Illustrative Math Alignment: Grade 6 Unit 7

Topic

Rationals and Radicals

Description

Horizontal asymptotes describe the behavior of a graph as x approaches infinity or negative infinity. For example, the equation y = (x + 1) / (x - 2) has a horizontal asymptote at y = 1, demonstrating long-term behavior of the function. Focuses on the role of horizontal asymptotes in describing the long-term trends of rational functions.

Topic

Rationals and Radicals

Description

Horizontal asymptotes describe the behavior of a graph as x approaches infinity or negative infinity. For example, the equation y = (x + 1) / (x - 2) has a horizontal asymptote at y = 1, demonstrating long-term behavior of the function. Focuses on the role of horizontal asymptotes in describing the long-term trends of rational functions.

Topic

Rationals and Radicals

Description

The index of a radical specifies the root being taken, such as the square root (index 2) or cube root (index 3). Examples include √25 for index 2 and the cube root of 27 for index 3. Defines and applies the concept of indices in radicals, foundational for understanding roots and powers.

Topic

Rationals and Radicals

Description

The index of a radical specifies the root being taken, such as the square root (index 2) or cube root (index 3). Examples include √25 for index 2 and the 3√27 for index 3. Defines and applies the concept of indices in radicals, foundational for understanding roots and powers.

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

Topic

Rationals and Radicals

Description

Integers are a subset of rational numbers including positive, negative, and zero values, represented by {..., -3, -2, -1, 0, 1, 2, 3, ...}. This emphasizes their role in forming the basis of discrete mathematics. Introduces integers as a key component of number systems, aiding comprehension of rational numbers.

Topic

Rationals and Radicals

Description

Integers are a subset of rational numbers including positive, negative, and zero values, represented by {..., -3, -2, -1, 0, 1, 2, 3, ...}. This emphasizes their role in forming the basis of discrete mathematics. Introduces integers as a key component of number systems, aiding comprehension of rational numbers.

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

Topic

Rationals and Radicals

Description

Inverse variation describes a non-linear relationship where the product of two variables is a constant k. For instance, xy = k is graphically represented by a hyperbola, illustrating how one variable inversely depends on the other. Describes inverse variation as a key relationship in algebra, foundational for understanding non-linear equations.

Topic

Rationals and Radicals

Description

Inverse variation describes a non-linear relationship where the product of two variables is a constant k. For instance, xy = k is graphically represented by a hyperbola, illustrating how one variable inversely depends on the other. Describes inverse variation as a key relationship in algebra, foundational for understanding non-linear equations.

Topic

Rationals and Radicals

Description

Irrational numbers cannot be expressed as ratios of integers. Examples include pi and √2, demonstrating their importance in representing real-world measurements and non-terminating decimals. Highlights the significance of irrational numbers in advanced mathematics and real-world applications.

Topic

Rationals and Radicals

Description

Irrational numbers cannot be expressed as ratios of integers. Examples include pi and √2, demonstrating their importance in representing real-world measurements and non-terminating decimals. Highlights the significance of irrational numbers in advanced mathematics and real-world applications.

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

Topic

Rationals and Radicals

Description

Another definition of irrational numbers emphasizes their non-representability as fractions. Examples include e and √3, reinforcing their role in mathematical constants and transcendental equations. Reinforces the understanding of irrational numbers by presenting additional examples and contexts.

Topic

Rationals and Radicals

Description

Another definition of irrational numbers emphasizes their non-representability as fractions. Examples include e and √3, reinforcing their role in mathematical constants and transcendental equations. Reinforces the understanding of irrational numbers by presenting additional examples and contexts.

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

Topic

Rationals and Radicals

Description

The laws of rational exponents extend the rules of exponents to fractional powers. For instance, 2(1/2) * 2(1/2) = 2(1/2 + 1/2) = 2, demonstrating the connection between exponents and roots. Explains the fundamental laws governing rational exponents, linking them to both roots and powers.

Topic

Rationals and Radicals

Description

The laws of rational exponents extend the rules of exponents to fractional powers. For instance, 2(1/2) * 2(1/2) = 2(1/2 + 1/2) = 2, demonstrating the connection between exponents and roots. Explains the fundamental laws governing rational exponents, linking them to both roots and powers.

Topic

Rationals and Radicals

Description

Asymptotes for a rational function are the values a function approaches but does not reach. An example is the function f(x) = 1 / x, where the graph has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. This concept explains the behavior of rational functions at extreme values, helping understand their graphs and limits.

Topic

Rationals and Radicals

Description

Asymptotes for a rational function are the values a function approaches but does not reach. An example is the function f(x) = 1 / x, where the graph has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. This concept explains the behavior of rational functions at extreme values, helping understand their graphs and limits.

Topic

Rationals and Radicals

Description

Multiplication of rational expressions involves multiplying the numerators and denominators of each expression. For example, (x / (x + 1)) * ((x - 1) / 3) simplifies to (x * (x - 1)) / (3 * (x + 1)), illustrating its application in simplifying complex expressions. Covers the multiplication of rational expressions, essential for algebraic simplifications and solving equations.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

An oblique asymptote occurs when, for large values of x, the y-values of the graph approach but do not touch the graph of a linear function. An example is given where y = (x3 - 2x2 + 1) / (x2 - 2x - 3) approaches y = x as x approaches infinity. This term provides insight into the behavior of rational functions, essential for graphing and understanding limits in radical and rational contexts.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

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.