Illustrative Math Alignment: Grade 6 Unit 9

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

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This is the transcript for the video of same title. Video contents: In this program, the TI-Nspire is used to explore the nature of quadratic functions. Examples ranging from space travel and projectile motion provide real-world examples for discovering algebraic concepts. All examples are solved graphically. The teacher's guide provides all keystrokes shown in the video, as well as providing support for TI-84 users. Algebra teachers looking to integrate hand-held technology and visual media into their instruction will benefit greatly from this series. Concepts explored: Quadratic functions and equations, standard form, graphing quadratic equations, solving quadratic equations graphically

This is the transcript for the video of same title. Video contents: In this Investigation we explore quadratic functions and their graphs.

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This is the transcript for the video of same title. Video contents: In this Investigation we use a quadratic model to explore the path of a rocket into space.

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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 for the video of same title. Video contents: Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video introduces students to systems of linear equations in two or three unknowns. To solve these systems, the host illustrates a variety of methods: four involve the TI-Nspire (spreadsheet, graphs and geometry, matrices and nSolve) and two are the classic algebraic methods known as substitution and elimination, also called the linear combinations method. The video ends with a summary of the three possible types of solutions. Concepts explored: equations, linear equations, linear systems

This is the transcript for the video of same title. Video contents: In this Investigation we solve a linear system.

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This is the transcript for the video of same title. Video contents: In this Investigation we use matrices and the elimination method to solve a linear system.

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This is the transcript for the video of same title. Video contents: In this Investigation we get a historical overview of equations.

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This is the transcript for the video of same title. Video contents: In this Investigation we solve linear and quadratic equations.

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This is the transcript for the video of same title. Video contents: The Citibank Tower in New York City presents some unique design challenges. In addition it has to cope with a problem that all tall structure have to deal with: heat loss. By managing the ratio of surface area to volume, a skyscraper can effective manage heat loss.

This is the transcript for the video of same title. Video contents: Ever since the mathematics of the Babylonians, equations have played a central role in the development of algebra. Written and hosted by internationally acclaimed mathematics educator Dr. Monica Neagoy, this video traces the history and evolution of equations. It explores the two principal equations encountered in an introductory algebra course -- linear and quadratic -- in an engaging way. The foundations of algebra are explored and fundamental questions about the nature of algebra are answered. In addition, problems involving linear and quadratic equations are solved using the TI-Nspire graphing calculator. Algebra teachers looking to integrate hand-held technology and visual media into their instruction will benefit greatly from this series.

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/2 + 1/3. The solution involves finding a common denominator, which is 6, and then adding the fractions to get 5/6. To solve this, we multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 3/3) + (1 * 2/2) = 3/6 + 2/6. Then, we add the numerators while keeping the common denominator: (3 + 2)/6 = 5/6.

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1/4 - 1/5. The solution involves finding a common denominator, which is 20, and then subtracting the fractions to get 1/20. We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 5/5) - (1 * 4/4) = 5/20 - 4/20. Then, we subtract the numerators while keeping the common denominator: (5 - 4)/20 = 1/20.

