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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.
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 and the
- 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 =
, where (1, 1) and (2, 2) are two different points on the line.
- Slope of a line can be found by the expression =
Common Misconceptions or Errors
- Students may invert the -and -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 -and -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.
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.
- 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 1A linear relationship is given in the table below. Determine the slope of the relationship.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
Teaching Idea
Video/Audio/Animation
STEM Lessons - Model Eliciting Activity
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.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
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.
Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.
Students are asked to derive the general equation of a line containing the origin.
Students are asked to derive the general equation of a line with a y-intercept of (0, b).
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.
Students are asked to graph a proportional relationship, given a table of values, and find and interpret the slope.
Students are asked to write a function to model a linear relationship given its graph.
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.
Students are asked to use similar triangles to explain why the slope is the same regardless of the points used to calculate it.
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.
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
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.
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.
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.
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
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
Original Student Tutorials
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
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
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
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
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
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
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 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
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
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.
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
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.
Type: Problem-Solving Task
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.
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
This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Video/Audio/Animation
"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
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
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
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.
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 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
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.
Type: Problem-Solving Task
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.
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
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.
Type: Problem-Solving Task
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.
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
This task asks the student to understand the relationship between slope and changes in x- and y-values of a linear function.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
