Loyce Engler
AMDM Lesson Plan 02/18, 21, 22, 23
Topic: Using Functions in Models and Decision Making: Cyclical Functions Objective: Students model real-‐world data using cyclical or sinusoidal models. Students interpret cyclical models in the context of the situation. Students use cyclical models to make predictions and draw conclusions. Students discuss various types of limitations that occur in models, including problems with extrapolating outside the data with models that fit the data but do not adhere to known principles or natural laws. Prerequisite Skills: • Making a scatterplot with or without technology • Analyzing regression models • Understanding characteristics of linear functions Resources and Materials: Graphing calculator, colored pencils Vocabulary amplitude, cyclical model, frequency, parameter, period, regression model, scatterplot, sinusoidal function Engage: Student Activity Sheet 4: Length of Daylight • Old Ideas: Scatterplots, regression, intersection points, transformations • New Ideas: Sinusoidal function, sinusoidal regression, period, amplitude Explore:
V.B Student Activity Sheet 4: Length of Daylight
Opening the Lesson Using a globe, ask students to locate the following: Houston, Texas; Philadelphia, Pennsylvania; Winnipeg, Manitoba, Canada; and Porto Alegre, Brazil. Ask students to identify the locations in terms of latitude and longitude. Have students describe the seasons in the first three cities based on where they are on the globe, mentioning that in the next activity, they will look at seasons in the fourth city . Framing Questions • Which city would you expect to be the warmest during the summer? have the longest days during the summer? be the coldest during the winter? have the longest days during the winter? Note Consider having students work Part A [Houston] as a class, and jigsaw Parts B and C [Philadelphia and Winnipeg, respectively], perhaps also including Quito [0o latitude] or other cities from Extension Question 8 at the end of Student Activity Sheet 4. For the jigsaw, students work in groups on different cities and present them to the class on transparencies laid on top of one another to note any patterns. Part A Question 1: Whole-class discussion. 1. After students have discussed the Framing questions, distribute Student Activity Sheet 4. 2. Introduce the activity by having a student read the introductory paragraphs or by summarizing the information. 3. Pose the Introductory question to students and allow them a minute or two to record their thoughts on the activity sheet. Allow some time for a short whole-‐class discussion. The accuracy of students’ responses at this point is not as important as their reasoning. Affirm all reasonable responses, knowing that students will have the opportunity to revisit their predictions after analyzing some data. 4. This activity is broken into four parts. The first three parts analyze and compare data for cities at three different latitudes in the Northern Hemisphere. The last part connects the sinusoidal regression models to the general form of a sinusoidal function and explains the different parameters in terms of the context of the problem.
5. Ask students to work through all of Part A, pausing at the end for a short whole-‐class debriefing. 6. In Part A, students generate graphs and analyze data for Houston, a city near 30°N latitude. 7. Students will use the blank graph for Length of Daylight for Cities. The axes are already labeled, so students need to plot only the points for the ordered pairs in the table. 8. Students plot the length of daylight (in minutes) according to the day number. The date and length of daylight in hours and minutes are provided for reference points. 9. It is important that students make a paper-‐and-‐pencil scatterplot for this activity, since they will see important points (such as maxima, minima, and intersections) more easily on the paper graph than on the calculator screen or other projection device. Question 2: Students work in pairs or small groups. 1. Students use their graphing calculators to make a scatterplot similar to the one they drew on paper in Question 1. The graphing calculator is necessary so that students can more easily develop a regression model describing the relationship between the length of daylight and the day of the year. 2. The sinusoidal regression model calculates a function that can be used to predict the length of daylight for any given day of the year. The data provided are for 2009 and include rounding to the nearest minute. Because of rounding and other astronomical nuances in the data, the regression model will not exactly match the data in the table. Therefore, answers in subsequent questions will be close to key values and dates but may not be exactly the same. 3. The calculator usually returns regression models in the form y = Asin(Bx – C) + D. Typically, precalculus and calculus textbooks present the general form of a sinusoidal equation to be y = Asin[B(x – C)] + D. The differences in the two forms are subtle but important. In both forms, B represents the angular frequency of the sinusoidal function. The difference lies in what C represents in each form. In the first form (generated by the calculator), C is the product of the angular frequency and the horizontal shift of the parent function, y = sinx. In the second form, B is factored so that C represents only the horizontal shift of the parent function. You may need to assist students with factoring B from the calculator-‐generated regression equation if choosing to bring to the class a discussion of the horizontal shift. This discussion can be modified or omitted, depending on the needs and previous experience of your students. Question 3: Students work in pairs or small groups. 1. For this question, students are being asked to generate a sinusoidal regression model from the data they entered into their calculator. There are many ways to do this for different styles of graphing calculator. 2. Be sure that students record their regression model in the Summary Table so that they are better able to make comparisons in subsequent parts of the activity. Rounding to the nearest hundredth is sufficient for representing these comparisons in the table. However, for making predictions and calculating particular values (such as maximum or minimum values), students should use the exact values generated by their calculator. Some calculator models allow for pasting in the exact values. See the AMDM calculator supplements posted on the Moodle sites for support from different calculator manufacturers (http://amdmsupport.org). 3. Some graphing calculators include values for r, a correlation coefficient, when computing a regression equation. It is important to remind students that r describes only the strength of a linear fit. For nonlinear regression models, the interpretations are not as clean as with linear models. For the purposes of this course, consider avoiding the use of r as a correlation coefficient to evaluate the strength of nonlinear models. Questions 4 and 5: Students work in pairs or small groups. 1. The purpose of Question 4 is to verify that the regression model describes the data. Using graphs is one way to do so, and students will use the graph in subsequent questions to identify key features of
