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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 7 Unit 5

Rational Number Arithmetic

Lesson 11: Dividing Rational Numbers

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topics
Definition--Rationals and Radicals--Rational Functions Definition--Rationals and Radicals--Rational Functions Rational Functions

Topic

Rationals and Radicals

Definition

Rational functions are functions that are the ratio of two polynomials.

Description

Rational Functions are a key concept in the study of Rational Numbers, Expressions, Equations, and Functions. These functions are the ratio of two polynomials, such as 

$$f(x) = \frac{P(x)}{Q(x)}$$

where P(x) and Q(x) are polynomials. Understanding rational functions involves analyzing their behavior, including identifying asymptotes, intercepts, and discontinuities. For example, the function 

$$f(x) = \frac{1}{x}$$

Rational Functions and Equations
Definition--Rationals and Radicals--Rational Numbers Definition--Rationals and Radicals--Rational Numbers Rational Numbers

Topic

Rationals and Radicals

Definition

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero.

Rational Functions and Equations
Definition--Rationals and Radicals--Rationalizing a Radical Definition--Rationals and Radicals--Rationalizing a Radical Rationalizing a Radical

Topic

Rationals and Radicals

Definition

Rationalizing a radical is the process of eliminating radicals from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing a Radical is an important technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves eliminating radicals from the denominator of a fraction, which simplifies the expression and often makes it easier to work with or compare to other expressions. For example, to rationalize the denominator of 

$$\frac{1}{\sqrt{3}}$$

Radical Expressions
Definition--Rationals and Radicals--Rationalizing the Denominator Definition--Rationals and Radicals--Rationalizing the Denominator Rationalizing the Denominator

Topic

Rationals and Radicals

Definition

Rationalizing the denominator is the process of eliminating radicals or complex numbers from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing the Denominator is a crucial technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves removing radicals or complex numbers from the denominator of a fraction, which simplifies the expression and often makes it easier to evaluate or compare with other expressions. For example, to rationalize the denominator of 

Radical Expressions
Definition--Rationals and Radicals--Simplifying a Radical Expression Definition--Rationals and Radicals--Simplifying a Radical Expression Simplifying a Radical Expression

Topic

Rationals and Radicals

Definition

Simplifying a radical expression involves reducing the expression to its simplest form by factoring the radicand and removing any perfect square factors (for square roots) or perfect cube factors (for cube roots).

Description

Simplifying a Radical Expression is a fundamental skill in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves reducing a radical expression to its simplest form, which often makes it easier to work with and understand. For example, simplifying 

$$\sqrt{18}$$

results in 

Radical Expressions
Definition--Rationals and Radicals--Simplifying a Rational Expression Definition--Rationals and Radicals--Simplifying a Rational Expression Simplifying a Rational Expression

Topic

Rationals and Radicals

Definition

Simplifying a rational expression involves reducing the fraction to its lowest terms by factoring both the numerator and denominator and canceling common factors.

Description

Simplifying a Rational Expression is a crucial skill in the study of Rational Numbers, Expressions, Equations, and Functions. This process involves reducing a rational expression to its simplest form by factoring both the numerator and denominator and canceling common factors. For example, simplifying 

$$\frac{x^2 - 1}{x - 1}$$

results in x + 1 for 

Rational Expressions
Definition--Rationals and Radicals--Square Root Definition--Rationals and Radicals--Square Root Square Root

Topic

Rationals and Radicals

Definition

The square root of a number is a value that, when multiplied by itself, gives the number. It is denoted by the radical symbol √.

Radical Expressions
Definition--Rationals and Radicals--Vertical Asymptote Definition--Rationals and Radicals--Vertical Asymptote Vertical Asymptote

Topic

Rationals and Radicals

Definition

A vertical asymptote is a vertical line that the graph of a function approaches but never reaches as the input values get closer to a certain point.

