**Instructional programs from prekindergarten through grade
12 should enable all students to—**

**Understand numbers, ways of representing numbers,
relationships among numbers, and number systems **

**Pre-K–2 Expectations:** In pre-K through grade 2 all
students should–

- count with understanding and recognize "how
many" in sets of objects;
- 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.

**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;
- recognize and generate equivalent forms of
commonly used fractions, decimals, and percents;
- 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.

**Grades 6–8 Expectations**: In grades 6–8 all students
should–

- work flexibly with fractions, decimals, and
percents to solve problems;
- 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;
- understand and use ratios and proportions to
represent quantitative relationships;
- develop an understanding of large numbers and
recognize and appropriately use exponential, scientific, and calculator
notation;
- use factors, multiples, prime factorization, and
relatively prime numbers to solve problems;
- develop meaning for integers and represent and
compare quantities with them.

**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;
- 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.

**Understand
meanings of operations and how they relate to one another **

**Pre-K–2 Expectations**: In pre-K 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;
- understand situations that entail multiplication
and division, such as equal groupings of objects and sharing equally.

**Grades 3–5 Expectations**: In grades 3–5 all students
should–

- understand various meanings of multiplication
and division;
- understand the effects of multiplying and
dividing whole numbers;
- 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.

**Grades 6–8 Expectations**: In grades 6–8 all students
should–

- understand the meaning and effects of arithmetic
operations with fractions, decimals, and integers;
- 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.

**Grades 9–12 Expectations: **In grades 9-12 all students
should–

- judge the effects of such operations as
multiplication, division, and computing powers and roots on the magnitudes of
quantities;
- develop an understanding of properties of, and
representations for, the addition and multiplication of vectors and matrices;
- develop an understanding of permutations and
combinations as counting techniques.

**Compute fluently and make reasonable estimates **

**Pre-K–2 Expectations**: In pre-K 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.

**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;
- develop and use strategies to estimate the
results of whole-number computations and to judge the reasonableness of such
results;
- develop and use strategies to estimate
computations involving fractions and decimals in situations relevant to
students' experience;
- use visual models, benchmarks, and equivalent
forms to add and subtract commonly used fractions and decimals;
- 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.

**Grades 6–8 Expectations**: In grades 6–8 all students
should–

- 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;
- develop and analyze algorithms for computing
with fractions, decimals, and integers and develop fluency in their use;
- develop and use strategies to estimate the
results of rational-number computations and judge the reasonableness of the
results;
- develop, analyze, and explain methods for
solving problems involving proportions, such as scaling and finding equivalent
ratios.

**Grades 9–12 Expectations**: In grades 9–12 all students should–

- 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.