Stewart J. College Algebra: Concepts and Contexts

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I. Properties of Linear Data

One of the main ways of finding patterns in data is to look at the differences between

consecutive data points. In this exploration we assume that the inputs are the equally

spaced numbers

0, 1, 2, 3, . . .

We want to find properties of the outputs which guarantee that there is a linear func-

tion that exactly models the data. So let’s consider the linear function

1. Find the outputs of the linear function corresponding to the

inputs 0, 1, 2, 3, . . . .

2. Use your answers to Question 1 to find the first differences of the outputs.

3. Use your answers to Questions 1 and 2 to confirm that the entries in the first

difference table below are correct. Use the pattern you see in the table to fill in

the columns for the inputs 4, 5, and 6.

f 112- f 102= _____

,f 122- f 112= _____,f 132- f 122= _____

f 102= _____

,f 112= _____,f 122= _____,f 132= _____

f 1x 2= b + mx

f 1x2= b + mx

4. A data set is given in the following table.

x 0123456

+ 2

+ 3

First difference —

mm m

table 1

First differences for f 1x 2= b + mx

x 01 234567

y 5 9 13 17 21 25 29 33

First difference —

(a) Fill in the entries for the first differences.

(b) Observe that the first differences are constant, so there is a linear function

that models the data. To find b and m, let’s compare this data table to

Table 1. Comparing the output corresponding to the input 0 in each of

these tables, we conclude that

b  __________.

f 1x 2= b + mx

234 CHAPTER 2

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Comparing the first differences in each of these tables, we conclude that

m ⫽ __________.

So a linear function that models the data is

f(x) ⫽ ____ ⫹ ____ x

II. Linear Patterns

1. A piece of wire is bent as shown in the figure. You can see that one cut through

the wire produces four pieces and two parallel cuts produce seven pieces.

(a) Complete the table for the number of pieces produced by parallel cuts, and

then calculate the first differences.

(b) Are the first differences constant? Is there a linear pattern relating the

number of cuts to the number of pieces? If so, find b and m by comparing

the entries in this table with the corresponding ones in Table 1.

b ⫽ __________ m ⫽ __________.

Find a linear function that models the pattern in the data:

f(x) ⫽ ____ ⫹ ____ x

2. Mia wants to make a square patio made of gray tile with a red tile border on

three sides. The figure shows a possible patio for which the gray part is a

square of side 2.

Number of cuts x 01234

Number of pieces y

First difference —

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(a) Complete the table for the number of red border tiles needed when the

gray part is a square of side x, and then calculate the first differences.

(b) Are the first differences constant? If so, find b and m by comparing the

entries in this table with the corresponding ones in Table 1.

b  m __________ m  __________.

Use these equations to find b. Now find a linear function that models the

pattern in the data:

f(x)  ____  ____ x

3. An amphitheater has several rows of seats, with 30 seats in the first row, 32 in

the second row, and so on.

(a) Complete the table for the number of seats in each row, and then calculate

the first differences.

(b) Are the first differences constant? If so, find b and m by comparing the

entries in this table with the corresponding ones in Table 1; then find a

linear function that models the pattern in the data:

f(x)  __________

4. Let’s make triangles out of dots as shown in the figure.

Stage

1 36 10

236 CHAPTER 2

Square of side x 1234

Number of border tiles y

First difference

Row number x 12345

Number of seats y

First difference —

(a) Complete the table for the number of dots in each triangle, and then

calculate the first differences.

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Bridge Science

OBJECTIVE To experience the process of collecting data and then analyzing the

data using linear regression.

If you want to build a bridge, how can you be sure that it will be strong enough to

support the cars that will drive over it? You wouldn’t want to just build one and hope

for the best. Mathematical models of the forces the bridge is expected to experience

can help us determine the strength of the bridge before we actually build it.

The most famous bridge collapse in modern history is that of the first Tacoma

Narrows bridge in Washington State. From the day of its opening on July 1, 1940,

the bridge exhibited strange behavior. On windy days it would sway up and down;

drivers would see approaching cars disappear and reappear as moving dips and

humps formed in the roadway. The locals nicknamed the bridge “Galloping Gertie.”

On the day of its collapse, November 7, 1940, there was a particularly strong wind,

and the bridge began to sway violently. Leonard Coatsworth, a driver stranded on the

bridge described it this way:

Just as I drove past the towers, the bridge began to sway violently from side to side. Before

I realized it, the tilt became so violent that I lost control of the car. . . . On hands and knees

most of the time, I crawled 500 yards or more to the towers. . . . Safely back at the toll

plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.

