Illustrative Math Alignment: Grade 8 Unit 3

Topic

Linear Functions

Description

The term is "Solution to a Linear Equation," defined as finding the value(s) of a variable that make the equation true. An example includes solving 2x + 5 = 15 step-by-step to find x = 5. This term explains how linear equations are solved, demonstrating the logical process of isolating variables.

Topic

Linear Functions

Description

The term is "Proportional Relationship," defined as a type of linear relationship where two variables maintain a constant ratio. Represented as y = kx, where k is the constant ratio. This term establishes the foundation for understanding direct variation and proportionality in linear relationships.

Topic

Linear Functions

Description

The term is "Scatter Plot," defined as a graph displaying individual data points as dots, where each dot represents the values of two variables. Used to examine relationships and identify patterns or trends. This term provides a graphical tool to visualize and analyze linear relationships in data.

Topic

Linear Functions

Description

The term is "Least Squares Regression Line," defined as the best fit to a set of data points on a scatter plot. It minimizes the sum of the squares of the vertical distances between the data points and the line. This term links linear functions to data analysis, providing a method to model relationships in data.

Topic

Linear Functions

Description

The term is "Linear Regression," defined as a statistical method used to analyze the relationship between two variables by fitting a linear equation to observed data points. An example includes using LinReg to calculate slope (m), intercept (b), and r-squared values. This term links linear functions with statistical applications, particularly in modeling and predictions based on data.

Topic

Linear Functions

Description

The term is "Correlation Coefficient," defined as a numerical measure quantifying the strength and direction of the linear relationship between two variables, ranging from -1 to 1. A scatterplot and calculation examples are provided to illustrate this concept. This term connects linear functions to data analysis, showing how linear relationships are quantified in statistics.

Topic

Linear Functions

Description

The term is "Linear Expression," defined as an algebraic expression consisting of at least one term where the variable has an exponent of 1. Examples include x, 2x - 3, and -4x + 8. This term establishes the foundation of linear algebra by focusing on the simplest forms of linear relationships.

Topic

Linear Functions

Description

The term is "Inverse of a Linear Function," defined as the function f-1(x) such that its graph is the mirror image of f(x) across the line y = x. An example is f(x) = -3x + 3 and its inverse f-1(x) = -1/3 * x + 1. This term introduces the concept of function inverses, showing symmetry in linear functions.

Topic

Linear Functions

Description

The term is "Piecewise Linear Functions," defined as special functions composed of linear function segments. Examples include step functions and absolute value functions. This term extends the scope of linear functions to piecewise and segmented applications, showcasing broader versatility.

Topic

Linear Functions

Description

The term is "Increasing Linear Function," defined as a linear function with a positive slope. The graph of the function slants up from left to right, represented as y = mx + b with m > 0. This term complements the concept of decreasing linear functions by focusing on positive slope behavior.

Topic

Linear Functions

Description

The term is "Discrete Linear Functions," defined as discontinuous functions made up of linear function parts. Examples include step functions, illustrated with a piecewise graph. This term extends the concept of linearity to piecewise or segmented functions, bridging the gap between continuous and discrete representations.

Topic

Linear Functions

Description

The term is "Graphs of Linear Functions," defined as a linear function in slope-intercept form y = mx + b that crosses the y-axis at (0, b) and has a slope of m. An example graph shows f(x) = 4x + 3 crossing the y-axis at (0, 3). This term visualizes how linear functions are represented on a graph, providing insight into slope and y-intercept properties.

Topic

Linear Functions

Description

The term is "Linear Binomial," defined as a linear polynomial consisting of two terms. Examples include x + 1, -3x + 2, and (x + 2)(x - 2) = x^2 - 4, demonstrating how linear binomials can factor into higher-order polynomials. This term builds on linear expressions by focusing on binomial cases, important for algebraic manipulation.

Topic

Linear Functions

Description

The term is "Coefficient of the Linear Term," defined as the numerical factor of the variable in a linear expression, equation, or function. Examples provided include x + 3, where the coefficient is 1; y = -3x + 4, where the coefficient is -3; and 7x - 6 = 15, where the coefficient is 7. This term introduces the concept of coefficients in linear expressions, forming the foundation for understanding how variables scale in linear equations.

