Illustrative Math Alignment: Grade 6 Unit 1

In this episode of Algebra Applications, students explore the following scenario: Scientists use conic sections to map the trajectories of spacecraft in flight. The setting for this episode is a planned future flight to Mars. As the ship travels from Earth to Mars, parabolic, circular, and elliptical paths are explored. In the process students will learn the difference between a quadratic relation and a quadratic function.

This is part of a collection of videos from the Algebra Applications video series on the topic of Functions and Relations.

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.

The relationship between slope and grade in cycling is explored. Go on a tour of Italy through the mountains of Tuscany and apply students' understanding of slope.

Scientists use conic sections to map the trajectories of spacecraft in flight. The setting for this episode is a planned future flight to Mars. As the ship travels from Earth to Mars, parabolic, circular, and elliptical paths are explored. In the process students will learn the difference between a quadratic relation and a quadratic function.

The liftoff and pre-orbital path of the rock is described by a parabola. Students explore the properties of parabolas from the standpoint of the parametric equations that describe the horizontal and vertical directions of motion for the rocket.

The path of a rocket orbiting the Earth can be modeled with the equation of a circle. Students explore the quadratic relation and the parametric equations that can be used to model the path of a spacecraft orbiting Earth.

The planets orbiting the sun follow elliptical paths. In fact, the trajectory of a spacecraft traveling to Mars would also be elliptical. Students explore these various ellipses.

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.

In this Investigation we explore the definition of a Relation. This video is Segment 1 of a 4 segment series related to Functions and Relations. Segments 1 and 2 are grouped together.

In this Investigation we look at functions. This video is Segment 3 of a 4 segment series related to Functions and Relations. Segments 3 and 4 are grouped together.

Topic

Functions and Relations

Definition

A constant function is a function that always returns the same value, no matter the input.

Description

The constant function is one of the simplest types of functions in mathematics, expressed as 

f(x) = c

Topic

Functions and Relations

Definition

A continuous function is a function that does not have any breaks, holes, or gaps in its domain.

Description

Continuous functions are fundamental in calculus and mathematical analysis because they allow for the application of limits, derivatives, and integrals. A function f(x) is continuous if, for every point 𝑐 c in its domain, 

lim x→c ​ f(x) = f(c)

Topic

Functions and Relations

Definition

A decreasing function is a function where the value of the function decreases as the input increases.

Description

Decreasing functions are important in mathematics because they describe scenarios where an increase in one variable leads to a decrease in another. This is mathematically represented as 

f(x1) > f(x2) for any 𝑥1 < 𝑥2 ​

Topic

Functions and Relations

Definition

A dependent variable is a variable whose value depends on one or more other variables.

Topic

Functions and Relations

Definition

A discontinuous function is a function that has one or more points where it is not continuous.

Topic

Functions and Relations

Definition

Distorting a function horizontally involves stretching or compressing the graph of the function along the x-axis.

Description

Horizontal distortions of functions are significant because they alter the input values while maintaining the overall shape of the graph. This is mathematically represented as 

f(kx)

where k is a constant. If k > 1, the function compresses horizontally, and if 0 < 𝑘 < 1, it stretches. 

Topic

Functions and Relations

Definition

Distorting a function vertically involves stretching or compressing the graph of the function along the y-axis.

Description

Vertical distortions of functions are significant because they alter the output values while maintaining the overall shape of the graph. This is mathematically represented as 

kf(x)

Topic

Functions and Relations

Definition

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Topic

Functions and Relations

Definition

An even function is a function that satisfies the condition f(x) = f(−x) for all x in its domain.

Description

Even functions are important in mathematics because they exhibit symmetry about the y-axis. This property is useful in various fields, including physics and engineering, where symmetry simplifies analysis and problem-solving. For example, the function 

f(x)=x2

Topic

Functions and Relations

Definition

A function is a relation that uniquely associates each element of a set with exactly one element of another set.

Topic

Functions and Relations

Definition

A function machine is a conceptual tool used to understand how functions work by visualizing inputs and outputs.

Description

The function machine is a useful educational tool that helps students grasp the concept of functions by visualizing the process of converting inputs into outputs. It emphasizes the idea that a function takes an input, processes it according to a specific rule, and produces an output. For example, if the function is 

f(x) = x + 2

Topic

Functions and Relations

Definition

Function notation is a way to represent functions in the form f(x), where f denotes the function and x denotes the input variable.

Description

Function notation is a standardized way to write functions, making it easier to understand and communicate mathematical relationships. It is widely used in algebra, calculus, and other branches of mathematics. For example, the notation 

f(x) = x2 

Topic

Functions and Relations

Definition

A function table is a table that lists input values and their corresponding output values for a given function.

Description

Function tables are useful tools in mathematics for organizing and analyzing the relationship between inputs and outputs of a function. They help in visualizing how a function behaves and in identifying patterns. For example, a function table for 

f(x) = 2x + 1

Topic

Functions and Relations

Definition

Graphs of functions are visual representations of the relationship between input values and output values of a function.

Description

Graphs of functions are essential tools in mathematics for visualizing how a function behaves. They provide a clear picture of the relationship between the input and output values, making it easier to analyze and interpret the function. For example, the graph of 

f(x) = x2 

Topic

Functions and Relations

Definition

Graphs of relations are visual representations of the relationship between two sets of values, not necessarily functions.