To see a video of the bridge collapse, search YouTube for Tacoma Narrows

bridge collapse.

(b) Are the first differences constant? Is there a linear pattern relating the

number of dots in a triangle and the number of dots in its base?

Bridge swaying

Bridge collapsing

Triangle number x 123456

Number of dots y

First difference —

AP Images

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238 CHAPTER 2

In this exploration we perform an experiment on simple paper bridges, gener-

ating data on the strength of the bridges. We then use linear regression to analyze

the data.

I. Setting up the Experiment

In this experiment you will construct bridges out of paper and use pennies as weights

to determine how strong each bridge is.

You will need:

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21 pieces of paper, each 11 by 4.25 inches (half of a regular sheet of paper

cut lengthwise)

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100 pennies (two rolls)

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2 books that have the same thickness

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1 small paper cup

Procedure:

1. Fold up a 1-inch strip on both long sides of each piece of paper. These paper

“U-beams” will be used to construct the bridges.

2. Suspend one or more U-beams (nested inside each other if you are using more

than one) across the space between the books, as shown in the figure, and

center the paper cup on the bridge. There should be about 9 inches of space

between the books.

3. One by one, put pennies into the cup until the bridge collapses. We’ll call the

number of pennies that it takes to make the bridge collapse the load strength

of the bridge.

II. Collecting the Data

1. Determine the load strength for bridges that are 1, 2, 3, 4, and 5 layers thick.

Make sure you use new U-beams to build each bridge, since U-beams from

collapsed bridges will be weaker than new ones.

2. Complete the following table with the load strength data for your bridges.

Layers in bridge 0123456

Load strength

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III. Analyzing the Data

1. Make a scatter plot of your data. Do the data follow an approximately linear

trend?

Pennies

1234

ers

Correlation and Causation

OBJECTIVE To understand real-life examples in which two variables are

mathematically correlated but changes in one variable do not necessarily cause

changes in the other.

If two variables are correlated, it does not necessarily mean that a change in one vari-

able causes a change in the other. For example, the mathematician John Allen Paulos

points out that shoe size is strongly correlated to mathematics scores among school

children. Does this mean that we should stretch children’s feet to improve their math-

ematical ability? Certainly not—both shoe size and math skills increase indepen-

dently with age. We refer to “age” here as the hidden variable; increasing age is the

real reason that shoe size and mathematical ability increase simultaneously. Here are

some more examples of the hidden variable phenomenon.

Do churches cause murder? In the 1980s several studies used census data to

show that the more churches a city has, the more murders occur in the city each year.

With tongue in cheek, the authors claimed to have proved that the presence of

churches increases the prevalence of murders. While the data were correct, the con-

clusion was obviously nonsense. What is the hidden variable here? Larger cities tend

2. Use a graphing calculator to find the regression line for your data.

3. Graph the regression line on your scatter plot above.

4. What is the slope of the regression line, and what does it tell us?

5. Use the regression line to predict the load strength of a bridge with six layers.

6. Build a bridge with six layers, and find its load strength. How does this

compare to the prediction from the regression line?

Regression line:y = _______________

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240 CHAPTER 2

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Linear Functions and Models

Do video games cause violent behavior? Some studies have shown that play-

ing violent video games and aggressive behavior are strongly correlated. While

it is certainly possible that violent video games might cause desensitization to ac-

tual aggressive behavior, could there be a hidden variable here as well? Some

people are innately more violent than others, quicker to lose their temper or lash

out. Perhaps these inborn tendencies cause both an interest in violent video

games and a tendency toward real-life violence. More research is needed to set-

tle these questions.

It is important not to jump to conclusions. Correlation and causation are not the

same thing. Correlation is a useful tool for bringing important cause-and-effect rela-

tionships to light, but to prove causation, we must explain the mechanism by which

one variable affects the other. For example, the link between smoking and lung can-

cer was observed as a correlation long before medical science determined how the

toxins in tobacco smoke actually cause lung cancer.

I. Hidden Variables

1. A public health student gathers data on bottled water use in her state. Her data

indicate that households that use bottled water have healthier children than

households that don’t. She concludes that drinking bottled water instead of tap

water helps to prevent childhood diseases. Do you think her conclusion is

valid, or is there likely to be a hidden variable that accounts for this correla-

tion? Write a short paragraph to explain your reasoning.

2. The residents of a seaside town have noticed that on days when the local ice

cream parlor is busy, a lot of people go swimming in the ocean. They wonder

whether going swimming causes people to crave ice cream or whether eating

ice cream makes people want to go swimming. Which alternative is correct?

