Getting Started 
Misconception/Error The student is unable to correctly write a function that models the given relationship. 
Examples of Student Work at this Level The student writes an incorrect function or does not write a function at all.

Questions Eliciting Thinking Can you restate the situation described in the problem? What information are you given?
How would you describe the relationship between the amount Juan has saved and the number of weeks he has been saving?
What do you think 'write a function' means?
How could you represent â€ś$35 per weekâ€ť if x is equal to the number of weeks Juan has been saving? 
Instructional Implications Assist the student in first identifying the relevant variables and quantities given in the problem and then verbally describing their relationship. Guide the student to translate his or her verbal description into an equation using the notation given in the problem. GuideÂ the student in using the equation to calculate the amount of money Juan will have after 10 weeks and the amount of time it will take Juan to save a total of $1000. Help the student differentiate between values of the independent and dependent variables. Model using function notation appropriately.
Provide the student with verbal descriptions of realworld examples of functions and have the student create tables of functional values. Ask the student to describe patterns in the tables. Encourage the student to observe not only a recursive relationship but also the explicit relationship between inputs and outputs. Ask the student to write explicit equations that model the relationships among the variables in problem descriptions and to use function notation whenÂ writing equations.
If necessary, review function notation. Consider implementing the MFAS task What Is the Function Notation? (FIF.1.2) . 
Moving Forward 
Misconception/Error The student correctly writes a function that models the given relationship but does not understand how to use the equation to find values of at least one of the variables. 
Examples of Student Work at this Level The student correctly writes the function as f(x) = 35x + 300 or f(x) = 300 + 35x. When attempting to determine how much Juan will have saved after 10 weeks, the student:
 Substitutes 10 for x but is unable to correctly calculate f(10).
 Multiplies $35 by 10 and says that Juan has $350.
When asked to determine the value of x for which f(x) = 1000, the student:
 Calculates f(1000).
 Divides $1000 by $35 and says it will take 28.6 weeks.
 Uses a guess and test method to find a value of x such that f(x) = 1000.

Questions Eliciting Thinking Did you use the function that you wrote to answer the other questions?
Explain to me how you found your answer to the second question. How did you get $350? Is that all the money Juan had saved?
Explain to me how you found your answer to the third question. Why did you divide 1000 by 35?
Why did you substitute 1000 for x? What does x represent? What does f(x) represent?
I see that you checked to see if Juan could save $1000 in 20 weeks, but how did you determine that x = 20? 
Instructional Implications Show the student how to use his or her equation to calculate the amount of money Juan will have after 10 weeks and the amount of time it will take Juan to save a total of $1000. Help the student to differentiate between values of the independent and dependent variables. Model using function notation appropriately.
Provide the student with additional examples of algebraic representations of functions both with and without context. Use function notation to ask the student to calculate functional values. For example, given the function g(x) = 5x â€“ 8, ask the student to calculate g(2) or to find x such that g(x) = 20. Be sure the student understands the difference between these two problems.
If necessary, review function notation. Consider implementing the MFAS task What Is the Function Notation? or Evaluating a Function (FIF.1.2). 
Almost There 
Misconception/Error The student correctly writes a function that models the given relationship but uses function notation incorrectly or not at all. 
Examples of Student Work at this Level The student can correctly answer questions two and three but does not use function notation correctly. For example, the student:
 Writes f = 300 + 350x, fx instead of f(x), 10(f) instead of f(10), or f(1000) instead of 1000.
 Uses different variables to represent the same value (e.g., uses both x and w to represent the number of weeks).
 Does not use function notation.

Questions Eliciting Thinking What is function notation? Which symbol represents the number of weeks? Which symbol represents the total amount of money that Juan has saved?
What does f(10) represent? What would f(1000) represent? 
Instructional Implications Review function notation emphasizing the meaning of the symbols x, f, f(x), and f(10). Explain that the choice of symbols to name the function and to represent the independent variable is arbitrary but does need to be used consistently throughout the problem. Provide explicit feedback to the student concerning any errors in notation and guide the student to correct them. Provide continued opportunities to use function notation.
Consider implementing the MFAS task What Is the Function Notation? (FIF.1.2). 
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 writes the function as f(x) = 35x + 300 or f(x) = 300 + 35x, substitutes 10 for x into the equation and determines Juan will have saved $650, substitutes $1000 for f(x), and determines that it will take 20 weeks for Juan to save $1000.
The student may make a slight mistake but is able to selfcorrect [e.g., f(x) = 650 instead of f(10) = 650].

Questions Eliciting Thinking For question two you wrote f(x) = 650. How should this be written in order to show the input value?
What do you think the graph of this function looks like? How do you know?
How would the graph change if Juan saved more money each week? Less money each week?
How would the graph change if Juanâ€™s initial amount was more? Less? 
Instructional Implications Provide the student with opportunities to write equations that define more complex functions and to write equations to define functions when given their graphs. 