Number

Procedural fluency in operations with real numbers and strategic competence in approximation are grounded in an understanding of place value. The rules of arithmetic govern operations on numbers and extend to operations in algebra:
 * Numbers can be added in any order with any grouping and multiplied in any order with any grouping.
 * Adding 0 and multiplying by 1 both leave a number unchanged.
 * All numbers have additive inverses, and all numbers except zero have multiplicative inverses.
 * Multiplication distributes over addition.

Subtraction and division are defined in terms of addition and multiplication, so are also governed by these rules. The place value system bundles units into 10s, then 10s into 100s, and so on, providing an efficient way to name large numbers. Subdividing in a similar way extends this to the decimal system, which provides an address system for locating all real numbers on the number line with arbitrarily high accuracy. Place value is the basis for efficient algorithms, reducing much computation to single-digit arithmetic. Mental computation strategies also make opportunistic use of the rules of arithmetic, as when the product 5×177×2 is computed at a glance to obtain 1770, rather than methodically working from left to right. An estimate may be more appropriate than an exact value, for example, when you want to know the number of calories in a meal. Often a result is reported using fewer digits than were calculated. A mature number sense includes having rules of thumb about how much accuracy is appropriate and understanding that accuracy to more than a few decimal places often takes substantial effort. Estimation and approximation are also useful in checking calculations. Rational numbers represented as fractions can be located on the number line by seeing them as numbers expressed in different units; for example, 3/5 is 3 units, where each unit is 1/5. However, rational numbers do not fill out the number line. There are also irrational numbers, such as π or √2. Each point on the number line then corresponds to a real number that is either rational or irrational. //Connections to Expressions, Functions and Coordinates.// The rules of arithmetic govern the manipulations of expressions and functions. Two perpendicular number lines define the coordinate plane.

Students understand that:
 * Core Concepts**
 * 1) The real numbers include the rational numbers and are in one-to-one correspondence with the points on the number line.
 * 2) Quantities can be compared using division, yielding rates and ratios.
 * 3) A fraction can represent the result of dividing the numerator by the denominator; equivalent fractions have the same value.
 * 4) Place value and the rules of arithmetic form the foundation for efficient algorithms.

Students can and do:
 * Core Skills**
 * 1) Compare numbers and make sense of their magnitude.
 * 2) Know when and how to use standard algorithms, and perform them flexibly, accurately and efficiently.
 * 3) Use mental strategies and technology to formulate, represent and solve problems.
 * 4) Solve multi-step problems involving fractions and percentages.
 * 5) Use estimation and approximation to solve problems.