Illustrative Math Alignment: Grade 7 Unit 5

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

A perfect cube is a number or polynomial that is the cube of an integer or linear expression. Examples include 125 = 53 and the polynomial (x + 1)3 = x3 + 3x2 + 3x + 1. Understanding perfect cubes is essential for simplifying expressions and solving radical equations involving cube roots.

Topic

Rationals and Radicals

Description

A perfect cube is a number or polynomial that is the cube of an integer or linear expression. Examples include 125 = 53 and the polynomial (x + 1)3 = x3 + 3x2 + 3x + 1. Understanding perfect cubes is essential for simplifying expressions and solving radical equations involving cube roots.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

A perfect square is a number or polynomial that is the square of an integer or linear expression. Examples include 64 = 82 and the polynomial (x - 1)2 = x2 - 2x + 1. Perfect squares are foundational in factoring, completing the square, and simplifying radical expressions.

Topic

Rationals and Radicals

Description

A perfect square is a number or polynomial that is the square of an integer or linear expression. Examples include 64 = 82 and the polynomial (x - 1)2 = x2 - 2x + 1. Perfect squares are foundational in factoring, completing the square, and simplifying radical expressions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

The principal square root is the non-negative square root of a non-negative real number. Examples include √36 = 6 and √81 = 9, excluding their negative counterparts. Clarifies the distinction between principal and negative roots, which is fundamental when solving radical equations.

Topic

Rationals and Radicals

Description

The principal square root is the non-negative square root of a non-negative real number. Examples include √36 = 6 and √81 = 9, excluding their negative counterparts. Clarifies the distinction between principal and negative roots, which is fundamental when solving radical equations.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Radical equations involve solving expressions with radicals by raising both sides to a power. An example solves √(x + 2) = x, resulting in x = 2 or x = -1. This term demonstrates how radicals can be manipulated algebraically to find solutions, integrating concepts of nth roots and square roots.

Topic

Rationals and Radicals

Description

Radical equations involve solving expressions with radicals by raising both sides to a power. An example solves √(x + 2) = x, resulting in x = 2 or x = -1. This term demonstrates how radicals can be manipulated algebraically to find solutions, integrating concepts of nth roots and square roots.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

A radical expression includes terms under a radical sign, such as √2 or x + √(b2 - a). Examples illustrate simple and complex radical forms. Defines and illustrates the structure of radical expressions, serving as a basis for simplification and operations.

Topic

Rationals and Radicals

Description

A radical expression includes terms under a radical sign, such as √2 or x + √(b2 - a). Examples illustrate simple and complex radical forms. Defines and illustrates the structure of radical expressions, serving as a basis for simplification and operations.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

A radical function includes an independent variable under the radical sign, such as y = √x or y = 4√(x - 2) + 3. Examples showcase square root, cube root, and higher-order radical functions. Connects radicals with functions, demonstrating their graphical representation and behavior.

Topic

Rationals and Radicals

Description

A radical function includes an independent variable under the radical sign, such as y = √x or y = 4√(x - 2) + 3. Examples showcase square root, cube root, and higher-order radical functions. Connects radicals with functions, demonstrating their graphical representation and behavior.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

A complex number is a non-real number expressed in forms such as a + bi, r e(iθ), or r cos(θ) + i r sin(θ). Examples include 2 + 5i, 12 e(iπ/4), and 5 cos(60°) + i 5 sin(60°). Complex numbers extend the number system to include imaginary units, enabling solutions to equations like x2 = -1.

Topic

Rationals and Radicals

Description

A complex number is a non-real number expressed in forms such as a + bi, r e(iθ), or r cos(θ) + i r sin(θ). Examples include 2 + 5i, 12 e(iπ/4), and 5 cos(60°) + i 5 sin(60°). Complex numbers extend the number system to include imaginary units, enabling solutions to equations like x2 = -1.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Simplifying a radical expression involves factoring out terms to find whole number factors. Examples simplify √360 into 6 * √10 and √288 into 12 * √2. Provides techniques for simplifying radicals, a key skill in manipulating and solving radical expressions.

