Illustrative Math Alignment: Grade 8 Unit 9

Overview

Overview

Overview

The collection of definitions on the topic of Linear Functions from Media4Math is an invaluable resource for students and educators alike. This comprehensive set includes essential terms such as slope, y-intercept, linear function, and rate of change. Understanding these terms is crucial for mastering the concept of linear functions, which are foundational in algebra and essential for advanced mathematical studies.

Overview

Overview

A set of resources for the TI-Nspire, TI-Nspire CX, and the Nspire App.

In this episode of Algebra Applications, students explore the dramatic events of 2008 related to the mortgage crisis. Brought about principally through mortgage defaults, the effect on the overall economy was severe. Yet, this situation offers an ideal case study for the exploration of key concepts in data analysis and probability. By exploring these questions students get a front row seat to the historical events of the world’s largest economy.

This is part of a collection of videos from the Algebra Applications video series on the topic of Data Analysis.

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 Nspire applications, and model the seamless connection among various problem representations. Concepts explored: statistics, data analysis, regression analysis.

In this TI Nspire tutorial, the Spreadsheet and Statistics windows are used to create a probability simulation of a coin toss. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

In this TI Nspire tutorial, the Spreadsheet and Statistics windows are used to create a probability simulation of tossing two coins. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

In this episode of Algebra Applications, students explore the dramatic events of 2008 related to the mortgage crisis. Brought about principally through mortgage defaults, the effect on the overall economy was severe. Yet, this situation offers an ideal case study for the exploration of key concepts in data analysis and probability. By exploring these questions students get a front row seat to the historical events of the world's largest economy.

In this introductory segment students learn about the mortage crisis of 2008. In the process they get a brief introduction to what a mortgage is.

The time value of money is at the basis of all loans. Students learn about the key factors that determine monthly mortgage payments and use the TI-Nspire to create an amortization table. This table is used throughout the rest of the program to explore different scenarios.

Having learned the general features of a mortgage, students learn the specifics of a subprime mortgage. With this comes the notion of a credit score, and with credit scores come the probabilities for a loan default. Students use the amortization table to run probability simulations to determine possible loan defaults on subprime mortgages.

Another factor in the mortgage crisis was the use of adjustable rate mortgages. Students run a number of scenarios to test adjustable rate mortgages, while also taking into account the state of the housing market during the time of the mortgage crisis.

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 Nspire applications, and model the seamless connection among various problem representations. Concepts explored: statistics, data analysis, regression analysis.

In this Investigation we explore uncertainty and randomness. This video is Segment 1 of a 4 segment series related to Data Analysis and Probability. Segments 1 and 2 are grouped together.

In this Investigation we look at real-world data involving endangered wolf populations. This video is Segment 3 of a 4 segment series related to Data Analysis and Probability. Segments 3 and 4 are grouped together.

In this TI Nspire tutorialthe Spreadsheet and Statistics windows are used to create a probability simulation of tossing two dice. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

In this TI Nspire tutorial, the Spreadsheet window is used to create a random number table. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

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Video Transcripts

Topic

Linear Functions

Definition

A constant function is a linear function of the form f(x) = b, where b is a constant. The graph of a constant function is a horizontal line.

Description

Constant functions are a fundamental concept in linear functions. They represent scenarios where the output value remains unchanged, regardless of the input value. This is depicted graphically as a horizontal line, indicating that the function's rate of change is zero.

In real-world applications, constant functions can model situations where a quantity remains steady over time. For example, a flat fee service charge that does not vary with usage can be represented as a constant function.

Topic

Linear Functions

Definition

Converting from standard form to slope-intercept form involves rewriting a linear equation from the form Ax + By = C to the form y = mx + b, where m is the slope and b is the y-intercept.

Description

Converting linear equations from standard form to slope-intercept form is a key skill in algebra. This conversion allows for easier graphing and interpretation of the equation's slope and y-intercept.

Topic

Linear Functions

Definition

A decreasing linear function is a linear function where the slope is negative, indicating that as the input value increases, the output value decreases.

