Implementing the Rule of Four

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Implementing the Rule of Four

• Module 0

This project is sponsored, in part, by a grant
from the National Science Foundation NSF DUE 06
32883. Any opinions, findings, and conclusions or
recommendations expressed in this material are
those of the author(s) and do not necessarily
reflect the views of the National Science
Foundation.
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Goals of the Module

• This module will provide examples of how faculty
  should employ the rule of four in the teaching
  and learning of mathematics especially with
  regard to college algebra.

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Contents

• Part 1 Linear Models
• Fuel in a generator
• Part 2 Polynomial Models
• Price of a mile-high snack
• Gas mileage

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The Rule of Four

• Have participants read the document
• The Rule of Four
• Discuss the rule of four and the 12 pathways and
  what they mean.
• Discuss the final figure, showing the shift to
  contextually based problems that motivate the
  mathematics.

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Example 1 Linear Models

• Electric generators are used in a variety of
  applications. Often they provide power for signs
  or equipment when normal electrical service is
  not available. They are also found in homes and
  used in case of power failure.
• These units usually run on gasoline and have a
  relatively small tank in which the fuel is stored.

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The Scenario

• Have participants read The Scenario for the
  electric generators.
• Discuss the scenario and define the problem.
  Have participants respond to the following
  questions
• Is the data sufficient?
• Is the data appropriate?
• Is the data reliable?
• What would you have done differently?
• Organize the data and decide how it should be
  presented graphically.

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Results
Participants shouldcreate a scatter plot as the
most appropriate wayof representing the data
graphically. However, remind them that getting to
this point with all the conversation is a
critical component of building the students
problems solving skills. Now ask, what
information does this data provide?
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Conclusions
Have participantsfind a linear modelfor the
data in whatever way they wish.Ask what the
slope of this model means. Find the intercepts
and have participants write the meaning of each
intercept using mathematical terminology.
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Reasonableness Does the model make
sense?Identify any problems with the model?
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Reasonableness Does the model make
sense?Identify any problems with the
model?See video generator_model
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Back to the problem

• The problem though is to find out at what rate
  the generator uses fuel. This data indicates that
  the generator uses fuel at a rate of about 40
  minutes per gallon.
• How do we turn this information into something
  that is useful?

NOTE All too often, math teachers talk about
models as if they are precise carrying out
coefficients and constants to several decimal
places when in reality, all we are looking for is
a reasonable, working model. Thus, rounding the
rate to 40 minutes per gallon is realistic and
reasonable and good enough for what we want.
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Model Gallons Used
Possible Questions If the team usually works for
5 hours before taking a lunch break, about how
many gallons should they put into the generator
to last until lunch? The team filled the
generator one morning, worked 5 hours and then
took a lunch break. After lunch, they needed to
complete the project and planned to work until
dark or about another 6 hours. How much fuel do
they need to add after lunch to last until dark?
N(t) 1.5 t
Discuss this model and what it tells us regarding
the scenario. Discuss questions like those
provided.
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Example 2 Number of Super-Snacks Sold

• Many airlines now offer food for purchase. A
  new airline conducted an experiment to attempt to
  determine the price they should charge customers
  for a super-snack.
• Have students read the scenario and determine how
  they might represent the data.

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The Data

• Students may determine that a scatter plot of the
  data is appropriate. (below)

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Have them determine a model and then define the
slope in terms of the variables.
An appropriate model seems to be Sold
-4(price) 33 Thus, -4 represents the change in
the number sold relative to the price or, you
sell four less snacks every time you raise the
price one dollar. However, this data doesnt tell
us anything about the real question revenue.
Discuss how to find the revenue and then
represent that data with a scatter plot.
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Price vs. Revenue
Discuss the trend you see in this data and
determine an appropriate function that might
model the data.
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Quadratic Model
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Cubic Model
Revenue Cubic Model
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Conclusions

• The cubic model seems like a better model
• y -1.6988x3 17.016x2 - 50.605x 104.6
• Or
• Revenue -1.7(price)3 17(price)2 51(price)
  105
• Using this model, if snacks are sold for 4.50
  each, the company is likely to maximize revenue.
• Have students discuss the results. In particular,
  discuss the results in terms of practicality. A
  price of 4.50 will require change be given back.
  Sometimes, a model provides information but the
  implementation might cause one to choose a less
  than perfect solution. Going with a price of 5
  per snack makes more sense.

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Example 3 Gas Mileage

• Have students read the scenario and discuss their
  experiences regarding situations similar to the
  data.
• Discuss the trends in the data and how the data
  can be represented graphically.

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Graphical RepresentationWhat is an appropriate
function to model the data?
M P G
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A Model Discuss why it is necessary to use so
many decimal places in the coefficients.Does
the model make sense? Explain.
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See the video trip
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Questions

• When Papa drives, he averages 77 mph. When
  Grandma drives, she averages 60 mph on the
  stretch down US 64 East.
• How much money do they save (one way) when
  Grandma drives? (Let gasoline be 3 per gallon.)
• How much time do they save (one way) when Papa
  drives?