build to the number line and properties of operations;
(F) evaluate the reasonableness of sums and differences
of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring
to the same whole; and
(G) represent fractions and decimals to the tenths
or hundredths as distances from zero on a number line.
(4) Number and operations. The student applies mathematical
process standards to develop and use strategies and methods for whole
number computations and decimal sums and differences in order to solve
problems with efficiency and accuracy. The student is expected to:
(A) add and subtract whole numbers and decimals to
the hundredths place using the standard algorithm;
(B) determine products of a number and 10 or 100 using
properties of operations and place value understandings;
(C) represent the product of 2 two-digit numbers using
arrays, area models, or equations, including perfect squares through
15 by 15;
(D) use strategies and algorithms, including the standard
algorithm, to multiply up to a four-digit number by a one-digit number
and to multiply a two-digit number by a two-digit number. Strategies
may include mental math, partial products, and the commutative, associative,
and distributive properties;
(E) represent the quotient of up to a four-digit whole
number divided by a one-digit whole number using arrays, area models,
or equations;
(F) use strategies and algorithms, including the standard
algorithm, to divide up to a four-digit dividend by a one-digit divisor;
(G) round to the nearest 10, 100, or 1,000 or use compatible
numbers to estimate solutions involving whole numbers; and
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