Illustrative Math Alignment: Grade 6 Unit 9

This is the Teacher's Guide that accompanies Algebra Nspirations: Quadratic Functions.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This is the Teacher's Guide that accompanies Algebra Nspirations: Variables and Equations.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This is the Teacher's Guide that accompanies Algebra Nspirations: Inequalities.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This is the Teacher's Guide that accompanies Algebra Nspirations: Functions and Relations.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This Teacher's Guide provides an overview of the 28 worked-out examples that show how to graph Rational Functions.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

This Teacher's Guide provides an overview of the 28 worked-out examples that show how to simplify a variety of rational expressions.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

Linear Expressions, Equations, and Functions

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.

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

This is the transcript for the video of same title. Video contents: Why do rivers meander instead of traveling in a straight line? In going from point A to point B, why should a river take the circuitous route it does instead of a direct path? Furthermore, what information can the ratio of the river's length to its straight-line distance tell us? In this segment the geological forces that account for a river's motion are explained. In the process, the so-called Meander Ratio is explored. Students construct a mathematical model of a meandering river using the TI-Nspire. Having built the model, students then use it to generate data to find the average of many Meander Ratios. The results show that on average the Meander Ratio is equal to pi.

This is the transcript for the video of same title. Video contents: What are the two meanings of statistics? What does it really mean that an event has a 50% probability of occurring? Why are data analysis and probability always taught together? Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video answers these questions and addresses fundamental concepts such as the law of large numbers and the notion of regression analysis. Both engaging investigations are based on true stories and real data, utilize different TI-Nspire iPad Applications, and model the seamless connection among various problem representations. Concepts explored: statistics, data analysis, regression analysis

This is the transcript for the video of same title. Video contents: In this Investigation we explore uncertainty and randomness.

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

This is the transcript for the video of same title. Video contents: In this Math Lab a hands-on probability activity involving coins is explored.

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

This is the transcript for the video of same title. Video contents: Almost everyone has an intuitive understanding that exponential growth means rapid growth. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video builds on students' intuitive notions, explores exponential notation, and analyzes properties of exponential function graphs, with the help of TI-Nspire features such as sliders and graph transformations. Using exponential functions to model finance applications and a Newton??s law of cooling problem further help students build a solid foundation for these fundamental algebraic concepts. Concepts explored: functions, exponents, exponential functions

This is the transcript for the video of same title. Video contents: In this Investigation we explore the properties of exponents and exponential graphs.

This is the transcript for the video of same title. Video contents: In this Investigation we look at exponential growth and decay models.

This is the transcript for the video of same title. Video contents: Functions are relationships between quantities that change. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video explores the definition of a function, its vocabulary and notations, and distinguishes the concept of function from a general relation. Multiple representations of functions are provided using the TI-Nspire, while dynamic visuals and scenarios put them into real-world contexts. Concepts explored: functions, relations, equations, quadratic functions, linear functions, multiple representations

This is the transcript for the video of same title. Video contents: In this Investigation we explore the definition of a Relation.

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

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

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

This is the transcript for the video of same title. Video contents: Used in just about any industry, inequalities, like equations, are fundamental building blocks of algebra. Written and hosted by internationally acclaimed mathematics educator Dr. Monica Neagoy, this video explores inequalities -- concepts, properties, solutions, and notations -- connects them to real-world contexts, and uses the TI-Nspire to make the algebra meaningful. The focus of this program is on linear inequalities in one and two variables. Concepts explored: Equations, inequalities

This is the transcript for the video of same title. Video contents: In this Investigation we explore linear inequalities in one variable.

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

This is the transcript for the video of same title. Video contents: In this Investigation we look at linear inequalities in two variables.

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

This is the transcript for the video of same title. Video contents: In this program, internationally acclaimed mathematics educator Dr. Monica Neagoy, explores the nature of linear functions through the use TI graphing calculators. Examples ranging from air travel, construction, engineering, and space travel provide real-world examples for discovering algebraic concepts. All examples are solved algebraically and then reinforced through the use of the TI-Nspire. Algebra teachers looking to integrate hand-held technology and visual media into their instruction will benefit greatly from this series. Concepts explored: Standard form, slope-intercept form, point-slope form, solving linear equations

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.

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

This is the transcript for the video of same title. Video contents: In this Investigation we look at a linear regression for carbon dioxide emission data.

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

This is the transcript for the video of same title. Video contents: This video begins with the historical invention of logarithms that forever changed the world of computation, until the advent of calculators more than 300 years later. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, it proceeds to derive the properties of logs, examine logarithmic functions and graphs, and finally explore the well-known Richter logarithmic scale. Concepts explored: functions and inverse functions, logarithms, exponential functions, logarithmic functions

This is the transcript for the video of same title. Video contents: In this Investigation we look at properties of logarithms.

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

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

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

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.

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

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.

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

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.

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

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

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

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.

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

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.

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

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

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

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

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

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).