# The Book Shelf

Reading time is over. Ryan and Zak help put books away on the shelf. Ryan puts eight books on the book shelf. Zak puts six books on the book shelf. Ryan thinks he put a greater number of books on the book shelf than Zak. Is Ryan correct? Show and tell how you know.

## 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, five frames, ten frames, place value strips, 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 only diagramming eight books for Ryan and six books for Zak would not work to solve the task. The student is not able to state a mathematical answer.

#### Reasoning & Proof

#### Apprentice

The student shows some correct conceptual reasoning. The student correctly indicates a set of eight books and a set of six books. The student is not able to compare the two amounts to determine which boy puts a greater number of books on the book shelf.

#### Communication

#### Novice

The student does not use any mathematical terms.

#### Connections

#### Novice

The student does not make a mathematically relevant observation about their solution.

#### Representation

#### Apprentice

The student's diagram of eight books and six books works to solve part of the task but the student is not able to apply the diagram to communicate that Ryan puts a greater number of books on the book shelf.

#### Overall Achievement Level:

#### Novice

#### Apprentice 1

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of diagraming eight and six books works to solve the task. The student's answer, "I made 8 and I made 6 and 8 is bigger so it is yes," is correct.

#### Reasoning & Proof

#### Practitioner

The student shows understanding of the underlying mathematical concepts of the task. The student correctly shows a set of eight and a set of six. The student compares the two groups to determine that "8 is bigger."

#### Communication

#### Apprentice

The student correctly uses the mathematical term *diagram*.

#### Connections

#### Novice

The student does not make a mathematically relevant observation about their solution.

#### Representation

#### Apprentice

The student's diagram of eight books and six books is appropriate but not accurate. The student does not label Ryan or Zak and does not label the books.

#### Overall Achievement Level:

#### Apprentice

The overall achievement level for this piece of student work falls under Exemplars exception to the rule category. If a student has all Apprentice scores or above, but a Novice in "Connections," the student may still receive an achievement level score of Apprentice. To learn more about Exemplars scoring, please refer to the section of your dashboard called "Tools for Success" and click on the link for “Using the Assessment Rubric.”

#### Apprentice 2

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of diagraming eight books for Ryan and six books for Zak to determine that Ryan puts a greater number of books on the book shelf works to solve the task. The student's answer, "Yes," is correct.

#### Reasoning & Proof

#### Practitioner

The student shows understanding of the mathematical concepts of the task. The student correctly shows a set of eight and a set of six. The student compares the two groups to determine which boy has the greater number of books.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *diagram*, *key*, *more*.

#### Connections

#### Novice

The student does not make a mathematically relevant observation about their solution.

#### Representation

#### Practitioner

The student's diagram is appropriate and accurate. The scribing provides the labels for Ryan and Zak. A key defines the books.

#### Overall Achievement Level:

#### Apprentice

The overall achievement level for this piece of student work falls under Exemplars exception to the rule category. If a student has all Apprentice scores or above, but a Novice in "Connections," the student may still receive an achievement level score of Apprentice. To learn more about Exemplars scoring, please refer to the section of your dashboard called "Tools for Success" and click on the link for “Using the Assessment Rubric.”

#### Practitioner 1

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of diagraming eight books for Ryan and six books for Zak to determine that Ryan puts a greater number of books on the book shelf works to solve the task. The student's answer, "Ryan has greater number books than Zak so it is yes for Ryan," is correct.

#### Reasoning & Proof

#### Practitioner

The student shows understanding of the mathematical concepts of the task. The student correctly shows a set of eight and a set of six. The student compare the two groups to determine that eight is greater than six.

#### Communication

#### Practitioner

The student correctly uses the mathematical term *greater number* from the task. The student also correctly uses the terms *diagram*, *key*, *rectangle*, *more*, *ten frames*.

#### Connections

#### Practitioner

The student makes mathematical observations about their solution. The student observes, "The rectangle is a book." The students uses a new strategy, ten frames, to determine that Ryan has "more books."

#### Representation

#### Practitioner

The student's diagram is appropriate and accurate. A key defines the books and the student correctly labels Ryan's books and Zak's books. The student's ten frames are appropriate and accurate. The student uses the same key for the books and correctly labels Ryan's books and Zak's books.

