Illustrative Math Alignment: Grade 7 Unit 5
Rational Number Arithmetic
Lesson 14: Solving Problems with Rational Numbers
Use the following Media4Math resources with this Illustrative Math lesson.
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Definition--Rationals and Radicals--Rationalizing a Radical | Rationalizing a RadicalTopicRationals and Radicals DefinitionRationalizing 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. DescriptionRationalizing 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 |
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Definition--Rationals and Radicals--Rationalizing the Denominator | Rationalizing the DenominatorTopicRationals and Radicals DefinitionRationalizing 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. DescriptionRationalizing 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 |
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Definition--Rationals and Radicals--Simplifying a Radical Expression | Simplifying a Radical ExpressionTopicRationals and Radicals DefinitionSimplifying 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). DescriptionSimplifying 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 |
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Definition--Rationals and Radicals--Simplifying a Rational Expression | Simplifying a Rational ExpressionTopicRationals and Radicals DefinitionSimplifying a rational expression involves reducing the fraction to its lowest terms by factoring both the numerator and denominator and canceling common factors. DescriptionSimplifying 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 |
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Definition--Rationals and Radicals--Square Root | Square RootTopicRationals and Radicals DefinitionThe 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 |
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Definition--Rationals and Radicals--Vertical Asymptote | Vertical AsymptoteTopicRationals and Radicals DefinitionA 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. DescriptionVertical 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 |
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Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 1 | Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 1TopicGeometric Models |
Applications of Surface Area and Volume and Rational Functions and Equations |
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Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2 | Google Earth Voyager Story: The Geometry of Sustainable Architecture, Part 2TopicGeometric Models |
Surface Area, Volume and Rational Functions and Equations |
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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 ResourcesTo see the complete library of Instructional Resources , click on this link. |
Rational Expressions and Rational Functions and Equations |
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INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Rational Functions | INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Rational Functions
In this Slide Show, use the Desmos graphing calculator to explore rational functions. To see the complete collection of Desmos Resources click on this link. Note: The download is a PPT file. This is part of a collection of Desmos tutorials on a variety of math topics. To see the complete collection of these resources, click on this link.Library of Instructional ResourcesTo see the complete library of Instructional Resources , click on this link. |
Rational Functions and Equations |
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INSTRUCTIONAL RESOURCE: Math Examples 23 | INSTRUCTIONAL RESOURCE: Math Examples--Graphs of Rational Functions
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 ResourcesTo see the complete library of Instructional Resources , click on this link. |
Rational Functions and Equations |
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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 ResourcesTo see the complete library of Instructional Resources , click on this link. |
Rational Expressions and Rational Functions and Equations |
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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 ResourcesTo see the complete library of Instructional Resources , click on this link. |
Rational Expressions and Rational Functions and Equations |
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Math Clip Art--Function Concepts--Graphs of Functions and Relations--Rational Function Graph | Math Clip Art--Function Concepts--Graphs of Functions and Relations--Rational Function GraphTopicFunctions DescriptionThis clip art depicts a rational function graph. Rational functions are quotients of polynomials and can have a wide variety of shapes depending on the degrees of the numerator and denominator. They generally pass the vertical line test and are thus functions, though they may have discontinuities. |
Rational Functions and Equations |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 1 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 1TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 10 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 10TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 11 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 11TopicExponents DescriptionExample 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 12 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 12TopicExponents DescriptionExample 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 13 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 13TopicExponents DescriptionExample 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 14 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 14TopicExponents DescriptionExample 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 15 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 15TopicExponents DescriptionExample 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 2 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 2TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 3 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 3TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 4 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 4TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 5 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 5TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 6 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 6TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 7 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 7TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 8 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 8TopicExponents DescriptionShows 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 |
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Math Example--Exponential Concepts--Integer and Rational Exponents--Example 9 | Math Example--Exponential Concepts--Integer and Rational Exponents--Example 9TopicExponents DescriptionShows 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 |
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Math Example--Graphical Solutions to Rational Equations--Example 1 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 10 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 2 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 3 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 4 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 5 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 6 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 7 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 8 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
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Math Example--Graphical Solutions to Rational Equations--Example 9 | This is part of a collection of math examples that show how to use graphical techniques to solve rational equations.
Note: The download is an image file.
Related ResourcesTo see additional resources on this topic, click on the Related Resources tab.Create a Slide ShowSubscribers can use Slide Show Creator to create a slide show from the complete collection of math examples on this topic. To see the complete clip art collection, click on this Link. To learn more about Slide Show Creator, click on this Link:AccessibilityThis resource can also be used with a screen reader. |
Rational Functions and Equations |
