MA.8.AR.3.2

Given a table, graph or written description of a linear relationship, determine the slope.

Clarifications

Clarification 1: Problem types include cases where two points are given to determine the slope.

Clarification 2: Instruction includes making connections of slope to the constant of proportionality and to similar triangles represented on the coordinate plane.

General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Algebraic Reasoning
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Slope

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 7, students determined the constant of proportionality in a proportional relationship. In grade 8, students are determining the slope of a linear relationship from a given table, graph, or written relationship. In Algebra 1, students will write a two-variable linear equation from a graph, written description, or a table to represent relationships between quantities in mathematical and real-world context.
  • Students identified the unit rate or the constant of proportionality in prior grade levels. This benchmark is the first one that references the slope, which represents a constant rate of change, and this is not the same as a constant of proportionality unless the relationship goes through the origin.
  • Instruction includes interpreting the meaning and value of slope in real-world context.
  • Understanding slope can be introduced through a graph and the change in value of the y and the x
  • To introduce the concept to students, use at least two points on a graph in quadrant one.
  • 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.”
  • Students should have experience utilizing a slope formula to determine the slope between two points on a line.
    • Slope of a line can be found by the expression m = y1 - y2 / x1 - x2, where (x1, y1) and (x2, y2) are two different points on the line.

 

Common Misconceptions or Errors

  • Students may invert the x-and y-values when calculating slope. To address this misconception, students should represent the relationship visually.

 

Strategies to Support Tiered Instruction

  • 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 providing students with graph paper with grid lengths larger than 1 centimeter and using appropriate scaling of the axes to allow for students to see the unit rate more easily.

 

Instructional Tasks

Instructional Task 1 (MTR.6.1)
Mr. Elliot needs to drain his above ground pool before the winter. The graph below represents the relationship between the number of gallons of water remaining in the pool and the number of hours that the pool has drained. Determine the slope and explain what it means in this situation.
graph represents the relationship between the number of gallons of water remaining in the pool and the number of hours that the pool has drained.

Instructional Task 2 (MTR.4.1, MTR.7.1)
Jack and Jill are selling gallons of water that are sold in different size pails. Jack charges $1.75 for every 2 gallons of water a pail holds. Additionally, he charges a $2 service fee. Jill's prices can be modeled with the graph shown.
Graph
  • Part A. Identify the slope of Jack's relationship. Explain what it means.
  • Part B. Identify the slope of Jill's graph and explain what it means.
  • Part C. Graph Jack’s prices on the same graph as Jill.
  • Part D. Whose has the better deal, Jack or Jill? Explain.

 

Instructional Items

Instructional Item 1
A linear relationship is given in the table below. Determine the slope of the relationship.
Table

 

*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.2: Given a table or graph of a linear relationship, identify the slope.

Related Resources

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

Formative Assessments

Deriving Lines - 2:

Students are asked to derive the general equation of a line with a y-intercept of (0, b).

Type: Formative Assessment

Slope Triangles:

Students are asked to use similar triangles to explain why the slope is the same regardless of the points used to calculate it.

Type: Formative Assessment

Deriving Lines - 1:

Students are asked to derive the general equation of a line containing the origin.

Type: Formative Assessment

Profitable Functions:

Students are asked to write a function to model a linear relationship given its graph.

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

Construction Function:

Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.

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

Speed Reading:

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

Type: Formative Assessment

Lesson Plans

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

A Day at the Park:

In this activity, students investigate a set of bivariate data to determine if there is a relationship between concession sales in the park and temperature. Students will construct a scatter plot, model the relationship with a linear function, write the equation of the function, and use it to make predictions about values of variables.

Type: Lesson Plan

You Can Plot it! Bivariate Data:

Students create scatter plots, calculate a regression equation using technology, and interpret the slope and y-intercept of the equation in the context of the data. This review lesson relates graphical and algebraic representations of bivariate data.

