Getting Started 
Misconception/Error The student cannot explain what the expression represents within the context of the problem. 
Examples of Student Work at this Level When explaining the meaning of the expression, the student:
 References only â€śthe cost,â€ť â€śhow much he sold,â€ť or â€śthe number of flavors sold.â€ť
 Says the expression represents â€śthe table.â€ť
 Explains that â€śit is the cost for one of each item on the cartâ€ť or â€śit adds everything on the menu.â€ť
 Defines the words quantity and price.
 Explains that â€śvariables represent quantityâ€ť and â€śnumbers represent price.â€ť
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Questions Eliciting Thinking What do the variables represent in each expression? What do the numbers represent?
What is each term calculating? What is the meaning of the value you get by adding all these together?
How is the second expression different from the first? Can you explain these differences?
Where did the 4.5 come from? What about the four? 
Instructional Implications Review the terms variable, coefficient, like terms, expression and equivalent expression as needed. Guide the student to determine the meaning of each term in the two expressions. Provide a value for a and ask the student to calculate a dollar amount for each a term and explain the meaning of that value. If needed, model explaining the meaning of each term (e.g., 1.75a represents the amount earned when a Mango Pops are sold, or 4.5a represents the amount earned by the items for which a units are sold). Repeat the process with b and c. Ask the student to explain the relationship between the two expressions using appropriate mathematical vocabulary. Next, ask the student to total values for each expression and describe the meaning of the sums. Guide the student to interpret the usefulness of each expression and explain why each expression might be preferred depending on the userâ€™s purpose. 
Making Progress 
Misconception/Error The student cannot explain how the different forms of the expression are related. 
Examples of Student Work at this Level The student explains what the second expression represents, but does not reference its relationship to the first expression. The student:
 References the â€ślookâ€ť rather than the context of the expressions writing:
 The second one is shorter.
 They both have variables and prices.
 They are not equivalent because â€śthere are fewer termsâ€ť or â€śthey changed the prices of the itemsâ€ť in the second expression.
 â€śIn the first expression, the aâ€™s, bâ€™s and câ€™s are all split up; in the second expression they are all together.â€ť
 Gives a mathematically incorrect explanation saying:
 They are equivalent because â€śthey both equal 10.75â€ť (or 10.75abc) when all the terms are added.
 States that the second expression â€ścombines like termsâ€ť or â€ścombines the variablesâ€ť without clearly explaining the context or how it relates to the first expression.
 Explains where the new coefficients came from (e.g., â€ś4.5a is 1.75 and 2.75 added togetherâ€ť) but not their meaning.Â

Questions Eliciting Thinking What do the numbers and variables represent in the second expression? Where did the numbers come from?
What does it mean for expressions to be equivalent? Is it possible for expressions to look different and still be equivalent?
What are like terms? How are terms combined in an expression?
What does 4.5a represent? What does 4b represent?
What does the first expression show that the second does not? Are you able to tell the price of each item using the second expression? 
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 â€śmathematically equivalentâ€ť and â€ślooks the same.â€ť Guide the student to give a justification for equivalence by referencing properties of operations.
Guide the student to explain the meaning of each term in the second expression and how that term was obtained from the first expression. Encourage the student to use mathematically precise vocabulary such as like terms and coefficients. Next, ask the student to compare like terms in the first expression and explain the differences in their meanings. Proceed to discuss the two expressions and ask the student to draw conclusions about why each expression might be preferred depending on the userâ€™s purpose. 
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 that the first expression represents the total cost of ice cream sold including the price and quantity of each item sold. When comparing the two versions of the expression, the student may explain that the first expression shows the product of each itemâ€™s cost and quantity sold. In the second expression, items that sold the same quantity are combined so there are fewer terms. 
Questions Eliciting Thinking How might it be useful to have the expression written in simplified form (with like terms combined) as it is in the second expression?
Which expression would be better to use to calculate total earnings for the day?
Which expression would be better to keep track of sales of each item over the whole week? 
Instructional Implications Give a quantity for each of a, b, and c and have the student determine the total sales for the day using each expression. Have the student explain which expression is preferred for this purpose and why.
Ask the student to prove that 1.75a + 2.75a = 4.5a by applying the Distributive Property. 