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?
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.
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 prespecified manipulative, or
go through a predetermined set of steps.
The student
decided to use a trialanderror 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.