Type: Lesson Plan

Basketball - it's a tall man's sport - or is it?:

The students will use NBA player data to determine if there is a correlation between the height of a basketball player and his free throw percentage. The students will use technology to create scatter plots, find the regression line and calculate the correlation coefficient.

Basketball is a tall man's sport in most regards. Shooting, rebounding, blocking shots - the taller player seems to have the advantage. But is that still true when shooting free throws?

Type: Lesson Plan

Triangle Mid-Segment Theorem:

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

Scatter Plots:

This lesson is an introduction to scatterplots and how to use a trend line to make predictions. Students should have some knowledge of graphing bivariate data prior to this lesson.

Type: Lesson Plan

Hand Me Your Data:

Students will gather and use data to calculate a line of fit and the correlation coefficient with their classmates' height and hand size. They will use their line of fit to make approximations.

Type: Lesson Plan

Fit Your Function:

Students will make a scatter plot and then create a line of fit for the data. From their graph, students will make predictions and describe relationships between the variables. Students will make predictions, inquire, and formulate ideas from observations and discussions.

Type: Lesson Plan

Beginning Linear Functions:

This is a simple lesson used to describe the concept of slope to algebra students. 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

Scatter Plots and Correlations:

Students create scatter plots, and lines of fit, and then calculate the correlation coefficient. Students analyze the results and make predictions. This lesson includes step-by-step directions for calculating the correlation coefficient using Excel, GeoGebra, and a TI-84 Plus graphing calculator. Students will make predictions for the number of views of a video for any given number of weeks on the charts.

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

Cat Got Your Tongue?:

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

Doggie Data: It's a Dog's Life:

Students use real-world data to construct and interpret scatter plots using technology. Students will create a scatter plot with a line of fit and a function. They describe the relationship of bivariate data. They recognize and interpret the slope and y-intercept of the line of fit within the context of the data.

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

Slope and y-Intercept of a Statistical Model:

Students will sketch and interpret the line of fit and then describe the correlation of the data. Students will determine if there’s a correlation between foot size and height by collecting 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

The Linear Function Connection:

The students will compare two linear functions that have been represented in different ways (equation, table, graph, verbal description). They will be able to find and compare the rate of change, or slope, of the function from any of the representations.

Type: Lesson Plan

Designing a Skateboard Kicker Ramp:

In this lesson students will design a "Skateboard Kicker Ramp" to discover that slope of similar triangles is the same at any two distinct points.  Students will model with mathematics the concept of slope by looking at the pattern set by similar triangles.

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

Hailey’s Treehouse: Similar Triangles & Slope:

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Type: Original Student Tutorial

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

Functions, Sweet Functions:

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

Type: Original Student Tutorial

It's a Slippery Slope!:

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

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.

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.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

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

Video/Audio/Animation

Slope:

"Slope" is a fundamental concept in mathematics. Slope of a linear function is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation

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.

Construction Function:

Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.

Deriving Lines - 1:

Students are asked to derive the general equation of a line containing the origin.

Deriving Lines - 2:

Students are asked to derive the general equation of a line with a y-intercept of (0, b).

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.

Interpreting Slope:

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

Profitable Functions:

Students are asked to write a function to model a linear relationship given its graph.

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.

Slope Triangles:

Students are asked to use similar triangles to explain why the slope is the same regardless of the points used to calculate it.

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.

Speed Reading:

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

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.

Functions, Sweet Functions:

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

Hailey’s Treehouse: Similar Triangles & Slope:

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Scatterplots Part 4: Equation of the Trend Line:

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.

Original Student Tutorials Mathematics - Grades 9-12

It's a Slippery Slope!:

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

Student Resources

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

Original Student Tutorials

Hailey’s Treehouse: Similar Triangles & Slope:

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Type: Original Student Tutorial

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

Functions, Sweet Functions:

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

Type: Original Student Tutorial

It's a Slippery Slope!:

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

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.

Type: Original Student Tutorial

Video/Audio/Animation

Slope:

"Slope" is a fundamental concept in mathematics. Slope of a linear function is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation

Parent Resources

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