the relationship between length of daylight and day of year. 2. Another way to verify the fit of a regression model is to use the function table feature, if possible, of the graphing calculator. Compare the values of the dependent variable that are calculated by the regression model to the actual data in the table. 3. For Question 5, students sketch their regression model onto their paper-‐and-‐pencil scatterplot. It may be helpful to color code the graphs using colored pencils. Question 6: Students work in pairs or small groups. Whole-class discussion. 1. Astronomical Check: Generally, the day with the greatest length of daylight is the summer solstice, and the day with the least length of daylight is the winter solstice. For 2009, the summer solstice is June 20 and the winter solstice is December 21. These dates vary slightly from year to year due to the fact that Earth’s orbit around the sun actually takes 365.25 days. 2. At the end of Part A, pause to be sure that students understand the following: • how to interpret the scatterplot, and • what the maximum and minimum values of the length of daylight represent. Part B Questions 1–4: Students work in pairs or small groups. 1. In Part B, students repeat the process that they used in Part A to analyze the data for Philadelphia. As with Part A, students can proceed through the activity in pairs or small groups or you can lead the class through the activity step by step while students work in pairs or small groups. Either way, pause at the end of Part B for a short debriefing before moving on to Part C. 2. Using the same graph, have students repeat their procedure for creating a scatterplot of the length of daylight by day of the year for Philadelphia, a city at 40°N latitude. It is helpful to use a second colored pencil for the scatterplot. Question 5: Students work in pairs or small groups. 1. Have students pause to note the similarities and differences in the regression models they recorded on the Summary Table. At this point, they may only notice that parameter A is different for the two models, but the other three parameters are very close to each other. These similarities and differences are explored in greater detail in Part D. Questions 6 and 7: Students work in pairs or small groups. 1. Because of rounding errors and other differences in the data set, the day for Philadelphia’s minimum length of daylight is different than the day for Houston’s minimum length of daylight, and neither of these is the winter solstice. Question 6 is important because it sets the stage for students to compare Philadelphia and Houston in Question 7. Reflection Question 8: Whole-class discussion. 1. Students revisit their predictions from the Introductory question at the beginning of Student Activity Sheet 4. Students whose predictions are affirmed should be able to explain why their predictions are correct. Students whose predictions are refuted should be able to explain why their predictions are incorrect. 2. Pause after the Reflection question to establish a common understanding that the maximum length of daylight for Philadelphia is greater than the maximum length of daylight for Houston. Questions 9 and 10: Students work in pairs or small groups. 1. Technology: Students can use their graphing calculator to find the coordinates of the two intersection points of the regression models for Houston and Philadelphia. Depending on their prior
experiences, students may struggle with isolating each intersection point. 2. For Question 10, students should recognize that each intersection point represents a day when Houston and Philadelphia have the same length of daylight. Astronomically, those days are the spring and vernal (autumnal) equinoxes. Reflection Question 11: Students work in pairs or individually. 1. Allow students to generalize when Houston or Philadelphia has the greater or lesser length of daylight. 2. Have students use their paper-‐and-‐pencil graphs to more easily see which graph has greater y-‐ values for which time interval. Use the intersection points to create time intervals. Be sure students notice that after Day 365, the cycle repeats itself for the next year. Questions 12 and 13: Students work in pairs or individually. 1. Data for the approximate latitudes of Houston, Philadelphia, and Winnipeg are found at the beginning of Student Activity Sheet 4. Question 14: Whole-class discussion. 1. This question provides a transition from Part B to Part C, when students repeat their modeling process for data describing the length of daylight by day for Winnipeg. 2. Pause for a short debriefing to be sure that students have common understanding about the following: • What happens to the length of daylight during the summer for cities that are farther north? (The length of daylight increases.) • What happens to the length of daylight during the winter for cities that are farther north? (The length of daylight decreases.) • Why do you think this is so? (Because of the tilt of the Earth’s axis, cities closer to the equator have lengths of daylight that vary less throughout the year. For cities that are closer to the North Pole, there is more variation between the minimum and maximum lengths of daylight.) Part C Questions 1–4: Students work in pairs or small groups. 1. In Part C, students add a scatterplot and regression model for Winnipeg to their collection. As with Parts A and B, students can proceed through the activity or you can lead the class through the activity step by step. Pause at the end of Part C for a short debriefing before moving on to Part D. 2. Using the same graph, have students repeat the procedure for creating a scatterplot of length of daylight by day of year for Winnipeg, a city at 50°N latitude. It is helpful to use a third colored pencil for the scatterplot. Question 5: Students work in pairs or individually. 1. Have students pause to note the similarities and differences in the three regression models they recorded on the Summary Table. As with the previous models, three of the four parameters are very close in value, but the fourth parameter (A) increases for cities that are located farther north. 2. With three graphs on the same grid, students may notice that all three graphs intersect at or near the same points and that the graph for Winnipeg has the highest peak (maximum value) in the middle and the lowest dips (minimum value) at the tails (or ends of the graph). Question 6: Students work in pairs or individually. 1. Because of rounding errors and other differences in the data set, the day for Winnipeg’s minimum length of daylight is different than the day for Houston’s and Philadelphia’s minimum length of daylight, and none of these is the winter solstice. Question 6 is important because it sets the stage for
