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:
Expert

The student’s strategy of using number lines and multiplication 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, “Both Joel and Sabrina are correct exactley,” is correct. The student also uses an alternate strategy of a chart and division. The student also brings prior knowledge of mixed fractions to the task.

Reasoning & Proof:
Expert

The student demonstrates understanding of the underlying concepts of the task. The student correctly uses multiplication to differentiate multiplicative comparison from additive comparison. The student justifies her/his answer by using an alternate strategy of a chart and division to arrive at the same answer.

Communication:
Expert

The student correctly uses the mathematical terms inches, foot, yard from the task. The student also correctly uses terms number line, distance, key, “tabe,” (table), division, even. The student correctly uses the mathematical notation ", ', 3 1/3, 3 12/36.

Connections:
Expert

The student makes the mathematical connections, “Sabrina was only 24 inches away from Joel’s car roll,” “or 2 feet away from Joel’s car roll,” and, “all numbers are even.” The student uses arrows to point to the even numbers 48”, 4’, 72”, 6’. The student makes the Expert connection of verifying that her/his answer is correct by using division and a chart. The student states, “I get 4 and 2 again for times so it is correct.” The student also determines that ten feet is equivalent to 3 1/3 yards and 120 inches is equivalent to 3 and 1/3 yards.

Representation:
Expert

The student’s use of number lines is appropriate to the task and accurate. Each number line is titled and correctly labeled. 72 and 48 inches are correctly indicated. The student’s chart is also appropriate to the task and accurate. All labels are indicated and the entered data is correct. The student uses her/his chart to verify that the data on the number lines and her/his answer is correct. ### Score Card:Student #1

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#### Overall Achievement Level:

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