Illustrative Math Alignment: Grade 8 Unit 8

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?

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.

Topic

Rational Numbers

Description

This video demonstrates how rational numbers are plotted on a number line, showing positive and negative values sequentially. Examples include ecosystem measurements and thermometers interpreted as number lines. It highlights the infinite nature and precise placement of rational numbers in real-world scenarios.

Topic

Rational Numbers

Description

The video explains absolute value as the distance between two points on a number line. Key examples include altitude changes of kites and submersibles. Techniques involve counting increments or using subtraction. It emphasizes absolute value's role in understanding distances irrespective of direction.

Topic

Rational Numbers

Description

This video presents methods for comparing and ordering rational numbers using number lines. It emphasizes arranging numbers from least to greatest and graphing positions. Real-world applications include submersible depths and relative point positioning.

Topic

Rational Numbers

Description

This video introduces techniques for adding rational numbers. Examples include altitude changes and submersible descents. It covers adding mixed numbers, decimals, and opposites, showcasing the relationship between rational numbers and their negative counterparts, with applications in various contexts.

Topic

Rational Numbers

Description

This video explores subtracting rational numbers, with examples like temperature ranges and vertical distances. Techniques highlight the relationship between subtraction and addition of negatives. Applications include interpreting distances and temperature differences.

Topic

Rational Numbers

Description

The video covers multiplying rational numbers with examples like drilling depths and financial credits. It explains how repeated addition is modeled as multiplication. The mathematical proof demonstrates that multiplying rational numbers yields another rational number.

Topic

Rational Numbers

Description

This video explains dividing rational numbers using examples like sharing pizzas and complex fraction simplifications. It emphasizes rewriting terms to align with the rational number definition and addresses why dividing rational numbers results in another rational number.

Topic

Rational Functions

Description

In this Math Lab explore rational numbers that can be written as repeating decimals.

This video provides an in-depth exploration of the concept of rational functions and their applications. By understanding the mathematical intricacies and observing real-world examples, students can connect abstract concepts to practical scenarios.

Watch this video about rational numbers. (The transcript is included.)

Video Transcript

In your study of integers you saw that dividing integers sometimes resulted in a non-integer quotient. In this video we look at a class of numbers called rational numbers, which also includes the integers.

Let’s start with an example.

Watch this video about rational numbers. (The transcript is included.)

Video Transcript

In your study of integers you saw that dividing integers sometimes resulted in a non-integer quotient. In this video we look at a class of numbers called rational numbers, which also includes the integers.

Let’s start with an example.

Watch this video about rational numbers. (The transcript is included.)

Video Transcript

In your study of integers you saw that dividing integers sometimes resulted in a non-integer quotient. In this video we look at a class of numbers called rational numbers, which also includes the integers.

Let’s start with an example.

Watch this video about rational numbers. (The transcript is included.)

Video Transcript

In your study of integers you saw that dividing integers sometimes resulted in a non-integer quotient. In this video we look at a class of numbers called rational numbers, which also includes the integers.

Let’s start with an example.

Watch this video about rational numbers. (The transcript is included.)

Video Transcript

In your study of integers you saw that dividing integers sometimes resulted in a non-integer quotient. In this video we look at a class of numbers called rational numbers, which also includes the integers.

Let’s start with an example.

In this TI-Nspire CX tutoria llearn how to program a function for solving quadratic equations.

In this TI-Nspire CX tutorial, the Graph Window is used to calculate the slope of the line between two points. The text window is used to generate a formula linked to coordinate values. Note: This video does not contain audio.

In this TI-Nspire CX tutorial, the Graph Window and Spreadsheet Window are used to calculate the slope of the line between two points. The coordinates of the points are linked to variables, which are then used in the Spreadsheet Window to calculate the slope. The split-screen, dynamically linked windows allow you to manipulate the points, while getting an updated value for the slope. Note: This video does not include audio.

In this Nspire CX tutorial, randomly create two points in a Spreadsheet Window. Graph them as a scatterplot. Connect the two points using the Line tool, then measure the slope of the line. Creating new random coordinates will yield a new line with a new slope. Note: This video does not include audio.

In this TI-Nspire CX tutorial, use the Geometry tools to construct a line whose measurements for the rise and run can be used to calculate the slope. Note: This video does not contain audio.