students to compare the three cities in Question 7. Question 7: Students work in pairs or individually. 1. Technically, the three graphs do not all intersect at the same points; they intersect near one another. On the paper-‐and-‐pencil graph, students see that the three graphs intersect near one another, and it is close enough for them to decide that it is at the same two points. For the purposes of this activity, that is enough. Those two points correspond to Day 77 (March 18) and Day 267 (September 24), which are very close to the spring equinox and vernal (autumnal) equinox, respectively. Questions 8 and 9: Students work in pairs or small groups. 1. Give students the opportunity to respond to Questions 8 and 9. 2. Once students have recorded their thoughts, facilitate a whole-‐ class discussion of the two questions. 3. Question 8 asks students to make predictions about the length of daylight in summer and winter in Seward, Alaska, which is 10° latitude farther north than Winnipeg. This exercise helps prepare students to generalize the relationship between a city’s latitude and patterns in the length of daylight in Question 9. Whole-class discussion. 4. After a discussion of Question 9, students should understand that if a city is farther from the equator, the maximum and minimum values for the length of daylight in a year are more extreme than the values for cities that are closer to the equator. In other words, the maximum length of daylight is longer for cities farther from the equator (even though the maximum occurs on the same day for all cities, June 21), and the minimum length of daylight is shorter for cities farther from the equator (even though the minimum occurs near the same day for all cities, December 21). Reflection Question 10: Students work in pairs or small groups. Whole-class discussion. 1. Students are asked to relate this scenario and the resulting data and graphical representations to the Singapore Flyer application that they worked with in Unit IV, “Using Recursion in Models and Decision Making.” 2. Facilitate a class discussion of their responses, as time allows. This comparison of different scenarios is relevant not just because one deals with heights and the other lengths of daylight. Each should better support this new type of data and application in which sinusoidal regression can be applied as readily as linear and other nonlinear regression tools can be. Part D Questions 1 and 2: Whole-class discussion. 1. Part D provides students an opportunity to make connections among their regression models, the parent function for a sinusoidal curve, and the situation of the problem. Depending on the experience level of students and the amount of time spent on Parts A–C, Part D could become an optional extension activity. If you chose not to address the factored form earlier, Part D could be a place to have a discussion about the factored form of the function. 2. Questions 1 and 2 connect the amplitude of the regression models to the situation. The amplitude (A) is the only parameter that changed among the three models. 3. Have students answer Question 1. Be sure that they connect the quotient of 113.5 to the amplitude of the regression model. 4. Have students answer Question 2. Give them some time to do the computations with a calculator and compare their results to the regression models on the Summary Table. Questions 3 and 4:
Students work in pairs or individually. 1. Questions 3 and 4 connect the frequency and period of the regression models to the situation. The period for each model is 365, the number of days in one year. The frequency is 2π divided by 365. 2. Have students answer Question 3. They should notice that the result, rounded to the nearest hundredth, is the same as the frequency for all three cities. 3. Have students answer Question 4. Be sure that they connect the period of 365 days to the length of a year, which is the time it takes Earth to complete one orbit around the sun. Question 5: Students work in pairs or individually. 1. Question 5 connects the regression models to a horizontal shift of the sine parent function. 2. Have students answer Question 5. The horizontal translation by a factor related to Day 77 (the spring equinox) occurs because the sine parent function begins at (0, 0). To use a sine function to model the length of daylight, the starting point of the sine function needs to be shifted 77 units to the right. Question 6: Students work in pairs or individually. 1. Question 6 connects the vertical translation of the parent function to the situation. 2. Have students answer Question 6. The average length of daylight is 12 hours, and the maximum and minimum values vary, according to the amplitude, in an up-‐and-‐down cycle from this number. Question 7: Students work in pairs or individually. 1. Question 7 helps answer why three of the four parameters do not change from city to city whereas the fourth parameter, the amplitude, does. 2. Have students answer Question 7. The values of B (frequency), C (horizontal translation), and D (vertical translation) all include information that is true for Earth in general—number of days in a year, defining a beginning point as the spring equinox, and the number of minutes in 12 hours—and that does not depend on a specific location. 3. The fourth parameter, A (amplitude), depends on the city’s latitude. The farther a city is from the equator (the closer it is to the pole), the greater the amplitude, because the difference between the maximum and minimum lengths of daylight is greater as you move toward the pole. Extension Question 8: Students work in pairs or individually. 1. Have students investigate different locations. Have a gallery walk so that each group can share its findings.