Getting Started 
Misconception/Error The student does not understand the meaning of the term equivalent. 
Examples of Student Work at this Level The student makes decisions based on whether or not the expressions â€ślook likeâ€ť the original expression in terms of placement of signs and numbers, e.g., where the five is located or where the negative is located in the fraction.
Student explanations may include:
 â€śit doesnâ€™t matchâ€ť or â€śitâ€™s dividing one and four, not five and 20â€ť
 â€śneither the 20 nor the five is negativeâ€ť or â€śthe numbers are in the wrong placeâ€ť
 â€śthe number four is not even in the problemâ€ť or â€śit is not a fractionâ€ť
 â€śno parenthesisâ€ť or â€śthereâ€™s too many negative signsâ€ť
 â€śit cannot be a fractionâ€ť or â€śit doesnâ€™t match upâ€ť
 â€śthe 20 shouldnâ€™t be negativeâ€ť or â€śthere should only be one negativeâ€ť or â€śitâ€™s kind of the sameâ€ť

Questions Eliciting Thinking What do you mean that it â€śmatches?â€ť
What does it mean for two fractions to be equivalent? Can you give me an example of two fractions that are equivalent? 
Instructional Implications Provide instruction on the meaning of equivalent as it applies to expressions that are equal in value. Provide opportunities for the student to evaluate expressions containing quotients of rational numbers and to determine its equivalency.
Provide instruction on the meaning of equivalent as it applies to fractions. Be sure the student understands that p/q means p Ă· q and â€“(p/q) = (â€“p)/q = p/(â€“q). Expose the student to effective strategies of his or her classmates to determine equivalency of expressions containing rational numbers. 
Moving Forward 
Misconception/Error The student makes errors with the order of division. 
Examples of Student Work at this Level The student interprets 5 Ă· 20 as 20 Ă· (5) or the student interprets Â as 4 Ă· 1.

Questions Eliciting Thinking What is the difference between 5 Ă· 20 and 20 Ă· 5?
What is the difference between 5 Ă· 20 and 20 Ă· (5)?
How would you rewrite Â as a division problem?
How would you rewrite 5 Ă· 20 as a fraction? 
Instructional Implications Guide the student to understand the order of division implied in simple division statements and in fractions. Give the student a general statement to follow such as â€śp Ă· q = p/q and that means how many times q divides into p.â€ť Also address writing each form using a standard long division symbol, since that is the method most students will use to evaluate fractions or convert them to decimals. Ask the student to redo the problems on the Quotients of Integers worksheet. Provide feedback as needed.
Note: The Quotients of Integers worksheet is editable and can be rewritten with new integers and the problems rearranged to give the student further practice. 
Almost There 
Misconception/Error The student uses a correct strategy but makes some sign errors. 
Examples of Student Work at this Level The student explains the equivalency of most expressions correctly, but has trouble with the use of three negatives within one fraction (d).
The student interprets fractions of the form Â as â€“a Ă· (â€“b).
Â Â Â
The student writes 5 Ă· 20 as a fraction and uses a crossmultiplication strategy to check for equivalency, but makes some errors when applying the strategy to fractions with multiple negatives.

Questions Eliciting Thinking What did you find difficult about (d)? How did you determine whether the sign of the answer is positive or negative?
What does it mean for a fraction to be negative?
When using crossmultiplication to check for equivalency, what did you do with the negatives in each fraction? 
Instructional Implications Provide guidance on how to evaluate a fraction with three negatives, such as (d). Many students learn rules for multiplication and division of pairs of negative integers but do not realize that these rules must be generalized when there are more than two negatives. Guide the student to systematically consider what happens when two, three, four, five, or more negative numbers are multiplied or divided, and help the student to develop a general rule for the sign of the result.
Be sure the student understands that . Ask the student to reconsider his or her responses to (b) and (e), if the meaning of the sign was misinterpreted in these problems. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level
 It is equivalent because it also equals Â or 0.25; 5 Ă· 20 simplifies to , which can also be written as .
 It is not equivalent because 5 Ă· 20 = Â but .
 It is not equivalent because Â does not equal 4.
 It is equivalent because the two negatives in the parentheses make the fraction positive, so the negative on the outside of the parentheses makes the whole expression negative.
 It is equivalent because the negative of a fraction can be in the numerator or the denominator or out in front of the whole fraction (those are all equal).
 It is not equivalent because 5 Ă· 20 is negative but Â is positive.

Questions Eliciting Thinking What would happen if the expression in (d) had only one negative sign in the parentheses? Would the answer change depending on whether the negative were in the numerator or the denominator?
What are some other fractions that are equivalent to 5 Ă· 20? Can you write one that uses the integer 100 somewhere in the fraction? 
Instructional Implications Ask the student to evaluate more complex expressions involving operations on rational numbers. Guide the student to apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers. 