Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

**Number:**MAFS.8.F.2

**Title:**Use functions to model relationships between quantities. (Major Cluster)

**Type:**Cluster

**Subject:**Mathematics - Archived

**Grade:**8

**Domain-Subdomain:**Functions

## Related Standards

## Related Access Points

## Access Points

## Related Resources

## 3D Modeling

## Educational Software / Tool

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

## Problem-Solving Tasks

## Professional Development

## Student Center Activity

## Teaching Idea

## Tutorials

## Unit/Lesson Sequences

## Virtual Manipulatives

## Worksheet

## Student Resources

## Original Student Tutorials

Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.

Type: Original Student Tutorial

Learn to construct a function to model a linear relationship between two quantities and determine the slope and y-intercept given two points that represent the function with this interactive tutorial.

Type: Original Student Tutorial

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Explore how to interpret the slope and y-intercept of a linear trend line when bivariate data is graphed on a scatterplot in this interactive tutorial.

This is part 5 in 6-part series. Click below to open the other tutorials in the series.

- Scatterplots Part 1: Graphing
- Scatterplots Part 2: Patterns, Associations and Correlations
- Scatterplots Part 3: Trend Lines
- Scatterplots Part 4: Equation of the Trend Line
- Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

- Scatterplots Part 1: Graphing
- Scatterplots Part 2: Patterns, Associations and Correlations
- Scatterplots Part 3: Trend Lines
- Scatterplots Part 5: Interpreting the Equation of the Trend Line
- Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Type: Problem-Solving Task

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Type: Problem-Solving Task

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Type: Problem-Solving Task

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary and illustrated solutions are included.

Type: Problem-Solving Task

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

In this tutorial, students will compare linear functions from a graph. Students should have an understanding of slope and rate of change before reviewing this tutorial.

Type: Tutorial

This tutorial shows how to compare linear functions that are presented in both a table and graph. Students should have an understanding of rate of change before viewing this video.

Type: Tutorial

Students will compare linear functions presented in a graph and in a table. Students should have a strong understanding of rate of change before viewing this tutorial.

Type: Tutorial

In this video, you will practice finding the slope of a line from data in a table, and interpret what the slope means in the problem.

Type: Tutorial

In this video, you will use a linear graph to determine the y-intercept (starting point) and slope (rate of change), as well as interpret what these mean in the given scenario.

Type: Tutorial

In this tutorial, you will look at several real-world examples of linear graphs and interpret the relationship between the two variables.

Type: Tutorial

In this video, you will practice writing the slope-intercept form for a line, given a table of x and y values.

Type: Tutorial

Given two points on a line, you will find the slope and the y-intercept. You will then write the equation of the line in slope-intercept form.

Type: Tutorial

Given the slope of a line and a point on the line, you will write the equation of the line in slope-intercept form.

Type: Tutorial

Students will learn how to determine an equation by checking solutions. Students will be given a table and 4 linear equations and they will have to determine which equation created the table.

Type: Tutorial

In this video, you will practice writing the equations of lines in slope-intercept form from graphs. You will then practice graphing lines from equations in slope-intercept form.

Type: Tutorial

Given a graph, we will be able to find the equation it represents.

Type: Tutorial

## Virtual Manipulatives

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

## Parent Resources

## Problem-Solving Tasks

This task provides a unique application of modeling with mathematics. Also, students often think that time must always be the independent variable and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.

Type: Problem-Solving Task

This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

Type: Problem-Solving Task

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Type: Problem-Solving Task

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Type: Problem-Solving Task

This task would be especially well-suited for instructional purposes. Students will benefit from a class discussion about the slope, y-intercept, x-intercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.

Type: Problem-Solving Task

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary and illustrated solutions are included.

Type: Problem-Solving Task

## Teaching Idea

This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)

Type: Teaching Idea

## Virtual Manipulative

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative