Miles of Words
In this task you are asked to read a passage from
a magazine article and then use mathematics to assess the reasonableness
of its claim that forty thousand words were uttered in a 200 mile
train journey.
The following appeared in The New Yorker, October 17,
1994:
I met Dodge on an Amtrak train in Union Station, Washington,
in January of 1993…He came into an empty car and sat down beside
me, explaining that the car would before long fill up. It did. He
didn’t know me from Chichikov, nor I him…Two hundred miles
of track lie between Union Station and Trenton, where I got off,
and over that distance he uttered about forty thousand words. After
I left him, I went home and called a friend who teaches Russian
literature at Princeton University, and asked her who could help
me assess what I had heard,….
Discuss in detail the statement:
“over that distance he uttered about forty thousand
words.”
Is this statement reasonable?
Why or why not? Show all of your calculations and explain your
reasoning.

This task helps answer these things about students’
understanding:
 Given a specific question based on a selection
from a written text, can students figure out what information from the
text is relevant and what mathematics is needed to answer the question?
(Here the mathematics is about rate relationships.)
 Can students work with the mechanics of these
rate relationships and arrive at correct results that answer the given
question?
In short, the task requires students to
(1) formulate and set up a problem from a given context, and then (2)
solve the problem.
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 
This work sample illustrates a standardsetting
performance for the following parts of the standards:

l

Geometry and
Measurement Concepts: Use quotient measures that give “per
unit” amounts. 
m 
Geometry
and Measurement Concepts: Understand unit conversions. 
a 
Function and
Algebra Concepts: Model given situations with formulas and functions,
and interpret given formulas and functions in terms of situations. 
d 
Function and
Algebra Concepts: Work with rates of many kinds. 
a 
Problem Solving
and Mathematical Reasoning: Formulation. 
b 
Mathematical
Skills and Tools: Use a variety of methods to estimate the values
of quantities met in applications. 
e 
Mathematical
Communication: Present mathematical ideas effectively. 


To get to the mathematical heart of the task,
students need to make reasonable estimates of the rate of speed s of a
train (in miles per hour) and the rate r of normal speech (in words per
minute). Using these estimates, students need to:
(i) Find the time T required to travel a given
distance D at the estimated rate of speed s, using the relationship
T = .
(ii) Find the number of words N that can be
spoken in that time T at the estimated rate r, using the relationship
N = rT.
Combining (i) and (ii) gives the formula N =
(D), expressing the number
of words N in terms of the estimated rate of speed s, the estimated rate
of speech r, and the given distance D. Since s, r, and D are known, the
formula can be used to see if the 40,000 words mentioned in the article
is reasonable.
(Interestingly, the quotient of
the rates r and s is itself a rate, “words per mile.” Other
students working on this task made use of this rate in their analysis.)
Students also need to make appropriate unit
conversions: the time T they find will be in hours, and they will have
to convert this to minutes before they use it to find the rate in “words
per minute.”
As individual exercises, (i) and (ii) above
would be too simple for high school. But the “Miles of Words”
task requires students to do more than work these as routine exercises.
Students must formulate the problem from the context, make estimates,
set up their own version of (i) and (ii), and then combine them. What
is being assessed in the task is this whole process.

l
Geometry and Measurement Concepts: The student
uses quotient measures, such as speed,…that give “per
unit” amounts….
m
Geometry and Measurement Concepts: The student
understands…unit conversions….
The student
immediately followed the computation “
= 5.71…hours” with a multiplication by the conversion
factor “60 minutes per hour,” and immediately followed
this with “= 342.857…minutes.” The calculations are
correct, but this use of a conversion factor in a train of equalities
is not ideal. It is clearer to keep the unit conversions separate
from the other calculations. In fact, the student did keep the unit
conversion separate below when using the conversion factor “60
sec/min.”
a
Function and Algebra Concepts: The student
models given situations with formulas and functions, and interprets
given formulas and functions in terms of situations.
d
Function and Algebra Concepts: The student
works with rates of many kinds, expressed numerically [and] symbolically….
The
student found the time of travel from the formula (time) =
The
student found the speaking rate required to support the claim from
the formula .
a
Problem Solving and Mathematical Reasoning:
Formulation. Given the basic statement of a problem situation, the
student:
• fills out the formulation of a definite
problem that is to be solved;
• extracts pertinent information from the situation as a basis
for working on the problem;
• asks and answers a series of appropriate questions in pursuit
of a solution and does so with minimal “scaffolding” in
the form of detailed guiding questions.
The response shows that the student read
the written passage from the article, focused on what is relevant
to the given question, and formulated and solved a particular problem
involving rates in order to answer this question. The work involved
is very different from solving a fully formulated mathematics problem.
b
Mathematical Skills and Tools: The student
uses a variety of methods to estimate the values, in appropriate
units, of quantities met in applications….
The
student suggested and supported an estimate for the rate of speed
of the train.
The
student concluded that a speaking rate of 2 words per second is
too fast to be reasonable. This is puzzling, since rates of 3 words
per second are commonly judged to be representative of actual speech.
Yet, the student gathered data on which to base this opinion.

e
Mathematical Communication: The student represents
mathematical ideas effectively…in writing.
The response gives a clear indication of what the student
did to solve the problem, and of the result.
The
response does not have a consistent approach to the number of significant
digits used. The estimate given of a train’s average speed
(about 35 mph) is very rough (perhaps ± as much as 20 mph),
but the time is reported later as 342.857 minutes. After carrying
out exact calculations with this number, the result is appropriately
rounded up to 2 words/second. It would have been more reasonable
to use only one significant digit in all calculations.
There are two misspellings (“recieved”
in the first line, “acctual” at the end of the first paragraph)
and a punctuation error (“trains” should have an apostrophe),
but these do not detract from communicating the meaning.

