Getting Started 
Misconception/Error The student cannot relate the expressions to specific aspects of the context of the problem. 
Examples of Student Work at this Level The student is unable to explain how Britâ€™s expression relates to the perimeter of the rectangle. The student:
 Focuses on trying to find a value for x.
 Does not give a specific explanation.
 Does not indicate an understanding that x represents the width and 3x + 2 represents the length of the rectangle.
Additionally, the student does not recognize the equivalence of the two expressions and is unable to interpret 8x + 4 in the problem context.

Questions Eliciting Thinking Why do you need to find the value of x? Do you have enough information to solve for x?
What does it mean to combine like terms? What terms can be combined in the first expression?
What do the variables represent in each expression? What do the numbers represent?
How is Abbeyâ€™s expression different from Britâ€™s? Can you explain the differences? 
Instructional Implications Make sure the student understands the terms expression and equivalent expression. Include an explanation of the rationale for writing expressions in equivalent form. Verify that the student understands the difference between mathematically equivalent and looks the same. Guide the student to give a justification for equivalence by referencing properties of operations (e.g., use the Distributive Property to justify combining like terms).
If needed, review how the perimeter of a rectangle is calculated. Then explain that x and 3x + 2 are expressions that represent the width and length of the rectangle. Guide the student to observe that Britâ€™s expression shows the sum of the lengths of the sides of the rectangle and is a representation of the rectangleâ€™s perimeter. Assist the student in rewriting Britâ€™s expression as 8x + 4. Explain the relationship between the two expressions using appropriate mathematical vocabulary. Ensure the student understands that the two expressions are equivalent.
Guide the student to interpret the expression 8x + 4 in terms of x, the width of the rectangle. If needed, model explaining that the perimeter can be found by multiplying the width by eight and adding fourÂ because x represents the width of the rectangle.
Provide additional opportunities for the student to rewrite expressions in equivalent forms. Ask the student to explain the relationship between the quantities represented and how the different forms can reveal different information about the problem context. 
Moving Forward 
Misconception/Error The student cannot identify equivalent expressions. 
Examples of Student Work at this Level The student can explain how Britâ€™s expression represents the perimeter of the rectangle but does not recognize that this expression is equivalent to 8x + 4.

Questions Eliciting Thinking What do the numbers and variables represent in the second expression? Where did the numbers come from?
Can you simplify the first expression by combining like terms? What is the result? 
Instructional Implications Assist the student in rewriting Britâ€™s expression as 8x + 4. Explain the relationship between the two expressions using appropriate mathematical vocabulary. Ensure the student understands that the two expressions are equivalent.
Guide the student to interpret the expression 8x + 4 in terms of x, the width of the rectangle. If needed, model explaining that the perimeter can be found by multiplying the width by eight and adding fourÂ because x represents the width of the rectangle.
Provide additional opportunities for the student to rewrite expressions in equivalent forms. Ask the student to explain the relationship between the quantities represented and how the different forms can reveal different information about the problem context. 
Almost There 
Misconception/Error The student is unable to interpret 8x + 4 in the context of the problem. 
Examples of Student Work at this Level The student:
 Only says that 8x + 4 can be used to find the perimeter.
 Explains how to find the perimeter of a rectangle.

Questions Eliciting Thinking What does x represent in this problem?
What operations are suggested by the expression 8x + 4? What is this expression telling you to do to the width of the rectangle? 
Instructional Implications Guide the student to interpret the expression 8x + 4 in terms of x, the width of the rectangle. Provide a value of x, and have the student use the expression 8x + 4 to calculate the perimeter of the rectangle. Ask the student to describe what the expression indicates about using the width to calculate the perimeter. Have the student compare 8x + 4 to x + (3x + 2) + x + (3x + 2). Ask the student what the latter expression conveys about the rectangle that 8x + 4 does not.
Ask the student to use the expression 3x + 2 to describe the length of the rectangle in terms of the width. 
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 explains:
 Since the expression shows the sum of the two widths (x) and two lengths (3x + 2), the expression represents the perimeter of the rectangle.
 The two expressions are equivalent. By using the Commutative and Associative Properties of Addition, x + (3xÂ + 2) + x + (3xÂ + 2) = (xÂ + 3x + x + 3x) + (2 + 2) = 8x + 4.
 Abbey's expression shows that an alternate way to find the perimeter is to multiply the width by 8 and add 4 since 8x + 4 is equivalent toÂ xÂ + (3xÂ + 2) +Â xÂ + (3xÂ + 2).

Questions Eliciting Thinking What is an advantage of using Abbey's expression to find the perimeter?
Could you have found the perimeter of this rectangle if you only knew its width?
Why is x + 3x + x + 3x = 8x and not 6x or ? What are the coefficients and why are the coefficients added and not multiplied? 
Instructional Implications Provide a value for x, the width of the rectangle, and have the student determine the perimeter using Britâ€™s and Abbeyâ€™s expressions. Have the student explain which expression is preferred for this purpose and why. Then ask the student to find the length of the rectangle and explain his or her strategy.
Consider using the MFAS task Explain Equivalent Expressions (7.EE.1.2). 