Getting Started 
Misconception/Error The student does not understand the meaning of function notation. 
Examples of Student Work at this Level Rather than rewriting the formula using function notation, the student:
 Rewrites the formula as V = sÂ·sÂ·s.
 Rewrites the formula using different variables.
 Explains the meaning of V and s without rewriting the formula using function notation.

Questions Eliciting Thinking What is function notation? Can you write an example of function notation?
Suppose for some function f, fÂ (3) = 6. Can you explain what this means? 
Instructional Implications Review the concept of a function and provide instruction on function notation. Give the student more experience with functions and function notation by presenting functions in a variety of ways, using verbal descriptions, graphs, algebraic rules, and tables. Use function notation to ask the student to find values of the independent and dependent variables [e.g., given a verbal description, graph, algebraic rule, or table of some function f, ask the student to find fÂ (5) or findÂ x such that fÂ (x) = 2].
Provide opportunities for the student to interpret statements that use function notation in terms of a context.
Consider implementing MFAS task Cell Phone Battery Life (FIF.1.2). 
Moving Forward 
Misconception/Error The student uses function notation incorrectly. 
Examples of Student Work at this Level The student attempts to use function notation but uses it incorrectly. For example, the student:
 Interchanges the independent and dependent variables in the function notation, writing .
 Introduces an unnecessary variable name writing , , or .

Questions Eliciting Thinking Why did you change the formula? If you want to rewrite the formula using function notation, what part of the formula will need to be rewritten?
Which variable in the formula represents the input? Which is output?
What does fÂ (x) mean? 
Instructional Implications Review function notation emphasizing the meaning of symbols such as x, f, and fÂ (x). Explain that the choice of symbol to represent the independent variable is arbitrary but does need to be used consistently throughout the algebraic representation of the function. Model rewriting other equations and formulas using function notation. Vary the choice of symbols but emphasize that the basic structure of the equation or formula should remain unchanged. Provide continued opportunities to use function notation and to interpret statements that use function notation in terms of a context. 
Almost There 
Misconception/Error The student correctly rewrites the equation using function notation but is unable to explain the meaning of the symbols. 
Examples of Student Work at this Level The student rewrites the formula correctly using function notation, for example, as Â or . However, when explaining the meaning of the notation, the student:
 Indicates how the notation is read rather than what it means.
 Explains that V(s) is the same as y or f(x).
 Explains the meaning of s and V rather than s and V(s).

Questions Eliciting Thinking Can you explain the meaning of the symbols you used? Why did you use V ? s? f? x? What does each represent? 
Instructional Implications Review function notation emphasizing the meaning of symbols such as s, V and VÂ (s). Model explaining the meaning of the symbols used in function notation, that is, explain that if ,Â then V names the function, s represents a value of the independent variable, and VÂ (s) represents the value of the dependent variable that corresponds to s. Be sure the student understands how to read function notation and interpret it in context. Provide continued opportunities to use function notation. 
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 Â and explains that s represents the length of a side of a cube and V(s) represents the volume of the cube when the length of a side is given by s or that the volume of a cube is a function of the length of a side of the cube. 
Questions Eliciting Thinking Does it matter which letters or symbols are chosen when writing function notation?
In the context of this function, what would V(5) represent?
Suppose V(a) = 100. Without calculating a, can you describe what it represents? 
Instructional Implications Ask the student to rewrite other twovariable formulas, such as the formulas for the circumference and area of a circle, using function notation.
Provide the student with examples of the misuse of function notation, such as fÂ (x) = y + 3; for some function g, g = x â€“ 8; or . Ask the student to explain eachÂ notational error and correct it. 