Illustrative Math Alignment: Grade 7 Unit 5

Subtract rational numbers in this 20-flash card set. The values of the numerators and denominators are in the range -10 to 10.

Note: The download is the teacher's guide for using Media4Math's Quizlet Flash Cards.

Related Resources

Subtract rational numbers in this 20-flash card set. The values of the numerators and denominators are in the range -10 to 10.

Note: The download is the teacher's guide for using Media4Math's Quizlet Flash Cards.

Related Resources

Subtract rational numbers in this 20-flash card set. The values of the numerators and denominators are in the range -10 to 10.

Note: The download is the teacher's guide for using Media4Math's Quizlet Flash Cards.

Related Resources

Topic

Rationals and Radicals

Description

Addition of rational expressions involves combining rational expressions to create a new rational expression, often requiring a common denominator. Examples include 1 / x + 1 / 2x = 3 / 2x and 1 / (x + 1) + 1 / (x - 1) = 2x / (x2 - 1). This term introduces the concept of combining rational expressions, which is fundamental to operations involving fractions with polynomial expressions.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

The geometric mean is the nth root of the product of n terms. Examples include the geometric mean of 3 and 12, √(3 × 12) = 6, and the geometric mean of 3, 4, and 12, ∛(3 × 4 × 12) ≈ 5.24. Geometric mean connects multiplication and roots, commonly used in statistics, geometry, and proportional reasoning.

Topic

Rationals and Radicals

Description

The geometric mean is the nth root of the product of n terms. Examples include the geometric mean of 3 and 12, √(3 × 12) = 6, and the geometric mean of 3, 4, and 12, ∛(3 × 4 × 12) ≈ 5.24. Geometric mean connects multiplication and roots, commonly used in statistics, geometry, and proportional reasoning.

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

Topic

Rationals and Radicals

Description

Graphs of radical functions have a characteristic shape where the domain is limited to values resulting in a positive radicand. For y = √x, the domain is x >= 0, and the range is y >= 0. This highlights the inherent properties of square root functions. Introduces the concept of radical graphs and their domain/range constraints, setting a foundation for graph behavior analysis.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

Graphs of rational functions often feature asymptotes, which are lines the graph approaches but never touches. For example, f(x) = 1 / x has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain and range exclude zero. Explains the unique characteristics of rational function graphs and the significance of asymptotes in defining their behavior.

Topic

Rationals and Radicals

Description

Graphs of rational functions often feature asymptotes, which are lines the graph approaches but never touches. For example, f(x) = 1 / x has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain and range exclude zero. Explains the unique characteristics of rational function graphs and the significance of asymptotes in defining their behavior.

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.