Description

Decreasing linear functions are important in understanding how variables inversely relate to each other. The negative slope signifies a reduction in the dependent variable as the independent variable increases.

Real-world examples include depreciation of assets over time or the decrease in temperature as altitude increases. These functions help model scenarios where an increase in one quantity results in a decrease in another.

Topic

Linear Functions

Definition

Direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other, expressed as y = kx, where k is the constant of variation.

Description

Direct variation is a fundamental concept in linear functions, illustrating how one variable changes proportionally with another. The constant of variation, k, represents the rate of change.

In real-world scenarios, direct variation can model relationships such as speed and distance, where distance traveled varies directly with time at a constant speed. Understanding this concept is crucial in fields like physics and engineering.

Topic

Linear Functions

Definition

Equations of parallel lines are linear equations that have the same slope but different y-intercepts, indicating that the lines never intersect.

Description

Understanding equations of parallel lines is crucial in geometry and algebra. Parallel lines have identical slopes, which means they run in the same direction and never meet.

In real-world applications, parallel lines can model scenarios such as railway tracks or lanes on a highway, where maintaining a consistent distance is essential.

Topic

Linear Functions

Definition

Equations of perpendicular lines are linear equations where the slopes are negative reciprocals of each other, indicating that the lines intersect at a right angle.

Description

Equations of perpendicular lines are significant in both geometry and algebra. The negative reciprocal relationship between their slopes ensures that the lines intersect at a 90-degree angle.

In real-world applications, perpendicular lines are found in various structures, such as the intersection of streets or the corners of a building, where right angles are essential.

Topic

Linear Functions

Definition

An identity function is a linear function of the form f(x) = x, where the output is equal to the input for all values of x.

Description

The identity function is a basic yet crucial concept in linear functions. It represents a scenario where the input value is always equal to the output value, graphically depicted as a 45-degree line passing through the origin.

In real-world applications, the identity function can model situations where input and output are directly proportional and identical, such as converting units of the same measure. This also introduces the concept of identity, which is fundamental to mathematics.

Topic

Linear Functions

Definition

An increasing linear function is a linear function where the slope is positive, indicating that as the input value increases, the output value also increases.

Description

Increasing linear functions are essential in understanding how variables positively relate to each other. The positive slope signifies an increase in the dependent variable as the independent variable increases.

Real-world examples include income increasing with hours worked or the rise in temperature with the increase in daylight hours. These functions help model scenarios where an increase in one quantity results in an increase in another.

Topic

Linear Functions

Definition

A line of best fit is a straight line that best represents the data on a scatter plot, showing the trend of the data points.

Description

The line of best fit is a crucial concept in statistics and data analysis. It helps in identifying the trend and making predictions based on the data.

In real-world applications, the line of best fit is used in various fields such as economics, biology, and engineering to analyze trends and make forecasts. For example, it can be used to predict future sales based on past data.

Topic

Linear Functions

Definition

Linear equations in standard form are written as Ax + By = C, where A, B, and C are constants, and A and B are not both zero.

Description

Linear equations in standard form are a fundamental representation of linear functions. They provide a way to express linear relationships in a general form.

Topic

Linear Functions

Definition

A linear function is a function that can be graphed as a straight line, typically written in the form y = mx + b, where m is the slope and b is the y-intercept.

Description

Linear functions are one of the most fundamental concepts in mathematics. They describe relationships where the rate of change between variables is constant, represented graphically as a straight line. This simplicity makes them a central topic in algebra and calculus.

Topic

Linear Functions

Definition

Linear function tables display the input-output pairs of a linear function, showing how the dependent variable changes with the independent variable.

Description

Linear function tables are useful tools for understanding and analyzing linear relationships. They provide a clear way to see how changes in the input (independent variable) affect the output (dependent variable).

Topic

Linear Functions

Definition

Point-slope form of a linear equation is written as y − y1 ​ = m(x−x1​ ), where m is the slope and (x1 ​ ,y1​) is a point on the line.

