# MA.8.AR.3.5

Given a real-world context, determine and interpret the slope and y-intercept of a two-variable linear equation from a written description, a table, a graph or an equation in slope-intercept form.

### Examples

Raul bought a palm tree to plant at his house. He records the growth over many months and creates the equation h=0.21m+4.9, where h is the height of the palm tree in feet and m is the number of months. Interpret the slope and y-intercept from his equation.

### Clarifications

Clarification 1: Problems include conversions with temperature and equations of lines of fit in scatter plots.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Intercept
• Linear Equation
• Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students solved real-world problems involving proportional relationships. In grade 8, students interpret the slope and $y$-intercept of a two-variable linear equation within a real-world context when given a written description, a table, a graph or an equation. In Algebra 1, students will solve mathematical and real-world problems that are modeled by linear functions, and will interpret key features of the graph in terms of the context.
• The purpose of this benchmark is to focus on interpreting the slope and $y$-intercept in a real-world context using information from a table, graph or written description.
• Students identify the rate of change (slope) and initial value ($y$-intercept) from tables, graphs, equations or verbal descriptions. Students recognize that if the value $x$ = 0 is in a table, the $y$-intercept is the corresponding $y$-value. Otherwise, the $y$-intercept can be found by substituting a point and the slope into the slope-intercept form of the equation and solving for the $y$-intercept. The slope can be determined by finding the ratio between the change in two $y$-values and the change between the two-corresponding $x$-values.
• Using graphs, students identify the $y$-intercept as the point where the line crosses the $y$-axis and the slope as the vertical change divided by the horizontal change. In a linear equation, the coefficient of $x$ is the slope and the constant is the $y$-intercept. Students should have practice with equations in formats other than $y$ = $m$$x$ + $b$, such as $y$ = $a$$x$ + $b$ or $y$ = $b$ + $m$$x$.
• Instruction includes using a variety of vocabulary to make connections to real-world concepts and future courses. To describe the slope, one can say either “the vertical change divided by the horizontal change” or “rise over run.”
• In contextual situations, the $y$-intercept is generally the starting value or the value in the situation when the independent variable is 0.
• The slope is the rate of change that occurs in the problem. Rates of change can often occur over years. In these situations it is helpful for the years to be "converted" to the number of years since the start year.
• For example, the years of 1960, 1970, and 1980 could be represented as 0 for 1960, 10 for 1970 and 20 for 1980.
• Students use the slope and $y$-intercept to write a linear function in the form $y$ = $m$$x$ + $b$.
• Students should remember to interpret the line of fit within the context of the data provided by the scatter plot (MA.8.DP.1.3). The line of fit is meant to understand the general trend of data, but it might not be able to explain everything about it.
• For mastery of this benchmark, it is not the expectation to compare slopes or $y$-intercepts of two linear equations in two variables.
• Instruction includes learning about linear relationships within other content areas. Students should recognize that the conversion between Fahrenheit and Celsius represents a linear relationship, but not a proportional one. Memorization of the formulas is not an expectation of the benchmark.
• The formula for converting Fahrenheit to Celsius is: $C$ = $\frac{\text{5}}{\text{9}}$($F$ − 32).
• The formula for converting Celsius to Fahrenheit is: $F$= $\frac{\text{9}}{\text{5}}$$C$ + 32.

### Common Misconceptions or Errors

• Students may incorrectly identify the slope and $y$-intercept.
• Students may incorrectly interpret the slope and $y$-intercept.
• The misconceptions of this benchmark may develop for some students based on the real-world context of the problems presented. To address this misconception, scaffold questions to help students understand the context.

### Strategies to Support Tiered Instruction

• Teacher supports students who incorrectly identify the values for the slope and $y$-intercept by providing opportunities for students to notice patterns between a given value for $b$, a line graphed on the coordinate plane, and a given equation of the same line.
• Teacher supports students who incorrectly calculate the slope by inverting the change in $y$ and the change in $x$ using error analysis tasks, in which the expression  is incorrectly written as , and have students find and correct the error.
• Teacher co-creates an anchor chart naming the slope and $y$-intercept of a given line and then discusses where to start when graphing the line.
• Teacher provides graphs and equations of several linear equations then co-illustrates connections between the slopes and $y$-intercepts of each line to the corresponding parts of each equation using the same color highlights.
• Instruction includes utilizing a three-read strategy. Students read the problem three different times, each with a different purpose.
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?

