Getting Started 
Misconception/Error The student is unable to correctly complete the problem and cannot explain what the problem means with regard to the product of two negative numbers. 
Examples of Student Work at this Level The student incorrectly evaluates the initial expression and puts that answer in the blank while ignoring the 10.
The student writes 4 in the blank because â€ś4 is the answer.â€ť
The student says 2 times 3 equals negative six and puts that answer in the blank. The student does not check that 10 + (6) does not equal 4.
The student is unable to give any explanation about multiplying negative numbers. 
Questions Eliciting Thinking How did you determine your answer?
Where do you suppose the 10 came from?
What is 2(3)? 
Instructional Implications If needed, review addition of integers and the result of multiplying a negative by a positive number. Model evaluating the original expression using order of operations {e.g., 2[5 + (3)] = 2(2) = 4}. Next, model using the Distributive Property to evaluate the expression, [e.g., 2(5 + 3) = 2(5) + 2(3) = 10 + 6 = 4]. Explain and justify each step.
Suggest to the student that sometimes mathematics can be best understood in terms of previously learned mathematical definitions, properties and theorems. Characterize each of the following: integer addition, the Distributive Property, and that the product of a positive number and a negative number is always a negative as ideas that have been previously learned and established. Be sure the student understands that the result of multiplying two negatives has not yet been established. Demonstrate that based on what he or she already knows about operations on integers, 2[5 + (3)] = 2(2) = 4, so 2[5 + (3)] must equal 4. Point out that 2[5 + (3)] can be evaluated in a different way, (e.g., using the Distributive Property). By using the Distributive Property, 2[5 + (3)] = 2(5) + 2(3) = 10 + ____. Since it has already been demonstrated that 2[5 + (3)] must equal 4, then 10 + ____ must also equal 4. Since 10 + 6 = 4, then 2(3) must equal positive six.
Provide additional opportunities to explain why the product of two negatives is positive, and to use this result to complete computational problems. 
Making Progress 
Misconception/Error The student can complete the problem but is unable to explain what it means with regard to the product of two negative numbers. 
Examples of Student Work at this Level The student identifies the number six as the number that should go in the blank. With regard to explaining the implications of this for multiplying two negatives:
 The student provides an explanation that is unrelated to the context or makes no sense.
 The student describes a mnemonic device or restates a procedural rule.
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Questions Eliciting Thinking How did you determine what number goes in the blank?
You determined that 2(3) = 6. Why did it have to be a positive six? Could negative six go in the blank?
Are there any other models, reasons, properties, or patterns that would explain why the product of two negatives is a positive? 
Instructional Implications Suggest to the student that sometimes mathematics can be best understood in terms of previously learned mathematical definitions, properties and theorems. Characterize each of the following: integer addition, the Distributive Property, and that the product of a positive number and a negative number is always a negative as ideas that have been previously learned and established. Be sure the student understands that the result of multiplying two negatives has not yet been established. Demonstrate that, based on what he or she already knows about operations on integers, 2[5 + (3)] = 2(2) = 4, so 2[5 + (3)] must equal 4. Point out that 2[5 + (3)] can be evaluated in a different way, e.g., using the Distributive Property. By using the Distributive Property, 2[5 + (3)] = 2(5) + 2(3) = 10 + ____. Since it has already been demonstrated that 2[5 + (3)] must equal 4, then 10 + ____ must also equal 4. Since 10 + 6 = 4, then 2(3) must equal positive six.
Give the student another computational problem such as 4(6 + 9) and ask the student to use the above reasoning to explain why the product of two negatives must be a positive.
Provide additional opportunities to explain why the product of two negatives is positive, and to use this result to complete computational 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 The student says, â€śWe know that the final answer has to be 4 so I put six in the blank, since 10 + 6 = 4. This means that 2(3) has to be positive six. So, a negative times a negative needs to be a positive.â€ť 
Questions Eliciting Thinking How did you determine what number goes in the blank?
How do you suppose we knew the final answer to the problem had to be 4? Is there another way to calculate 2[5 + (3)]?
Do you think that the product of two negatives is always a positive? 
Instructional Implications Consider pairing the student with a Making Progress classmate to explain why the product of two negatives is positive.
Give the student another computational problem involving rational numbers such as Â and ask the student to use this problem as the basis for explaining why the product of two negatives must be a positive. 