Getting Started 
Misconception/Error The student is unable to correctly write an equation that expresses one quantity in terms of another. 
Examples of Student Work at this Level The student may correctly identify and describe the dependent and independent variables given the equation written. However, the student writes an equation that does not represent the relationship between the variables. For example, the student:
 Interchanges the variables and writes h = 10D.
 Writes an equation in one variable.
 Introduces a third variable and writes an incorrect equation.

Questions Eliciting Thinking What is the relationship among distance, rate, and time? Can you explain how your equation relates distance to time?
Can you write an equation that uses both variables? 
Instructional Implications If needed, clarify the difference among the following: expressions, equations in onevariable, and equations in twovariables. Use the context of distance, rate, and time to illustrate the differences (e.g., 10h is an expression, 50 = 10h is an equation in one variable, and D = 10h is an equation in twovariables).
Review the relationship among the variables distance, rate, and time (i.e., d = rt). Use realworld examples to assist the student in understanding this relationship. For example, explain that if one travels at an average rate of 70 mph on the highway for 2 hours, then the distance travelled is 70 x 2 = 140 miles. Provide additional opportunities for the student to write equations in either one or twovariables in the context of distance, rate, and time.
Consider implementing the CPALMS Lesson Plan Expressions, Phrases and Word Problems, Oh My! (ID 47911). Also, consider using the MFAS task Writing Expressions (6.EE.1.2). 
Moving Forward 
Misconception/Error The student is unable to identify or explain the dependent and independent variables. 
Examples of Student Work at this Level The student writes the equation as D = 10h but:
 Identifies h as the dependent variable and D (the distance ridden) as the independent variable.
 States both variables are dependent (or independent).
 Identifies the independent and dependent variables but cannot explain why the distance ridden depends on the amount of time ridden.
 Provides a general description of dependent and independent variables without referring to the problem context.

Questions Eliciting Thinking What does dependent mean? What does independent mean?
What is a dependent variable? How can you tell which variable is dependent?
What is an independent variable? How can you tell which variable is independent? 
Instructional Implications Review the terms dependent variable and independent variable. Clarify that both the dependent and independent variables change, though in different ways. Change in the dependent variable depends on change in the independent variable. Explain that typically the value of the independent variable is freely chosen but the value of the dependent variable is calculated for particular values of the independent variable. It might be helpful to describe these variables in terms of an inputoutput system (where the independent variable is the input and the dependent variable is the output).
Explain that if the equation had been written in the equivalent form Â then the roles of the variables change, that is, D is now the independent variable and h is the dependent variable. The number of hours spent riding is now dependent on the distance that is ridden.
Explain that in equations such as a = 3 + b, the isolated variable is generally considered to be the dependent variable (output). Ask the student to distinguish dependent from independent variables given equations, written descriptions, and realworld situations. 
Almost There 
Misconception/Error The student makes a small error in writing one of the equations. 
Examples of Student Work at this Level The student:
 Introduces a new variable instead of using one given in the problem.
 Initially rewrites the second equation as D = hr but can revise the equationÂ with minimal prompting.

Questions Eliciting Thinking What do the variables that you used represent? Are these the ones given in the problem?
You left an important value out of your equation. Can you correct your equation? 
Instructional Implications Provide feedback to the student concerning any errors made and allow the student to revise his or her work. Provide additional opportunities to write equations that express one quantity in terms of another. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student writes the equation D = 10h. The student identifies h (number of hours) as the independent variable because the number of hours can be freely chosen and D (distance) as the dependent variable because its value depends on the number of hours.

Questions Eliciting Thinking How did you decide which was the dependent variable?
What might be possible values of h? What might be possible values of D?
Can you think of another realworld situation that could be modeled using an independent and a dependent variable? 
Instructional Implications Ask the student to complete additional problems in which he or she must identify and define the independent variable, identify and define the dependent variable, and then represent the dependent variable in terms of the independent variable.
Consider implementing MFAS task Grinding Equations (6.EE.3.9). 