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Number and Operations Standard
Instructional programs from prekindergarten through
grade 12
should enable all students to—
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Understand numbers,
ways of representing numbers, relationships among numbers, and number
systems
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
| • |
count
with understanding and recognize "how many" in sets of objects;
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| • |
use multiple models to develop initial understandings of place
value and the base-ten number system; |
| • |
develop understanding of the relative position and magnitude
of whole numbers and of ordinal and cardinal numbers and their
connections; |
| • |
develop a sense of whole numbers and represent and use them
in flexible ways, including relating, composing, and decomposing
numbers; |
| • |
connect number words and numerals to the quantities they represent,
using various physical models and representations; |
| • |
understand
and represent commonly used fractions, such as 1/4, 1/3, and
1/2.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
| • |
understand the place-value structure of the base-ten number
system and be able to represent and compare whole numbers and
decimals; |
| • |
recognize equivalent representations for the same number and
generate them by decomposing and composing numbers; |
| • |
develop
understanding of fractions as parts of unit wholes, as parts
of a collection, as locations on number lines, and as divisions
of whole numbers; |
| • |
use models, benchmarks, and equivalent forms to judge the size
of fractions; |
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recognize and generate equivalent forms of commonly used fractions,
decimals, and percents; |
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explore numbers less than 0 by extending the number line and
through familiar applications; |
| • |
describe classes of numbers according to characteristics such
as the nature of their factors.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
| • |
work flexibly with fractions, decimals, and percents to solve
problems; |
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compare and order fractions, decimals, and percents efficiently
and find their approximate locations on a number line; |
| • |
develop meaning for percents greater than 100 and less than
1; |
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understand and use ratios and proportions to represent quantitative
relationships; |
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develop an understanding of large numbers and recognize and
appropriately use exponential, scientific, and calculator notation;
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use factors, multiples, prime factorization, and relatively
prime numbers to solve problems; |
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develop meaning for integers and represent and compare quantities
with them.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
| • |
develop a deeper understanding of very large and very small
numbers and of various representations of them; |
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compare and contrast the properties of numbers and number systems,
including the rational and real numbers, and understand complex
numbers as solutions to quadratic equations that do not have
real solutions; |
| • |
understand vectors and matrices as systems that have some of
the properties of the real-number system; |
| • |
use
number-theory arguments to justify relationships involving
whole numbers.
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Understand meanings
of operations and how they relate to one another
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
| • |
understand various meanings of addition and subtraction of whole
numbers and the relationship between the two operations; |
| • |
understand the effects of adding and subtracting whole numbers;
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understand situations that entail multiplication and division,
such as equal groupings of objects and sharing equally.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
| • |
understand various meanings of multiplication and division;
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understand the effects of multiplying and dividing whole numbers;
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| • |
identify
and use relationships between operations, such as division as
the inverse of multiplication, to solve problems; |
| • |
understand and use properties of operations, such as the distributivity
of multiplication over addition.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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understand the meaning and effects of arithmetic operations
with fractions, decimals, and integers; |
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use the associative and commutative properties of addition and
multiplication and the distributive property of multiplication
over addition to simplify computations with integers, fractions,
and decimals; |
| • |
understand and use the inverse relationships of addition and
subtraction, multiplication and division, and squaring and finding
square roots to simplify computations and solve problems.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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judge the effects of such operations as multiplication, division,
and computing powers and roots on the magnitudes of quantities;
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develop an understanding of properties of, and representations
for, the addition and multiplication of vectors and matrices;
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| • |
develop an understanding of permutations and combinations
as counting techniques.
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Compute fluently
and make reasonable estimates
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
| • |
develop and use strategies for whole-number computations, with
a focus on addition and subtraction; |
| • |
develop fluency with basic number combinations for addition
and subtraction; |
| • |
use
a variety of methods and tools to compute, including objects,
mental computation, estimation, paper and pencil, and calculators.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
| • |
develop fluency with basic number combinations for multiplication
and division and use these combinations to mentally compute
related problems, such as 30 50; |
| • |
develop fluency in adding, subtracting, multiplying, and dividing
whole numbers; |
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develop and use strategies to estimate the results of whole-number
computations and to judge the reasonableness of such results;
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develop and use strategies to estimate computations involving
fractions and decimals in situations relevant to students' experience;
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use visual models, benchmarks, and equivalent forms to add and
subtract commonly used fractions and decimals; |
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select appropriate methods and tools for computing with whole
numbers from among mental computation, estimation, calculators,
and paper and pencil according to the context and nature of
the computation and use the selected method or tools.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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select appropriate methods and tools for computing with fractions
and decimals from among mental computation, estimation, calculators
or computers, and paper and pencil, depending on the situation,
and apply the selected methods; |
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develop and analyze algorithms for computing with fractions,
decimals, and integers and develop fluency in their use; |
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develop and use strategies to estimate the results of rational-number
computations and judge the reasonableness of the results; |
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develop, analyze, and explain methods for solving problems involving
proportions, such as scaling and finding equivalent ratios.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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develop
fluency in operations with real numbers, vectors, and matrices,
using mental computation or paper-and-pencil calculations for
simple cases and technology for more-complicated cases. |
| • |
judge
the reasonableness of numerical computations and their results.
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