**The Number System**

**6th Grade**

**8th Grade**

CCSS.Math.Content.6.NS.C.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.

CCSS.Math.Content.6.NS.C.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.

CCSS.Math.Content.6.NS.C.6a

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.

CCSS.Math.Content.6.NS.C.6b

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.

CCSS.Math.Content.6.NS.C.6c

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

CCSS.Math.Content.6.NS.C.7

Understand ordering and absolute value of rational numbers.

CCSS.Math.Content.6.NS.C.7a

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

CCSS.Math.Content.6.NS.C.7b

Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.

CCSS.Math.Content.6.NS.C.7c

Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

Understand what positive and negative numbers are.

Understand the concept

of base 10.

The meaning

of 0.

Whole Numbers

What is a rational

number.

How to use a number line

with positive numbers.

Points on a number line.

Coordinate axes.

What is a coordinate plane.

Understand what are positive

and negative numbers.

Understand how to use

a number line.

Understand what 0

means.

Understand the oposites

of each number.

Understand what positive

and negative numbers are.

Know what are coordinate plane is

and the different quadrants.

Know what the X and

Y axis are.

Can plot a point on a number line

and coordinate plane.

Understand what an

integer is.

Understand what a rational

number is.

Be able to draw a

number line.

Know how to point points on a

number line and coordinate plane.

Know what real numbers are.

Know what rational numbers are.

Understand what absolute

value means.

Understand the symbols

< , > , and =.

Know what is a statement

of inequality is.

Know how to plot a point on a

number line.

Understand what number a point

is on a number line.

Understand what a rational

number is.

Know what positive

and negative numbers are.

Know what the degree

symbol means in terms

of temperature.

Understand what absolute

value means.

Understand what a rational

number is.

Understand how to

graph a number line.

Know what the difference is

between positive and

negative numbers.

Understand how money relates to

mathematics.

CCSS.Math.Content.6.NS.B.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)..

CCSS.Math.Content.6.NS.B.3

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

CCSS.Math.Content.6.NS.B.2

Fluently divide multi-digit numbers using the standard algorithm.

CCSS.Math.Content.6.NS.A.1

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

CCSS.Math.Content.6.NS.C.7d

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.

CCSS.Math.Content.6.NS.C.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.

**7th Grade**

CCSS.Math.Content.7.NS.A.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.

CCSS.Math.Content.7.NS.A.1

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.

CCSS.Math.Content.7.NS.A.1b

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.

CCSS.Math.Content.7.NS.A.1c

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.

CCSS.Math.Content.7.NS.A.1d

Apply properties of operations as strategies to add and subtract rational numbers.

Understand how to calculate the greatest common factor.

Whole Numbers

Understand how to calculate the least common multiple.

Understand the distributive property.

Add, subtract, multiply, and divide multidigit whole numbers

Understand decimals.

Understand the standard algorithim.

Division of single-digit numbers.

The Standard Algorithim

Understand Quotients

Understand Fractions

Understand Word Problems.

Understand graphing points.

Understand the quadrants of a graph.

Understand absolute values.

Understand the distance formula.

CCSS.Math.Content.7.NS.A.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

CCSS.Math.Content.7.NS.A.2a

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.

Understand the difference between a rational and irrational number.

Fractions include a numerator and denominator. A fraction expresses part of a whole.

Understand how to multiply and divide fractions.

Know that all fractions are rational numbers.

Know Properties of Operations:

Commutative Property of Multiplication (a * b = b * a)

Associative Property of Multiplication (a * (b * c) = (a * b) * c)

Distributive Property (a * (b + c) = a * b + a * c)

Multiplicative Identity Property (a * 1 = a)

Multiplicative Inverse Property ( a * (1 / a) = 1)... a cannot = 0

Zero Property (a * 0 = 0)

Know the difference between a negative number and positive number on the number line.

When multiplying signed numbers:

two numbers with same sign = positive

two numbers with different sign = negative

Fractions are used in everyday life: recipes - measuring

Know order of operations.

CCSS.Math.Content.7.NS.A.2b

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 real-world contexts.

Know distributive property

Know integers are whole numbers

Know rules for rational numbers: can be an integer, fraction, terminating decimal, or repeating decimal

Know meaning of divisor, quotient, dividend and remainder

Know if divisor or denominator is zero, it is undefined

CCSS.Math.Content.7.NS.A.2c

Apply properties of operations as strategies to multiply and divide rational numbers.

Know Properties of Operations:

Commutative Property of Multiplication (a * b = b * a)

Associative Property of Multiplication (a * (b * c) = (a * b) * c)

Distributive Property (a * (b + c) = a * b + a * c)

Multiplicative Identity Property (a * 1 = a)

Multiplicative Inverse Property ( a * (1 / a) = 1)...a cannot = 0

Zero Property (a * 0 = 0)

When multiplying and dividing fractions: common denominators are not needed. Convert the fraction to an improper fraction

When multiplying fractions: numerator times numerator, denominator times denominator; reduce fraction

When dividing fractions: change division sign into multiplication. invert the second fraction and multiply numerator times numerator, denominator times denominator

Know meaning of inverse fraction: substituting denominator for numerator and numerator for denominator

Know how to multiply and divide decimals

CCSS.Math.Content.7.NS.A.2d

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.

Know meaning of divisor, quotient, dividend and remainder

Know difference between a rational and irrational number

When dividing signed numbers:

two numbers with same sign = positive

two numbers with different sign = negative

CCSS.Math.Content.7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

Know order of operations

Know the four operations: addition, subtraction, multiplication, division

CCSS.Math.Content.8.NS.A.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.

Know what an irrational number is: decimals that never terminate or repeat... pi

Know how to divide and multiply decimals

Know how to indicate a repeating pattern in a decimal

CCSS.Math.Content.8.NS.A.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., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Know how to use a number line

All real numbers can be plotted on the number line

Know an irrational number is a real number

Know that square roots can be irrational or rational

Know how to identify a square root and a radical number

Know order of operations

Understand absolute value.

Understand comparisons.

Understand addition and subtraction.

Rational Numbers

Understand line diagrams.

Understand opposite quantities.

Understand the value 0.

Understand how math applies to real world situations.

Understand absolute value.

Understand a number line with a positive and negative direction.

Understand combining opposites.

Understand finding distances.

Rational Numbers

Understand how to use a number line.

Understand absolute value.

Understand inverses.

Understand properties of operations.

Rational Numbers