The graph below shows a scatter plot and its line of fit for data collected on the height and foot length of a sample of 10 male students.

• Part A. What does the graph indicate about the relationship between foot length and height?
• Part B. The equation of the line of fit is $f$ = 1.5$h$ − 4.3, where $f$ is foot length in millimeters and $h$ is height in centimeters. Explain the meaning of the slope and the $f$-intercept of this equation in the context of the data.

### Instructional Items

Instructional Item 1
At Stay-a-While Coffee shop, they display their internet fees on a chart like the one shown below. Determine the slope for the relationship between the number of minutes, $x$, and the amount charged, $y$.

Instructional Item 2
Joshua adopted a puppy from a dog shelter. He records the puppy’s height over many months and creates the equation $h$= $\frac{\text{m}}{\text{5}}$ + 3, where $h$ is the height of the puppy, in feet, and $m$ is the number of months. Interpret the slope and $h$-intercept from his equation.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.AR.3.AP.5: Given a real-world context, identify the slope and y-intercept of a two-variable linear equation from a table, a graph or an equation in slope-intercept form.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Stretching Statistics:

Students are asked to interpret a specific solution and the y-intercept of a linear equation that describes a context.

Type: Formative Assessment

Developmental Data:

Students are asked to interpret the slope of a linear model.

Type: Formative Assessment

Smart TV:

Students are asked to determine the rate of change and initial value of a linear function given a table of values, and interpret the rate of change and initial value in terms of the situation it models.

Type: Formative Assessment

Drain the Pool:

Students are asked to determine the rate of change and initial value of a linear function when given a graph, and to interpret the rate of change and initial value in terms of the situation it models.

Type: Formative Assessment

Compare Slopes:

Students are asked to identify, describe and compare the slopes of two proportional relationships given the graph of one and the equation of the other.

Type: Formative Assessment

Proportional Paint:

Students are given a graph of a proportional relationship and asked to determine the unit rate of the relationship and compare it to the slope of the graph.

Type: Formative Assessment

Interpreting Slope:

Students are asked to graph a proportional relationship, given a table of values, and find and interpret the slope.

Type: Formative Assessment

Innovative Functions:

Students are asked to determine the rates of change of two functions presented in different forms (an expression and a table) and determine which is the greater rate of change within a real-world context.

Type: Formative Assessment

Competing Functions:

Students are asked to determine and interpret the initial values of two functions represented in different ways (equation and graph), and compare them.

Type: Formative Assessment

This House Is Mine!:

Students are asked to determine a specific value of two functions given in different forms (a graph and a verbal description) within a real-world context, and compare them.

Type: Formative Assessment

## Lesson Plans

Seeing the Slope:

This 5E lesson will build on students' prior knowledge of positive proportional relationships and graphing them. It will introduce students to negative values of slopes, which will lead to graphing negative proportional relationships. Students will discover properties of different values of slope and have the opportunity to practice graphing. This lesson is designed to be done in a 50-minute block.

Type: Lesson Plan

Compacting Cardboard:

Students investigate the amount of space that could be saved by flattening cardboard boxes. The analysis includes linear graphs and regression analysis along with discussions of slope and a direct variation phenomenon.

Type: Lesson Plan

Home Lines:

Students will create an outline of a room and write equations of the lines that contain the sides of the room. This lesson provides an opportunity to review and reinforce writing equations of lines (including horizontal and vertical lines) and to apply the relationship between the slopes of parallel and perpendicular lines.

Type: Lesson Plan

What does it mean?:

This lesson provides the students with scatter plots, lines of best fit and the linear equations to practice interpreting the slope and y-intercept in the context of the problem.

Type: Lesson Plan

Beginning Linear Functions:

This lesson is designed to introduce students to the concept of slope. Students will be able to:

• determine positive, negative, zero, and undefined slopes by looking at graphed functions.
• determine x- and y-intercepts by substitution, or by examining graphs.
• write equations in slope-intercept form and make graphs based on slope/y-intercept of linear functions.

Type: Lesson Plan

What's Slope got to do with it?:

Students will interpret the meaning of slope and y-intercept in a wide variety of examples of real-world situations modeled by linear functions.

