Getting Started 
Misconception/Error The student is unable to identify a pattern of association between the two variables. 
Examples of Student Work at this Level Rather than describing an association between the variables, the student:
 Says there is no association.
 Describes the pattern of change in one variable at only one level of the other variable (e.g., the start time preferences of only the eighth graders).
 Describes the pattern of change in one variable at each level of the other variable (e.g., the start time preferences at each grade level).
 Provides a summary statement about the start time preferences across all grade levels.

Questions Eliciting Thinking What do the numbers in the table mean? Do you see any patterns in the values in the table?
What percentage of sixth graders prefer an earlier start time to a later start time? Seventh graders? Eight graders? Do you notice any relationship between the grade levels and start time preference? 
Instructional Implications If needed, provide instruction on the structure of twoway frequency tables. Emphasize that the structure of the table must allow for all possible combinations of the levels of the variables to be represented. Guide the student to describe the dimensions of the table (e.g., two by three).
Assist the student in identifying the two variables described in the context (e.g., grade level and start time) and the number of levels of each variable [e.g., three grade levels (6th, 7th, and 8th) and two levels of start time (earlier and later)]. Guide the student in observing which way the percentages total to 100%, by columns or by rows. Ask the student to investigate possible patterns of association by observing the trend across grade levels.
Provide additional instruction on interpreting relationships between categorical variables in a twoway frequency table. Emphasize that for a relationship to exist between two categorical variables, there should be differences among the values described by the categories. A pattern in these differences suggests an association. Explain to the student that it is often difficult to detect patterns in raw data but converting the data to percentages may help. When this is done, the percentages can be calculated using the row totals or the column totals although two relative frequency tables can be constructed, one using row totals and one using column totals. Each table can be investigated to determine if there is any evidence of an association between the variables.
Provide the student with additional opportunities to investigate relationships between categorical variables presented in twoway frequency tables. Consider implementing the CPALMS Lesson Plan What’s Your Favorite Subject? (ID 42078). 
Making Progress 
Misconception/Error The student is unable to clearly and completely interpret an entry in the table. 
Examples of Student Work at this Level The student correctly describes the association between grade level and preference for school start time (as grade level increases, the percentage of students wanting an earlier start time increases). However, the student does not completely and clearly describe the meaning of the 25% in the second row.

Questions Eliciting Thinking Does the 25% represent a percentage of students who prefer a later start time or an earlier start time?
Does the 25% represent the percentage of all the students in the survey? How can you tell? 
Instructional Implications Provide feedback to the student regarding any error made or ways in which responses could be improved. Ask the student to interpret other values given in the table. Provide the student with additional opportunities to investigate relationships between categorical variables presented in twoway frequency tables. Consider implementing the MFAS task Siblings and Pets (8.SP.1.4). 
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 identifies the pattern of association and interprets the meaning of the data in context:
 As grade level increases, the percentage of students wanting an earlier start time increases.
 The 25% in the second row indicates that 25% of the seventh graders surveyed prefer a later start time.

Questions Eliciting Thinking How does the data in the table support your conclusion?
Can you tell from the data how many students were surveyed? Explain.
If 200 students were surveyed, how many students would be in each category of the table?
Can this data be used to make an inference about elementary or high school student preferences? Explain. 
Instructional Implications Ask the student to interpret other values given in the table.
Provide raw data for the student to organize into a twoway frequency table. Have the student calculate row and column totals and look for patterns of association between the variables. 