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The task
Students were given the following task:

Connect all points with segments.


How many segments are needed to connect:
5 points? 6 points? 8 points? 10 points? 30 points? 100 points? n points?

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 work sample illustrates a standard-setting
performance for the following parts of the standards:
b Number and Operation Concepts: Use the inverse operation to determine unknown quantities in equations.
a Function and Algebra Concepts: Discover, describe, and generalize patterns, and represent them with variables and expressions.
b Function and Algebra Concepts: Represent relationships.
a Problem Solving and Mathematical Reasoning: Implementation.
d Problem Solving and Mathematical Reasoning: Mathematical reasoning.
a Mathematical Communication: Use mathematical language and representations with appropriate accuracy.
b Mathematical Communication: Organize work, explain a solution orally and in writing, and use other techniques to make meaning clear to the audience.

What the work shows
This student’s work provides clear evidence of the strategies used for solving the problem and for the development of the solution in stages. This student’s work exhibits logical thinking in establishing a pattern in order to generalize a formula.

b Number and Operation Concepts: The student uses the inverse operation to determine unknown quantities in equations.
The student stated that the inverse of doubling is dividing by 2—a key understanding for the solution presented.

a Function and Algebra Concepts: The student discovers, describes, and generalizes patterns…and represents them with variables and expressions.
The student generalized the relationship between the number of points and the number of line segments.

b Function and Algebra Concepts: The student represents relationships….
The student expressed the number of segments as a function of the number of points.
The proper use of parentheses demonstrates the student’s knowledge of the distributive property.

b Problem Solving and Mathematical Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution….
The student used “making a diagram” and “finding a pattern” as strategies to reach a solution.

d Problem Solving and Mathematical Reasoning: Mathematical reasoning. The student demonstrates mathematical reasoning….
The student made a conjecture to find a formula and defined a plan for finding the formula. Further, the student recognized that the product of the number of vertices and the number of line segments from each vertex is double the total number of line segments.
The student stated that the inverse of doubling is dividing by 2, and therefore, was able to arrive at the general formula for determining the number of line segments.

a Mathematical Communication: The student uses mathematical language and representations with appropriate accuracy….

b Mathematical Communication: The student organizes work, explains…a solution orally and in writing, and uses other techniques to make meaning clear to the audience.
The student uses clear semi-concrete representation (the diagram) followed by semi-abstract (numerical) representation in making the transition to the abstract (the formula: ).

The minor error (“that’s is” on the first page) does not detract from the overall quality of the work.