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 Page 1 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 © 2003 Teachers Teaching With Technology Page 2 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 © 2003 Teachers Teaching With Technology Page 3 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 Page 4 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 © 2003 Teachers Teaching With Technology Page 5 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 © 2003 Teachers Teaching With Technology Page 6 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 PTE: Algebra © 2003 Teachers Teaching With Technology Page 7 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 © 2003 Teachers Teaching With Technology Page 8 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 Page 9
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