How Much Daylight is There?
Concepts
• Periodicity
• Graphical characteristics of the sine function
• Scatterplots
• Sine regression
Materials
• Student activity sheet “How Much Daylight is There?”
• TI-83 Plus/SE
Introduction
The number of daylight hours depends upon many factors. Based on experience, you know that
north of the equator there are more daylight hours during the summer months than there are during
the winter months. This recurring situation leads to the conclusion that there is a regular pattern or
period associated with the number of daylight hours. In this activity you will use the TI-83 Plus/SE
to investigate some of the factors affecting how many hours of daylight an indicated city will have
for a given day of the year.
PTE: Algebra
© 2003 Teachers Teaching With Technology
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How Much Daylight Is There?
Student Activity Sheet
Problem 1:
The number of hours of daylight varies each day in a year. For example, the number of daylight
hours for San Diego, California is given in Table 1.
Table 1: Daylight Hours by Date for San Diego, CA
Date
Day of
Number of
Date
the
Daylight
Year
Hours
January 5
5
9.69
July 5
January 15
15
9.84
July 31
January 27
27
10.10
August 13
February 4
35
10.32
August 29
February 14
45
10.64
September 16
March 17
76
11.84
September 21
March 21
80
12.00
October 12
April 1
91
12.45
October 28
April 15
105
13.00
November 3
May 6
127
13.74
November 25
May 20
141
14.08
December 6
June 11
162
14.37
December 21
June 21
172
14.40
Day of
the
Year
186
212
225
241
259
264
285
301
307
329
340
355
Number of
Daylight
Hours
14.32
13.83
13.44
12.83
12.14
11.94
11.09
10.52
10.34
9.81
9.67
9.60
Use the EDIT option on the STAT menu and enter the data for Day of the Year into L1 and
Number of Daylight Hours into L2, as indicated below. Create a scatterplot in the STAT PLOT
menu. Use ZOOMSTAT from the ZOOM menu to get the best view.
1. What is the relationship between the day of the year and the number of daylight hours for
that day? Use a graphical representation to answer the question.
PTE: Algebra
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How Much Daylight Is There?
Use CALC from the STAT menu to determine the regression equation that best describes the
data. Copy the regression equation into Y1 on the grapher menu and graph the regression
function and the scatterplot on the same coordinate plane.
2. What algebraic (mathematical) model would best describe the graphical pattern you observed
in the previous question?
3. Two students, Pat and Chris, said that both a quadratic equation and a quartic equation model
the data. Explain to Pat and Chris whether or not these models are appropriate when
extrapolating information for the next year.
Problem 2:
Latitude is another factor that affects the number of daylight hours for any day in a year. Table 2
gives the date, day of the year, latitudes, and number of daylight hours for San Diego, California,
Montreal, Ontario, and Fairbanks, Alaska for March 21, June 21, September 21, and December
21.
Table 2: Number of Daylight Hours
Number of Daylight Hours
Date
Day of
the
Year
San Diego, CA
(320 42' 53" N)
Montreal, QE
(450 30' 33" N)
Fairbanks, AK
(640 48' 00" N)
March 21
80
12.0
12.0
12.0
June 21
172
14.4
15.6
20.3
September 21
264
11.8
11.9
11.8
December 21
355
9.6
8.4
3.7
4. What is the significance of the days listed in Table 2?
5. Compare each date’s number of daylight hours for each city. Describe whether or not this is
what you expected? Why?
6. Find three other cities in North America that you think would have approximately the same
number of daylight hours for each given day as the cities in Table 2. Why do you think your
chosen cities would have the same number of daylight hours?
PTE: Algebra
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How Much Daylight Is There?
Problem 3:
The number of daylight hours for a given day is approximately the same from year to year. This
indicates periodic behavior for the number of daylight hours with respect to a given day in a year
for a specific latitude. The equation that generates the data for this pattern is given by
2π
H = 12 + F sin((
)(t − 80)) , where H is the number of daylight hours, F is a factor based on the
365
latitude, and t is the number of the day in the year. Table 3 gives the latitudes and the number of
daylight hours for May 15 for selected cities.
