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.
 Only explains the meaning of one or two cell entries.
 Makes an incorrect comparative statement about cell entries.
 Provides a general description of the two variables on which data was collected.

Questions Eliciting Thinking What are the two variables on which data was collected?
What do the numbers in the table mean?
What does each denominator represent? What does each numerator represent?
Would it help you to write each ratio in percent form?
Can you tell from the table, how many students have siblings? How many students do not have any siblings?
Out of the students who do not have siblings, can you tell how many of them do not have pets? Out of the students who have siblings, how many of them have pets?
Do you notice any relationship between having a pet and having a sibling? 
Instructional Implications If needed, provide instruction on the structure of twoway frequency tables. Guide the student to describe the dimensions of the table (e.g. two by two). Emphasize that the structure of the table must allow for all possible combinations of the levels of the variables to be represented.
Assist the student in identifying the two variables described in the context (e.g., having a sibling and having a pet) and the number of levels of each variable (e.g., two levels of siblings and two levels of pets). Guide the student in observing which totals were used to determine the relative frequencies, the column or the row totals. Ask the student to investigate possible patterns of association by observing the trend across rows.
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 columns 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 having a sibling and having a pet (students who have siblings are less likely to have pets; students who have no sibling are more likely to have pets). However, the student does not completely and clearly describe the meaning of in the table.

Questions Eliciting Thinking How can you tell if the data is calculated by column or row?
How did you determine the association between having a sibling and having a pet?
What does the five in the numerator represent? What does the 75 in the denominator represent? 
Instructional Implications Provide assistance interpreting relative frequencies for data in twoway tables. Remind the student that each ratio represents a part and a whole. Ask the student to explain in context (e.g., five represents the number of students who do not have a pet out of the total of 75 students who do not have a sibling).
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 School Start Time (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. The student explains:
 The students who do not have a sibling are more likely to have a pet (or the students who have a sibling are less likely to have a pet).
 The means that of the 75 students surveyed who do not have a sibling, five do not have a pet.

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.
Can you tell from the table how many students have a pet? Of these students, how many have a sibling?
If 500 people were surveyed, how many would be in each category of the table? How would you determine that? 
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. 