Illustrative Math Alignment: Grade 7 Unit 5

Topic

Rationals and Radicals

Definition

A radical function is a function that contains a radical expression with the independent variable in the radicand.

Description

Radical Functions are a vital part of Radical Numbers, Expressions, Equations, and Functions. These functions involve radicals, such as square roots or cube roots, with the independent variable inside the radical. For example, the function

$$f(x) = \sqrt{x}$$

Topic

Rationals and Radicals

Definition

The radical symbol (√) is used to denote the root of a number, such as a square root or cube root.

Description

The Radical Symbol is a fundamental notation in the study of Radical Numbers, Expressions, Equations, and Functions. This symbol (√) indicates the root of a number, with the most common being the square root. For example, the expression 

$$\sqrt{25}$$

Topic

Rationals and Radicals

Definition

The radicand is the number or expression inside the radical symbol that is being rooted.

Description

The Radicand is a key component in the study of Radical Numbers, Expressions, Equations, and Functions. It is the number or expression inside the radical symbol that is being rooted. For example, in the expression 

$$\sqrt{49}$$

Topic

Rationals and Radicals

Definition

Rational equations are equations that involve rational expressions, which are fractions containing polynomials in the numerator and denominator.

Description

Rational Equations are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These equations involve rational expressions, which are fractions containing polynomials in the numerator and denominator. Solving rational equations typically requires finding a common denominator, clearing the fractions, and then solving the resulting polynomial equation. For example, to solve 

$$\frac{1}{x} + \frac{1}{x+1} = \frac{1}{2}$$

Topic

Rationals and Radicals

Definition

A rational exponent is an exponent that is a fraction, where the numerator indicates the power and the denominator indicates the root.

Description

Rational Exponents are a crucial concept in the study of Rational Numbers, Expressions, Equations, and Functions. These exponents are fractions, where the numerator indicates the power and the denominator indicates the root. For example, the expression 

$$a^{m/n}$$

can be rewritten as 

$$\sqrt[n]{a^m}$$

Topic

Rationals and Radicals

Definition

Rational expressions are fractions in which the numerator and/or the denominator are polynomials.

Description

Rational Expressions are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions often involves factoring the polynomials and canceling common factors. For example, the rational expression 

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

can be simplified to x + 1, provided that 

$$x \neq 1$$

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 

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

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.

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 

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

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 

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 

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 

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 √.

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 

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

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

Library of Instructional Resources

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

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

Library of Instructional Resources

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

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

Library of Instructional Resources

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

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

—Press PREVIEW to see the tutorial—

In this lesson, students will explore the fundamentals of rational expressions, a key concept in algebra. They will learn how to identify, simplify, and evaluate rational expressions by applying their understanding of fractions and algebraic operations. This lesson introduces students to the structure of rational expressions, emphasizing their similarities to numerical fractions and their role in algebraic problem-solving.

Key concepts covered in this lesson include:

In this high school algebra lesson, students will learn how to simplify rational expressions by applying factoring techniques and identifying common factors. Rational expressions, which are algebraic fractions, can often be simplified by reducing common terms in the numerator and denominator. This lesson provides a step-by-step approach to recognizing and simplifying these expressions while reinforcing key algebraic concepts.

Key concepts covered in this lesson include:

In this high school algebra lesson, students will learn how to add and subtract rational expressions by finding common denominators and simplifying algebraic fractions. Mastering these operations is essential for working with complex rational equations and solving real-world problems involving algebraic fractions. This lesson builds on prior knowledge of simplifying rational expressions and applying fraction rules.

Key concepts covered in this lesson include:

In this high school algebra lesson, students will learn how to multiply and divide rational expressions by applying fraction rules and algebraic simplification techniques. Understanding these operations is essential for working with complex algebraic expressions and solving rational equations. This lesson builds on students’ knowledge of simplifying rational expressions and introduces step-by-step strategies for performing multiplication and division.

Key concepts covered in this lesson include:

In this high school algebra lesson, students will learn how to simplify complex rational expressions—fractions that contain other fractions in the numerator, denominator, or both. Mastering this topic is essential for working with advanced algebraic expressions and preparing for higher-level math courses, including calculus.

Key concepts covered in this lesson include:

In this lesson, students explore ratios and proportional relationships, building on their prior knowledge to deepen their understanding of proportional reasoning. They will analyze proportional relationships in tables, graphs, and equations while identifying the constant of proportionality and its significance in real-world applications.

Key concepts covered in this lesson include:

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.