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The task

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:

  1. 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.)

  2. 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.

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

This work sample illustrates a standard-setting
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.

Mathematics required by the task


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.

What the work shows

 

 

 

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.