Getting Started 
Misconception/Error The student does not understand how to use probability to determine an expected frequency. 
Examples of Student Work at this Level The student:
 Identifies the probability of rolling a sum of 10 but is unable to accurately calculate the expected frequency.
 Creates a proportion that does not correctly model the problem.
 Provides an incorrect answer without work or explanation.Â

Questions Eliciting Thinking What is the probability of getting a sum of 10?
What is probability? What does a probability of mean?
How can you use the probability to estimate how many times Olivia will get a 10 if she rolls 600 times?
Can you explain what you mean by ? 
Instructional Implications Explain to the student that the frequency of the occurrence of an event can be estimated based on a known probability (either theoretical or experimental). Guide the student to understand that a probability of means one expects to roll a sum of 10 one out of twelve times, so over 600 rolls, one would expect to roll a sum of tenÂ of 600 (or x 600 = 50) times. Ask the student to determine the probability of rolling a sum of six () and to use this probability to estimate the number of times one would expect to roll a sum of six in 600 turns.
Provide additional opportunities to estimate the frequency of an event based on a given probability.
Consider implementing CPALMS Lesson Plans A Roll of the Dice (ID 34343) or Marble Mania (ID 4732), to help students understand probability of simple events.
Consider implementing other MFAS tasks for standard 7.SP.3.6. 
Making Progress 
Misconception/Error The student does not understand the relationship between expected outcomes and longrun relative frequency. 
Examples of Student Work at this Level The student states that Olivia will obtain a sum of tenÂ exactly 50 times because:
 of 600 equals 50.
 The math was done correctly.
 That is the most likely thing to happen.
The student provides an answer that does not directly address the question. 
Questions Eliciting Thinking Do you think it is possible that what happens in reality could differ from a prediction based on a probability?
If the probability of rolling a sum of 10 is , does that mean you will always roll a sum of 10 one out of twelve tries? 
Instructional Implications Use manipulatives to demonstrate the relationship between probabilities, expected frequencies, and observed frequencies (e.g., the probability of getting â€śheadsâ€ť when flipping a coin is ). Ask the student to flip a coin two times and compare his or her results to the expected frequency ( x 2 = 1). Discuss how the small number of trials might account for any differences in the expected and observed frequencies. Explain probability in terms of what is expected to occur in the long run.
Conduct a simulation of the coin flipping experiment. Note the observed frequency of heads after 1, 2, 3, 4, 5, 10, 50, 100, and 1000 trials. Guide the student to observe that in the long run, the number of heads converges on the expected frequency. 
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 estimates that the sum of 10 will occur 50 out of 600 tries based on the calculation x 600 = 50. The student explains it is not likely Olivia will roll a sum of 10 exactly 50 times because:
 It is just an estimate based on the probability.
 The value is only theoretical.
 It is probable but not certain.Â

Questions Eliciting Thinking Is it possible that she could roll a sum of 10 only 20 times out of 600 tries? Why or why not?
What is the probability Olivia will not roll a sum of 10 on any given try? 
Instructional Implications Provide additional opportunities to explore the difference between expected frequencies and observed frequencies. For example, ask the student to determine the expected frequency of heads when tossing a fair coin. Then have the student toss a coin up to 100 times. Ask the student to record the cumulative observed frequencies of heads after each successive coin toss. HaveÂ the student calculate the expected frequencies for each number of coin tosses and compare them to the corresponding observed frequencies. Guide the student to observe the longrun behavior of the observed frequencies and how they tend to converge on the expected frequencies.
Consider implementing other MFAS tasks for standard 7.SP.3.6. 