Getting Started 
Misconception/Error The student is unable to explain why the product of a positive and a negative rational number is negative. 
Examples of Student Work at this Level The student finds the product of 2 and Â and offers no explanation.
The student states a procedural rule that the product of a positive and a negative is always negative.
The student describes a mnemonic device for remembering the result.

Questions Eliciting Thinking Can you tell me why the rule works?
What are we doing when we multiply? What does multiplication mean?
Can you illustrate the problem on a number line?
How is the problem affected by the negative sign? 
Instructional Implications Remind the student that multiplication is repeated addition using an example involving whole numbers, (e.g., 4 x 3 = 3 + 3 + 3 + 3). Ask the student to rewrite Â as a repeated addition problem. Assist the student, if necessary, in determining the sum. Use a drawing or a number line to help the student understand the relationship between multiplication and addition. Model explaining why Â is a negative number using the context of repeated addition.
Suggest to the student that sometimes mathematics can best be understood in terms of previously established mathematical definitions, properties, and theorems. Use prior knowledge of additive inverses and the Distributive Property to show why the product of a positive and a negative number should be negative. For example, given the problem 2 x (3) explain that 2(3 + 3) should equal zero since 3 and 3 are opposites and since 2 x 0 = 0. Use the Distributive Property to show that 2(3 + 3) = 2(3) + 2(3) = 2(3) + 6. In order for 2(3) + 6 to equal zero, 2(3) should be the opposite of 6. Therefore, 2(3) must equal 6.
Provide additional opportunities to explain why the product of a positive and negative rational number is negative, and how to use this result to complete computational problems. 
Making Progress 
Misconception/Error The student attempts an explanation, but it is incomplete or lacks proper mathematical terms. 
Examples of Student Work at this Level The student attempts to explain that the product of a positive and negative number should be negative in terms of patterns, [e.g., 2 x 3 = 6, 2 x 2 = 4, 2 x 1 = 2, 2 x 0 = 0, 2 x (1) = 2, 2 x (2) = 4].
The student attempts to explain using a number line but does not provide a complete or convincing argument.
The student suggests that multiplying a negative by two makes the result more negative.

Questions Eliciting Thinking What are you showing on the number line when you multiply 2 times ?
How do you know which way to move on the number line?
Can you explain why 2 x 3 results in a negative number? 
Instructional Implications Expose the student to explanations based on both (1) rewriting multiplication as repeated addition and (2) using prior knowledge of additive inverses and the Distributive Property to show that the product of a positive and a negative number should be negative. Allow the student to choose one of these explanations and apply it to the problem Â to explain why the product should be .
Provide additional opportunities to explain why the product of a positive and negative rational number is negative, 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 explains the result in terms of repeated addition. For example, the student says that .
The student uses the Additive Inverse and Distributive Properties to solve the problem. The student explains that Â should equal zero since Â and Â are opposites and since 2 x 0 = 0. The student uses the Distributive Property to show that . The student explains that in order for this to equal zero, Â should be the opposite of . Therefore,Â Â must equalÂ . 
Questions Eliciting Thinking How do you know that the sum of two negatives is a negative?
Would the result still be the same if the numbers were multiplied in the order ?
Does the Commutative Property of Multiplication still hold for the rational numbers?
How can additive inverses and the Distributive Property be used to explain the problem?
How would you solve a problem with three factors having different signs? 
Instructional Implications Consider pairing the student with a Making Progress classmate to explain why the product of a positive and a negative is negative.
If the student did not use prior knowledge of additive inverses and the Distributive Property to explain why the product must be negative, challenge the student to do so.
Model for the student how to justify why the product of two negatives is a positive. Consider implementing MFAS task Negative Times (7.NS.1.2). 