Illustrative Math Alignment: Grade 8 Unit 8

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.

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.

Topic

Rationals and Radicals

Description

The Zero of a Rational Function is the value of the input variable for which the rational function equals zero. For example, in f(x) = (x - 1)/(x + 1), the zero occurs at x = 1. This term highlights the connection between the numerator of a rational function and its roots, which are essential for solving equations involving rational expressions.

Topic

Rationals and Radicals

Description

The Zero of a Rational Function is the value of the input variable for which the rational function equals zero. For example, in f(x) = (x - 1)/(x + 1), the zero occurs at x = 1. This term highlights the connection between the numerator of a rational function and its roots, which are essential for solving equations involving rational expressions.

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.

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

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

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

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.

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.

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

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.

This is the transcript for the video of same title. Video contents: In this episode of Algebra Applications, students explore various scenarios that can be explained through the use of rational functions. Such disparate phenomena as submarines, photography, and the appearance of certain organisms can be explained through rational function models.

This is the transcript for the video of same title. Video contents: In spite of their massive size, submarines are precision instruments. A submarine must withstand large amounts of water pressure; otherwise, a serious breach can occur. Rational functions are used to study the relationship between water pressure and volume. Students graph rational functions to study the forces at work with a submarine.

This is the transcript for the video of same title. Video contents: All living things take up a certain amount of space, and therefore have volume. They also have a certain amount of surface area. The ratio of surface area to volume, which is a rational function, reveals important information about the organism. Students look at different graphs of these functions for different organisms.

This is the transcript for the video of same title. Video contents: The Hubble Telescope has transformed how we view the universe. We learn about the lens formula and how it is used in the construction of telescopes.

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What Are Rational Numbers?

This is the transcript that goes with the video segment entitled Video: Rational Numbers: Comparing and Ordering Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Dividing Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Multiplying Rational Numbers.

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This is the transcript that goes with the video segment entitled Video: Rational Numbers: Numerical Expressions with Rational Numbers.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Absolute Value.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Exponents.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Irrational Numbers.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers and Irrational Numbers.

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Distance Measures on a Number Line

This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Rational Numbers On the Cartesian Coordinate System.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Subtracting Rational Numbers.

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This is the transcript that goes with the video segment entitled Video Tutorial: Rational Numbers: Variable Expressions with Rational Numbers.

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What Are Rational Numbers?

This is the transcript for the TI-Nspire Mini-Tutorial entitled, Multiplying Rational Expressions.

To see the complete collection of video transcriptsy, click on this link.

Topic

Rational Numbers

Description

This video introduces rational numbers, defined as ratios of two integers where the denominator is non-zero. Key concepts include fractions, ratios, and integers as rational numbers. Applications involve classroom ratios and number line representation with positions like 5/4 and -2/4. It emphasizes rational numbers in everyday contexts, such as temperature comparisons using decimals.

Topic

Rational Numbers

Description

The video explores the interaction of rational numbers with exponents. Examples include expressing square roots as rational exponents and simplifying complex expressions. Applications in geometry and algebra highlight the versatility of these concepts for further mathematical studies.

Topic

Rational Numbers

Description

This video discusses evaluating numerical expressions with rational numbers, emphasizing the order of operations (PEMDAS). Examples include calculating areas of rectangles and finding the slope of a line using the Cartesian plane. It highlights parentheses' role in operations and real-world applications like geometry and measurements.

Topic

Rational Numbers

Description

The video explores variable expressions involving rational numbers. Examples include finding combined areas of rectangles using algebraic expressions and analyzing the trajectory of a ball using a formula with time as a variable. It demonstrates real-world applications of rational number operations in algebra and physics.

Topic

Rational Numbers

Description

This video focuses on graphing rational numbers on the Cartesian coordinate system, calculating slopes, and writing line equations. It explains conditions for positive, negative, zero, and undefined slopes using delta-y and delta-x. Applications include geometry and understanding coordinate relationships.

Topic

Rational Numbers

Description

The video explores rational and irrational numbers. Rational numbers include integers, fractions, and repeating decimals. Irrational numbers, such as the square root of non-perfect squares and pi, are non-terminating and non-repeating. Examples include Pythagorean triples and circle circumference ratios. Applications focus on distinguishing between the two.

Topic

Rational Numbers

Description

The video explores rational and irrational numbers. Rational numbers include integers, fractions, and repeating decimals. Irrational numbers, such as the square root of non-perfect squares and pi, are non-terminating and non-repeating. Examples include Pythagorean triples and circle circumference ratios. Applications focus on distinguishing between the two.