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The task
The teachers gave their students the following instructions:

Five people enter a room and introduce themselves to each other. If everyone shakes everyone else’s hand just once, what is the total number of handshakes that occurred?

For Samples 1, 3, and 4, the instructions were given in written form. For Sample 2, the instructions were given orally.

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

Sample 1
This problem was a non-routine problem for the students in all of the classrooms, which made this an appropriate problem solving task.

Sample 2 was completed in Spanish in a bilingual classroom. The translation was provided by the teacher.

Sample 3 was completed in Russian by an LEP student during pull-out Title VII program and after a series of follow-up lessons in basic probability and statistics. The translation was provided by the teacher.

Sample 4 was completed in Haitian Creole by an LEP student enrolled in a self contained bilingual class following an introductory lesson on probability and statistics. The translation was provided by the Haitian Creole calibration team.

These work samples illustrate standard-setting performances for the following parts of the standards:
b Function and Algebra Concepts: Build iterations of simple non-linear patterns.
f Statistics and Probability Concepts: Find all
possible combinations.
a Problem Solving and Reasoning: Formulation.
b Problem Solving and Reasoning: Implementation.
c Problem Solving and Reasoning: Conclusion.
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
b Function and Algebra Concepts: The student builds iterations of simple non-linear patterns….

The students built the solution by building the pattern.

f Statistics and Probability Concepts: The student finds all possible combinations and arrangements within certain constraints involving a limited number of variables.
The drawings throughout the samples show that the students found all the possible combinations of handshakes.

 

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.
Students decided to use diagrams, lists, and equations.
• uses previously learned strategies, skills, knowledge, and concepts to make decisions.
All of the samples demonstrate use of simple addition, as well as skill in drawing diagrams and making lists.
• 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.
In Samples 1, 2, and 3 the students used diagrams, lists, and equations.
• makes connections among concepts in order to solve problems.

All students made connections between simple number concepts and patterns 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.
In Samples 1 and 2 the students went beyond the problem by finding a rule that would work for any number of people. Although the students of Samples 3 and 4 accomplished the task assigned, they did not state the general rule.

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.

There are some grammatical and spelling errors in these work samples. For example, in Sample 1, the student misspelled “drawed” and “egzact.” In Sample 4, the student made some grammatical mistakes (e.g., the capitalization of “kantite” in the middle of a sentence) and misspelled a few words (e.g., “Jwen” instead of “Jwenn,” and “sink” instead of “senk”). However, these errors do not detract from the mathematics in the work. This was a class assignment and was not edited for spelling and grammar.