#### Overall Achievement Level:

#### Practitioner

#### Practitioner 2

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of gluing eight books for Ryan, six books for Zak, and drawing a line to match books to determine that Ryan puts a greater number of books on the book shelf works to solve the task. The student's answer, "Yes," is correct.

#### Reasoning & Proof

#### Practitioner

The student shows understanding of the mathematical concepts of the task. The student correctly shows a set of eight and a set of six. The student compares the two groups to determine that eight is greater than six.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *more than*, *rectangles*, *line*, *more*, *less*.

#### Connections

#### Practitioner

The student makes mathematical observations about their solution. The student defines the paper shapes as "rectangles." The student also states, "This means Ryan has two more books than Zak," and "They put 1, 2, 3 ...14 books on the book shelf."

#### Representation

#### Practitioner

The student's diagram using paper rectangles glued on their paper is appropriate and accurate. Ryan and Zak's books are correctly labeled and the books are defined in the scribing.

#### Overall Achievement Level:

#### Practitioner

#### Practitioner 3

### Scoring Rationale

#### Problem Solving

#### Practitioner

The student's strategy of diagraming eight books for Ryan and six books for Zak to determine that Ryan puts a greater number of books on the book shelf works to solve the task. The student's answer, "Yes," is correct.

#### Reasoning & Proof

#### Practitioner

The student shows understanding of the mathematical concepts of the task. The student correctly shows a set of eight and a set of six. The student compares the two groups to determine that eight is greater than six.

#### Communication

#### Practitioner

The student correctly uses the mathematical terms *diagram*, *greater than*, * more*, *number lines*.

#### Connections

#### Practitioner

The student make mathematical observations about their solution. The student states, "2 more," and "Ryan puts two more books on the shelf than Zak." The student uses a new strategy by indicating on number lines the amount of books each boy has. The student states, "Ryan has more books." The student does not link their number lines to the diagrams for the Expert level.

#### Representation

#### Practitioner

The student's diagram is appropriate and accurate. Ryan and Zak and their books are correctly defined in the scribing. The student's two number lines are appropriate and accurate. Labels are used to define Zak and Ryan and the books.

#### Overall Achievement Level:

#### Practitioner

#### Expert 1

### Scoring Rationale

#### Problem Solving

#### Expert

The student's strategy of diagraming the books Ryan and Zak put on the shelf and comparing each boy's total books to determine that Ryan is correct, works to solve the task. The student's answer, "yes," is correct. The student brings prior knowledge of greater than and less than notation and fair share/equal to the task.

#### Reasoning & Proof

#### Expert

The student shows understanding of the mathematical concepts of the task. The student shows correct totals of books for each boy and compares the two totals to conclude that Ryan is correct as he has a greater number of books. The student also uses conceptual thinking of the notation for greater than and less than. The student discovers that a total of 14 books can be a fair share if seven books are put on each shelf. The student also justifies that their answer is correct.

#### Communication

#### Expert

The student correctly uses the mathematical terms *diagram*, *key*, *more*, *pair*, *greater than*, *total*, *fair share*, *equal*, *number lines*, *ten frames*, *more than . The student correctly uses the mathematical symbolic notation 8 > 6, 6 < 8.*

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*#### Connections

#### Expert

The student makes mathematically relevant Practitioner observations about their solution. "I see Ryan has two more books than Zak. That is a pair," "This is 14. They have 14 books in all. That is total books," "Ryan needs two more for ten," and "Zak needs four more for ten." The student makes Expert connections. The student states, "8 > 6, This means eight is greater than six. My brother taught me." The student also states, "If they each put seven on it is a fair share of books. It is equal books." The student verifies their solution by using number lines and ten frames. The student states, "I did two number lines and two ten frames and Ryan always has more books than Zak."

#### Representation

#### Expert

The student's diagram is appropriate and accurate. A key defines the books and each boy's name is clearly labeled. The student's number lines are appropriate and accurate. The numbers are labeled "books," each boy's name is assigned to the correct number line, and the "jumps" are correct. The student's ten frames are appropriate and accurate. A label for books is used and each boy's name is clearly labeled. The student uses the number lines and ten frames to clarify that their answer is correct.

#### Overall Achievement Level:

#### Expert

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