Getting Started 
Misconception/Error The student is unable to correctly compute with fractions. 
Examples of Student Work at this Level The student uses an incorrect strategy for either adding, subtracting, multiplying, or dividing fractions. The student may also have difficulties computing with positive and negative numbers.
Â Â Â
Â
Â 
Questions Eliciting Thinking Can you explain your strategy for adding (subtracting, multiplying, or dividing) these fractions?
Can you explain why you need a common denominator to add and subtract fractions?
Does the fact that some of the fractions are negative alter your strategy? How would you have added (subtracted, multiplied, or divided) these fractions if both were positive? 
Instructional Implications If needed, review the concept of a fraction, specifically the meaning of both the numerator and the denominator. Provide models to illustrate equivalent fractions and review computational methods for creating sets of equivalent fractions. Review strategies for adding, subtracting, multiplying, and dividing fractions. Assist the student in understanding the need for a common denominator when adding and subtracting fractions. Once the student is proficient computing with fractions, review computing with integers. Then introduce computing with rational numbers. Provide opportunities for the student to work with rational numbers in a variety of contexts. Include rational numbers in both realworld and mathematical situations. 
Moving Forward 
Misconception/Error The student is unable to correctly compute with integers. 
Examples of Student Work at this Level The student can correctly add, subtract, multiply, and divide fractions but makes errors related to the signs of the fractions.
Â 
Questions Eliciting Thinking Can you explain your strategy for adding (subtracting, multiplying, or dividing) integers?
Does the fact that these numbers are fractions alter your strategy for computing with positive and negative numbers? How would you have added (subtracted, multiplied, or divided) these numbers if they were whole numbers with the same signs instead of fractions? 
Instructional Implications Review how to add, subtract, multiply, and divide integers. Then assist the student in applying strategies for working with integers to rational numbers. Provide additional opportunities to add, subtract, multiply, and divide rational numbers in both realworld and mathematical situations. 
Almost There 
Misconception/Error The student makes a computational or other minor error. 
Examples of Student Work at this Level The student appears to understand how to compute with rational numbers. However, the student:
 Makes an error when rewriting an improper fraction as a mixed number.
Â
 Makes an error when reducing fractions while multiplying.
Â
 Drops a negative symbol from the final answer.

Questions Eliciting Thinking I think you made an error in this problem. Can you find and correct it?
Should this product be positive or negative? Why? 
Instructional Implications If needed, assist the student in locating his or her error and ask the student to make corrections. Provide additional opportunities to add, subtract, multiply, and divide rational numbers in both realworld and mathematical situations. 
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 completes each problem with work coherently shown getting answers equivalent to:
1. 2. 3 3. 2 4.
Â Â Â
Â 
Questions Eliciting Thinking Are there other equivalent forms in which these answers could be written?
How is computing with positive and negative fractions the same as or different from computing with positive and negative whole numbers?
Do you know any other strategies for adding, subtracting, multiplying, or dividing fractions? 
Instructional Implications If the student did not write answers as fractions in lowest terms, ask the student to do so. Challenge the student to find, explain, and justify another correct strategy for dividing fractions.
Provide additional opportunities to add, subtract, multiply, and divide rational numbers in realworld situations. 