# On the Beach

Nick finds thirty brown shells and twelve pink shells on the beach. Kim finds twenty-two brown shells and some pink shells on the beach. Kim shows her shells to Nick. Nick says, "We have the same number of shells!" How many pink shells does Kim find on the beach? Show all your mathematical thinking.

## Plan

#### Underlying Mathematical Concepts

#### Possible Problem-Solving Strategies

#### Formal Mathematical Language and Symbolic Notation

A student may independently select a printed number line, number chart, ten frames, graph paper, etc. as they work on a task.

### Possible Solutions

## Solutions tabs

### Assess

### Anchor Papers

#### Novice 1

### Scoring Rationale

#### Problem Solving

#### Novice

The student's strategy of diagramming thirty-nine circles for Nick and eighteen circles for Kim does not work to solve the task. The student's answer,"12," is not correct.

#### Reasoning & Proof

#### Novice

No correct reasoning is evident. A key does not define the O and the number of circles does not match the number of shells found by Nick and Kim.

#### Communication

#### Novice

The student does not use any mathematical terms.

#### Connections

#### Novice

The student does not attempt a mathematically relevant connection.

#### Representation

#### Apprentice

The student attempts a diagram. It appears that the circles represent shells and the student divides them between Nick and Kim. The number of circles does not match the numbers in a correct solution and no key was used to define the circle.

#### Overall Achievement Level:

#### Novice

#### Apprentice 1

### Scoring Rationale

#### Problem Solving

#### Apprentice

The student's strategy of diagramming the brown and pink shells for Nick and Kim would work to solve the task, but the student diagrams only twenty-nine brown shells for Nick and twenty-one brown shells for Kim and does not provide a key. The student's answer, "20 pink shells," is not supported by the student's diagram and is not considered correct.

#### Reasoning & Proof

#### Apprentice

Some correct reasoning is present. The student omits a key to define the B, P, [change], O so one has to fill in a number of gaps to determine if the student is applying correct reasoning to the task.

#### Communication

#### Apprentice

The student correctly uses the mathematical term *most*.

#### Connections

#### Apprentice

The student attempts the mathematically relevant observation, "B is the most of them." The student never identifies that B represents brown shells so does not achieve the Practitioner performance level.

#### Representation

#### Apprentice

The student's diagram of the brown and pink shells for Kim is appropriate to the problem but not accurate. A key is not provided to identify each color of shell. It appears that one brown shell for Nick and one brown shell for Kim is missing.

#### Overall Achievement Level:

#### Apprentice

#### Practitioner 1

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of diagramming Nick's brown and pink shells for forty-two shells and twenty brown shells for twenty-two shells, and then determining that Kim has twenty pink shells, works to solve the task. The student's answer, "20 pink shells," is correct.

#### Reasoning & Proof

#### Practitioner

The student's solution is constructed with adequate mathematical basis. The student demonstrates understanding of the underlying concepts of addition and subtraction to find results unknown.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *diagram*, *key*, *number line*, *most*.

#### Connections

#### Practitioner

The student makes the mathematically relevant observation, "Nick has the most brown shells."

#### Representation

#### Practitioner

The student's diagram of the brown shells for Kim and Nick and the pink shells for Nick is appropriate to the task and accurate. A key defines the brown and pink shells. The student's number line is appropriate and accurate with the shells labeled.

#### Overall Achievement Level:

#### Practitioner

#### Expert 1

### Scoring Rationale

#### Problem Solving

#### Expert

The student's strategy of creating a table to record the number of brown and pink shells Nick and Kim find on the beach works to solve the task. The student's answer, "20," is correct. The student verifies her/his answer with a diagram.

#### Reasoning & Proof

#### Expert

The student's solution is constructed with adequate mathematical basis. The student demonstrates understanding of the underlying concepts of addition and subtraction to find results unknown. The student uses a diagram to verify her/his answer.

#### Communication

#### Expert

The student correctly uses the mathematical terms *table*, *total*, *even*, *dozen*, *diagram*, *key*, *array*. The student uses the mathematical notation *3 x 10*.

#### Connections

#### Expert

The student makes the mathematically relevant Practitioner connections, "allways even shells," "Nick has a dozen pink shells," and, "They have a total of 84 shells." The student makes the Expert connection by using a diagram to verify that her/his answer is correct. The student states, "The answer is the same so it is correct."

#### Representation

#### Expert

The student's table is appropriate and accurate. All labels are provided and the data is correct. The student uses a diagram to verify her/his answer. The diagram is accurate and a key defines the brown and pink shells.