Getting Started 
Misconception/Error The student does not describe a context involving an unknown quantity or a quantity that can vary. 
Examples of Student Work at this Level The student describes a context in which:
 A specific quantity is unit such as cup of butter.
 The variable represents the result of a specific calculation involving as a factor such as x = (25).
The student does not describe a context but states that:
 x is â€śthe variableâ€ť or â€śthe unknown.â€ť
 x is â€śthe number you multiply by .â€ťÂ

Questions Eliciting Thinking What quantity in your story is unknown?
How did you incorporate a factor of into your story?
What does it mean to multiply a quantity by ? What does that do to the quantity?
Can you think of a quantity from real life that you might want to multiply by ? 
Instructional Implications Review the concept of a variable. Provide scenarios in which an unknown quantity can be represented by a variable such as â€śthe number of miles that I run each day.â€ť Emphasize that the variable stands for a number and is not a name or label. Then present another quantity that is described in terms of the variable (e.g., â€śEach day Mike runs twice as many miles as I do.â€ť). Guide the student to represent the new quantity in terms of the variable (e.g., as 2x). Frequently remind the student that the variable represents a specific quantity, so 2x represents twice that quantity. Then ask the student to use the expression to calculate a specific quantity (e.g., ask, 'If I ran 5 miles today, how far did Mike run?'). Model using the expression to calculate Mikeâ€™s distance.
Provide many mathematical and realworld contexts that describe unknown quantities that can be represented by variables and variable expressions. Ask the student to clearly define the variable as a quantity and to write expressions for other quantities described in terms of the variable. For example, â€śAndrew went to the store to buy some cheese crackers. He divided the cheese crackers equally into sandwich bags and gave a bag to each of four friends in math class.â€ť Ask the student to describe the unknown quantities (e.g., the total number of crackers purchased and the number of crackers per bag), to represent one of the unknown quantities with a variable (e.g., x is the number of crackers per bag), and to represent the other unknown quantity in terms of x (e.g., 4x represents the total number of crackers).
Directly address misconceptions about variables such as:
 A variable is a single missing number that must be identified or invented. Clarify that a variable can refer to a single number or a set of numbers depending on context. For example, in the equation 8 = 2x, x has only one possible value. In the expression 2x where x is the number of helium atoms and two refers to the number of protons per helium atom, x could represent any whole number.
 Variables can be equated to a label or other nonnumerical value. For example, given the situation, â€śSally has s spoons then buys two more,â€ť many students will declare s is â€śspoons.â€ť Emphasize that this is an insufficient description of the variable. A sufficient description must include the fact that the variable represents a number, and when appropriate, the unit/label and context for the number. In the previously given example, s could be described as the number of spoons with which Sally started.
Consider implementing otherÂ MFAS tasks and/or CPALMS lesson plans Decoding Word PhrasesTranslating Verbal Phrases to Variable Expressions (ID 28322), Let's Translate!! (ID 55214), and Expressions, Phrases and Word Problems, Oh My! (ID 47911).

Moving Forward 
Misconception/Error The student is unable to clearly interpret variables and variable expressions as quantities. 
Examples of Student Work at this Level The student describes a context in which x is an unknown quantity or one that can vary. However, the student is unable to clearly interpret either x or x as a quantity. For example, the student says:
 At a shipping company, it takes a day for each shipload to be shipped. The student then says that x represents â€śeach shipload,â€ť and x represents â€śhalf a day per shipload.â€ť
 Each person at dinner receives half a fish. The student then says that x represents â€śevery person,â€ť and x means â€śeveryone that comes gets half a fish.â€ť
 Ms. B has half as many pages of stickers as Coach G. The student then says that x represents the number of pages that Coach G. has but is unable to describe what x represents.
 If x is the number of hot dogs, x means of a hot dog.Â

Questions Eliciting Thinking Can you describe x as a quantity?
What does it mean to multiply a quantity by ? What does that do to the quantity?
What does x mean in your story? 
Instructional Implications Guide the student to describe the meaning of the variable as a quantity (e.g., the number of loads that needs to be shipped) and to consult the context of the problem to specifically interpret the meaning of x (e.g., if it takes day per load to ship, it will take x days to ship x loads).
Provide many mathematical and realworld contexts that describe unknown quantities that can be represented by variables and variable expressions. Ask the student to clearly define the variable as a quantity and to write expressions for other quantities described in terms of the variable.
Consider implementing otherÂ MFAS tasks and/or CPALMS lesson plans Decoding Word PhrasesTranslating Verbal Phrases to Variable Expressions (ID 28322), Let's Translate!! (ID 55214), and Expressions, Phrases and Word Problems, Oh My! (ID 47911). 
Almost There 
Misconception/Error The studentâ€™s description is unclear or incomplete. 
Examples of Student Work at this Level The student describes x and x as specific quantities, but the initial description of the context is unclear or not consistent with the description of x and x. For example, the student:
 Describes a story situation that cannot be represented by x.
 Describes a story in which is being added to, subtracted from, or divided by x.
 Introduces additional quantities or variables that cannot be represented by x.
 Describes a situation in which the significance of the variable is not consistent through all four questions.Â

Questions Eliciting Thinking Is your story consistent with the way you described x and x?
You mentioned another variable (or quantity) in your story. Was that really needed? Does it make sense to take of that quantity?
Did you maintain the same description of x when you answered each of the questions? 
Instructional Implications Provide feedback to the student regarding his or her responses. Ask the student to revise the response, so the initial story is consistent with the description of x and x. Provide additional expressions, such as 5x or x + 8, and ask the student to create a realworld context that can be represented by the expression and to describe the meaning of both the variable and the expression in context.
Consider implementing otherÂ MFAS tasks and/or CPALMS lesson plans Decoding Word PhrasesTranslating Verbal Phrases to Variable Expressions (ID 28322), Let's Translate!! (ID 55214), and Expressions, Phrases and Word Problems, Oh My! (ID 47911).
Consider implementing the MFAS task Writing RealWorld Expressions. 
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 describes a situation in which a quantity is unknown and interprets x in terms of that quantity. For example, the student says, â€śHalf of each P.E. class is made up of girls. If x represents the number of students in one of the P.E. classes, x represents the number of girls in that class.â€ť The student is able to explain what it would mean for x to equal four. For example, the student says, â€śOne P.E. class has four students. Since half of four is two, there are two girls in that class.â€ť 
Questions Eliciting Thinking What values might x have in the situation you described?
What if x is equal to 26? What would x equal?
Are there any kinds of numbers that x could not represent in your problem? For example, can x be an odd number, a negative number, or a fraction? 
Instructional Implications Challenge the student to write realworld contexts for variable expressions with greater complexity (e.g., x â€“ 7). 