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Implementing the Rule of Four
Module 0This 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.
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.
Contents
Part 1 Linear Models
Fuel in a generator
Part 2 Polynomial Models
Price of a mile-high snack
Gas mileage
The Rule of Four
Have participants read the documentThe Rule of FourThe 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.
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.
The ScenarioHave 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.
ResultsParticipants 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?
Generator
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Gallons of Fuel
Tim
e (m
inu
tes)
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.
Generator
y = 43.5x - 34.5
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Gallons of Fuel
Tim
e (m
inu
tes)
Reasonableness – Does the model make sense?
Identify any problems with the model?
y = 43.5x - 34.5
-100
-50
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-2 0 2 4 6 8 10
Reasonableness – Does the model make sense?Identify any problems with the
model?
See video: generator_model
Back to the problemThe 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.
Model – Gallons Used
Discuss this model and what it tells us regarding the scenario. Discuss questions like those provided.
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?
Model - Gallons Used
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Time (hours)
Gal
lons
N(t) = 1.5 t
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.
The DataStudents may determine that a scatter plot of the data is appropriate. (below)
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Price
# S
old
Have them determine a model and then define the slope in terms of
the variables.
y = -4.0463x + 32.704
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Price
# S
old
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 doesn’t tell us anything about the real question: revenue. Discuss how to find the revenue and then represent that data with a scatter plot.
Price vs. RevenueRevenue
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Price
Re
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Discuss the trend you see in this data and determine an appropriate function that might model the data.
Quadratic Model
y = -5.9264x2 + 47.121x - 27.26
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Cubic Model
Chart Title
y = -1.6988x3 + 17.016x2 - 50.605x + 104.6
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Price
Rev
enu
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Revenue – Cubic Model
ConclusionsThe cubic model seems like a better model:
y = -1.6988x3 + 17.016x2 - 50.605x + 104.6Or
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.
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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Ave Speed
M P
H
Graphical Representation
What is an appropriate function to model the data?
M P
G
A Model – Discuss why it is necessary
to use so many decimal places in the coefficients.
Does the model make sense?
Explain.y = 9E-05x3 - 0.0149x2 + 0.7029x + 16.364
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Ave Speed
M P
H
See the video: trip
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?
