Illustrative Math Alignment: Grade 6 Unit 9

This is a collection of clip art images that show different ratios based on zoo animal populations.

This is a collection of clip art images that show different ratios based on zoo animal populations.

This is a collection of clip art images that show different ratios based on zoo animal populations.

Topic

Addition and Subtraction

Description

Definition: Inverse Operations refers to the relationship between addition and subtraction, where one operation undoes the other. For example, 12 + 5 - 5 = 12 and 17 - 3 + 3 = 17. This concept highlights the interplay between addition and subtraction, crucial for problem-solving and algebraic thinking.

Topic

Addition and Subtraction

Description

Definition: A Math Operation refers to the use of plus or minus signs in addition or subtraction expressions. Examples include 9 + 5 = 14 and 12 - 7 = 5. Defining operations ensures clarity in interpreting and solving mathematical expressions.

Topic

Addition and Subtraction

Description

Inverse Operations refers to the relationship between addition and subtraction, where one operation undoes the other. For example, 12 + 5 - 5 = 12 and 17 - 3 + 3 = 17. This concept highlights the interplay between addition and subtraction, crucial for problem-solving and algebraic thinking.

Topic

Addition and Subtraction

Description

A Math Operation refers to the use of plus or minus signs in addition or subtraction expressions. Examples include 9 + 5 = 14 and 12 - 7 = 5. Defining operations ensures clarity in interpreting and solving mathematical expressions.

Topic

Rationals and Radicals

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

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

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

The cube root of a number, when multiplied by itself three times, equals the given number. Examples include ∛27 = 3, ∛125 = 5, and ∛8 = 2. Cube roots are essential for solving equations involving cubic powers and for understanding radical expressions.

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

Topic

Rationals and Radicals

Description

Division of rational expressions involves dividing one rational expression by another, resulting in a new rational expression. Examples include 1 / (2x2) ÷ 1 / (3x) = 3 / 2x and 2 / (x + 1) ÷ (3 / (x - 1)) = 2(x - 1) / 3(x + 1). This operation is crucial for manipulating and simplifying rational expressions in algebra.

Topic

Rationals and Radicals

Description

An extraneous solution is a false solution discarded during the solving process. For example, solving √(x + 1) = x - 5 gives x = 3 or x = 8, but x = 3 is extraneous as it does not satisfy the original equation. This concept highlights the need to verify solutions in equations involving radicals or rational expressions.

Topic

Rationals and Radicals

Description

Factoring a radical expression involves extracting whole number factors. For example, √36x2 = 6x or solving x = (-6 ± √(36 - 16)) / 4, which simplifies to -3 ± √5 / 2. Factoring radicals is a key step in simplifying expressions and solving equations with square roots.

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

Topic

Rationals and Radicals

Description

The fourth root of a number, when multiplied by itself four times, equals the given number. Examples include ∜16 = 2 and ∜81 = 3. Fourth roots extend the concept of radicals to higher powers, useful in algebraic and geometric contexts.

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

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

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

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.

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.