Topic

Rationals and Radicals

Description

Simplifying a radical expression involves factoring out terms to find whole number factors. Examples simplify √360 into 6 * √10 and √288 into 12 * √2. Provides techniques for simplifying radicals, a key skill in manipulating and solving radical expressions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

The Radical Symbol is used to find a specified root of a number or expression. Examples include square roots (√2) and cube roots (³√5). It connects to fractions by identifying roots in rational forms. Introduces the concept of radicals and their importance in mathematical expressions.

Topic

Rationals and Radicals

Description

The Radical Symbol is used to find a specified root of a number or expression. Examples include square roots (√2) and cube roots (³√5). It connects to fractions by identifying roots in rational forms. Introduces the concept of radicals and their importance in mathematical expressions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Radicand refers to the number or expression under the radical symbol, such as 5 in √5. This is fundamental for identifying what part of a fraction or term is being rooted. Builds on the radical concept by focusing on the specific part of the radical expression being evaluated.

Topic

Rationals and Radicals

Description

Radicand refers to the number or expression under the radical symbol, such as 5 in √5. This is fundamental for identifying what part of a fraction or term is being rooted. Builds on the radical concept by focusing on the specific part of the radical expression being evaluated.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rational Equations are equations involving fractions where both sides include variables. These often involve steps to solve for variables, like x in (2x + 6) / (x - 3) = 5. Extends the use of fractions by incorporating variables, creating a bridge between fractions and algebra.

Topic

Rationals and Radicals

Description

Rational Equations are equations involving fractions where both sides include variables. These often involve steps to solve for variables, like x in (2x + 6) / (x - 3) = 5. Extends the use of fractions by incorporating variables, creating a bridge between fractions and algebra.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

A Rational Exponent rewrites radicals without symbols, like √2 as 2(1/2). This demonstrates how radicals are expressed in exponent notation in equations. Connects radicals to exponent notation, providing a versatile way to express roots.

Topic

Rationals and Radicals

Description

A Rational Exponent rewrites radicals without symbols, like √2 as 2(1/2). This demonstrates how radicals are expressed in exponent notation in equations. Connects radicals to exponent notation, providing a versatile way to express roots.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rational Expressions are ratios of two polynomials, such as (x + 1) / (x - 1). They demonstrate how polynomials can form fractions with numerical denominators. Demonstrates how polynomials form the basis for more complex fractional expressions.

Topic

Rationals and Radicals

Description

Rational Expressions are ratios of two polynomials, such as (x + 1) / (x - 1). They demonstrate how polynomials can form fractions with numerical denominators. Demonstrates how polynomials form the basis for more complex fractional expressions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rational Functions are ratios between two polynomial functions, such as f(x) = x / (x + 1). They showcase how algebra creates equations that are fractions and functions. Highlights how functions can also be expressed as ratios, blending fractions and algebraic functions.

Topic

Rationals and Radicals

Description

Rational Functions are ratios between two polynomial functions, such as f(x) = x / (x + 1). They showcase how algebra creates equations that are fractions and functions. Highlights how functions can also be expressed as ratios, blending fractions and algebraic functions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rational Numbers are fractions expressed as a/b, such as 1 / 4 or 2 / 3. These are critical in understanding all fraction constructions. Provides a foundational understanding of fractions as ratios of integers, essential for all rational concepts.

Understanding this video is essential for grasping the fundamentals of Rationals and Radicals. The mathematical concepts explored here form the basis for further exploration and application in related fields.

Topic

Rationals and Radicals

Description

Rational Numbers are fractions expressed as a/b, such as 1 / 4 or 2 / 3. These are critical in understanding all fraction constructions. Provides a foundational understanding of fractions as ratios of integers, essential for all rational concepts.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rationalizing a Radical removes roots by using conjugates such as (√3) * (√3). The process connects radicals to polynomials and simplifies rational expressions. Shows how to simplify radical expressions, making them more usable in equations and calculations.

