Sabrina and Joel are measuring how far their cars roll down a ramp. Sabrina's car rolls forty-eight inches. Joel's car rolls seventy-two inches. Sabrina says her car rolls four times as far as one foot. Joel says his car rolls two times as far as one yard. Who is correct, Sabrina or Joel? Show all your mathematical thinking.

Aligned to the 4.OA.A.2 Standard

### Possible Solutions

#### 4.OA.A.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

## Assess Student Performance Click to enlarge
##### 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 using a chart and division to determine that Sabrina’s car does roll four times as far as one foot and Joel’s car does roll two times as far as one yard works to solve the task. The student’s answer, “Joel is correct,” and, “Sabrina is correct,” is correct.

Reasoning & Proof:
Practitioner

The student demonstrates understanding of the underlying concepts of the task. The student correctly uses division to differentiate multiplicative comparison from additive comparison.

Communication:
Apprentice

The student correctly uses the mathematical term "in." (inch) from the task. The student does not include any other mathematical terms in her/his solution.

Connections:
Apprentice

The student attempts a mathematical connection by deciding to determine the sum of the inches traveled by the two cars. The student makes a computational error that leads to an incorrect connection of “110 in. – how much they roll together.”

Representation:
Practitioner

The student’s use of a chart is appropriate to the task and accurate. Each column is labeled correctly and all entered data is correct. ### Score Card:Student #1

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#### Overall Achievement Level:

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Unassigned ### Score Card:Student #3

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