Or does a hidden variable cause both phenomena? Explain your answer.

3. A study investigating the connection between dietary fat and cancer in

various countries came up with the data shown in the table, relating daily

fat intake (in grams) and annual cancer death rates (in deaths per 100,000

population).

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240 CHAPTER 2

More churches . . . More crime?

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to have more of everything, including both churches and murders, so the hidden vari-

able is population size.

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SECTION 2.6

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Linear Equations: Getting Information from a Model 241

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(a) Make a scatter plot of the data. Do they appear to be linearly correlated?

Country Fat (g/day) Cancer deaths

Austria 120 18

Denmark 160 23

El Salvador 40 1

Greece 95 7

Hungary 105 14

Norway 130 17

Poland 92 10

Portugal 74 13

United States 150 20

(b) Does it seem reasonable to conclude that a high-fat diet causes cancer?

What else would you need to know to settle the question?

mechanism has been discovered to explain how eating fat would cause

cancer. Can you think of any hidden variable that would account for this

correlation?

4. A researcher wants to test the hypothesis that paving over ground surfaces

causes an increase in fecal bacteria in groundwater. He studies six creeks in

Maryland and gathers the following data relating the percentage of each

creek’s watershed that is covered with pavement and the average fecal

coliform bacteria count (in units/100 mL).

(a) Make a scatter plot of the data on the coordinate plane provided on the

next page. Do the data appear to be linearly correlated?

(b) Can you think of any reasons why the presence of pavement might

increase the bacteria count?

increases fecal bacteria counts. Can you think of a hidden variable that

might cause both increases?

Deaths

50 100 150

Fat (

/da

)

Paved Bacteria

7% 18

9% 22

14% 40

18% 65

21% 80

22% 92

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5. From your own experience or your reading, think of two variables that are

correlated but for which the correlation is caused by some third hidden

variable. Write a short paragraph describing the correlation and explaining

how the hidden variable is the true reason for the correlation.

II. Are We Measuring the Right Thing?

Just because not every correlation is evidence of causation doesn’t mean that we

should be too skeptical about studies that use correlation to link two variables. But

sometimes people get the direction of causation wrong; instead of concluding cor-

rectly that A causes B, they conclude wrongly that B causes A. Here are some exam-

ples of this type of thinking.

1. Some diet books recommend that people who want to lose weight should not

drink diet soda, because studies have shown that diet soda drinkers tend to be

heavier than regular soda drinkers. Does this really mean that drinking diet

soda causes weight gain? How do you explain these studies?

2. Inhabitants of the New Hebrides used to believe that body lice improved

their health. They correctly observed that healthy people had lice and sick

people didn’t. Do you think their belief was reasonable? What do you think

might account for their observation? (Note that lice don’t like high body

temperatures.)

Bacteria

100

4 8 12 16 20 24

Percent paved

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Fair Division of Assets

OBJECTIVE To understand the real-life problem of distributing assets fairly among

several claimants and how linear equations can help solve such problems.

When companies or individuals cannot meet their financial obligations they can

sometimes file for bankruptcy. Under the Chapter 7 bankruptcy rules, all debts are

terminated, and the company’s assets are divided among its creditors. The primary

goal of the legal process of dividing the assets among the claimants is fairness. This

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process can be quite complicated because there are different ways of interpreting

“fairness.” One of the largest bankruptcies in U.S. history was that of Worldcom in

2002. The telecommunications giant, which operated the world’s largest Internet

network, had $107 billion in assets and $41 billion in debt, owed to over 1000 cred-

itors. As you can imagine, restructuring such a huge debt among so many creditors

is a complicated process.

In this Exploration we see how algebra can be used to analyze different possi-

ble ways of dividing assets fairly among several claimants.

I. Dividing Assets Fairly

When high-tech Company C goes bankrupt, it owes $120 million to Company A and

$480 million to Company B. Unfortunately, Company C has only $300 million in as-

sets. How should the court divide these assets between Companies A and B? Here

are some possible ways:

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Companies A and B divide the assets equally.

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Companies A and B share the losses equally.

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Companies A and B each get the same fixed percentage of what they are

owed.

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Companies A and B get an amount proportional to what they are owed.

Let’s explore these alternatives to see which is fairest.

1. Suppose Companies A and B divide the $300 million equally.

(a) How much does each company receive?

A gets: __________ B gets: __________

(b) After the assets have been distributed, what is the net loss for each

company?

A’s net loss: __________ B’s net loss: __________

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