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Topic: The Number System 

Common Core Mathematics 6-8

 Cluster  Compute fluently with multi-digit numbers and find common factors and multiples.
  Grade 6
   6.NS.2.Fluently divide multi-digit numbers using the standard algorithm.
   6.NS.3.Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
   6.NS.4.Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
 Cluster  Apply and extend previous understandings of numbers to the system of rational numbers.
  Grade 6
   6.NS.5.Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
   6.NS.6.Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
   6.NS.6.a.Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
   6.NS.6.b.Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
   6.NS.6.c.Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
   6.NS.7.a.Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
   6.NS.7.d.Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
   6.NS.8.Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
 Cluster  Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
  Grade 7
   7.NS.1.Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
   7.NS.1.a.Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
   7.EE.1.b.Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
   7.NS.1.c.Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
   7.NS.1.d.Apply properties of operations as strategies to add and subtract rational numbers.
   7.NS.2.Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
   7.NS.2.a.Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
   7.NS.6.b.Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing realworld contexts.
   7.NS.6.c.Apply properties of operations as strategies to multiply and divide rational numbers.
   7.NS.6.d.Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
   7.NS.3.Solve real-world and mathematical problems involving the four operations with rational numbers.[1Computations with rational numbers extend the rules for manipulating fractions to complex fractions]
 Cluster  Know that there are numbers that are not rational, and approximate them by rational numbers.
  Grade 8
   8.NS.1.Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
   8.NS.2.Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., ƒÎ2). For example, by truncating the decimal expansion of ã2, show that ã2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.