Illustrative Math Alignment: Grade 7 Unit 5

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

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.

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.

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

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 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.

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

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

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.

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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This is the transcript for the video of same title. Video contents: In this Investigation we look at linear models for objects moving at a constant speed.

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This is the transcript for the video of same title. Video contents: After briefly reviewing the concept of inverse variation, this video explores Boyle??s law, a real world example of an inversely proportional relationship between pressure and volume of a gas. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, it goes on to examine similarities and differences among rational functions and numbers. Finally, it takes a look at rational functions graphs and ends with a delightful example merging Euclidean and analytic geometry, thanks to the TI-Nspire technology. Concepts explored: functions, rational expressions, rational functions, asymptotes

This is the transcript for the video of same title. Video contents: In this Investigation we look at an application of rational functions: Boyle's Law.

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This is the transcript for the video of same title. Video contents: In this Investigation we look at graphs of rational functions.

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This is the transcript that goes with the video segment entitled Video Tutorial: Ratios and Rates: Converting Measurement Units.

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This is the transcript that goes with the video segment entitled Video Tutorial: Ratios and Rates: Rates and Slopes of Lines.

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This is the transcript that goes with the video segment entitled Video Tutorial: Ratios and Rates: Rates from Data.

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This is the transcript that goes with the video segment entitled Video Tutorials: Ratios and Rates: Ratios and Decimals.

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This is the transcript that goes with the video segment entitled Video Tutorials: Ratios, Proportions, and Percents: Calculating Percents.

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What Are Ratios?

A ratio is the relationship between two or more quantities among a group of items.

This is the transcript for the TI-Nspire Mini-Tutorial entitled, Exploring Rational Function Graphs with Sliders.

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Learn the basics of inverse variations. Graphical and algebraic representations of inverse variations are shown, as well as examples of such functions.

Topic

Ratios

Description

This video demonstrates calculating rates from data sets, focusing on patterns like distance-time relationships and wages. Examples include determining car speeds, hourly wages, and unit costs of gasoline. Data tables are used to visualize and compute rates.

Topic

Ratios

Description

Rates are linked to slopes in linear functions. The video explores calculating rates of change for graphs of speed, savings growth, and loan repayment. It highlights using the slope formula to interpret and solve practical problems.

Topic

Ratios

Description

This video covers converting units using rates, with examples like speed conversion, currency exchange, and calculating seconds in a year. It emphasizes multiplication by conversion rates to transition between units effectively.

Topic

Ratios

Description

Ratios are connected to percentages in this video. Examples include finding percentages of items in collections. It develops a formula for converting part-to-whole ratios into percentages.

Relevance to the Topic: This video is a crucial resource for understanding the concept of Ratios. It delves into mathematical foundations by exploring how ratios function in real-life scenarios, such as scaling, comparisons, or visual representations. The mathematical principles demonstrated include proportional reasoning, equivalence, and fraction comparisons.

Topic

Ratios

Description

This video illustrates converting ratios to percents using visual aids like area models and grids. Examples include determining percentages of colored eggs, fruit types, and combinations of colored lights.

Topic

Ratios

Description

The video explores working with ratios involving decimals, such as finding unit costs or recycling rates. Scientific notation and multi-step conversions are used to calculate speeds of spacecraft like Voyager I.

Topic

Ratios

Description

The video focuses on unit rates, where the denominator equals one. It includes practical examples like finding the cost per pound of bananas or determining hourly wages. Applications extend to conversions and scaling calculations for various real-world tasks.

Topic

Topic: Inequalities

Topic

Rational Functions

Description

Explains submarine pressure and volume relationships using rational functions, illustrating depth impacts on vessel integrity and scuba safety.

Relevance: This video provides a practical perspective on Rational Functions, making abstract concepts more accessible through real-world applications.

Topic

Rational Functions

Description

Examines surface area to volume ratios in animals using rational functions, connecting these ratios to evolutionary adaptations in different climates.

Relevance: This video provides a practical perspective on Rational Functions, making abstract concepts more accessible through real-world applications.

Topic

Rational Functions

Description

Demonstrates the optics of the Hubble telescope through rational functions, analyzing lens properties and asymptotes for image clarity.

Relevance: This video provides a practical perspective on Rational Functions, making abstract concepts more accessible through real-world applications.

Topic

Rational Functions

Topic

Rational Functions

Description

This segment explores rational functions, focusing on direct and inverse variation. Key terms include constant of proportionality and rational function. Boyle's Law serves as a central example, showcasing the relationship between pressure and volume of a gas. Data plotting and regression analysis with TI-Nspire help identify mathematical models. Applications highlight scientific principles from physics and chemistry.

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.

Topic

Rational Functions

Description

This part examines graphs of rational functions like y = a/x, including their asymptotic behavior and properties. Key terms include hyperbola, vertical asymptote, and horizontal asymptote. TI-Nspire is used to analyze effects of parameter changes, transitions between quadrants, and graphical intersections. Applications include real-world scenarios and mathematical exploration of function behaviors.

Topic

Rational Functions

Description

In this Math Lab explore the graphs of rational functions.

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.

In this TI Nspire tutorial, the Graph window is used to create a slider-based graph of a rational function. This video supports the TI-Nspire Clickpad and Touchpad.