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/x + 1/3. The solution involves finding a common denominator, which is 3x, and then adding the fractions to get (x + 3)/3x. We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 3/3) + (1 * x/x) = 3/(3x) + x/(3x). Then, we add the numerators while keeping the common denominator: (3 + x)/(3x) = (x + 3)/(3x).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1/x - 1/2. The solution involves finding a common denominator, which is 2x, and then subtracting the fractions to get (x - 2)/2x. We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 2/2) - (1 * x/x) = 2/(2x) - x/(2x). Then, we subtract the numerators while keeping the common denominator: (2 - x)/(2x) = (x - 2)/(2x).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/(2x) + 1/5. The solution involves finding a common denominator, which is 10x, and then adding the fractions to get (2x + 5)/(10x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * 5)/(2x * 5) + (1 * 2x)/(5 * 2x) = 5/(10x) + 2x/(10x). Then, we add the numerators while keeping the common denominator: (5 + 2x)/(10x).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1/(4x) - 1/7. The solution involves finding a common denominator, which is 28x, and then subtracting the fractions to get (7 - 4x)/(28x), which simplifies to (4x - 7)/(28x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * 7)/(4x * 7) - (1 * 4x)/(7 * 4x) = 7/(28x) - 4x/(28x). Then, we subtract the numerators while keeping the common denominator: (7 - 4x)/(28x).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/(8x) + 1/8. The solution involves finding a common denominator, which is 8x, and then adding the fractions to get (x + 1)/(8x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * x)/(8x * x) + (1 * x)/(8 * x) = 1/(8x) + x/(8x). Then, we add the numerators while keeping the common denominator: (1 + x)/(8x).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/(9x) - 1/3. The solution involves finding a common denominator, which is 9x, and then subtracting the fractions to get (1 - 3x)/(9x), which simplifies to (3x - 1)/(9x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * x)/(9x * x) - (3 * x)/(3 * x) = 1/(9x) - 3x/(9x). Then, we subtract the numerators while keeping the common denominator: (1 - 3x)/(9x).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1/(2x) + 1/(5x). The solution involves finding a common denominator, which is 10x, and then adding the fractions to get 7/(10x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * 5)/(2x * 5) + (1 * 2)/(5x * 2) = 5/(10x) + 2/(10x). Then, we add the numerators while keeping the common denominator: (5 + 2)/(10x) = 7/(10x).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/(8x) - 1/(3x). The solution involves finding a common denominator, which is 24x, and then subtracting the fractions to get -5/(24x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (1 * 3)/(8x * 3) - (1 * 8)/(3x * 8) = 3/(24x) - 8/(24x). Then, we subtract the numerators while keeping the common denominator: (3 - 8)/(24x) = -5/(24x).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 2/(3x) + 4/(5x). The solution involves finding a common denominator, which is 15x, and then adding the fractions to get 22/(15x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (2 * 5)/(3x * 5) + (4 * 3)/(5x * 3) = 10/(15x) + 12/(15x). Then, we add the numerators while keeping the common denominator: (10 + 12)/(15x) = 22/(15x).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 4/(7x) - 3/(8x). The solution involves finding a common denominator, which is 56x, and then subtracting the fractions to get 11/(56x). We multiply each fraction by the appropriate unit fraction to create equivalent fractions with the common denominator: (4 * 8)/(7x * 8) - (3 * 7)/(8x * 7) = 32/(56x) - 21/(56x). Then, we subtract the numerators while keeping the common denominator: (32 - 21)/(56x) = 11/(56x).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1/(x + 1) + 1/3. The solution involves finding a common denominator, which is 3(x + 1), and then adding the fractions to get (x + 4)/(3(x + 1)). We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 3)/(x + 1) * 3 + (1 * (x + 1))/(3 * (x + 1)) = 3/(3(x + 1)) + (x + 1)/(3(x + 1)). Then, we add the numerators while keeping the common denominator: (3 + x + 1)/(3(x + 1)) = (x + 4)/(3(x + 1)).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1/(x + 3) - 1/4. The solution involves finding a common denominator, which is 4(x + 3), and then subtracting the fractions to get (-x + 1)/(4(x + 3)). We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 4)/(x + 3) * 4 - (1 * (x + 3))/(4 * (x + 3)) = 4/(4(x + 3)) - (x + 3)/(4(x + 3)). Then, we subtract the numerators while keeping the common denominator: (4 - (x + 3))/(4(x + 3)) = (-x + 1)/(4(x + 3)).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1 / (x - 5) + 1 / 6. The solution involves finding a common denominator, which is 6(x - 5), and then adding the fractions to get (x + 1) / (6(x - 5)). We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 6) / ((x - 5) * 6) + (1 * (x - 5)) / (6 * (x - 5)) = 6 / (6(x - 5)) + (x - 5) / (6(x - 5)). Then, we add the numerators while keeping the common denominator: (6 + x - 5) / (6(x - 5)) = (x + 1) / (6(x - 5)).

Topic

Rational Expressions

Description

This example illustrates how to combine the rational expressions 1 / (x - 7) + 1 / 8. The solution involves finding a common denominator, which is 8(x - 7), and then adding the fractions to get (x + 1) / (8(x - 7)). We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 8) / ((x - 7) * 8) + (1 * (x - 7)) / (8 * (x - 7)) = 8 / (8(x - 7)) + (x - 7) / (8(x - 7)). Then, we add the numerators while keeping the common denominator: (8 + x - 7) / (8(x - 7)) = (x + 1) / (8(x - 7)).

Topic

Rational Expressions

Description

This example demonstrates how to combine the rational expressions 1 / (x - 9) - 1 / 10. The solution involves finding a common denominator, which is 10(x - 9), and then subtracting the fractions to get (-x + 19) / (10(x - 9)). We multiply each fraction by the appropriate factor to create equivalent fractions with the common denominator: (1 * 10) / ((x - 9) * 10) - (1 * (x - 9)) / (10 * (x - 9)) = 10 / (10(x - 9)) - (x - 9) / (10(x - 9)). Then, we subtract the numerators while keeping the common denominator: (10 - (x - 9)) / (10(x - 9)) = (19 - x) / (10(x - 9)) = (-x + 19) / (10(x - 9)).

Topic

Rational Expressions

Topic

Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

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Rational Expressions

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / x. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with branches in the first and third quadrants, illustrating the characteristic shape of this basic rational function.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / x. The image showcases both the table of x-y coordinates and the resulting graph. The graph is a hyperbola with branches in the second and fourth quadrants, illustrating how the negative sign affects the function's shape compared to the previous example.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x). The image presents both the table of x-y coordinates and the resulting graph. Interestingly, the graph is identical to that of y = -1 / x, showing a hyperbola with branches in the second and fourth quadrants. This example demonstrates how different forms of rational functions can produce the same graph.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x). The image showcases both the table of x-y coordinates and the resulting graph. Notably, the graph is identical to that of y = 1 / x, displaying a hyperbola with branches in the first and third quadrants. This example further illustrates how different forms of rational functions can produce the same graph.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (x + 1). The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with points marked at specific coordinates, illustrating how the addition of a constant in the denominator affects the graph's position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (x + 1). The image showcases both the table of x-y coordinates and the resulting graph. The graph displays a hyperbola with labeled points on the curve, illustrating how the negative sign in the numerator and the constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (-x + 1). The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with specific points highlighted, illustrating how the negative sign and constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (x - 1). The image showcases both the table of x-y coordinates and the resulting graph. The graph features a hyperbola with marked points on its curve, illustrating how the constant in the denominator affects the graph's position.