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The task
The teacher gave students the prompt (the diagrams were the size of tangram pieces) illustrated here.

Circumstances of performance
This sample of student work was produced under the following conditions:
- alone in a group
- in class as homework
- with teacher feedback - with peer feedback
timed - opportunity for revision

Students had previously worked with tangrams. They had begun to study area relationships and fractions.

This student was preparing a New Standards Elementary Mathematics Portfolio and was aware of the requirements for the portfolio. In providing evidence of problem solving work, students using New Standards Mathematics Portfolios must show that they can go beyond the problem and make connections, extensions or generalizations (
c).
This work sample illustrates a standard-setting performance for the following parts of the standards:
d Arithmetic and Number Concepts: Describe and compare quantities by using simple fractions.
e Geometry and Measurement Concepts: Solve problems by showing relationships between and among figures.
a Problem Solving and Reasoning: Formulation.
b Problem Solving and Reasoning: Implementation.
c Problem Solving and Reasoning: Conclusion.
e Mathematical Skills and Tools: Refer to geometric shapes and terms correctly.
b Mathematical Communication: Show mathematical ideas in a variety of ways.
c Mathematical Communication: Explain solutions to problems clearly and logically.
What the work shows
d Arithmetic and Number Concepts: The student describes and compares quantities by using concrete and real world models of simple fractions; that is:
• finds simple parts of wholes.
The student explained and showed in a model that there can be four equal (parallelogram) parts within the whole (parallelogram).
• uses drawings, diagrams, or models to show what the numerator and denominator mean, including when adding like fractions, e.g., + , or when showing that is more than .
The student used two models to show that one small parallelogram is equivalent to one out of four equal parts of the whole parallelogram, i.e., that “Tracy’s shape was not a of Terri’s shape.”

e Geometry and Measurement Concepts: The student solves problems by showing relationships between and among figures, e.g., using congruence and similarity….
The student explained and modeled that “…one big shape can equal (i.e., be congruent to) four little of the same (i.e., a similar) shape.”

a Problem Solving and Reasoning: Formulation. Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.
Given a problem situation with many possible approaches, the student decided what materials to use (pattern blocks and tiles) and what approach to follow.
• uses previously learned strategies, skills, knowledge, and concepts to make decisions.

The student used knowledge about the relationship between simple parts and wholes to decide who was correct in the dispute.
• uses strategies, such as using manipulatives or drawing sketches, to model problems.

b Problem Solving and Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches to solving problems and uses or learns approaches that other people use, as appropriate.
The student used fractional and geometric strategies to solve the problem.
• makes connections among concepts in order to solve problems.
The student made connections between number and geometry concepts to solve the problem.
• solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution.

c Problem Solving and Reasoning: Conclusion. The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student:…
• explains how the problem is similar to other problems he or she has solved….
The student went beyond the problem by illustrating that the same approach and concepts can be used in a different situation.

e Mathematical Skills and Tools: The student refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle,…parallelogram….
The student referred correctly to the parallelogram, which is not mentioned by name in the task.

b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers,…pictures….


c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in both written and oral work.
The solution is explained clearly.
The conclusion is supported by diagrammatic evidence
.