Getting Started 
Misconception/Error The student writes or graphs the ordered pairs incorrectly. 
Examples of Student Work at this Level The student:
 Writes and graphs an ordered pair not given in the problem.
 Uses the grade level in place of the number of teachers.
 Writes the ordered pairs correctly but makes errors in graphing them.

Questions Eliciting Thinking How is this ordered pair [e.g., (4, 81)] related to the problem?
What ratios were you asked to graph? Did you write out the ratios before you graphed them?
Can you show me how you graphed these ordered pairs? 
Instructional Implications Clarify why the grades of the teachers and students do not need to be considered. Encourage the student to consider which quantities are being compared to avoid writing ordered pairs using the wrong information. Suggest underlining or circling the words related to the ratio.
Provide direct instruction on graphing ordered pairs in the coordinate plane. Model how to write ordered pairs from the given ratios, and have the student attempt to graph the pairs. Provide another set of ratios and ask the student to write and graph the ordered pairs. Consider implementing the CPALMS Lesson Plan Bomb the Boat â€“ Sink the Teacherâ€™s Fleet! (ID 48848).
Review what it means for two variables to be proportionally related. Introduce the concept of the constant of proportionality and its role in describing the relationship between variables that are proportionally related. Show the student both an example of a graph of proportionally related data and an example of a graph of linearly related data that is not proportional. For each relationship, ask the student to identify several ordered pairs and determine the constant of proportionality (when possible). Guide the student to observe that the graph of the proportionally related data is linear and passes through the origin. Use the equation of the proportional relationship to explain why the graph must contain the point (0, 0). 
Moving Forward 
Misconception/Error The student does not demonstrate an understanding of proportionality. 
Examples of Student Work at this Level The student writes and graphs each ordered pair correctly but says the quantities are proportionally related because:
 â€śThe more amount of teachers they have the more students.â€ť The student appears to confuse a proportional relationship with a positive correlation.
 â€śYou can divide each one evenly.â€ť By â€śevenly,â€ť the student means the quotients are whole numbers.

Questions Eliciting Thinking What does proportional mean?
How can you determine if two quantities are proportionally related?
What is the constant of proportionality? 
Instructional Implications Review what it means for two variables to be proportionally related. Introduce the concept of the constant of proportionality and its role in describing the relationship between variables that are proportionally related. Show the student both an example of a graph of proportionally related data and an example of a graph of linearly related data that is not proportional. For each relationship, ask the student to identify several ordered pairs and determine the constant of proportionality (when possible). Guide the student to observe that the graph of the proportionally related data is linear and passes through the origin. Use the equation of the proportional relationship to explain why the graph must contain the point (0, 0). 
Making Progress 
Misconception/Error The student does not determine if the given ratios are proportionally related but demonstrates an understanding of proportionality. 
Examples of Student Work at this Level The student rewrites:
 Three of the ratios in an equivalent form, graphs them, and recognizes that each is equivalent to 1:23. The student ignores the one ratio that is not equivalent to 1:23 and concludes the relationship is proportional.
 One ratio as a unit rate and generates other equivalent ratios which are graphed. The student ignores the other ratios in the problem and concludes the relationship graphed is proportional.Â

Questions Eliciting Thinking How did you determine the ordered pairs that you graphed?
Did you write ordered pairs for all four of the situations? Did you graph these ordered pairs?
What does the graph of proportionally related data look like?
What might the graph look like if the data is not proportional? 
Instructional Implications Explain to the student that the proportionality of the two quantities has not yet been determined. Guide the student to write ordered pairs for each of the given situations and graph these ordered pairs. Then ask the student to decide if the graph represents a proportional relationship. Consider changing the ratios and implementing this task again for further assessment. 
Almost There 
Misconception/Error The student determines proportionality without reference to the graph. 
Examples of Student Work at this Level The student rewrites each ratio as a unit rate and compares unit rates to determine proportionality. Upon questioning, the student does not demonstrate an understanding of the graph of a proportional relationship.

Questions Eliciting Thinking How did you determine the quantities are not proportional?
Is there another way to determine proportionality?
Can you use the graph to determine proportionality? 
Instructional Implications Focus instruction specifically on graphs of proportionally related data. Assist the student in observing that the graph of proportionally related data is always linear and contains the origin. Provide the student with a variety of ratio tables and their graphs. Ask the student to match each table with its graph. Include tables and graphs that do not represent proportionally related data and ask the student to identify the graphs of the proportional relationships. Consider using the MFAS task Graphs of Proportional Relationships (7.RP.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 correctly writes and graphs each ratio. The student references the graph and acknowledges that the points do not fall in a straight line which means that the number of teachers is not proportional to the number of students. The student may also calculate unit rates but demonstrates an understanding of the graph of a proportional relationship.

Questions Eliciting Thinking Why do the graphs of proportionally related quantities form a straight line?
What special point will the graph of a proportional relationship always contain?
What are some other ways to determine proportionality? 
Instructional Implications Challenge the student to find the one ratio that prevents the data from being proportional. Ask the student to rewrite this ratio so that it is proportional to the other three ratios. Then ask the student to write an equation to show the relationship between the number of teachers and the number of students.
Consider implementing the MFAS task Babysitting Graph (7.RP.1.2). 