Description

Vertical Asymptotes are a crucial concept in the study of Rational Numbers, Expressions, Equations, and Functions. They occur in rational functions when the denominator equals zero for certain input values, causing the function to approach infinity or negative infinity. For example, the function 

$$f(x) = \frac{1}{x-2}$$

Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Tutorial: Adding and Subtracting Rational Numbers INSTRUCTIONAL RESOURCE: Tutorial: Adding and Subtracting Rational Numbers INSTRUCTIONAL RESOURCE: Tutorial: Adding and Subtracting Rational Numbers

In this Slide Show, learn how to add and subtract rational numbers. Includes links to several Media4Math videos and a math game.

This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file.

Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

Rational Expressions and Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Math Examples--Rational Expressions INSTRUCTIONAL RESOURCE: Math Examples 44 INSTRUCTIONAL RESOURCE: Math Examples--Rational Expressions

The complete set of 28 examples that make up this set of tutorials.

This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file.

Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

Rational Expressions and Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Math Examples--Rational Functions in Tabular and Graph Form INSTRUCTIONAL RESOURCE: Math Examples 45 INSTRUCTIONAL RESOURCE: Math Examples--Rational Functions in Tabular and Graph Form

This set of tutorials provides 28 examples of rational functions in tabular and graph form.

This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file.

Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

Rational Expressions and Rational Functions and Equations
Tutorial Instructional Resource: Tutorial: What Are Rational Expressions? Instructional Resource: Tutorial: What Are Rational Expressions?

In this tutorial students learn about rational expressions and use the properties of fractions to combine them.

—Press PREVIEW to see the tutorial—

Rational Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 1 Math Clip Art--Number Systems--Rational Numbers 01 Math Clip Art--Number Systems--Rational Numbers 01

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 2 Math Clip Art--Number Systems--Rational Numbers 02 Math Clip Art--Number Systems--Rational Numbers 02

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 3 Math Clip Art--Number Systems--Rational Numbers 03 Math Clip Art--Number Systems--Rational Numbers 03

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 4 Math Clip Art--Number Systems--Rational Numbers 04 Math Clip Art--Number Systems--Rational Numbers 04

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 5 Math Clip Art--Number Systems--Rational Numbers 05 Math Clip Art--Number Systems--Rational Numbers 05

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 6 Math Clip Art--Number Systems--Rational Numbers 06 Math Clip Art--Number Systems--Rational Numbers 06

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 7 Math Clip Art--Number Systems--Rational Numbers 07 Math Clip Art--Number Systems--Rational Numbers 07

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 8 Math Clip Art--Number Systems--Rational Numbers 08 Math Clip Art--Number Systems--Rational Numbers 08

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 9 Math Clip Art--Number Systems--Rational Numbers 09 Math Clip Art--Number Systems--Rational Numbers 09

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 10 Math Clip Art--Number Systems--Rational Numbers 10 Math Clip Art--Number Systems--Rational Numbers 10

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Clip Art--Number Systems--Rational Numbers, Image 11 Math Clip Art--Number Systems--Rational Numbers 11 Math Clip Art--Number Systems--Rational Numbers 11

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 1 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 1 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 1

Topic

Exponents

Description

Shows Example 1 with the expression 32. The solution explains how to simplify by multiplying 3 by itself according to the exponent. Example 1: Simplify 32. Multiply 3 by itself two times: 3•3 = 9.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 10 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 10 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 10

Topic

Exponents

Description

Shows Example 10 with the expression (-3)3•54. The solution uses order of operations to evaluate each term before multiplying. Example 10: Simplify (-3)3•54. Evaluate each exponential term separately, then multiply: -27•625= -16,875.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 11 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 11 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 11

Topic

Exponents

Description

Example 11 shows how to simplify (1/2)2. The image illustrates multiplying 1/2 by itself due to the exponent of 2. Example 11: Simplify (1/2)2. Multiply one-half two times. Solution: (1/2) * (1/2) = 1/4.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 12 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 12 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 12

Topic

Exponents

Description

Example 12 shows how to simplify (1/3)4. The image illustrates multiplying 1/3 four times due to the exponent of 4. Example 12: Simplify (1/3)4. Multiply one-third four times. Solution: (1/3) * (1/3) * (1/3) * (1/3) = 1/81.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 13 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 13 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 13

