 Number and Operations

• Instructional programs from prekindergarten through grade 12 should enable each and every student to—

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems
• Understand meanings of operations and how they relate to one another
• Compute fluently and make reasonable estimates

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

Pre-K–2 Expectations: In pre-K through grade 2 each and every student 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.

• 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.

• 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.

• 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 each and every student 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.

• 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.

• 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.

• 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 each and every student 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.

• 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.

• 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.  