Getting Started 
Misconception/Error The student is not able to interpret conditional relative frequencies of data in a twoway table. 
Examples of Student Work at this Level The studentâ€™s explanation of conditional relative frequency is unclear or incorrect. For example, the student:
 Writes the same explanation for both conditional relative frequencies.
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 Converts each conditional relative frequency to a decimal or percent without further interpretation.
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 Incorrectly describes one or both relative frequencies.
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Questions Eliciting Thinking What is the meaning of conditional relative frequency?
The conditional relative frequencies are given as fractions. What does the numerator of each fraction represent? The denominator?
Where are the values of the denominator of each fraction located in the table? How can this location help you understand its meaning?
In the table, the value of 17 combines which two categories? What does the value of 50 represent in the table? Can you use the structure of a fraction to interpret how these two categories are related?
What does the value of 29 represent in the table? Can you use the structure of a fraction to interpret the relationship between 17 and 29? 
Instructional Implications Provide instruction on the structure of twoway frequency tables. Assist the student in identifying the two variables described in the context (e.g., gender and vegetarianism) and the number of levels of each variable [e.g., two levels of gender (male and female) and two levels of vegetarianism (vegetarian and nonvegetarian)]. Provide the student direct instruction onÂ interpreting marginal, joint, and conditional relative frequencies for data in twoway tables. Remind the student that a conditional frequency represents a number in a cell from the body of the table out of a row or column total. Ask the student to identify the cell containing 17 and its row total of 50. Guide the student to use the appropriate column and row titles and the context of the data to explain the meaning of . Do the same forÂ .
Provide practice writing marginal, joint, and conditional relative frequencies for data in twoway tables. Assist the student with interpreting marginal, joint, and conditional relative frequencies when given in fraction form. Tell the student to use the relationship between the numerator and denominator of a fraction to guide the word arrangement for a clear and concise response. Give feedback as necessary. 
Making Progress 
Misconception/Error The studentâ€™s interpretation lacks specificity. 
Examples of Student Work at this Level The student indicates an understanding of the conditional relative frequencies in the context of the data but omits an important piece of information in his or her interpretation. For example, the student:
 Does not reference the actual values in the conditional relative frequencies when describing them.
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 Does not explain the meaning of the numerator or the denominator.
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 Provides a correct interpretation but makes errors when converting fractions to percents.
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Questions Eliciting Thinking How did you determine your interpretation? Which part of the table did you use?
How does understanding the structure (the numerator and denominator) of a fraction help you answer these questions? What does the numerator of a fraction represent? The denominator?
Your answer is somewhat correct, but can you refine it or describe it more clearly and concisely?
What does the value of 17 represent in the table? Which value of 50 in the table do you think best relates to the value of 17? 
Instructional Implications Ask the student to identify the two categories related by the cell with an entry of 17. Next, ask the student to identify the values of 50 in the table and to choose the value in the same row or column as 17. Finally, remind the student to use wording that supports the meaning of each of these values when interpreting the conditional relative frequency.
Remind the student that writing a clear and concise response starts with using the wording of the question. For example, guide the student to describe the meaning of the relative frequencies as: The conditional relative frequency of Â is the number of (the category related to the numerator) out of (the category related to the denominator).
Provide practice writing the marginal, joint, and conditional relative frequencies for data in twoway tables. Assist the student with interpreting marginal, joint, and conditional relative frequencies when given in fraction form. Tell the student to use the relationship between the numerator and denominator of a fraction to guide the word arrangement for a clear and concise response. Give feedback as necessary. 
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 interprets the relative frequencies as follows:
 The conditional relative frequency of Â means that out of the 50 females surveyed, 17 of them are vegetarians.
 The conditional relative frequency of Â means that out of the 29 vegetarians surveyed, 17 of them are females.
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Questions Eliciting Thinking Can you interpret the meaning ofÂ ?Â ? What if the sample size had not been 100? Would that change how you determine the relative frequencies? 
Instructional Implications Provide the student with data presented in a twoway table with a sample size that is not easily converted to a percent and with unequal row totals and unequal column totals. Ask the student to consider how to calculate relative marginal and joint frequencies (i.e., what values should serve as denominators for each relative frequency). 