Topic

Exponents

Description

Example 13 shows how to simplify (-1/3)3. The image illustrates multiplying -1/3 three times due to the exponent of 3. Example 13: Simplify (-1/3)3. Multiply negative one-third three times. Solution: (-1/3) * (-1/3) * (-1/3) = -1/27.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 14 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 14 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 14

Topic

Exponents

Description

Example 14 shows how to simplify 2-1. The image illustrates rewriting 2-1 as the reciprocal of 2 raised to the first power. Example 14: Simplify 2-1. Negative exponents are written as reciprocals. Solution: 2-1 = 1/2.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 15 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 15 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 15

Topic

Exponents

Description

Example 15 shows how to simplify 2-2. The image illustrates rewriting 2-2 as the reciprocal of 2 raised to the second power. Example 15: Simplify 2-2. Negative exponents are written as reciprocals. Solution: 2-2 = 1/(22) = 1/4.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 2 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 2 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 2

Topic

Exponents

Description

Shows Example 2 with the expression 43. The solution explains the simplification by multiplying 4 three times. Example 2: Simplify 43. Multiply 4 by itself three times: 4•4•4 = 64.

In general, the topic of exponents involves understanding how repeated multiplication can be expressed more compactly. The examples provided in this collection allow students to see the step-by-step breakdown of how to simplify various exponential expressions, which can include positive and negative bases, fractional bases, and negative exponents.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 3 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 3 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 3

Topic

Exponents

Description

Shows Example 3 with the expression 54. The solution details multiplying 5 four times. Example 3: Simplify 54. Multiply 5 by itself four times: 5•5•5•5 = 625.

In general, the topic of exponents involves understanding how repeated multiplication can be expressed more compactly. The examples provided in this collection allow students to see the step-by-step breakdown of how to simplify various exponential expressions, which can include positive and negative bases, fractional bases, and negative exponents.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 4 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 4 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 4

Topic

Exponents

Description

Shows Example 4 with the expression 106. The solution simplifies by multiplying 10 six times. Example 4: Simplify 106. Multiply 10 by itself six times: 10•10•10•10•10•10 = 1,000,000.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 5 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 5 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 5

Topic

Exponents

Description

Shows Example 5 with the expression (-1)2. The solution explains the result by multiplying -1 by itself. Example 5: Simplify (-1)2. Multiply -1 by itself two times: (-1)•(-1) = 1.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 6 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 6 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 6

Topic

Exponents

Description

Shows Example 6 with the expression (-5)3. The solution demonstrates multiplying -5 three times. Example 6: Simplify (_5)3. Multiply -5 by itself three times: (-5)•(-5)•(-5) = -125.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 7 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 7 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 7

Topic

Exponents

Description

Shows Example 7 with the expression (-6)4. The solution explains multiplying -6 by itself four times. Example 7: Simplify (-6)4. Multiply -6 by itself four times: (-6)•(-6)•(-6)•(-6) = 1296.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 8 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 8 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 8

Topic

Exponents

Description

Shows Example 8 with the expression 23•32. The solution demonstrates using order of operations to simplify each term separately, then multiply. Example 8: Simplify 23•32. Evaluate each exponential term separately, then multiply: 8•9 = 72.

Numerical Expressions
Math Example--Exponential Concepts--Integer and Rational Exponents--Example 9 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 9 Math Example--Exponential Concepts--Integer and Rational Exponents--Example 9

Topic

Exponents

Description

Shows Example 9 with the expression (-1)2•43. The solution explains simplifying each term individually before multiplying. Example 9: Simplify (-1)2•43. Evaluate each exponential term separately, then multiply: 1•64 = 64.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 1 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 1 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 1

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 10 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 10 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 10

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 2 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 2 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 2

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 3 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 3 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 3

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
ComparingOrderingIntegersRationals--Example04.png Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 4 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 4

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 5 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 5 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 5

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 6 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 6 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 6

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 7 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 7 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 7

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 8 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 8 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 8

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 9 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 9 Math Example--Rational Concepts--Comparing and Ordering Integers and Rational Numbers--Example 9

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 1 Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 1 Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 1

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions
Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 10 Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 10 Math Example--Rational Concepts--Graphing Integers and Rational Numbers on a Coordinate Grid--Example 10

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Numerical Expressions