Description

The point-slope form is a versatile way to express linear equations, especially useful when you know a point on the line and the slope. It allows for quick construction of the equation of a line.

Topic

Linear Functions

Definition

Rate of change in a linear function is the ratio of the change in the dependent variable to the change in the independent variable, often represented as the slope m in the equation y = mx + b.

Description

Rate of change is a fundamental concept in understanding linear functions. It describes how one variable changes in relation to another, and is graphically represented by the slope of a line.

Topic

Linear Functions

Definition

Slope-intercept form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept.

Description

Slope-intercept form is one of the most commonly used forms of linear equations. It provides a clear way to understand the slope and y-intercept of a line, making it easier to graph and interpret.

In real-world applications, slope-intercept form is used in various fields such as economics, physics, and engineering to model linear relationships. For example, it can represent the relationship between cost and production levels in business.

Topic

Linear Functions

Definition

The equation of a line from two coordinates can be determined by finding the slope between the two points and using it in the point-slope form of a linear equation.

Description

Finding the equation of a line from two coordinates is a fundamental skill in algebra. It involves calculating the slope between the two points and then using one of the points to form the equation in point-slope or slope-intercept form.

Topic

Linear Functions

Definition

The x-intercept is the point where a graph crosses the x-axis, indicating the value of x when 𝑦 y is zero.

Description

The x-intercept is a key concept in understanding the behavior of linear functions. It represents the point where the function's output is zero, providing insight into the function's roots and behavior.

In real-world applications, the x-intercept can be used to determine break-even points in business, where revenue equals costs, or to find the time at which a process starts or stops.

Topic

Linear Functions

Definition

The y-intercept is the point where a graph crosses the y-axis, indicating the value of y when x is zero.

Description

The y-intercept is a fundamental concept in understanding the behavior of linear functions. It represents the initial value of the function when the input is zero, providing insight into the function's starting point.

In real-world applications, the y-intercept can be used to determine initial conditions in various scenarios, such as the starting balance in a bank account or the initial position of an object in motion.

In this Algebra Application, students examine the music industry, specifically the rise of Korean pop music, or K-Pop. Using industry data students examine tables and graphs of categorical data before identifying possible functional relationships. A mathematical model is developed to explore the relationship between viral videos and sales. Topics covered: Mathematical modeling, Linear regression, Categorical data, Functions.

In this Algebra Application, students examine the budding industry of commercial space travel. Using data from the Bureau of Transportation Statistics, students build a mathematical model from historical airfare data. This model is used to predict when commercial space travel will become affordable. This also includes a discussion of inflation-adjusted costs and the future value of money. Topics covered: Mathematical modeling, Linear regression, Exponential functions, The time value of money.

5/30/11. In this issue we look at the subject of taxation of tobacco products. Many states are using such taxes to meet budget shortfalls. What are the unintended consequences of these taxes?

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6/6/11. In this issue we look at the technology of Doppler Radar and explore its underlying mathematical structure.

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6/20/11. In this issue we look at the possibility of an NFL strike and what the issues are that keep players and owners from coming to agreement.

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7/4/11. In this issue we look at box office receipts for blockbuster movies as well as flops. There is a characteristic curve that results for either type and we develop some regression models based on the data.

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7/11/11. In this issue we look at the underlying factors that make Texas such a powerhouse for employment. Using data from the Bureau of Labor Statistics and the Census Bureau, we dig into the numbers.

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7/25/11. In this issue we look at the elite group of baseball players that have hit 3000 or more hits in their careers. The most recent is Derek Jeter, and based on the records of previous 3000+ hitters we explore whether he will make it to the Baseball Hall of Fame.

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8/29/11. In this issue we look at random path of a hurricane by developing a simulation for tracking the path of a hurricane.

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February 2012. In this issue of Math in the News, we analyze data on the price of gasoline.

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4/18/11. In this issue we look at income taxes. We analyze who pays what proportion of income taxes based on income level.

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