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?
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
standardsetting
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. 



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.
