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 
with manipulatives 
with calculator 
For each of the companies, the total phone
bill for a month is a linear function of the number of
calls, n, made in a month:
ESP total bill = $4.00 + $0.10 n
PQR total bill = $7.00 + $0.06 n
One function (ESP) has a smaller yintercept, the other (PQR) has
a smaller slope. As a result, the graphs of the two functions cross.
The nvalue of the crossing point can be found by solving a linear
equation:
$4.00 + $0.10 n = $7.00 + $0.06 n
The solution of the equation, n = 75 calls, is the critical number
for this task.
Whether a family’s calls are above or below this number determines
which company is cheaper:
If n < 75 calls, ESP is a better deal, since its cost graph lies
below that for PQR.
a
Number and Operation Concepts: The student
uses addition, subtraction, multiplication, and division in forming
and working with numerical and algebraic expressions.
c
Number and Operation Concepts: The student
has facility with the mechanics of operations as well as understanding
of their typical meaning and uses in applications.
e
Number and Operation Concepts: The student represents numbers in
decimal form.
The
student computed the cost for each family correctly by multiplying
and adding. The student correctly chose the lesser cost as the better
buy.
o
Geometry and Measurement Concepts: The student
represents geometric curves and graphs of functions in standard
coordinate systems.
The
student used a line graph to compare the two payment plans. The
accuracy of the graph is demonstrated by the labeling of the axes,
using appropriate scales, using the vertical axis to represent the
range (monthly cost), accurately plotting the points, and carefully
drawing and labeling the lines representing each payment plan.
a
Function and Algebra Concepts: The
student models given situations with formulas and functions, and
interprets given formulas and functions in terms of situations.
b
Function and Algebra Concepts: The student
describes, generalizes, and uses basic types of functions: linear,
exponential, power, rational, square and square root, and cube and
cube root.
The
student created an equation for each telephone service in words
and in symbols; this was a linear function. The use of the phrase
“amount of calls” by the student is imprecise, when “number
of calls” is what is meant. The student made a careless error
in the formula for plan PQR in the base fee, using $4 instead of
$7. Earlier, the correct value was used to compute the cost of plan
PQR.
i
Function and Algebra Concepts: The student
represents functional relationships in formulas, tables, and graphs,
and translates between pairs of these.
The student
graphed the two functions from the table of values.
The table
accurately reflects the cost of each plan and the step of 10 calls
per month was a reasonable choice.
The work
demonstrates the student’s understanding of the relationship
between the costs of the two plans. Specifically, the student wrote:
“…ESP has a cheaper price…until you make about 80
calls.”
c
Problem Solving and Mathematical Reasoning:
Conclusion. The student provides closure to the solution process,
that is, the student formulates generalizations of the results obtained.
The student’s
letter makes it clear that for more than 75 calls, PQR was the cheaper
plan. The statement at the end of the letter “My amount…gradually
increased” is ambiguous as a description of the scale on the
horizontal axis.
a
Mathematical Skills and Tools: The student
carries out numerical calculations and symbol
manipulations effectively, using mental computations, pencil and
paper, or other technological aids, as appropriate.
The student
performed many calculations to create a suitable table of values.
f
Mathematical Skills and Tools: The student
uses the number line and Cartesian coordinates in the plane….
The student
plotted the table of values and drew the lines.
g
Mathematical Skills and Tools: The student
creates and interprets graphs of many kinds, such as function graphs….
The student’s
letter demonstrates the understanding that the point of intersection
of the two lines
(75, $10.50) represents the number of monthly calls that would cost
the same in either plan. This is a refinement of the student’s
earlier observation from the table of values that at “…about
80 calls” PQR becomes the better choice.
The first
sentence reverses the logic used: indeed, the better deal depends
on the amount of calls.
The point
of intersection is actually (75, $11.50). Nonetheless, the letter
clearly states the important result.
Note: In connecting the discrete points of the graph, the straight
lines can be used to identify a trend and thereby make an informed
decision; they cannot be used to find the cost of anything other
than an integral number of calls.

This work sample illustrates a standardsetting
performance for the following parts of the standards:

a 
Number and
Operation Concepts: Use addition, multiplication, and division
in forming and working with numerical and algebraic expressions. 
c 
Number and
Operation Concepts: Have facility with the mechanics of operations
as well as understanding of their typical meaning and uses in
applications. 
e 
Number and
Operation Concepts: Represent numbers in decimal form. 
o 
Geometry and
Measurement Concepts: Represent geometric curves and graphs
of functions in
standard coordinate systems. 
a 
Function and
Algebra Concepts: Model given situations with formulas. 
b 
Function and
Algebra Concepts: Use basic types of functions. 
i 
Function and
Algebra Concepts: Represent functional relationships. 
c 
Problem Solving
and Mathematical Reasoning: Conclusion. 
a 
Mathematical
Skills and Tools: Carry out numerical calculations effectively. 
f 
Mathematical
Skills and Tools: Use the number line and Cartesian coordinates
in the plane and in space. 
g 
Mathematical
Skills and Tools: Create and interpret graphs of many kinds. 
c 
Mathematical Communication:
Organize work and present mathematical procedures and results
correctly. 
d 
Mathematical Communication:
Communicate logical arguments clearly, showing sensibility and
validity. 
c
Mathematical Communication: The student
organizes work and presents mathematical procedures and results
clearly, systematically, succinctly, and correctly.
d
Mathematical Communication: The student
communicates logical arguments clearly, showing why a result makes
sense and why the reasoning is valid.
The student arranged the work in a clear
sequence and labeled all the work appropriately. The student concluded
with a logical and clearly worded statement about which telephone
plan is better and justified that conclusion correctly through reference
to the graph.
The minor grammatical errors (e.g., “at” instead of “to”
in the first paragraph of the letter) do not detract from the overall
quality of the work.