Topic

Linear Functions

Description

The term is "Constant Term," defined as the numerical term of a linear expression, equation, or function. Examples include x + 5, where the constant is 5; y = 5x + 2, where the constant is 2; and 3x - 2y = 10, where the constant is 10. This term explains the role of constants in linear equations, showing how they affect the y-intercept in graphing linear functions.

Topic

Linear Functions

Description

The term is "Decreasing Linear Function," defined as a linear function with a negative slope. The graph of the function slants down from left to right, represented as y = mx + b with m < 0. This term explores a specific behavior of linear functions, illustrating how slope affects the direction of a graph.

Topic

Linear Functions

Description

The term is "Identity Function," defined as a linear function of the form y = x, where input values equal the output values. The graph is a line of slope 1. This term explains the simplest linear function and its unique property where the input equals the output.

Topic

Linear Functions

Description

The term is "Slope-Intercept Form," defined as the equation of a linear function written in a form that easily identifies the slope (m) and y-intercept (b). Represented as y = mx + b. This term focuses on the most common representation of linear equations, simplifying analysis and graphing.

Topic

Linear Functions

Description

The term is "Linear Function Table," defined as an input-output table where the output values have a common difference. An example shows a table with x-values increasing by 1 and f(x)-values increasing by 2. This term ties numerical patterns to linear functions, highlighting their consistent rate of change.

Topic

Linear Functions

Description

The term is "Point-Slope Form," defined as a formula used to find the equation of a line in slope-intercept form for a given set of coordinates and slope. The formula is y - y1 = m(x - x1). This term provides a useful tool for transitioning between graph-based data and algebraic representations of linear functions.

Topic

Linear Functions

Description

The term is "The Equation of a Line from Two Coordinates," defined as deriving the equation of a line in slope-intercept form from two given points. The slope is calculated as m = (y2 - y1) / (x2 - x1), and the equation is written as y - y1 = m(x - x1). This term builds the connection between points on a graph and their corresponding linear equation, demonstrating practical derivations.

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 systems of equations. Such disparate phenomena as profit and loss, secret codes, and ballistic missile shields can be explored through systems of equations.

This is the transcript for the video of same title. Video contents: Profit and loss are the key measures in a business. A system of equations that includes an equation for income and one for expenses can be used to determine profit and loss. Students solve a system graphically.

This is the transcript for the video of same title. Video contents: Secret codes and encryption are ideal examples of a system of equations. In this activity, students encrypt and decrypt a message.

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This is the transcript for the video of same title. Video contents: A ballistic missile shield allows you to shoot incoming missiles out of the sky. Mathematically, this is an example of a linear-quadratic system. Students graph such a system and find the points of intersection between a line and a parabola.

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.

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

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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: In this Investigation we look at a linear regression for carbon dioxide emission data.

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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: 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.

This is the transcript that goes with the video segment entitled Video: Linear Functions: Negative Slope, Negative y-Intercept.

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This is the transcript that goes with the video segment entitled Video: Linear Functions: Negative Slope, Positive y-Intercept.

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This is the transcript that goes with the video segment entitled Video: Linear Functions: Negative Slope, Zero y-Intercept.

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This is the transcript that goes with the video segment entitled Video: Linear Functions: Positive Slope, Negative y-Intercept.

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This is the transcript that goes with the video segment entitled Video: Linear Functions: Positive Slope, Positive y-Intercept.

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This is the transcript that goes with the video segment entitled Video: Linear Functions: Positive Slope, Zero y-Intercept.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Negative Slope, Coordinates in Quadrant I.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Negative Slope, Coordinates in Quadrant II.

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This is the transcript that goes with the video segment entitled Video Tutorial: Negative Slope, Coordinates in Quadrant III.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Negative Slope, Coordinates in Quadrant IV.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Negative Slope, Coordinates in Quadrants I and II.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Negative Slope, Coordinates in Quadrants III and IV.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Positive Slope, Coordinates in Quadrant I.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Positive Slope, Coordinates in Quadrant II.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Positive Slope, Coordinates in Quadrant III.

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This is the transcript that goes with the video segment entitled Video Tutorial: Slope Formula: Positive Slope, Coordinates in Quadrant IV.

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