Table 3: Latitudes and Daylight Hours for Selected Cities on May 15
Number of Daylight Hours for May 15
City
Latitude
(Day #135)
Adelaide Island, Antarctic
650 S
3.9
Commodora, Argentina
450 S
9.1
Santiago, Chile
330 S
9.7
Belo Horizonte, Brazil
0
20 S
10.9
Entebbe, Uganda
00
12.0
Cairo, Egypt
300 N
13.4
Shenyang, China
420 N
14.5
Barrow, Alaska
710 N
21.7
7. What is the value of F (the latitude factor) for each city? Round answers to the nearest tenth.
City
F
Adelaide Island, Antarctic
Commodora, Argentina
Santiago, Chile
Belo Horizonte, Brazil
Entebbe, Uganda
Cairo, Egypt
Shenyang, China
Barrow, Alaska
8. Write an equation for each city’s number of daylight hours in the form
2π
H = 12 + F sin((
)(t − 80)) , where H is the number of daylight hours, F is a factor based
365
on the latitude, and t is the number of the day in the year.
PTE: Algebra
© 2003 Teachers Teaching With Technology
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How Much Daylight Is There?
City
Daylight Hours Equation
Adelaide Island, Antarctic
Commodora, Argentina
Santiago, Chile
Belo Horizonte, Brazil
Entebbe, Uganda
Cairo, Egypt
Shenyang, China
Barrow, Alaska
9. Draw a scatterplot using latitude as the input variable and determine the relationship between
a city’s latitude and its latitude factor, F. As an example, for Barrow, Alaska, the equation is
2π
H = 12 + 11.952sin((
)(t − 80)) .
365
Enter latitude in L1 and the number of daylight hours in L2. Although a cubic regression is
illustrated below, experiment with finding the regression equation that gives the best
regression coefficient and the best fit.
a)
Describe the relationship you observe.
b)
What algebraic model would best describe the data?
Enter the latitude factor, F, in L3. Compute the best regression equation using L2 (number
of daylight hours) and L3 (latitude factor), and then set up PLOT2 in STAT PLOT for L2 and
L3.
c)
What is the relationship between the number of daylight hours, H, and the latitude
factor, F? Use the number of daylight hours as the input variable.
PTE: Algebra
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How Much Daylight Is There?
Teacher Notes
Introduction
The number of daylight hours depends upon many factors. Based on experience, you know that
north of the equator there are more daylight hours during the summer months than there are during
the winter months. This recurring situation leads to the conclusion that there is a regular pattern or
period associated with the number of daylight hours.
In this activity students will use the TI-83 Plus/SE to investigate some of the factors affecting how
many hours of daylight an indicated city will have for a given day of the year. Students will need
to use the Internet, almanac, etc. to find the latitude degrees and number of daylight hours for
various cities in North America and the World. A TI-83 Plus/SE Viewscreen for classroom display
will be helpful.
Instructions
1. Lead students in using the TI-83 Plus/SE to create a scatterplot for the Day of the Year and
Number of Daylight Hours in San Diego data found in Table 1: Daylight Hours by Date for
San Diego, CA. Use a TI-83 Plus/SE Viewscreen for classroom display. Ask students to
examine their scatterplot and to respond to question #1 from Problem 1.
Use the EDIT option on the STAT menu and enter the data for Day of the Year into L1 and
Number of Daylight Hours into L2, as indicated below. Setup a scatterplot in the STAT PLOT
menu. Use ZOOMSTAT from the ZOOM menu to get the best view.
Engage students in a discussion of their responses. A common initial student response is that a
quadratic relationship exists between the day of the year and the number of daylight hours.
Point out that although the quadratic and quartic regression equations appear to be appropriate
guesses, neither equation would be reasonable when extrapolating into the next year. This will
help guide their thinking toward investigating a periodic function.
2. Lead students in using the TI-83 Plus/SE to use CALC from the STAT menu to determine the
sine function that best describes the data. Copy the SinReg equation into Y1 on the grapher
menu and graph the regression function and the scatterplot on the same coordinate plane.
PTE: Algebra
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How Much Daylight Is There?
3.
4.
5.
6.
7.
It may be helpful to have students calculate regression coefficients for both quadratic and
quartic fits (0.920 and 0.999, respectively) to illustrate the importance of considering other
factors when finding the curve of best fit.
Ask students explain in writing to Pat and Chris whether or not both a quadratic equation and a
quartic equation model are appropriate when extrapolating information for the next year.
Answer: Based on the data, both a quadratic equation and a quartic equation reach a
maximum only once. Their graphs suggest that after the maximum is attained, all other
subsequent values will be lower than the previous ones.
Have students examine Table 2: Number of Daylight Hours. Ask them what is the
significance of the days listed in Table 2 and have them record their answer on the student
activity sheet. Note: Cities that have approximately the same latitude will have similar numbers
of daylight hours.
