Justine has 6 blue shirts, 2 red shirts, and 5 green shirts in her closet. On Monday, she randomly chooses a blue shirt. She does not put the blue shirt back in the closet.

Step A

What is the probability that Justine will randomly select a green shirt on Tuesday?

Step B

Use what you know about probability to explain how you determined the probability that Justine will randomly select a green shirt on Tuesday. Use words, numbers, and/or symbols in your explanation.

Step A is scored 0 (Incorrect) or 1 (Correct) and assesses 5.B.2.a.
Step B is scored with a 3 point (0, 1, 2) rubric and assesses Processes of Mathematics.

Note: Seventeen "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: This response is irrelevant to the problem. A discussion of the order in which Justine chooses colored shirts demonstrates no understanding of the process used to determine her probability of choosing a green shirt on Tuesday.

Score for Sample Student Response #2:

Annotation for Step B, Using the Rubric: This response is completely incorrect. Information given in the prompt is merely restated. The idea of "looking at the word problem" does not indicate a strategy for solving this problem.

Score for Sample Student Response #3:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy to determine probability is indicated. The explanation supports the given answer of 1/12, but reveals a logical flaw. The response supports determining the probability of choosing any one shirt on Tuesday, but does not demonstrate understanding of how the different color choices affect that probability.

Score for Sample Student Response #4:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy for finding odds, not probability is demonstrated. The explanation reveals a logical flaw: "you put the 5 green shirts over the rest of the shirts and you get 5/7."

Score for Sample Student Response #5:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. A reasonable strategy for finding probability is indicated. However, the explanation reveals a flaw in reasoning: "I determined my answer by looking at how many green shirts Justine had... Then I took the total of all the colored shirts... I turned it into a fraction." The idea that Monday's shirt was chosen and not replaced is not addressed.

Score for Sample Student Response #6:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The application of a reasonable strategy for determining probability is indicated. The explanation of the process used to calculate this probability is only partially developed: "I added all of the shirts together... then I subtracted 1." Supportive information for both how the numerator was determined and how the 5 and 12 were placed in the given fraction are not provided.

Score for Sample Student Response #7:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the probability was determined is partially developed: "I added the number of shirts and she has five green shirts." Supportive information for both how the denominator was determined and how the 5 and 12 were placed in the given fraction are not provided.

Score for Sample Student Response #8:

Annotation for Step B, Using the Rubric: This response demonstrates a complete understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the probability was determined is clear and developed: "there were 15 shirts, then...There were 14 in all and 5 green." Appropriate supportive numbers are provided: "5 out of 14." The minor arithmetic error in calculating that Justine had 15 shirts on Monday does not detract from the complete understanding of the process used to determine probability in the context of the problem.

Score for Sample Student Response #9:

Annotation for Step B, Using the Rubric: This response demonstrates a complete understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the probability was determined is clear and developed: "adding all of the numbers... subtracted 1... took the number of green shirts... 5:12." The labeled and numbered diagram of shirts at the bottom of the response provides appropriate support and could stand alone as a clear explanation in symbolic form.

Score for Sample Student Response #10:

Annotation for Step B, Using the Rubric: This response demonstrates a complete understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the probability was determined is clear and developed: "adding all the shirts... subtracting one... there were 5 green shirts... that leaves me with 5 out of 12." Appropriate supportive numbers are provided for this explanation. (Note that 5 out of 12, 5/12, or 5:12 are acceptable answers for Step A. Even thought 41.5% is incorrect, the student has demonstrated an understanding of how to express the probability in three other ways. This item is not assessing conversion of the fractions to percents.)

Score for Sample Student Response #11:

Annotation for Step B, Using the Rubric: This response demonstrates a complete understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the probability was determined is clear and developed: "I put the amount of green shirts over 12... I turned the fraction into a percent." The connection to the strategy of finding a percent is clear and stated explicitly. The explanation of how "12" was determined is only implied, but this does not detract from the overall understanding of the mathematical processes used to solve this problem. Appropriate supportive numbers for how the faction was converted to a percent are provided. Compare to Sample Student Response #10.

Score for Sample Student Response #12:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. A partial application of a strategy to determine probability is indicated. The explanation of how to determine the probability is logically flawed. The new total number of shirts is determined correctly: "I counted all the shirts. I took away the shirt that she didn't put back." The process of dividing "12 by 3" is incorrect in the context of the problem. Appropriate supportive numbers are not provided. Compare to Sample Student Response #3.

Score for Sample Student Response #13:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The application of a reasonable strategy for determining probability is indicated. The explanation of the process used to calculate this probability is only partially developed: "[I drew] a picture and work[ed] my way down." Supportive information for how the number of each shirt that remained in the closet is provided. However, no supportive information for how either number in the probability was determined is provided. Compare to Sample Student Response #6.

Score for Sample Student Response #14:

Annotation for Step B, Using the Rubric: This response demonstrates a complete understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated. The explanation of how the denominator of the probability was determined is clear, developed, and logical: "She already took one blue shirt so...5 [blue]...2 reds and 5 greens. That makes 12... 12-total amount of shirts." Also, the explanation of how the numerator was determined is clear: "5 - all 5 green shirts." Appropriate supportive information and numbers are provided: "6-1=5... 5+2+5=12." The labeled example at the bottom of the response provides appropriate support and could stand alone as a clear explanation in numeric form. Compare to Sample Student Response #9.

Score for Sample Student Response #15:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. A reasonable strategy for finding probability is indicated. However, the explanation reveals a flaw in reasoning: "I subtracted from 5 & then added everything up... I put the # of green... over the total #." The number of green remains 5, while the number of blue decreases from 6 to 5. Compare to Sample Student Response #5.

Score for Sample Student Response #16:

Annotation for Step B, Using the Rubric: This response is irrelevant to the problem. The explanation does not support the answer given in Step A. Compare to Sample Student Response #1.

Score for Sample Student Response #17:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The application of a reasonable strategy for determining probability in the context of the problem is indicated but reveals a logical flaw. The explanation supports picking a blue shirt on Tuesday, not a green shirt. In addition, the explanation of how the probability was determined is only partially developed: "I put the number of blue shirts... over the number of shirts in all." Supportive information for how the denominator was determined is not provided. Compare to Sample Student Response #7.

Score 2

The response demonstrates a complete 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, developed, and logical.
• Connections and/or extensions made within mathematics or outside of mathematics are clear.
• 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 partially developed, logically flawed, or missing.
• Connections and/or extensions made within mathematics or outside of mathematics are partial or overly general, or flawed.
• 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