Topic

Functions and Relations

Definition

An increasing function is a function where the value of the function increases as the input increases.

Topic

Functions and Relations

Definition

An independent variable is a variable that represents the input or cause and is manipulated to observe its effect on the dependent variable.

Topic

Functions and Relations

Definition

An odd function is a function that satisfies the condition f(−x) = −f(x) for all x in its domain.

Topic

Functions and Relations

Definition

Parametric equations are a set of equations that express the coordinates of the points of a curve as functions of a variable called a parameter.

Topic

Functions and Relations

Definition

The range of a function is the set of all possible output values (y-values) that the function can produce.

Description

Understanding the range of a function is crucial in mathematics because it defines the scope of possible outputs. The range is determined by the function's rule and the domain. For example, the range of the function 

f(x) = x2 

Topic

Functions and Relations

Definition

A recursive function is a function that calls itself in its definition.

Description

Recursive functions are important in mathematics and computer science because they provide a way to solve problems by breaking them down into simpler sub-problems. They are defined by a base case and a recursive case. For example, the factorial function 

f(n) = n⋅f(n−1) with f(0)=1 

Topic

Functions and Relations

Definition

Reflecting a function involves flipping the graph of the function over a specified axis.

Description

Reflecting functions is significant in mathematics because it helps in understanding the symmetry and transformations of functions. A function can be reflected over the x-axis or y-axis, changing its orientation. For example, reflecting the function 

f(x) = x2

Topic

Functions and Relations

Definition

A relation is a set of ordered pairs, where each element from one set is paired with an element from another set.

Topic

Functions and Relations

Definition

Translating a function involves shifting the graph of the function horizontally, vertically, or both, without changing its shape.

Description

Translating functions is significant in mathematics because it helps in understanding how functions behave under shifts. A function can be translated horizontally by adding or subtracting a constant to the input, and vertically by adding or subtracting a constant to the output. For example, translating the function 

f(x) = x2 

horizontally by 2 units results in 

Topic

Functions and Relations

Definition

The vertical line test is a method used to determine if a graph represents a function by checking if any vertical line intersects the graph more than once.

Description

The vertical line test is important in mathematics because it helps in identifying whether a given graph represents a function. If a vertical line intersects the graph at more than one point, then the graph does not represent a function. This test is used in various fields, including computer science for validating functions in programming and in mathematics for analyzing graphs. For example, the graph of 

Topic

Functions and Relations

Definition

Visual models of functions are graphical representations that illustrate the relationship between input and output values of functions.

Description

Visual models of functions are important in mathematics because they provide a clear and intuitive way to understand how functions behave. These models include graphs, tables, and diagrams that show the relationship between input and output values. For example, the graph of 

f(x) = x2 

Topic

Functions and Relations

Definition

Visual models of relations are graphical representations that illustrate the relationship between two sets of values, not necessarily functions.

Topic

Functions and Relations

Definition

Visualization of the dependent variable involves graphically representing the outcomes or responses that depend on the independent variable.

Topic

Functions and Relations

Definition

Visualization of the independent variable involves graphically representing the variable that is manipulated to observe its effect on the dependent variable.

In this Slide Show, learn what it means to evaluate a function.

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In this Slide Show, learn about domain and range.

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In this Slide Show, learn what a function is.

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Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

In this Slide Show, learn what a recursive function is.

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To see the complete library of Instructional Resources , click on this link.

In this Slide Show, learn about funciton notation in the context of evaluating or transforming a function.

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Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

Topic

Functions

Description

This image serves as a title card for a collection of 9 math clip art images on function graphs. It features a pencil on a graph showing a curve intersecting the axes, labeled with points A and B. This visual representation introduces the concept of functions and their graphical depiction, setting the stage for a deeper exploration of function graphs.

Topic

Functions

Description

This image presents a graph with points labeled (a, b), (c, d), (e, f), and (g, h) without connecting lines. It illustrates that a function can be graphed as a set of x-y coordinates, emphasizing the discrete nature of data points in a function.

Topic

Functions

Description

This image showcases a graph with points labeled (a, b), (c, d), (e, f), and (g, h) connected by line segments. It demonstrates that a function can be graphed as a set of x-y coordinates with the segments connecting them, illustrating a step towards continuous function representation.

Topic

Functions

Description

This image depicts a grid with a straight line connecting four points labeled (a, b), (c, d), (e, f), and (g, h). It illustrates that a function can be graphed as a set of x-y coordinates with a line connecting them all, representing a linear function.

The straight line passing through all the labeled points introduces the concept of linear functions. This visualization helps students understand how a continuous straight line can represent a function, with each point on the line corresponding to an input-output pair. It's an excellent tool for discussing linear relationships, slope, and the equation of a line.

Topic

Functions

Description

This image shows a grid with a curve passing through four points labeled (a, b), (c, d), (e, f), and (g, h). It demonstrates that a function can be graphed as a set of x-y coordinates with a curve connecting them all, representing a non-linear function.

Topic

Functions

Description

This image presents a grid with four points labeled (a, b), (c, d), (e, f), and (g, h), with vertical dashed lines passing through each point. It illustrates the Vertical Line Test, a crucial concept in determining whether a graph represents a function.

The vertical dashed lines passing through each point introduce the Vertical Line Test, which states that a graph represents a function if any vertical line intersects the graph at most once. This visualization helps students understand a fundamental property of functions: for each input value (x-coordinate), there is only one output value (y-coordinate).