Answer: The days are the Vernal Equinox (March 21), the Summer Solstice (June 21), the
Autumnal Equinox (September 21), and the Winter Solstice (December 21). There are
approximately 12 hours of daylight on the Vernal and Autumnal Equinoxes; the greatest
number of daylight hours on the Summer Solstice and the least number on the Winter Solstice.
Engage students in a discussion of their responses.
Ask students to compare each date’s number of daylight hours for each city and to describe
whether or not this is what they expected and why.
Answer: The largest number of daylight hours occurs on June 21. Daylight hours for March
21 and September 21 are approximately the same. The fewest number of daylight hours occurs
on December 21.
Have students find three other cities in North America with approximately the same number of
daylight hours for each given day as the cities in Table 2. Ask them to explain in writing on the
student activity sheet why they think their chosen cities would have the same number of
daylight hours.
Answer: Many correct responses are possible. If cities have approximately the same latitude,
then their temperatures should be approximately the same.
Based on the information found in Table 3: Latitudes and Daylight Hours for Selected
Cities on May 15, have students determine the latitude factor, rounded to the nearest tenth, for
each city and complete the table found on the student activity sheet. Caution students that the
calculator should be in radian mode.
Answer:
F
City
Adelaide Island, Antarctic
-9.98
Commodora, Argentina
-3.57
Santiago, Chile
-2.83
Belo Horizonte, Brazil
-1.35
Entebbe, Uganda
0.0
Cairo, Egypt
1.72
Shenyang, China
3.08
Barrow, Alaska
11.95
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How Much Daylight Is There?
Note that the F factors for cities in the southern hemisphere are the negative of F factors for
cities in the northern hemisphere with similar latitudes.
2π
)(t − 80)) , where H is the
8. Ask students to write an equation in the form H = 12 + F sin((
365
number of daylight hours, F is a factor based on the latitude, and t is the number of the day in
the year for each city’s number of daylight hours. Have students record their response in the
table found on the student activity sheet.
Answer:
City
Daylight Hours Equation
Adelaide Island, Antarctic
H = 12 − 9.98sin((
2π
)(135 − 80))
365
Commodora, Argentina
H = 12 − 3.37sin((
2π
)(135 − 80))
365
Santiago, Chile
H = 12 − 2.83sin((
2π
)(135 − 80))
365
Belo Horizonte, Brazil
H = 12 − 1.35sin((
2π
)(135 − 80))
365
Entebbe, Uganda
H = 12 + 0.0sin((
Cairo, Egypt
H = 12 + 1.72sin((
2π
)(135 − 80))
365
Shenyang, China
H = 12 + 3.08sin((
2π
)(135 − 80))
365
Barrow, Alaska
H = 12 + 11.952sin((
2π
)(135 − 80))
365
2π
)(135 − 80))
365
9. Have students create a scatterplot representing a city’s latitude and its latitude factor. Use
latitude as the input variable and determine the relationship between a city’s latitude and its
2π
)(t − 80)) .
latitude factor, F. As an example, for Barrow, Alaska, H = 12 + 11.952sin((
365
Enter latitude in L1 and the number of daylight hours in L2. Although a cubic regression is
illustrated below, ask students to experiment with finding the regression equation that gives the
best regression coefficient and the best fit.
Have students use their observations to answer questions #9a-b on the student activity sheet.
PTE: Algebra
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How Much Daylight Is There?
a) Describe the relationship you observe.
Answer: As the latitude increases, the number of daylight hours appears to increase.
b) What algebraic model would best describe the data?
Answer: The cubic regression is the best algebraic model. See equation above.
Engage students in a discussion of their responses.
Have students compute the best regression equation using number of daylight hours and
latitude factor and record their response to question #9c on the student activity sheet. Enter the
latitude factor, F, in L3. Compute the best regression equation using L2 (number of daylight
hours) and L3 (latitude factor), and then set up PLOT2 in STAT PLOT for L2 and L3.
c) What is the relationship between the number of daylight hours, H, and the latitude factor,
F? Use the number of daylight hours as the input variable.
Answer: The cubic curve appears to fit the data points the best.
Assessment
The following journal entry prompts are suggested:
What did you learn from this activity?
What questions do you still have related to the mathematics content in this activity?
What questions do you still have related to the technology used in this activity?
Teacher Reflection
• What is the significance of the investigated comparisons? What other comparisons could be
investigated? Would they be meaningful?
• What mathematical concepts can be introduced and/or strengthened with this activity?
• How can the Internet be used to gather data and/or to verify computed results?
• What different teaching strategies need to be used to best guide students into the discoveries?
PTE: Algebra
© 2003 Teachers Teaching With Technology
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