Getting Started 
Misconception/Error The student is unable to interpret any part of the expression including the likely meaning of n. 
Examples of Student Work at this Level The student:
 Attempts to simplify the expression Â but offers no interpretation of its parts.
 Evaluates the expression for one or two values of n but does not reach a conclusion about what the expression represents.
 Offers an incorrect explanation such as â€śn represents the number of circles.â€ť

Questions Eliciting Thinking What kind of symbols are in each diagram? Do you see a pattern in the sequence of diagrams?
What does the expression calculate? How many parts are there to the expression?
How do you think you might use the expression to calculate the number of symbols in each diagram? 
Instructional Implications Ask the student to reread the problem to find the explanation of what the expression calculates. Have the student evaluate the expression for n = 1 through n = 5. Guide the student to verify that the expression can be used to calculate the number of symbols in each diagram. Explain that n represents the dimension of each square array. Then have the student write out the sequence of numbers that describes the number of diamonds and the sequence of numbers that describes the number of dots in each diagram. Show the student how the two parts of the expression are related to each of these sequences. Then ask the student to explain how the entire expression can be used to calculate the number of symbols in each diagram.
Give the student more practice interpreting algebraic expressions that represent simple sequences. For example, ask the student to explain how the expression n(n + 1) calculates the number of dots in each diagram in a sequence of rectangular arrays of the form 1 x 2, 2 x 3, 3 x 4, 4 x 5, â€¦. 
Making Progress 
Misconception/Error The student systematically explores the diagrams and their relationship to the expression but has difficulty composing a complete and clear explanation. 
Examples of Student Work at this Level The student labels the figures as â€śn = 1, n = 2, n = 3, â€¦â€ť and indicates that (n  1)^{2} represents or corresponds to the dots and (2n â€“ 1) represents or corresponds to the diamonds.
Upon questioning, the student indicates an understanding of how each part of the expression relates to the number of diamonds and dots in each diagram but the studentâ€™s written explanation is unclear or incomplete. 
Questions Eliciting Thinking How would you explain what n represents?
How would you explain what (n  1)^{2} represents?
How would you explain what (2n â€“ 1) represents?
Can you write your explanation on your paper? 
Instructional Implications Guide the student to systematically answer each question in the problem. Encourage the student to compose answers and verbalize them before writing them on his or her paper. Have the student rewrite his or her responses and provide feedback.
Give the student more practice interpreting algebraic expressions that represent quantities in terms of a context. Use formulas from a variety of contexts (e.g., finance, science, and geometry). Allow the student to work with a partner in order to compare interpretations and explanations before finalizing them on paper.
Share examples of complete, correct, and wellwritten responses prepared by other students.
Â 
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 defines n as the diagram number or the dimension of each square array. The student explains that (n  1)^{2} representsÂ the number of circles in each diagram while (2n â€“ 1) representsÂ the number of diamonds in each diagram. So, the expression Â representsÂ the total number of symbols in each diagram.
Note: Upon questioning, the student whose work is shown above explained why the expression results in the number of symbols in each diagram. 
Questions Eliciting Thinking Can you simplify this expression? How does the simplified version of the expression relate to the diagrams? What does the original expression show that the simplified version obscures? 
Instructional Implications Challenge the student to alter the sequence in some systematic way and then find an expression that can be used to calculate the number of symbols in each diagram. Have the student share his or her sequence and expression with other students, so they can provide interpretations of the expression in context. 