Type: Lesson Plan

This lesson uses real-world examples to practice interpreting the slope and y-intercept of a linear model in the context of data. Students will collect data, graph a scatter plot, and use spaghetti to identify a line of fit. A PowerPoint is included for guidance throughout the lesson and guided notes are also provided for students.

Type: Lesson Plan

If the line fits, where's it?:

In this lesson students learn how to informally determine a "best fit" line for a scatter plot by considering the idea of closeness.

Type: Lesson Plan

Scatter Plots at Arm's Reach:

This lesson is an introductory lesson to scatter plots and line of best fit (trend lines). Students will be using small round candy pieces to represent different associations in scatter plots and measure each other's height and arm span to create their own bivariate data to analyze. Students will be describing the association of the data, patterns of the data, informally draw a line of best fit (trend line), write the equation of the trend line, interpret the slope and y-intercept, and make predictions.

Type: Lesson Plan

Scrambled Coefficient:

Students will learn how the correlation coefficient is used to determine the strength of relationships among real data. Students use card sorting to order situations from negative to positive correlations. Students will create a scatter plot and use technology to calculate the line of fit and the correlation coefficient. Students will make a prediction and then use the line of fit and the correlation coefficient to confirm or deny their prediction.

Students will learn how to use the Linear Regression feature of a graphing calculator to determine the line of fit and the correlation coefficient.

The lesson includes the guided card sorting task, a formative assessment, and a summative assessment.

Type: Lesson Plan

Slippery Slopes:

This lesson will not only reinforce students understanding of slope and y-intercept, but will also ensure the students understand how it can be modeled in a real world situation. The focus of this lesson is to emphasize that slope is a rate of change and the y-intercept the value of y when x is zero. Students will be able to read a problem and create a linear equation based upon what they read. They will then make predictions based upon this information.

Type: Lesson Plan

Linear Statistical Models:

In this lesson, students will learn how to analyze data and find the equation of the line of best fit. Students will then find the slope and intercept of the best fit line and interpret the meaning in the context of the data.

Type: Lesson Plan

Line of Fit:

Students will graph scatterplots and draw a line of fit. Next, students will write an equation for the line and use it to interpret the slope and y-intercept in context. Students will also use the graph and the equation to make predictions.

Type: Lesson Plan

Don't Mope Over Slope:

This is an introductory lesson designed to help students have a better understanding of the interpretation of the slope (rate of change) of a graph.

Type: Lesson Plan

In the Real World:

This resource provides a Lesson Plan for teaching students how to analyze real-world problems to look for clearly identified values and determine which of them is a constant value and which of them is subject to change (will increase or decrease per unit of time, weight, length, etc.). The students will also be taught how to determine the correct units for each value in an equation written in slope-intercept form.

Type: Lesson Plan

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph eachÂ to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Exploring Slope Intercept Form with Graphs and Physical Activity:

Students will work in pairs and compose three different linear equations in slope intercept form. They will discover and describe how different values for the slope and y-intercept affect the graph. After graphing lines on graph paper, they will do a physical activity involving graphing.

Type: Lesson Plan

Graphing Equations on the Cartesian Plane: Slope:

The lesson teaches students about an important characteristic of lines: their slope. Slope can be determined either in graphical or algebraic form. Slope can also be described as positive, negative, zero, or undefined. Students get an explanation of when and how these different types of slopes occur. Finally, students learn how slope relates to parallel and perpendicular lines. When two lines are parallel, they have the same slope and when they are perpendicular their slopes are negative reciprocals of one another.

Prerequisite knowledge: Students must know how to graph points on the Cartesian plane. They must be familiar with the x- and y-axis on the plane in both the positive and negative directions.

Type: Lesson Plan

Movie Theater MEA:

In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.

Type: Lesson Plan

Scatter plots, spaghetti, and predicting the future:

Students will construct a scatter plot from given data. They will identify the correlation, sketch an approximate line of fit, and determine an equation for the line of fit. They will explain the meaning of the slope and y-intercept in the context of the data and use the line of fit to interpolate and extrapolate values.

Type: Lesson Plan

## Original Student Tutorials

Constructing Functions From Two Points:

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

Scatterplots Part 6: Using Linear Models :

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

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

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

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.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Slope and Deep Sea Sharks:

Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Type: Perspectives Video: Professional/Enthusiast

DVD Profits, Variation 1:

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

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.

