Getting Started 
Misconception/Error The student is not able to determine the relationship between independent and dependent variables from the table. 
Examples of Student Work at this Level The student cannot write a correct equation because he or she does not understand the relationship between the variables.

Questions Eliciting Thinking How are the numbers on the left side of the table related to the numbers on the right side of the table? Do you notice a pattern? Does your pattern work for every pair of numbers in the table?
Will your equation work for every input value? How can you check?
Does one variable increase (or decrease) when the other variable increases (or decreases)?
What is an independent variable? What is a dependent variable?
What happens to the test grade when the number correct on the test changes? Can you determine how much the grade changes for each right question?
What happens to the number of diagonals when the number of sides changes? Can you relate the amount of change in the number of sides to the change in the number of diagonals? 
Instructional Implications Provide the student with some completed tables of values and guide him or her to recognize trends or patterns. Have the student determine whether the value of the dependent variable increases or decreases as the value of the independent variable increases. Then encourage the student to analyze the relationship more closely and describe the actual amount of change by referring to numerical values and operations. Show the student how to check each row and be certain the mathematical relationship is consistent for every input value.
Consider implementing the CPALMS Unit FunctionsDay Trips (National Security Agency) (ID 4008), a threelesson unit that explores patterns in tables.
Clarify the difference between expressions and equations. Have the student compare 4n to 4n = g. Explain to the student how the equation shows the relationship between the variables. Guide the student to write equations to model the relationship between the variables shown by the tables. 
Moving Forward 
Misconception/Error The student is unable to write equations that accurately model the relationships shown by the tables. 
Examples of Student Work at this Level The student recognizes the relationship between independent and dependent variables but does not write an accurate equation. Instead, the student:
 Records the operation instead of the complete equation.
 Writes an expression instead of the complete equation.

Questions Eliciting Thinking Can you write an equation that uses both variables?
Is this an equation? Can you turn this expression into an equation?
What is the difference between an equation and an expression?
Which variable is independent? Which variable is dependent? How is the dependent variable related (mathematically) to the independent variable? 
Instructional Implications Ask the student to identify the independent and dependent variables and to verbally describe the relationship between them. Guide the student to describe precisely, in mathematical terms, how to calculate a value of the dependent variable from a value of the independent variable. Then assist the student in mathematically representing this relationship with an equation.
Provide additional problem contexts that can be modeled by an equation in two variables. Encourage the student to analyze the relationship using a table or graph and then write an equation that reflects the relationship between the variables. 
Almost There 
Misconception/Error The student is not able to explain clearly how the equation shows the relationship between the independent and dependent variables. 
Examples of Student Work at this Level The student’s explanation is unclear or imprecise.
The student provides no explanation.

Questions Eliciting Thinking Which variable is independent (dependent)? How are the independent and dependent variables related in the table?
What is the meaning of each change in the independent variable?
What does it mean when n increases by one? What happens to g when n increases by one?
How come you have to subtract three from the number of sides? What happens to d each time s changes? 
Instructional Implications Review the terms dependent variable and independent variable. Clarify that both the dependent and independent variables change, though in different ways. Change in the dependent variable depends on change in the independent variable. Explain that typically the value of the independent variable is freely chosen, but the value of the dependent variable is calculated for particular values of the independent variable. It might be helpful to describe these variables in terms of an inputoutput system (where the independent variable is the input and the dependent variable is the output).
Explain that in equations such as a = 3 + b, the isolated variable is generally considered to be the dependent variable (output). Ask the student to distinguish dependent from independent variables given in equations, written descriptions, and realworld situations. 
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 equation 4n = g (or n · 4 = g) for the first problem and explains that each question is worth four points and the grade is determined by multiplying the number of questions answered correctly by four.
The student writes s – 3 = d for the second problem and explains that the number of diagonals is always three less than the number of sides and the equation is written to reflect this.

Questions Eliciting Thinking How many questions are on the test? How do you know?
How many questions would the student get right if he or she scored an 80 on the test?
Is it possible for n to equal 30? Why or why not?
Is it possible for s to equal 2 or 1? Why or why not? 
Instructional Implications Pair the student with a Moving Forward partner. Have the student explain the difference between an expression and an equation and how an equation can show the relationship between the dependent and independent variables.
Provide the student with a table of inputs and outputs where the meaning of the variables is not described. Challenge the student to write an equation to represent the relationship in the table and then create a realworld context that can be modeled by the equation.
Challenge the student to write an equation of the form mx + b = y from a table such as:
