# MAFS.6.EE.3.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
General Information
Subject Area: Mathematics
Domain-Subdomain: Expressions & Equations
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Represent and analyze quantitative relationships between dependent and independent variables. (Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

• Assessment Limits :
Items must involve relationships and/or equations of the form ???? = ???????? or ???? = ???? + ????. Numbers in items should not require students to perform operations with negative rational numbers or result in answers with negative rational numbers. Variables need to be defined.
• Calculator :

No

• Context :

Required

Sample Test Items (2)
• Test Item #: Sample Item 1
• Question: A graph of Evan’s bank account is shown. What are the dependent and independent variables? • Difficulty: N/A
• Type: MI: Matching Item

• Test Item #: Sample Item 2
• Question: The cost per bushel of corn at Farm A is constant. The graph shows the total cost, C, in dollars, of purchasing different numbers of bushels of corn, b, at a farm. In the table below, select the phrase that represents the independent variable, the phrase that represents the dependent variable, and the equation that represents the graph.

• Difficulty: N/A
• Type: SHT: Selectable Hot Text

## Related Courses

This benchmark is part of these courses.
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022 (current), 2022 and beyond)
2002040: M/J Comprehensive Science 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
2002050: M/J Comprehensive Science 1, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
2001010: M/J Earth/Space Science (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
2001020: M/J Earth/Space Science, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
2000010: M/J Life Science (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
2000020: M/J Life Science, Advanced (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022 (current), 2022 and beyond)
7820015: Access M/J Comprehensive Science 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 and beyond (current))
7912110: Fundamental Explorations in Mathematics 1 (Specifically in versions: 2013 - 2015, 2015 - 2017 (course terminated))
2001025: M/J STEM Astronomy and Space Science (Specifically in versions: 2015 - 2022 (current), 2022 and beyond)

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.6.EE.3.AP.9a: Write an equation using variables to represent two quantities where one variable represents the dependent variable and the second represents the independent variable.
MAFS.6.EE.3.AP.9b: Write an expression that illustrates the relationship between two variables from a provided table.

## Related Resources

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

## Assessments

Sample 4 - Sixth Grade Math State Interim Assessment:

This is a State Interim Assessment for sixth grade.

Type: Assessment

Sample 3 - Sixth Grade Math State Interim Assessment:

This is a State Interim Assessment for sixth grade.

Type: Assessment

## Formative Assessments

Table to Equation:

Students are asked to write an equation that represents the relationship between two variables and to explain how the equation reflects the relationship.

Type: Formative Assessment

Analyzing the Relationship:

Students are given an equation in context and are asked to use the equation to make a table of values and a graph. Students are also asked to explain how the equation shows the relationship between the independent and dependent variables.

Type: Formative Assessment

Bicycling Equations:

Students are asked to write an equation from a real-world context and identify and describe the independent and dependent variables.

Type: Formative Assessment

Grinding Equations:

Students are asked to write equations from real-world contexts and identify the independent and dependent variables.

Type: Formative Assessment

## Lesson Plans

Gather Data For Distribution by Programming an App:

This lesson allow students to gather, calculate, and plot data using both computer code and mathematical equations. In this lesson students will create a pedometer app to demonstrate the understanding of algorithms, components (such as buttons, textboxes, sensors, etc.), and If/Then statements. This lesson uses algebraic equations and random data to access the needed components to store data in a spreadsheet.

Type: Lesson Plan

Data Sets Represented in Computers:

This lesson shows how data can be represented by computers, in relation to everyday activities we may not be aware that we use computer. It gives an overview of graphing data by creating a histogram based on population data. Using the data collected, students will get a chance to hand write code to show what structure is needed for computers to collect, analyze and distribute such data. This lesson is lesson 1 of the Data Set and Deviation Statistics Unit and bridges statistical concepts of data collection, graphing and analysis with programming a computer using coding language while reinforcing foundational algebraic skills.

Type: Lesson Plan

Lightyear Rockets:

Students are asked to evaluate and test several rocket fin designs to determine the most effective design. After launch, the students are asked to test an additional design and also design their own rocket fin. Additionally, students will record and graph their results.

Type: Lesson Plan

From Tables to Graphs and Back!:

Students will match corresponding sets of tables, graphs and linear equations in order to deepen their understanding of multiple representations of the relationships between dependent and independent variables.

Type: Lesson Plan

Recognizing Proportional Relationships to Develop Sense of Scale:

This 90-minute lesson (15-minute pre-lesson, 60-minute lesson and 15-minute follow up lesson or homework) asks students to analyze proportional relationships to solve real world and mathematical problems. The examples use recipes, paint, and buildings. Students begin by working individually, then in pairs or threes, and then as a whole class. Student will need calculators, large sheets of paper to make a poster and the lesson materials.

Type: Lesson Plan

Travel Troubles:

This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency.

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

The Speeding Ticket (Part 1: Solving Linear Equations with One Variable):

"The Speeding Ticket" lesson uses real world application to create and solve linear equations and tables with two variables numerically, verbally, and algebraically. The student will also learn the relationship between the independent and dependent variables.

Type: Lesson Plan

The Physics of Land Yachting:

In this lesson, students will explore motion related to an object in terms of its change in position over time compared to a reference point.

Students will be given a variety of simple materials to create and test their very own land yachts to explore motion.

Type: Lesson Plan

## Original Student Tutorial

MacCoder’s Farm Part 3: If Statements:

Explore computer coding on the farm by using relational operators and IF statements to evaluate expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Chocolate Bar Sales:

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Teaching Idea

Walk This Way:

This challenges small groups of students (2-3) to apply their understanding of constant speed and constant acceleration to the task of interpreting and replicating position v. time and velocity-time graphs that display a variety of 1-D motion examples.

Type: Teaching Idea

## Tutorials

Dependent and Independent Variables Exercise:

In an equation with 2 variables, we will be able to determine which is the dependent variable, and which is the independent variable.

Type: Tutorial

Dependent and Independent Variables Exercise: Express the Graph as an Equation:

Given a graph, we will be able to find the equation it represents.

Type: Tutorial

## Unit/Lesson Sequence

Direct and Inverse Variation:

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

## STEM Lessons - Model Eliciting Activity

Lightyear Rockets:

Students are asked to evaluate and test several rocket fin designs to determine the most effective design. After launch, the students are asked to test an additional design and also design their own rocket fin. Additionally, students will record and graph their results.

Travel Troubles:

This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency.

## MFAS Formative Assessments

Analyzing the Relationship:

Students are given an equation in context and are asked to use the equation to make a table of values and a graph. Students are also asked to explain how the equation shows the relationship between the independent and dependent variables.

Bicycling Equations:

Students are asked to write an equation from a real-world context and identify and describe the independent and dependent variables.

Grinding Equations:

Students are asked to write equations from real-world contexts and identify the independent and dependent variables.

Table to Equation:

Students are asked to write an equation that represents the relationship between two variables and to explain how the equation reflects the relationship.

## Computer Science Original Student Tutorials

MacCoder’s Farm Part 3: If Statements:

Explore computer coding on the farm by using relational operators and IF statements to evaluate expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

## Student Resources

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

## Original Student Tutorial

MacCoder’s Farm Part 3: If Statements:

Explore computer coding on the farm by using relational operators and IF statements to evaluate expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Chocolate Bar Sales:

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Dependent and Independent Variables Exercise:

In an equation with 2 variables, we will be able to determine which is the dependent variable, and which is the independent variable.

Type: Tutorial

Dependent and Independent Variables Exercise: Express the Graph as an Equation:

Given a graph, we will be able to find the equation it represents.

Type: Tutorial

## Parent Resources

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