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

### Clarifications

**Fluency Expectations or Examples of Culminating Standards**

Adding, subtracting, multiplying, and dividing rational numbers is the culmination of numerical work with the four basic operations. The number system will continue to develop in grade 8, expanding to become the real numbers by the introduction of irrational numbers, and will develop further in high school, expanding to become the complex numbers with the introduction of imaginary numbers. Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic, fluency with rational number arithmetic should be the goal in grade 7.

**Subject Area:**Mathematics

**Grade:**7

**Domain-Subdomain:**The Number System

**Cluster:**Level 2: Basic Application of Skills & Concepts

**Cluster:**Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

**Date Adopted or Revised:**02/14

**Date of Last Rating:**02/14

**Status:**State Board Approved

**Assessed:**Yes

**Assessment Limits :**

7.NS.1.2a, 2b, and 2c require the incorporation of a negative value.**Calculator :**No

**Context :**Allowable

**Test Item #:**Sample Item 1**Question:**Joe and Scott equally share a pizza.

If Scott eats of his portion of the pizza, what fraction of the whole pizza does he eat?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 2**Question:**In Homestead, of the households own pets. Of the households with pets, have cats.

What fraction of the households in Homestead own cats?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 3**Question:**Sandy uses of a pound of raisins in each batch of raisin bread.

Yesterday, Sandy used of a pound of raisins. How many batches of raisin bread did sandy make yesterday?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 4**Question:**Springfield has an elevation of −150 feet. Greenville is 3 times as far below sea level as Springfield.What is Greenville’s elevation, in feet?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 5**Question:**An equation is shown, where z < 0.

x · y = z

A. Assume x > 0. Drag the point to the number line to identify a possible location for y.

B. Assume x < 0. Drag the point to the number line to identify a possible location for y.

**Difficulty:**N/A**Type:**GRID: Graphic Response Item Display

**Test Item #:**Sample Item 6**Question:**What is written as a decimal?

**Difficulty:**N/A**Type:**MC: Multiple Choice

## Related Courses

## Related Access Points

## Related Resources

## Assessments

## Educational Game

## Educational Software / Tool

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Problem-Solving Tasks

## Tutorials

## Video/Audio/Animation

## Virtual Manipulative

## MFAS Formative Assessments

Students are asked to evaluate expressions involving multiplication of rational numbers and use the properties of operations to simplify calculations.

Students are asked to use long division to convert four different fractions to equivalent decimals and to identify those that are rational.

Students are asked to describe a real-world context for a given expression involving the quotient of two rational integers.

Students are shown a problem that illustrates why the product of two negatives is a positive and are asked to provide a rationale.

Students are asked to describe a real-world context for a given expression involving the product of two rational numbers.

Students are given an integer division problem and asked to identify fractions which are equivalent to the division problem.

Students are asked to explain why the product of a positive and a negative rational number is negative.

## Original Student Tutorials Mathematics - Grades 6-8

Use mathematical properties to explain why a negative factor times a negative factor equals a positive product… instead of just quoting a rule with this interactive tutorial.

## Student Resources

## Original Student Tutorial

Use mathematical properties to explain why a negative factor times a negative factor equals a positive product… instead of just quoting a rule with this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

This interactive game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. Students can play individually or in teams.

Type: Educational Game

## Educational Software / Tool

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Software / Tool

## Problem-Solving Task

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Type: Problem-Solving Task

## Tutorials

This video shows some examples that test your understanding of what happens when positive and negative numbers are multiplied and divided.

Type: Tutorial

Students will learn how to convert difficult repeating decimals to fractions.

Type: Tutorial

This tutorial shows students how to convert basic repeating decimals to fractions.

Type: Tutorial

Students will learn how to convert a fraction into a repeating decimal. Students should know how to use long division before starting this tutorial.

Type: Tutorial

In this video, you will practice changing a fraction into decimal form.

Type: Tutorial

You will learn how multiplication and division problems give us a positive or negative answer depending on whether there are an even or odd number of negative integers used in the problem.

Type: Tutorial

In this tutorial, you will simplify expressions involving positive and negative fractions.

Type: Tutorial

In this tutorial, you will see how to simplify complex fractions.

Type: Tutorial

In this tutorial, you will evaluate fractions involving negative numbers and variables to determine if expressions are equivalent.

Type: Tutorial

In this tutorial, you will see how to divide fractions involving negative integers.

Type: Tutorial

In this tutorial you will practice multiplying and dividing fractions involving negative numbers.

Type: Tutorial

In this tutorial, you will learn rules for multiplying positive and negative integers.

Type: Tutorial

In this tutorial you will learn how to divide with negative integers.

Type: Tutorial

In this tutorial you will use the repeated addition model of multiplication to help you understand why multiplying negative numbers results in a positive answer.

Type: Tutorial

In this tutorial, you will use the distributive property to understand why the product of two negative numbers is positive.

Type: Tutorial

This tutorial demonstrates the number line method of multiplying integers. You will encounter four different combinations when multiplying integers: (1) Positive times positive, (2) Positive times negative, (3) Negative times negative, (4) Negative times positive. The lesson is available in video format, and there is a quiz for practice.

Type: Tutorial

The first fractions used by ancient civilizations were "unit fractions." Later, numerators other than one were added, creating "vulgar fractions" which became our modern fractions. Together, fractions and integers form the "rational numbers."

Type: Tutorial

When number systems were expanded to include negative numbers, rules had to be formulated so that multiplication would be consistent regardless of the sign of the operands.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Video/Audio/Animation

Any fraction can be converted into an equivalent decimal number with a sequence of digits after the decimal point, which either repeats or terminates. The reason can be understood by close examination of the number line.

Type: Video/Audio/Animation

## Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

## Parent Resources

## Problem-Solving Task

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Type: Problem-Solving Task

## Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative