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

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:
Practitioner

The student’s strategy of making a table to show small stones, large stones to 10 days, a running total of small stones, large stones, and total stones; and then finding the total number of stones that are needed to make Sarah’s wall works to solve this task. The student’s answer, “She collects 40 SS and 32 LS she collected on the tenth Day,” and, “Sarah will need 405 stones for her garage wall. She collects 220 SS and 185 LS,” is correct. The student also uses an alternate strategy of a graph for the first part of the task.

Reasoning & Proof:
Practitioner

The student’s solution is constructed with adequate mathematical basis. The student uses correct reasoning by generating numerical patterns for small and large stones and finding the running total of small and large stones Sarah uses for the wall. The student correctly uses a graph to support her/his thinking.

Communication:
Expert

The student correctly uses the mathematical terms total, tenth, day from the task. The student also correctly uses the mathematical terms table, key, running total, rule. The student correctly uses the symbolic notation 4n = SS and 3n + 2 = LS.

Connections:
Practitioner

The student uses a graph as a second strategy to show the number of small and large stones for each of 10 days. The student also generalizes two rules, “If it told me to find the rule for the SS it would be 4n = SS,” and, “If it told me to find the rule for the LS it would be 3n + 2 = LS.” The student defines the variables in her/his key. The student does not earn Expert credit for the rules because she/he does not use the rules to verify her/his answer or to find the number of small and large stones collected for any of the days on her/his table or other days.

Representation:
Practitioner

The student’s use of a table is appropriate and accurate. The student provides all necessary labels and a key for the second, third, fourth, fifth, and sixth columns. All entered data is correct. The student’s graph is also correct with the X and Y axis labeled and a key defines the · and x as large and small stones.

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