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Clarifications
Clarification 1: Instruction includes the use of standard form, slope-intercept form and point-slope form, and the conversion between these forms.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Linear Equation
- Slope
- -intercept
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students wrote linear two-variable equations in slope-intercept form from tables, graphs and written descriptions. In Algebra I, students write linear two-variable equations in all forms from real-world and mathematical contexts. In future courses, students will write systems and solve problems involving systems in three-variables and linear programming. Additionally, linear equations and linear functions are fundamental parts of all future high school courses.- Instruction includes making connections to various forms of linear equations to show
their equivalency. Students should understand and interpret when one form might be
more useful than other depending on the context.
- Standard Form
Can be described by the equation + = C, where , and are any rational number. This form can be useful when identifying the - and -intercepts. - Slope-Intercept Form
Can be described by the equation = + , where is the slope and is the -intercept. This form can be useful when identifying the slope and -intercept. - Point-Slope Form
Can be described by the equation − 1 = ( − 1), where (1, 1) are a point on the line and is the slope of the line. This form can be useful when a point on the line is given and the -intercept is not easily determinable.
- Standard Form
- Look for opportunities to point out the connection between linear contexts and constant rates of change.
- Problem types should include cases for vertical and horizontal lines.
Common Misconceptions or Errors
- Students may have difficulty identifying both variables from a context. Much of their work previously has involved univariate contexts. Place emphasis on asking students what is changing in each context. Help guide their thoughts to recognize bivariate contexts as having two “things” that change in tandem.
- Students may attempt to estimate intercepts in order to continue using a linear form they prefer for some contexts. Use these opportunities to address the need for precision in mathematics.
Strategies to Support Tiered Instruction
- Instruction includes strategies from MA.912.AR.1.2 on rearranging equations to help students when they convert from one form of two-variable linear equation to another.
- Teacher models opportunities to address the need for precision in mathematics when
determining intercepts.
- For example, a student could prefer to use slope-intercept form when writing two variable linear equations. If the given information, as shown below, is two points that do not include the y-intercept, then the student may only estimate the -intercept rather than determining it exactly. The student should realize that they could use point-slope form to write the equation without having to determine the -intercept.
- Instruction includes explicit questions such as “What is staying the same or constant?”,
“What is changing or varying?” or “Is there anything else varying?”
- For example, students are given the situation where a dog groomer charges $25 for a shampoo and hair cut plus $10 for each hour the dog stays at the groomer and are asked to write a linear two-variable equation that represents the total cost. The teacher can provide questions to help determine the constant value and two variables.
- Instruction includes discussions about what a particular coordinate point on a line means in the context of the problem. Teachers may ask, “What does the identified point represent in the context of the problem?” and “How does the change as increases/decreases?”
- Teacher models finding the slope by color coding the points.
- For example, to find the slope of a line passing through points (1 2) and (4 0), you would use
- For example, to find the slope of a line passing through points (1 2) and (4 0), you would use
- For students who need extra support in adding or subtracting integers, instruction
includes using two-colored counters or algebra tiles to model the operation.
- For example, given the expression 8 − (−2), students can use two-colored
counters to find the difference as shown.
- For example, given the expression 8 − (−2), students can use two-colored
counters to find the difference as shown.
- Instruction includes a graphic organizer to identify the key features. Based on the key
features identified, ask students which form of an equation would be the best.
- For example, given the graph below, students can use an organizer to fill in some of the information.
- Once the information is filled in, students can write an equation of the line and determine the rest of the key features.
Instructional Tasks
Instructional Task 1 (MTR.2.1, MTR.3.1, MTR.5.1)- Jamie bought a car in 2005 for $28,500. By 2008, the car was worth $23,700.
- Part A. What function type could model the given situation?
- Part B. What is the rate of change in the vehicle’s worth per year?
- Part C. Create a model that describes this situation.
- Use the graph below to answer the following questions.
- Part A. In order to write the equation that represents this line, what information do you need?
- Part B. Write a linear two-variable equation that represents the graph above. Justify why you choose the linear form to represent the graph.
- Part C. Write a real-world situation that could represent this graph.
Instructional Items
Instructional Item 1- Sharon is ordering tickets for an upcoming basketball game for herself and her friends. The
ticket website shows the following table of ticket options.
- Write a linear two-variable equation to represent the total cost of tickets.
Instructional Item 2
- Write a linear two-variable equation that represents the graph below.
*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
Tutorials
Video/Audio/Animations
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 explain the relationship between a given linear equation and both a point on its graph and a point not on its graph.
Students are asked to write an equation in three variables from a verbal description.
Note: This task may assess skills that exceed the general expectation for this mathematical concept at this grade level. The task is intended for students who have demonstrated mastery within the scope of instruction who may be ready for a more rigorous extensions of the content. As with all materials, ensure to gauge the readiness of students or adapt according to students needs prior to administration.
Students are asked to write a function to model the relationship between two variables described in a real-world context.
Students are asked to explain the relationship between a given linear equation and both a point on its graph and a point not on its graph.
Students are given a table of values and are asked to write a linear function.
Original Student Tutorials Mathematics - Grades 9-12
Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.
Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.
This part 6 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)
Learn how to write equations in two variables in this interactive tutorial.
Student Resources
Original Student Tutorials
Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.
This part 6 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn how to write equations in two variables in this interactive tutorial.
Type: Original Student Tutorial
Learn to determine the number of possible solutions for a linear equation with this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
Type: Problem-Solving Task
This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
Type: Problem-Solving Task
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement. Students are asked to verify a given linear equation from data in a table and interpret its key components in context.
Type: Problem-Solving Task
The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.
Type: Problem-Solving Task
In a geometric context, two distinct points ??1 and ??2 always determine a unique line in the Cartesian plane (this is one of the basic postulates of Euclidean geometry). Only the non-vertical lines, however, can be described by the graph of a function ??(??)=????+??. This task focuses on producing an explicit function ??(??) as long as the line is not vertical.
This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations to find the explicit equation of the line through two points (when that line is not vertical).
Type: Problem-Solving Task
The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.
Type: Problem-Solving Task
In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.
Type: Problem-Solving Task
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.
Type: Problem-Solving Task
The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.
Type: Problem-Solving Task
Tutorials
This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.
Type: Tutorial
In this video, you will learn about Rene Descartes, and how he bridged the gap between algebra and geometry.
Type: Tutorial
Video/Audio/Animations
When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?
Type: Video/Audio/Animation
The point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.
Type: Video/Audio/Animation
The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.
Type: Video/Audio/Animation
Parent Resources
Problem-Solving Tasks
This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
Type: Problem-Solving Task
This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
Type: Problem-Solving Task
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement. Students are asked to verify a given linear equation from data in a table and interpret its key components in context.
Type: Problem-Solving Task
The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.
Type: Problem-Solving Task
In a geometric context, two distinct points ??1 and ??2 always determine a unique line in the Cartesian plane (this is one of the basic postulates of Euclidean geometry). Only the non-vertical lines, however, can be described by the graph of a function ??(??)=????+??. This task focuses on producing an explicit function ??(??) as long as the line is not vertical.
This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations to find the explicit equation of the line through two points (when that line is not vertical).
Type: Problem-Solving Task
The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.
Type: Problem-Solving Task
In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.
Type: Problem-Solving Task
The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.
Type: Problem-Solving Task
The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.
Type: Problem-Solving Task
