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Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Intercept
- Slope
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 7, students translated between different representations of proportional relationships. In grade 8, students write an equation in slope-intercept form from a written description, a table, or a graph. In Algebra 1, students write a linear two-variable equation to represent a relationship given by a variety of mathematical and real-world contexts.- Point-slope form and standard forms are not expectations at this grade level.
- Instruction connects proportional relationships to support the generation of the equation = + . Helping students see how the linear equation is both the same and different from the proportional relationship will support the appropriate use of proportional thinking using the rate of change.
- Using an online dynamic graphing tool to explore how the graph changes as either the slope or the -intercept changes helps students visualize the coefficients and constants in the equation (MTR.4.1).
- Students should recognize in a table that the -intercept is the -value when is equal to 0. The slope can be determined by finding the ratio between the change in two -values and the change between the two corresponding -values.
- Using graphs, students identify the -intercept as the point where the line crosses the -axis and the slope as the vertical change divided by the horizontal change. In a linear equation, the coefficient of is the slope and the constant is the -intercept. Students need to be given the equations in formats other than = + , such as = + or = + .
- 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.”
- The instruction includes examples where the slope is positive or negative and the ??intercept is given as a positive or a negative in the equation.
- When providing a graph, be sure there are easily identifiable points for students to use in calculating the slope.
- Instruction allows students to make connections between the different representations of a linear relationship (MTR.2.1).
Common Misconceptions or Errors
- Students may incorrectly identify the values for the slope and -intercept.
- Students may incorrectly calculate the slope with a common error of inverting the change in and the change in .
Strategies to Support Tiered Instruction
- Teacher supports students who incorrectly identify the values for the slope and -intercept by providing opportunities to notice patterns between a given value for , 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 and the change in using error analysis tasks, in which the expression
is incorrectly written as 
, and has students find and correct the error. - 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.
- Teacher co-creates an anchor chart naming the slope and -intercept of a given line and then discusses where to start when graphing the line.
- Instruction incudes graphing various linear equations from a table and then discussing the pattern students notice in regard to the -intercept.
- Teacher provides students with graphs and equations of several linear equations then co-illustrates connections between the slopes and -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 -intercept in a table, graph and verbal description.
Instructional Tasks
Instructional Task 1 (MTR.7.1)Victoria owns a store that sells board games. She made the following graph to relate the number of board games she sells to her overall profits.
- Part A. Write an equation in slope-intercept form to describe this relation. Explain how to determine the equation.
- Part B. What is the meaning of the -intercept in the given context? What is the meaning of the slope in the given context?
Instructional Items
Instructional Item 1Write an equation that represents the graph shown.
Instructional Item 2
The table shown represents a linear relationship. Using the table, write an equation in slope-intercept form.
*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
MFAS Formative Assessments
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 write a function to model a linear relationship given its graph.
Students are asked to construct a function to model a linear relationship between two quantities given a table of values.
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.
Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.
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.
Student Resources
Original Student Tutorials
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
Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.
This is part 4 in 6-part series. Click below to open the other tutorials in the series.
- Scatterplots Part 1: Graphing
- Scatterplots Part 2: Patterns, Associations and Correlations
- Scatterplots Part 3: Trend Lines
- Scatterplots Part 5: Interpreting the Equation of the Trend Line
- Scatterplots Part 6: Using Linear Models
Type: Original Student Tutorial
Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
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
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 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
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
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
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 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
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
