Look at this number line.

Step A
Plot a point on the number line that represents the location of 2

Step B

• Explain why the point you plotted is correct. Use what you know about number lines in your explanation. Use words, numbers, and/or symbols in your explanation.
• Benjamin says that if he divides the number line between 2 and 3 into two equal parts, the location of the point 2 will change. Explain why Benjamin is correct or incorrect. Use what you know about number lines in your explanation. Use words, numbers, and/or symbols in your explanation.

Step A is scored 0 (Incorrect) or 1 (Correct) and assesses 1.C.1.a.
Step B is scored with a 4 point (0, 1, 2, 3) rubric and assesses Processes of Mathematics.

Note: 6 "Sample Student Responses" follow below. Each response appears on its own separate page and includes scoring information. The "Sample Student Responses" represent a range of score points.

Score for Sample Student Response #1:

Annotation for Step B, Using the Rubric: In this response, the student explains the first step in solving the problem, and by indicating a relevant starting point on the number line ("I went over to two"), this response demonstrates a minimal understanding and analysis of number lines.

Score for Sample Student Response #2:

Annotation for Step B, Using the Rubric: The student repeats the answer in the space for Step B, so no credit in Step B is derived from the second page of this response. However, the correctly plotted point on the number line in Step A is a partial explanation of the mathematical process used to solve the problem. Admittedly, the explanation uses no words, only symbols, but the symbol (the point) drawn on the number line is correct and relevant to the problem. Therefore, Step B is considered to demonstrate a minimal understanding and analysis of the problem.

Score for Sample Student Response #3:

Annotation for Step B, Using the Rubric: The response demonstrates a general understanding and analysis of the problem. In the first part of the response, the explanation for the mathematical process used to solve the problem ("put lines inbetween all of the smaller lines, each represents ; marked the first one") is partially developed, as the student never justifies why each line represents . The student simply lists the steps taken to solve the problem, without specifying why those steps were correct. In the second part of the response, the student fully justifies the underlying mathematics ("Because number lines never change the location of a number").

Score for Sample Student Response #4:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. The student describes the mathematical process of plotting the point ("2 wholes so go past 2 on the number line; put the point half way between 2 and 2") and logically justifies why that process is correct ("number line is split in 's; is half of "). However, the second part of the response is incorrect, ("point would need to be at 2 or a equivalent fraction; point's location will change").

Score for Sample Student Response #5:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The mathematical process used to solve the problem ("I put after the 2 but before the mark") comes with a justification that is fully developed and logical ("because it is less than ; first line shows "). In addition, the justification for the mathematical process used to solve the second part of the question is fully developed, logical, and clear, given the words ("the halfway point is still the same") and symbols (comparison of the two number lines drawn in Step B), which show that 2 is equivalent to 2 and that 2 is equivalent to 2.

Score for Sample Student Response #6:

Annotation for Step A, Using the Rubric: The point is plotted correctly.

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The mathematical process used to plot the point is explained ("2 is halfway between 2 and 2") and justified clearly and logically ("if 2 fractions have the same numerator and one's denominator is double the other fraction the one with the smaller denominator is 2 × the other fraction"). In the second part, the explanation for why the location of the point will not change is fully developed, logical, and clear ("even if the number line is changed; the value of 2 stays the same; point represents the value of 2; also constant."

Score 3

The response demonstrates a comprehensive understanding and analysis of a problem.

• Application of a reasonable strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is clear, fully developed, and logical.
• Connections and/or extensions made within mathematics or outside of mathematics are clear and stated explicitly.
• Supportive information and/or numbers are provided as appropriate. ³

Score 2

The response demonstrates a general understanding and analysis of a problem.

• Application of a reasonable strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is feasible, but may be only partially developed.
• Connections and/or extensions made within mathematics or outside of mathematics are partial or overly general, or may be implied.
• Supportive information and/or numbers are provided as appropriate. ³

Score 1

The response demonstrates a minimal understanding and analysis of a problem.

• Partial application of a strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is logically flawed or missing.
• Connections and/or extensions made within mathematics or outside of mathematics are flawed or missing.
• Supportive information and/or numbers may or may not be provided as appropriate.³

Score 0

The response is completely incorrect, irrelevant to the problem, or missing.⁴

Note 1:

Explanation refers to students' ability to communicate how they arrived at the solution for an item using the language of mathematics.

Note 2:

Justification refers to students' ability to support the reasoning used to solve a problem, or to demonstrate why the solution is correct using mathematical concepts and principles.

Note 3:

Students need to complete rubric criteria for explanation, justification, connections and/or extensions as cued for in a given problem.

Note 4:

Merely an exact copy or paraphrase of the problem will receive a score of "0".

Rubric Document Date: August 2003