Common Core State Standards for Mathematics
Grade 6
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers.
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.
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.
- 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.
- 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.
- 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.
Grade 7
The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
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.
- 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.
- 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.
- 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.
- Apply properties of operations as strategies to add and subtract rational numbers.
Ohio Mathematics Academic Content Standards (2001)
Number, Number Sense and Operations Standard
Benchmarks (3–4)
H.
Use relationships between operations, such as subtraction as the inverse of addition and division as the inverse of multiplication.
Benchmarks (5–7)
A.
Represent and compare numbers less than 0 through familiar applications and extending the number line.
I.
Use a variety of strategies, including proportional reasoning, to estimate, compute, solve and explain solutions to problems involving integers, fractions, decimals and percents.
Grade Level Indicators (Grade 3)
10.
Explain and use relationships between operations, such as:
a. relate addition and subtraction as inverse operations;
b. relate multiplication and division as inverse operations;
c. relate addition to multiplication (repeated addition);
d. relate subtraction to division (repeated subtraction).
Grade Level Indicators (Grade 5)
6.
Represent and compare numbers less than 0 by extending the number line and using familiar applications; e.g., temperature, owing money.
8.
Identify and use relationships between operations to solve problems.
Grade Level Indicators (Grade 6)
7.
Use simple expressions involving integers to represent and solve problems; e.g., if a running back loses 15 yards on the first carry but gains 8 yards on the second carry, what is the net gain/loss?
Grade Level Indicators (Grade 7)
6.
Simplify numerical expressions involving integers and use integers to solve real-life problems.
Principles and Standards for School Mathematics
Number and Operations Standard
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Expectations (3–5)
explore numbers less than 0 by extending the number line and through familiar applications;
Expectations (6–8)
develop meaning for integers and represent and compare quantities with them.
Understand meanings of operations and how they relate to one another
Expectations (3–5)
explore numbers less than 0 by extending the number line and through familiar applications;
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;
Expectations (6–8)
develop meaning for integers and represent and compare quantities with them.
Compute fluently and make reasonable estimates
Expectations (3–5)
explore numbers less than 0 by extending the number line and through familiar applications;
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;
Expectations (6–8)
develop meaning for integers and represent and compare quantities with them.
develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;