Getting Started 
Misconception/Error The student is unable to correctly write equations that express 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. However, the student writes an equation that does not represent the relationship between the variables. For example, the student:
 Writes the first equation as 1500n = q, 1500 â€“ 20n = c, or 1500 â€“ cn = q.
Â Â Â
Â
Â
 Writes the second equation as 200 + t = c, 200 â€“ t = c or 2t = c.
Â Â Â
Â
Â

Questions Eliciting Thinking Can you describe this problem in your own words? What are the two variables? How are they related? What operation represents what you described?
How would you use your equation to calculateÂ q? Does it make sense to multiply n by 1500? 
Instructional Implications Review vocabulary associated with various mathematical operations. Have the student make a chart or table that includes the operations and vocabulary that suggests each. Provide the student with opportunities to practice writing variable expressions from verbal descriptions.
If needed, clarify the difference between expressions and equations. Then guide the student through a series of exercises of increasing complexity to better develop the concept of variable expressions:
 Given a variable, ask the student to write and describe a variable expression. For example, given â€śa = a number of apples,â€ť the student might write, â€ś3a is three times the number of apples.â€ť
 Given a realworld situation, ask the student to define a variable and write a variable expression. For example, given â€śBruno had some apples and ate half of them,â€ť the student writes, â€śa = number of apples Bruno started withâ€ť and â€śa is the number he ate (or has left).â€ť
 Given a variable expression, ask the student to define the variable and write a realworld situation that matches the expression. For example, given â€ś6a,â€ť the student writes, â€śa = the number of apples in a bushelâ€ť and â€śthere are 6Â bushels that each contain a apples so 6a describes the total number of apples.â€ť
Extend work with variable expressions to equations in which a variable is set equal to a variable expression. Provide example equations within context and model analyzing the equation. For example, given h = 6a, identify that six refers to the number of bushels, a refers to the number of apples in a bushel, and h refers to the total number of apples in all bushels. Ask the student to perform similar analyses of given equations within context.
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 correctly writes each equation but:
 Does not correctly identify an independent or dependent variable.
Â Â Â
 Does not clearly explain the relationship between the independent and dependent variables.
Â Â Â

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 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 in equations, written descriptions, and realworld situations. 
Almost There 
Misconception/Error The student makes an easily corrected error when 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 2t â€“ 200 = c but can correct the error with minimal prompting.

Questions Eliciting Thinking What do the variables that you used represent? Are these the ones given in the problem?
You made an error in writing the second equation. Can you find and correct it? 
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 For the first problem, the student correctly writes the equation q = 1500 â€“ n and identifies q as the dependent variable. The student explains that the value of q, the quantity of coffee remaining in the bin, depends on n, the quantity of coffee beans removed.
For the second problem, the student correctly writes the equation c = 200 â€“ 2t and identifies the independent variable as the number of times the crank is turned. The student explains that t can take on any value while the amount of coffee remaining, c, depends on the value of t.

Questions Eliciting Thinking What might be possible values of n? What might be possible values of t?
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 other MFAS tasks for standard 6.EE.3.9. 