Getting Started 
Misconception/Error The student does not understand how to find the constant of proportionality and its significance in writing the equation. 
Examples of Student Work at this Level The student simply writes the ratio 160:40 and can go no further.
The student writes a numerical equation that relates 40 and 160.
The student writes an equation in one variable relating 40 and 160.

Questions Eliciting Thinking Can you write Â in lowest terms? What does this ratio mean in the context of this problem?
You wrote that 40 x 4 = 160. What is the significance of the four in this equation?
How many calories are in one gram of oatmeal?
What are the two variables in this problem? 
Instructional Implications Review the concept of ratio and encourage the student to use a ratio table to write and explore patterns in equivalent ratios. Guide the student to use multiplication (rather than repeated addition) to generate equivalent ratios. Point out that associated values in the table are related by a constant ratio and define this ratio as the constant of proportionality. Give the student additional ratio tables and ask him or her to calculate the constant of proportionality.
Guide the student to use the constant of proportionality to write an equation that models the relationship between variables that are proportionally related. Make explicit what the variables represent and how the constant of proportionality relates associated values of the two variables. Provide the student with tables, graphs, and verbal descriptions of proportionally related variables and ask the student to identify the constant of proportionality and use it to write an equation that models the relationship. 
Moving Forward 
Misconception/Error The student can find the constant of proportionality but is unable to write the equation. 
Examples of Student Work at this Level The student divides 160 by 40 to find the constant of proportionality but does not understand how to use it to write an equation that models the relationship between the number of calories and the serving size (in grams).
Â Â Â

Questions Eliciting Thinking Did you write an equation? Is four calories per gram an equation?
What two variables should the equation relate?
I see you found the constant of proportionality. Do you know how to use it to write an equation? 
Instructional Implications Guide the student to use the constant of proportionality to write an equation that models the relationship between variables that are proportionally related. Make explicit what the variables represent and how the constant of proportionality relates associated values of the two variables. Provide the student with tables, graphs, and verbal descriptions of proportionally related variables, and ask the student to identify the constant of proportionality and use it to write an equation that models the relationship. 
Almost There 
Misconception/Error The student writes a correct equation but is unable to define the variables. 
Examples of Student Work at this Level The student writes an equation such as y = 4x but is unable to explain what the variables represent. 
Questions Eliciting Thinking What does the four in your equation mean or represent?
What does the x in your equation represent? What does the y in your equation represent?
How could one use this equation? What could be calculated with it? 
Instructional Implications Model for the student writing equations that represent the relationship between proportionally related variables. Always include an explicit description of the quantities represented by the variables. Provide the student with additional tables, graphs, and verbal descriptions of proportionally related variables, and ask the student to identify the constant of proportionality and use it to write an equation that models the relationship. Require the student to explicitly describe the variables used in the equations. 
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 an equation such as y = 4x and explains that x represents the serving size in grams and y represents the number of calories.

Questions Eliciting Thinking What does the four in your equation mean or represent?
How could one use this equation? What could be calculated with it?
How many calories are in an 8 gram serving?
How large a serving would be needed for 200 calories? 
Instructional Implications Provide the student with additional tables, graphs, and verbal descriptions of proportionally related variables, and ask the student to identify the constant of proportionality and use it to write an equation that models the relationship.
Give the student another example of two proportionally related variables and have the student explore the relationship between the constant of proportionality, the unit rate, the equation, and its graph. 