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The task
The teacher gave the students the following instructions:
Suppose you were given a string that is sixteen inches long. If you cut or fold it in any two places, will it always make a triangle?

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
This sample was completed in Spanish in a bilingual classroom. The translation was provided by the teacher.

This work sample illustrates a standard-setting performance for the following parts of the standards:
d Geometry and Measurement Concepts: Use many types of figures.
e Geometry and Measurement Concepts: Solve problems by showing relationships between and among figures.
g Geometry and Measurement Concepts: Use basic ways of estimating and measuring the size of figures and objects in the real world.
i Geometry and Measurement Concepts: Select and use units for estimating and measuring length.
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 Geometry and Measurement Concepts: The student uses many types of figures (angles, triangles…) and identifies the figures by their properties….
The student used angles and triangles throughout this problem solving activity.
The student did not merely identify, but discovered a property for, triangles: “if the two sides add up to more than the bottom side, it will make a triangle.”

e Geometry and Measurement Concepts: The student solves problems by showing relationships between and among figures….
The student searched for and showed relationships among sides of triangles.
The student summarized relationships among sides of triangles in a general rule.

g Geometry and Measurement Concepts: The student uses basic ways of estimating and measuring the size of figures and objects in the real world, including length,…perimeter….

i Geometry and Measurement Concepts: The student selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as…length….

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.
The student decided to use a trial-and-error approach, which later became more systematic. The student also decided to use string, scissors, and a ruler.
• uses previously learned strategies, skills, knowledge, and concepts to make decisions.
The student used knowledge about triangles and measurement to develop the approach.
• 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 measurement strategies to solve the problem.
• makes connections among concepts in order to solve problems.
The student made connections between measurement 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:…
• makes the solution into a general rule that applies to other circumstances.
The student went beyond the problem by finding a rule that would work for making any triangle.


e Mathematical Skills and Tools: The student refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle,…side….

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

c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in both written and oral work.