Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
TASK 6.1.1: LINEAR VS. EXPONENTIAL MODELS
Solutions
Who will win the income race?
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
Men
9521
10038
11148
12088
12786
13821
14732
15726
16882
18711
20297
21689
22857
23891
25497
26365
27335
28313
28180
29556
29987
30874
31408
Women
5616
5872
6331
6791
7370
8117
8728
9257
10121
11071
12156
13259
14477
15292
16169
17124
17675
18531
18509
19752
20556
21272
22141
US Census, 1995
1. Build scatterplots with your calculator, setting the years 1970 – 1992 as 0 – 22.
Briefly discuss the data. Instruct participants to enter the data into their graphing calculators
and create two scatterplots, men’s median income over time and women’s median income
over time.
men
women
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
1
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
2
a. What makes data linear? Data fits a linear model if it has a constant rate of change.
b. Find successive differences. Successive differences do not yield exactly a constant rate of
change. Take an average of the differences to find an average rate of change. An
average of the differences for men’s income is $995 and for women’s income is $751.
The assumption here is to find first successive differences. Later in the institute we find
second and third successive differences and discuss their implications.
c. Use the differences to find a reasonable linear model for each scatter plot.
Using the starting income for the men of $9521 and an average rate of change of 995, a
linear model is m ( x ) = 9521 + 995x , where x represent years and m(x) gives median
salary. Then we adjusted the parameters to get about as many points on top of the scatter
plot as underneath the scatter plot. A guess and check trend line is
m ( x ) = 8750 + 1100x . (A calculator regression is m ( x ) = 9015 + 1084x .) Using the
starting income for the women of $5616 and an average rate of change of 751, a linear
model is w ( x ) = 5616 + 751x , where x represent years and w(x) gives median salary.
Then we adjusted the parameters to get about as many points on top of the scatter plot as
underneath the scatter plot. A guess and check trend line is w ( x ) = 4000 + 850x . (A
calculator regression is w ( x ) = 803 + 4475x .)
d. Use your models to predict the earnings in the year 2000. m ( 30 ) = 41, 750 and
w ( 30 ) = 29, 500 . The answers are based on the guess and check trend lines above.
2. What makes a function exponential?
a. Fill in the table. Participants fill in the second column by evaluating y = 3! 2 x .
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
b. Find successive differences in the third column. Participants fill in the third column with
differences. Again here we assume that participants are finding first differences.
c. Find successive quotients in the fourth column. Participants fill in the fourth column with
first consecutive quotients. Their filled-in table:
x
y = 3! 2 x
3
6
12
24
48
96
0
1
2
3
4
5
Differences
Quotients
3
6
12
24
48
2
2
2
2
2
d. Where does the 3 in y = 3! 2 x appear in the table? The value at x=0 is 3.
e. Where does the 2 from y = 3! 2 x appear in the table? The common quotient is 2.
3. Turn off your linear equations for the income data. Do the following.
a. Take successive quotients and find the percent increase for each year. The average
quotient for the men’s income data is about 1.056. The average quotient for the
women’s income data is about 1.064.
b. Use the quotients to find reasonable exponential models. Using the starting income
for the men of $9521 and an average quotient of 1.056, an exponential model is
m x = 9521 1.056 x . A calculator regression is m x = 10498 1.058x . Using the
()
()
starting income for the women of $5616 and an average quotient of 1.064, an
exponential model is w x = 5616 1.064 x . A calculator regression is
()
()
x
w x = 5883 1.068 .
c. Use your models to predict the earnings in the year 2000.Using the calculator
regression models we found predicted earnings for the year 2000, m 30 = 57,320
( )
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
4
( )
and w 30 = 42,512 . Compare these to the predicted earnings with the linear models
( )
( )
which were m 30 = 41,750 and w 30 = 29,500 .
4. Compare the two models: linear and exponential.
a. Using the linear model, will women’s earnings ever catch up with men’s? Why or why
not?
• Compare the men’s and women’s rates of change (slope). Which is greater? The
men’s income slope is higher.
• What does that mean for the predictions using the linear models? Using linear
models, men’s incomes are increasing faster than women’s. If the trend continues,
women’s incomes will never catch men’s incomes.
b. Using the exponential models, will women’s earnings ever catch up with men’s? Why or
why not?
• Compare the quotients for the men’s and women’s incomes. Which is greater? The
quotient of the women’s income is higher.
• What does that mean for the predictions using the exponential models? Using
exponential models, women’s incomes are increasing faster than men’s. If the trend
continues, women’s incomes will eventually catch and surpass men’s incomes.
c. Which do you think makes more sense? Why? Answers will vary. Note that often the
government reports data using percent increases and decreases.
5. Find the percentage of women’s median income to men’s median income. Create a scatter
plot of this percentage versus years. Based on this graph, will women’s earnings ever catch
up with men’s? Why or why not?
[-2, 22] [-0.1, 1]
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
5
A linear model of the data is y = .56 + .0059x .
•
•
What are the units of the slope? The ratio of men’s income to women’s income per year.
Based on this model, how long will it take for women’s median income to reach men’s
median income? 76 years. When x = 76, y ≈ 1, which means the incomes are
approximately the same.
Math notes
A strength of this task is that participants look at different mathematical models for the same data
and compare the two.
Teaching notes
Briefly discuss the data. Instruct participants to enter the data into their graphing calculators and
create two scatter plots, men’s median income over time and women’s median income over time.
Participants should work in groups and compare their conclusions to Exercise 1. In Exercise 2, if
they need help prompt students when working on 2b with questions such as “Do you think a
linear model will describe this data?” and expect answers such as “No, because the consecutive
differences are not constant.” As they work on 2c, the leader may prompt with questions such as
“What seems to be true about successive quotients for exponentials?” and expect answers such as
“it appears the quotients are constant.”
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
TASK 6.1.1: LINEAR VS. EXPONENTIAL MODELS
Who will win the income race?
Year
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
Men
9521
10038
11148
12088
12786
13821
14732
15726
16882
18711
20297
21689
22857
23891
25497
26365
27335
28313
28180
29556
29987
30874
31408
Women
5616
5872
6331
6791
7370
8117
8728
9257
10121
11071
12156
13259
14477
15292
16169
17124
17675
18531
18509
19752
20556
21272
22141
US Census, 1995
1. Build scatterplots with your calculator, setting the years 1970 – 1992 as 0 – 22.
a. What makes data linear?
b. Find successive differences.
c. Use the differences to find a reasonable linear model for each scatter plot.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
7
d. Use your models to predict the earnings in the year 2000.
2. What makes a function exponential?
a. Fill in the table.
x
y = 3 ! 2x
0
1
2
3
4
5
b. Find successive differences in the third column.
c. Find successive quotients in the fourth column.
d. Where does the 3 in y = 3! 2 x appear in the table?
e. Where does the 2 from y = 3! 2 x appear in the table?
3. Turn off your linear equations for the income data. Do the following.
a. Take successive quotients and find the percent increase for each year.
b. Use the quotients to find reasonable exponential models.
c. Use your models to predict the earnings in the year 2000.
4. Compare the two models: linear and exponential.
a. Using the linear model, will women’s earnings ever catch up with men’s? Why or why
not?
b. Using the exponential models, will women’s earnings ever catch up with men’s? Why or
why not?
c. Which do you think makes more sense? Why?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of
Exponentials and Logarithms; Task 6.1.1
5. Find the percentage of women’s median income to men’s median income. Create a scatter
plot of this percentage versus years. Based on this graph, will women’s earnings ever catch
up with men’s? Why or why not?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
8