Topic

Rationals and Radicals

Description

Rationalizing a Radical removes roots by using conjugates such as (√3) * (√3). The process connects radicals to polynomials and simplifies rational expressions. Shows how to simplify radical expressions, making them more usable in equations and calculations.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rationalizing an Expression involves removing radicals in the denominator by multiplying by a conjugate term. For example, 1 / (√2 + 1) is rationalized by multiplying by (√2 - 1). Focuses on simplifying expressions to remove radicals, aiding in creating standard forms for fractions.

Topic

Rationals and Radicals

Description

Rationalizing an Expression involves removing radicals in the denominator by multiplying by a conjugate term. For example, 1 / (√2 + 1) is rationalized by multiplying by (√2 - 1). Focuses on simplifying expressions to remove radicals, aiding in creating standard forms for fractions.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

The conjugate of a radical is a factor that eliminates radicals in an expression. For example, the conjugate of √2 is √2, and the conjugate of (1 + √2) is (1 - √2), leading to simplified results like 1 - 2 = -1. Conjugates are used to rationalize denominators and simplify expressions involving radicals.

Topic

Rationals and Radicals

Description

The conjugate of a radical is a factor that eliminates radicals in an expression. For example, the conjugate of √2 is √2, and the conjugate of (1 + √2) is (1 - √2), leading to simplified results like 1 - 2 = -1. Conjugates are used to rationalize denominators and simplify expressions involving radicals.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Rationalizing the Denominator removes radicals from the denominator by multiplying by its conjugate. For example, 1 / √2 becomes √2 / 2 after rationalization. Illustrates how rationalization is used to make fractions easier to interpret and solve.

Topic

Rationals and Radicals

Description

Rationalizing the Denominator removes radicals from the denominator by multiplying by its conjugate. For example, 1 / √2 becomes √2 / 2 after rationalization. Illustrates how rationalization is used to make fractions easier to interpret and solve.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

A Square Root is a number that, when multiplied by itself, equals a given number. Square roots can be written using the radical symbol or as a rational exponent. For example, √25 = 5 and 36(1/2) = 6. This definition includes properties such as √a * √a = a. This term connects to the topic by introducing square roots, which are frequently encountered in operations involving radicals and rational expressions.

Topic

Rationals and Radicals

Description

A Square Root is a number that, when multiplied by itself, equals a given number. Square roots can be written using the radical symbol or as a rational exponent. For example, √25 = 5 and 36^(1/2) = 6. This definition includes properties such as √a * √a = a. This term connects to the topic by introducing square roots, which are frequently encountered in operations involving radicals and rational expressions.

Topic

Rationals and Radicals

Description

The Square Root Property states that if a2 = b, then a = ±√b. Examples include 52 = 25, so √25 = ±5, and 62 = 36, so √36 = ±6. This property provides the basis for solving quadratic equations and simplifying radical expressions, tying back to the earlier concepts of radicals.

Topic

Rationals and Radicals

Description

The Square Root Property states that if a2 = b, then a = ±√b. Examples include 52 = 25, so √25 = ±5, and 62 = 36, so √36 = ±6. This property provides the basis for solving quadratic equations and simplifying radical expressions, tying back to the earlier concepts of radicals.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Subtraction of Rational Expressions involves subtracting one rational expression from another, resulting in another rational expression. The example simplifies 1/(x + 1) - 1/(x - 1) into -2/(x2 - 1) by finding a common denominator and combining the terms. This term supports the topic by illustrating how operations on rational expressions, like subtraction, can be performed systematically.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

A Vertical Asymptote is defined by the restricted values of x where the function approaches but does not touch. For example, in y = 1/(x - 2), the function is not defined at x = 2, resulting in a vertical asymptote at that point. This term relates to rational expressions by identifying points of discontinuity, which are critical in understanding the behavior of rational functions.