Sore Throats, Variation 2:

Students graph proportional relationships and understand the unit rate as a measure of the steepness of the related line, called the slope. Students will also treat slopes more formally when they graph proportional relationships and interpret the unit rate as the slope of the graph.

Who Has the Best Job?:

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Baseball Cards:

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.

Battery Charging:

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Modeling with a Linear Function:

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.

Downhill:

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.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Comparing Speeds in Graphs and Equations:

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Chicken and Steak, Variation 1:

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.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Peaches and Plums:

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Video Streaming:

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.

Chicken and Steak, Variation 2:

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.

Delivering the Mail:

This problem-solving task involves constructing a linear function and interpreting its parameters in a mail delivery context. It includes annotated solutions.

Distance Across the Channel:

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.

Equations of Lines:

This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.

Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

## Teaching Idea

Now That is a Dense Graph:

In this activity, the density of ethanol is found by graphical means. In the second part, the density of sodium thiosulfate is found, also by graphical means. The values found are then analyzed statistically.

Type: Teaching Idea

## STEM Lessons - Model Eliciting Activity

Movie Theater MEA:

In this Model Eliciting Activity, MEA, students create a plan for a movie theater to stay in business. Data is provided for students to determine the best film to show, and then based on that decision, create a model of ideal sales. Students will create equations and graph them to visually represent the relationships.

## MFAS Formative Assessments

Compare Slopes:

Students are asked to identify, describe and compare the slopes of two proportional relationships given the graph of one and the equation of the other.

Competing Functions:

Students are asked to determine and interpret the initial values of two functions represented in different ways (equation and graph), and compare them.

Developmental Data:

Students are asked to interpret the slope of a linear model.

Drain the Pool:

Students are asked to determine the rate of change and initial value of a linear function when given a graph, and to interpret the rate of change and initial value in terms of the situation it models.

Innovative Functions:

Students are asked to determine the rates of change of two functions presented in different forms (an expression and a table) and determine which is the greater rate of change within a real-world context.

Interpreting Slope:

Students are asked to graph a proportional relationship, given a table of values, and find and interpret the slope.

Proportional Paint:

Students are given a graph of a proportional relationship and asked to determine the unit rate of the relationship and compare it to the slope of the graph.

Smart TV:

Students are asked to determine the rate of change and initial value of a linear function given a table of values, and interpret the rate of change and initial value in terms of the situation it models.

Stretching Statistics:

Students are asked to interpret a specific solution and the y-intercept of a linear equation that describes a context.

This House Is Mine!:

Students are asked to determine a specific value of two functions given in different forms (a graph and a verbal description) within a real-world context, and compare them.

## Original Student Tutorials Mathematics - Grades 6-8

Constructing Functions From Two Points:

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.

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

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 6: Using Linear Models :

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

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

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Constructing Functions From Two Points:

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

Scatterplots Part 6: Using Linear Models :

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

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

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

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.

Type: Original Student Tutorial

DVD Profits, Variation 1:

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

Who Has the Best Job?:

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Modeling with a Linear Function:

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.

Downhill:

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.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Comparing Speeds in Graphs and Equations:

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Chicken and Steak, Variation 1:

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.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Peaches and Plums:

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Video Streaming:

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.

Chicken and Steak, Variation 2:

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.

Distance Across the Channel:

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.

Equations of Lines:

This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.

Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

DVD Profits, Variation 1:

In this task, students are asked to determine the unit price of a product under two different circumstances. They are also asked to generalize the cost of producing x items in each case.

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.

Sore Throats, Variation 2:

Students graph proportional relationships and understand the unit rate as a measure of the steepness of the related line, called the slope. Students will also treat slopes more formally when they graph proportional relationships and interpret the unit rate as the slope of the graph.

Who Has the Best Job?:

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Baseball Cards:

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.

Battery Charging:

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Modeling with a Linear Function:

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.

Downhill:

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.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Comparing Speeds in Graphs and Equations:

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Chicken and Steak, Variation 1:

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.

Kimi and Jordan:

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Peaches and Plums:

This task asks students to reason about the relative costs per pound of two fruits without actually knowing what the costs are. Students who find this difficult may add a scale to the graph and reason about the meanings of the ordered pairs. Comparing the two approaches in a class discussion can be a profitable way to help students make sense of slope.

Video Streaming:

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.

Chicken and Steak, Variation 2:

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.

Distance Across the Channel:

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.