Getting Started 
Misconception/Error The student is unable to devise an appropriate strategy to solve the problem. 
Examples of Student Work at this Level The student:
 Multiplies, adds, or subtracts the given quantities.
 Reverses the dividend and divisor to get an answer of Â and is unable to evaluate the reasonableness of this answer in the context of the problem.

Questions Eliciting Thinking What about the wording of the problem led you to multiply (or add or subtract)?
Can you draw a diagram to show what is happening in this problem? What math problem can you write to describe your diagram? 
Instructional Implications Guide the student to restate word problems in his or her own words and then draw a model or diagram to show how quantities in the problem are related. Assist the student in relating the model or diagram to a mathematical operation and translating the model into a mathematical problem. Give the student additional practice with a variety of word problems. Provide feedback as needed.
Provide additional instruction on adding, subtracting, multiplying, and dividing rational numbers. Begin with expressions that contain positive fractions and provide focused instruction on any operation with which the student struggles.
Guide the student to use the relationship between division and multiplication to develop the â€śinvert and multiplyâ€ť strategy in a meaningful way. Give the student additional practice with dividing fractions. 
Moving Forward 
Misconception/Error The student is unable to correctly divide fractions and mixed numbers. 
Examples of Student Work at this Level The student:
 Attempts to determine how many addends of comprise Â but errs in adding the fractions.Â
 Divides only the fractional portions of the numbers, then adds the whole number 3 to get an answer of 4.
 Multiplies by Â rather than its reciprocal, .
 Divides the numerators and then the denominators but rounds these quotients getting an answer of .
 Rewrites each fraction with a common denominator of 20 but says the denominator of the product is 20, rather than 400.
 Makes multiple errors in attempting to divide.

Questions Eliciting Thinking Why did you use that procedure? What do you remember about fraction multiplication and division?
How is division like subtraction? How is it different? Do you need to use common denominators when you multiply and divide? 
Instructional Implications Provide additional instruction on adding, subtracting, multiplying, and dividing rational numbers. Begin with expressions that contain positive fractions and provide focused instruction on any operation with which the student struggles.
Use a visual fraction model to explore a simpler example such asÂ . Guide the student to draw a rectangle that represents a whole and to partition the rectangle into fourths. Have the student shade in three of the fourths to model the fractionÂ . Then ask the student to further partition the rectangle into eighths. Interpret dividingÂ Â byÂ Â as finding the number of eighths in . Assist the student in identifying an eighth in the model and in counting the number of eighths in . Demonstrate that the â€śinvert and multiplyâ€ť strategy results in the same quotient, six.
Guide the student to use the relationship between division and multiplication to develop the â€śinvert and multiplyâ€ť strategy in a meaningful way. Give the student additional practice with dividing fractions.
See the Grades 6  8 Progressions for the State Standards in Mathematics (http://ime.math.arizona.edu/progressions) for a more thorough discussion of strategies for dividing rational numbers. 
Almost There 
Misconception/Error The student makes minor mathematical errors in completing the problem. 
Examples of Student Work at this Level In using division, the student:
 Computes a product incorrectly (e.g., Â ).
 Changes Â to an improper fraction incorrectly, (e.g., Â or Â rather than ).
 Changes the product of Â incorrectly to a mixed number.
The student makes a minor subtraction error when using the repeated subtraction method.
The student gives the full mathematical answer of 5Â rather than the contextual answer of five full bags.
Â

Questions Eliciting Thinking How did you get your answer? I think you made an error. Can you check your work or rework the problem to see if you get the same answer?
What does your answer mean in the context of this problem? 
Instructional Implications Provide direct feedback on any errors that the student might have made and allow the student to correct them. Change the quantities in the problem and have the student work the new problem. Provide feedback.
If necessary, review converting between mixed numbers and improper fractions. Give the student additional practice with these conversions.
If necessary, ask the student to reread the problem to be sure that he or she has answered the question asked. 
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 gives the answer of â€śfive complete bagsâ€ť either by:
 Completing the division problem Â to get five with a remainder of Â of a full bag.
 Using repeated subtraction of Â from Â (OR Â from Â ) to get five with a remainder of Â of a cup.
 Changing both quantities to decimals and using repeated subtraction of 0.75 from 3.80 to get five with a remainder of 0.05 of a cup.

Questions Eliciting Thinking Will there be any leftover? If so, how much?
What does the remainder mean in the context of the problem (bags or cups)? 
Instructional Implications Give the student practice solving various multistep problems with rational numbers.
Have the student use the remainder to find and describe both the quantity of leftover cups of trail mix (Â of a cup) and the portion of a bag that can be filled with the leftovers (Â of a bag). 