Getting Started 
Misconception/Error The student is unable to compute quotients of fractions. 
Examples of Student Work at this Level The student:
 Multiplies the fractions instead of dividing them.
 Rewrites the fraction division problem as an equivalent multiplication problem but does not multiply the denominators. Instead, the student uses the common denominator as the denominator of the quotient getting .
 Rewrites each fraction with a common denominator but divides numerators in the wrong order.

Questions Eliciting Thinking If you were asked to multiply Â and , would the product be the same as the quotient?
I see that you rewrote the division as an equivalent multiplication problem. How did you multiply the two fractions?
Is it necessary to find a common denominator when you are dividing fractions? If you do find one, how should you then complete the division? 
Instructional Implications Help the student develop a strategy for dividing fractions. Begin with simple problems such as Â Ă· Â and use a visual model to help the student make sense of the division and its quotient. Eventually, transition the student to using the relationship between division and multiplication to develop the â€śinvert and multiplyâ€ť strategy in a meaningful way (see the Progressions for State Standards in Mathematics, Grades 6 â€“ 8, The Number System which can be found at http://ime.math.arizona.edu/progressions).
Give the student a pairÂ of problems involving fractions, one multiplication and one division. Ask the student to determine a strategy that is appropriate for each problem. Compare the strategies and engage the student in a discussion of the justification of each strategy. 
Moving Forward 
Misconception/Error The student is unable to compute quotients of mixed numbers. 
Examples of Student Work at this Level The student:
 Divides each part of the mixed numbers separately (8 Ă· 4; 1 Ă· 1, 10 Ă· 5 getting a final answer of 2).
 Multiplies each part of the mixed numbers separately (8 x 4, 1 x 1, 10 x 5 getting a final answer of 32).
 Attempts to use an invert and multiply strategy without converting the mixed numbers to fractions.
 Rewrites the mixed numbers as fractions and then multiplies the fractions instead of dividing them.
 Multiplies the divisor by the reciprocal of the dividend.

Questions Eliciting Thinking Can you explain your strategy for dividing mixed numbers? Can you mathematically justify your strategy?
If you estimate the quotient, does your answer make sense compared to the estimate? 
Instructional Implications Be sure the student has developed an understanding of and proficiency with multiplying and dividing simple fractions. Then emphasize that once mixed numbers are rewritten as fractions, they can be multiplied and divided using the same strategies the student uses with simple fractions.
Allow the student to explore other strategies for dividing fractions and mixed numbers that make sense to the student (e.g., rewriting the divisor and the dividend with a common denominator and then dividing the numerators to find the quotient). Assist the student in evaluating and justifying the strategies. 
Almost There 
Misconception/Error The student makes minor errors when trying to compute quotients of fractions and mixed numbers. 
Examples of Student Work at this Level The student:

Questions Eliciting Thinking Can you read through your problem to look for any math errors?
What does it mean to write in lowest terms? Did you do that? How can you tell if a fraction is written in lowest terms?
Why did you put an equal sign between these expressions? Is there some way you can tell if they really are equivalent? What should you do with the second fraction when you change the operation to multiplication? 
Instructional Implications If needed, provide guidance on converting fractions to mixed numbers and writing fractions in lowest terms. Provide additional problems involving division of fractions and mixed numbers. Pair the student with another Almost There student, so they can compare strategies and answers and reconcile any differences.
Model writing mathematics correctly for the student. Provide immediate feedback if the student shows work in a mathematically incorrect manner (e.g., the student writes Â Ă· 5 when he or she meansÂ ). 
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 completes both division problems in a mathematically correct way, showing all work, and getting final answers of (1) Â and (2) Â or 1.

Questions Eliciting Thinking If you rewrite the fractions in lowest terms before you multiply, will you get the same answer?
Would you get a different answer if you reversed the dividend and divisor? Are the new answers related to the original answer in any way?
Can you change each of the original fractions into decimals and redo the problem using all decimals? Is your decimal answer equivalent to your fraction answer? How do you know? 
Instructional Implications Allow the student to explore other strategies for dividing fractions and mixed numbers that make sense to the student (e.g., rewriting the divisor and the dividend with a common denominator and then dividing the numerators to find the quotient). Ask the student to explain and justify the strategies. 