Getting Started 
Misconception/Error The studentâ€™s realworld description does not reflect a division problem or is the wrong division problem. 
Examples of Student Work at this Level The studentâ€™s description:
 Reflects a subtraction problem.
 Reflects a multiplication problem using three groups of 12.
 Does not reflect a division expression.
 Reflects the wrong division problem.
 Is contrived to end with an answer of 4.
 Is simply a restatement of the math expression in words.Â

Questions Eliciting Thinking Why did you choose this description for the expression? How is it related to division?
Why did you describe the steps in the expression? What does a â€śrealworldâ€ť description mean? 
Instructional Implications Model writing an example involving division of two positive values. Brainstorm specific vocabulary that indicates division, e.g., divided evenly, equally shared, or in equal installments. Ask the student to create a context for an expression such as 12 Ă· 3. Then have the student suggest a quantity that can be represented by a negative value. Challenge the student to create a context in which this value would need to be divided in order to answer a question.
Give the student a selection of similarlooking expressions of various operations (use the same numbers in each one) and a selection of associated word problems. Ask the student to match the expression with its related reallife word problem and to justify each match. 
Moving Forward 
Misconception/Error The studentâ€™s realworld description has a major conceptual error. 
Examples of Student Work at this Level The student describes an inappropriate context for 12, e.g., a negative number of friends or a negative number of fruit.
The description implies a division context without clear wording to support it.
Examples of wording at this level:
 Bob has 3 pieces of pie and 12 people.
 You owe three people a total of $12 dollars altogether.
 Billy Bob had 12 cookies less than Tom. Billy Bob shared his cookies with 3 friends.
 Billy gave 12 marbles away and divided them into 3 friends.
 There are 12 slices of pizza divided among 3 people.Â

Questions Eliciting Thinking What part of your description represents the negative, the 12, the three and the division?
What does it mean to have â€ś12 friends?â€ť Is there another way you can express that? 
Instructional Implications Give the student practice relating a wide variety of mathematical expressions to realworld descriptions to gain experience with contexts that give rise to each operation.
Engage the student in a discussion of reallife experiences that involve negative numbers in context. Discuss contexts where it is and is not appropriate to describe quantities with negative numbers.
Give the student a selection of similarlooking expressions of various operations (use the same numbers in each one) and a selection of associated word problems. Ask the student to match the expression with its related reallife word problem and to justify each match. 
Almost There 
Misconception/Error The student creates a logical context for the expression, but some details of the realworld reference are incomplete or unclear. 
Examples of Student Work at this Level The student does not make explicit the role of 12.
The studentâ€™s context for the quotient of 4 does not clearly represent a negative number.
The student does not supply an explicit reason for dividing by three.

Questions Eliciting Thinking What is the quotient of this expression? What does it mean in relation to the context you gave?
You said the quotient of 4 represents four groups. How is a group negative? 
Instructional Implications Provide specific feedback to the student concerning his or her response. Allow the student to revise the response in light of the feedback.
Give the student a set of division equations, each with a negative in a different position, e.g., dividend, divisor and quotient and an associated set of contextual word problems. Have the student match the word problems to equations, identifying how the negative is used in context within each. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level Realworld descriptions at this level may include descriptions such as:
 Total loss of twelve yards on three successive plays of the same yardage means they lost four yards on each play.
 Dropping twelve floors in an elevator while making only three stops means they averaged four floors down per stop.
 If a diver swam down twelve meters below the surface in three seconds, she averaged diving four meters per second.
 Your bank account has $12 in it, you add money in three $4 increments to get out of negative.

Questions Eliciting Thinking How is division related to subtraction? What subtraction problem would be equivalent to 12 Ă· 3? Would you need to word the description differently? 
Instructional Implications Provide opportunities for the student to develop realworld contexts for division that includes noninteger rational numbers, e.g., . 