Scoring Tasks

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Task: A Stone Wall

Sarah wants to build a stone wall along one side of her garage. Sarah collects stones from the field behind her house. The first day, Sarah collects four small stones and five large stones. The second day, Sarah collects eight small stones and eight large stones. The third day, Sarah collects twelve small stones and eleven large stones. If this pattern continues, how many small and large stones does Sarah collect on the tenth day? Sarah realizes that she now has enough small and large stones for her stone wall. How many small and large stones does Sarah collect for the stone wall? Show all your mathematical thinking.

Aligned to the 5.OA.B.3 Standard

More on the 5.OA.B.3 Standard

Possible Solutions

On the 10th day Sarah collects 40 small stones and 32 large stones. Sarah collects a total of 220 small stones and 185 large stones for the stone wall.

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5.OA.B.3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Assess Student Performance

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Assess Student #1’s work according to the following category:

No strategy is chosen, or a strategy is chosen that will not lead to a solution.

Little or no evidence of engagement in the task present.

A partially correct strategy is chosen, or a correct strategy for only solving part of the task is chosen.

Evidence of drawing on some relevant previous knowledge is present, showing some relevant engagement in the task.

A correct strategy is chosen based on the mathematical situation in the task.

Planning or monitoring of strategy is evident.

Evidence of solidifying prior knowledge and applying it to the problem-solving situation is present.

Note: The Practitioner must achieve a correct answer.

An efficient strategy is chosen and progress towards a solution is evaluated.

Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered.

Evidence of analyzing the situation in mathematical terms and extending prior knowledge is present.

Note: The Expert must achieve a correct answer.

Arguments are made with no mathematical basis.

No correct reasoning nor justification for reasoning is present.

Arguments are made with some mathematical basis.

Some correct reasoning or justification for reasoning is present.

Arguments are constructed with adequate mathematical basis.

A systematic approach and/or justification of correct reasoning is present.

Deductive arguments are used to justify decisions and may result in formal proofs.

Evidence is used to justify and support decisions made and conclusions reached.

No awareness of audience or purpose is communicated.

No formal mathematical terms or symbolic notations are evident.

Some awareness of audience or purpose is communicated.

Some communication of an approach is evident through verbal/written accounts and explanations.

An attempt is made to use formal math language. One formal math term or symbolic notation is evident.

A sense of audience or purpose is communicated.

Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response.

Formal math language is used to share and clarify ideas. At least two formal math terms or symbolic notations are evident, in any combination.

A sense of audience and purpose is communicated.

Communication of an approach is evident through a methodical, organize, coherent, sequenced and labeled response. Communication of an argument is supported by mathematical properties.

Formal math language and symbolic notation is used to consolidate math thinking and to communicate ideas. At least one of the two formal math terms or symbolic notations is beyond grade level.

No connections are made or connections are mathematically or contextually irrelevant.

A mathematical connection is attempted but is partially incorrect or lacks contextual relevance.

A mathematical connection is made. Proper contexts are identified that link both the mathematics and the situation in the task.

Some examples may include one or more of the following:

  • clarification of the mathematical or situational context of the task
  • exploration of mathematical phenomenon in the context of the broader topic in which the task is situated
  • noting patterns, structures and regularities

Mathematical connections are used to extend the solution to other mathematics or to a deeper understanding of the mathematics in the task.

Some examples may include one or more of the following:

  • testing and accepting or rejecting of a hypothesis or conjecture
  • explanation of phenomenon
  • generalizing and extending the solution to other cases

No attempt is made to construct a mathematical representation.

An attempt is made to construct a mathematical representation to record and communicate problem solving but is not accurate.

An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions.

An appropriate mathematical representation is constructed to analyze relationships, extend thinking and clarify or interpret phenomenon.

This Achievement Level demonstrates little or no evidence of understanding the math concepts of the task.

This Achievement Level has the broadest range, from a student who is just beginning to demonstrate mathematical understanding to a student who almost meets the standard.

This Achievement Level meets the standard and is proficient in understanding the underlying mathematics of the task. A correct solution must be achieved.

This Achievement Level goes beyond the standard and shows a deeper understanding of the underlying mathematics.

Exemplars has assessed Student #1’s work in the following way:

Problem Solving:
Apprentice

The student’s strategy of making a table to show the days, small stones, large stones, and total stones per day for 10 days works to solve the first part of the task. The student’s answer for this part of the task, “on the 10th day she collected 40 SS and 32 LS…” is correct. The student’s strategy of finding the total amount of stones per day but not finding the total amount of small and large stones for 10 days does not work to solve the second part of this task. The student’s answer, “…and all together she collected 72 stones,” is not correct.

Reasoning & Proof:
Apprentice

The first part of the student’s solution is constructed with adequate mathematical basis. The student generated two correct numerical patterns. The student’s reasoning for the second part of the task is partially correct. The student finds the total amount of stones per day in the fourth column but not the total number of stones for all 10 days.

Communication:
Practitioner

The student correctly uses the mathematical terms 10th (tenth) and day, from the task. The student also correctly uses the mathematical term key. The student does not earn credit for the term running total. The fourth column of the table only shows the total amount of small and large stones per day.

Connections:
Practitioner

The student solves the task and makes a mathematically relevant connection. The student states, “if she looked 11 days she would have 44 little stones and 35 large stones.”

Representation:
Apprentice

The student’s use of a table is appropriate but not accurate. The fourth column should indicate 30 total stones for the fourth day instead of 32 stones.

Score Card:
Student #1

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Student #4

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