Getting Started 
Misconception/Error The student is unable to describe realworld quantities that can be represented by negative rational numbers. 
Examples of Student Work at this Level The student correctly computes the answer, but describes no context. Instead, the student:
 States a rule learned for multiplying integers.
 Describes an inappropriate context for 12.5, e.g., a negative number of bacteria.

Questions Eliciting Thinking Can you think of something that can be described by the number 12.5? Would it make sense for it to be negative?
Can you think of anything that can be represented by a negative number?
Can you have a negative number of bacteria? 
Instructional Implications Brainstorm with the student a list of quantities that could be described by a negative rational number, (e.g., depth with regard to sea level, changes in altitude, loss of yardage in football, temperature or drops in temperature, and debits). Remind the student that zeroes can be added to the right of the decimal without changing the value, (e.g., 12.5 = 12.50). The extra decimal place may prompt the student to suggest owing money as a context. Next, compile a list of situations where multiplication can be applied, (e.g., four withdrawals from a bank account, four plays on the football field, or four drops in temperature). Guide the student to use a situation that makes sense to provide a context for the expression.
Model composing a complete response to the problem. (1) Determine a context and decide what is represented by 12.5, (e.g., â€śyards lost on a football playâ€ť). (2) Incorporate the need for multiplication into the context, (e.g., â€śon each of four consecutive plays, a football team lost 12.5 yardsâ€ť). (3) Devise a question, (e.g., â€śHow many yards did they lose in total?â€ť). (4) Explain what the answer means in the chosen context, (e.g., â€śBecause 4 x 12.5 = 50, the number 50 describes the change in yardageâ€ť).
Provide additional opportunities for the student to create realworld contexts for rational quantities and expressions involving rational numbers. 
Moving Forward 
Misconception/Error The student is unable to incorporate multiplication into their chosen context. 
Examples of Student Work at this Level The student contrives an appropriate context for the negative value, but does not incorporate multiplication correctly in the context.

Questions Eliciting Thinking How did you represent the multiplication in your description?
Can you make up a story for the expression 2 x 5?
What does it mean to multiply? What words can be used to indicate multiplication? 
Instructional Implications Model writing an example involving multiplication of two positive values. Brainstorm specific vocabulary that indicates multiplication, (e.g., four times greater than, twice as tall, half the size). Ask the student to create a context for expressions such as 2.5 x 6. 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 multiplied in order to answer a question.
Provide additional opportunities for the student to create realworld contexts for products of rational numbers. 
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 lacks complete knowledge of the context causing the realworld reference to be somewhat incorrect, (e.g., how to explain withdrawal of money from a bank account).
The student uses owing money in a correct context, but words the description of the problem in a way that does not require the use of the negative sign.

Questions Eliciting Thinking Do you know the meaning of deposit and withdrawal?
Can you review your problem and find any errors?
Does it make sense to say that you owe $12.50? 
Instructional Implications Conduct a mini lesson on banking. Using rational numbers, act out and record making deposits and withdrawals to a bank account. Model how to represent the product 4 (12.50) in this context using correct terminology and notation. Suggest other contexts for rational numbers and ask the student to use one that makes sense and write a realworld problem involving multiplication of rational numbers.
Pair the student with a Got It peer to conduct a partnered review of the response and to find and correct errors. 
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 describes a context in which the product of 12.5 and 4 has a clear and coherent meaning:
 Samantha went to the store to buy sandals. The price for each pair was $12.50, and she got four pairs. This means Samanthaâ€™s bank account decreased by $50.00 because 4 x 12.5 = 50.
 I think the 12.5 stands for feet. Every time the quarterback got the football, he was knocked 12.5 feet backwards. After four plays, the ball moved 50 feet and the coach was unhappy.Â

Questions Eliciting Thinking Can you think of a context for dividing two numbers? What do you think will happen when you divide two numbers having different signs?
Can you think of an example that does not involve owing money? 
Instructional Implications Challenge the student to create a problem in which the Distributive Property might be applied, (e.g., Joseph bought four items at $2.50 each and four at $10 each). Have the student translate the scenario into a mathematical expression and then calculate the answer in two different ways, using order of operations and using the Distributive Property.
Ask the student to use the knowledge of multiplication of integers to predict the outcome of dividing two rational numbers with opposite signs. Challenge the student to find a realworld context for division of rational numbers with opposite signs. 