# MA.8.AR.3.4

Given a mathematical or real-world context, graph a two-variable linear equation from a written description, a table or an equation in slope-intercept form.
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 graphed proportional relationships from a table, equation or a written description. In grade 8, students graph an equation from slope-intercept form from a written description, a table, a graph or an equation. In Algebra 1, students will graph a linear function when given a table, equation or written description.
• Point-slope form and standard forms are not expectations at this grade level.
• Review the concept of slope from MA.8.AR.3.2 for students who may need additional work to determine the slope and understand the meaning of slope.
• The instruction includes examples where the slope is positive or negative and the $y$-intercept is given as a positive or a negative in the equation.
• When introducing the benchmark, review graphing on the coordinate plane and determining appropriate scales for the graph.
• Instruction includes the understanding that a real-world context can be represented by a linear two-variable equation even though it only has meaning for discrete values. Discussing discrete values will prepare students to represent domain and range of real-world contexts in later courses.
• For example, if a gym membership cost \$10.00 plus \$6.00 for each class, this can be represented as $y$ = 10 + 6$c$. When represented on the coordinate plane, the relationship is graphed using the points (0,10), (1,16), (2,22), and so on.
• For mastery of this benchmark, students should be given flexibility to represent real-world contexts with discrete values as a line or as a set of points.

### Common Misconceptions or Errors

• Students may incorrectly identify the slope and $y$-intercept.
• When graphing, students may incorrectly graph the line by inverting the directions of the slope values.
• For example, if the slope is $\frac{\text{2}}{\text{3}}$, a student may think that 2 represents the change in the horizontal direction rather than the vertical direction.

### Strategies to Support Tiered Instruction

• Teacher supports students who incorrectly identify the values for the slope and $y$-intercept by providing opportunities 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 invert the $x$- and $y$-values when calculating slope by using real-world problems that students can relate to and helping students represent the relationship visually.
• Instruction includes supporting students who incorrectly graph the line by inverting the directions of the slope values. Students may incorrectly calculate the slope with a common error of inverting the change in $y$ and the change in $x$. Teachers can support students using error analysis tasks, in which the expression  is incorrectly written as .
• Instruction includes having students find the error and make corrections.
• Teacher supports students who incorrectly graph the slope of a given line through error analysis tasks, in which a line is incorrectly graphed by inverting the change in $y$ and the change in $x$ and then 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.
• Teacher co-creates a graphic organizer with students to include examples of positive and negative slope; the meaning of each variable in slope intercept form; and how to determine the slope and $y$-intercept in a table, graph and verbal description.
• Teacher provides instruction on creating an equation table to clear up the misconception of incorrectly graphing an equation on a coordinate plane.

• Teacher provides instruction on determining the slope and $y$-intercept when reading verbal description.

Brent wants to buy a 60" LED Smart TV. He opened a savings account and added money to the account every month. The table below shows the relationship between the number of months Brent has been saving and the total amount of money in his account.

• Part A. Graph the relationship on a coordinate plane.
• Part B. If the new Smart TV costs \$1500 and tax will be \$110, approximately how many more months does he need to save money in order to make the purchase?

Part A. Graph $y$ = .25$x$ − 3.5 on the coordinate plane.
Part B. Discuss with a partner your method of graphing the equation of the line.

### Instructional Items

Instructional Item 1
Graph $y$ = $x$ − 2 on the coordinate plane.

Instructional Item 2
Supplies for the car wash cost \$25. The booster club is charging \$10 per car. Graph the relationship between the amount of money earned and the number of cars washed.

Instructional Item 3
The table shown represents a linear relationship. Use the table to graph the relationship.

*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.4: Graph a two-variable linear equation from a table or an equation in slope-intercept form.

## Related Resources

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

## 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

## 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

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

Fundamental Property of Reflections:

This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.

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

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

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

The Speeding Ticket: Part 2 - Graphing Linear Functions:

This lesson allows the student to learn about dependent and independent variables and how to make the connection between the linear equation, a linear function, and its graph. The student will learn graphing relationships and how to identify linear functions.

Type: Lesson Plan

Students will learn to find the solutions to a system of linear equations, by graphing the equations.

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

When Will We Ever Meet?:

Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine whether lines are parallel, perpendicular, or neither by looking at the graph and the equation.

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

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.

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.

Introduction to Linear Functions:

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

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.

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.

## 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

## MFAS Formative Assessments

Interpreting Slope:

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

## Student Resources

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

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.

Introduction to Linear Functions:

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

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.

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.

## Parent Resources

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

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.

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.

Introduction to Linear Functions:

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

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.