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The task
Imagine that you represent a consumer group in our neighborhood and you are responsible for advising new families that move into the area as to which telephone company would cost less for the average number of telephone calls that the family will make per month. You have narrowed the search to two companies, ESP Telephone and PQR Telephone. They decide to price their services in the following manner:

ESP TELEPHONE PQR TELEPHONE
$4.00 base fee per month $7.00 base fee per month
$0.10 per local call $0.06 per local call

a) The O’Neal family makes about 80 local calls per month. Which service is better for them? Why?
b) The Jordan family makes about 50 local calls per month. Which service is better for them? Why?
c) i) FOR EACH TELEPHONE COMPANY, make an In-Out table (a table of values: number of calls and the monthly cost.)
ii) FOR EACH TELEPHONE PLAN, write an equation or statement or rule that expresses the relationship between the number of calls made and the total cost.
d) On the same set of axes, make a graph for the ESP Telephone Company and for the PQR Telephone Company.

Based on the responses above, write a letter to the consumer group with your recommendations about which plan a family should choose. Explain how you arrived at your recommendation.



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
with manipulatives with calculator

Mathematics required by the task
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 y-intercept, the other (PQR) has a smaller slope. As a result, the graphs of the two functions cross.

The n-value 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.

What the work shows
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 standard-setting
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.