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Advanced Functions and Modeling Applications of Power Functions Below is a collection of four data gathering activities for which an appropriate model is a power function. They include the period of a pendulum, area under the curve y x 2 , the number of balls that fit in a trough, and the intensity of light which is a CBL activity. These activities can be used in several ways. They are written so that each data collection is a stand-alone activity with a series of questions exploring the meaning of the model. In that format, each of the four activities can be used individually as teaching tools when studying power functions. As students develop and use these models it is important that they understand and can interpret the constants in the context of the scenario. These activities can also be used as a culminating experience after students have studied power functions. As such, the teacher might allow different groups of students to perform an experiment and then analyze the data. Students could prepare a written report of their analysis, rather than merely answering questions. This report could be used for assessing students’ understanding of the unit. A sample set of student directions for a project report is also provided. These directions should be modified to reflect the individual teacher’s expectations for presentation, word processing, graphs, etc. AFM DATA PROJECT Due Date: You will prepare a report in which you analyze the data you collected and your group. With your partner(s), prepare a short paper discussing how you gathered the data, the equation of the model you develop. Also examine the residuals and use them in your discussion of the quality of your model. The report must be word-processed and should “hang together”. It should not read as a list. Graphs should be imported into your document from the calculator. The report should include the following items. An introductory paragraph describing in your own words how the data were collected. A table of the data you collected should be provided. Be sure to include information about units of measurement. Use your calculator to find the equation of an appropriate model. Tell me why you have chosen this model. Include a graph that shows the curve superimposed on the scatterplot of the data. Be sure to indicate the scale. (hand written labels are ok) Provide a residual plot. Based on information obtained from the residual plot, discuss whether or not your model is appropriate. Interpret the constants in your model. For example, suppose you found a model for data that was about the area of circles and the model equation was A 3.24 x 2.1 . You might explain that the equation of the area of a circle is given by A r 2 . Therefore, since 3.24 is close to , it is a reasonable coefficient. We expect the power of the independent variable to be 2 since we are looking at area. The value 2.1 is also close, so our model closely resembles the theoretical model for the area of a circle. Answer any questions that are part of the problem statement. Be sure to write in complete sentences and proofread your report, especially the math. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Pendulum Activity Teacher notes Materials Needed to Collect Data: Pendulum Meter stick or tape measure Stopwatch A pendulum can be made by attaching a weight, such as several metal washers, to a string. If you swing a long pendulum, it takes more time to complete one swing than if you swing a short pendulum. This experiment allows you to investigate the relationship between the length of a pendulum and the length of time it takes to complete one swing. The ordered pairs will be (length of pendulum, length of time to complete one swing). Scientists call this relationship Hooke’s Law. Data Collection Instructions: Vary the length of the string by about 5 or 10 centimeters from one trial to the next, and measure how the period of the pendulum (time to complete one swing across and back) changes. To measure the period, students should hold one end of the string stable, pull the weight slightly (about 10 to 20 ) to one side, let the weight make 10 complete swings, record the time, and then divide the time by 10. Collect 8 to10 data points. Be sure to include some long lengths as well as short lengths. It is particularly important to collect data for small values of the length of the pendulum so you might want to collect times for lengths of the string such as 5 cm, 10 cm, 15 cm, 20 cm, and then 30 cm, 40 cm, 50 cm, etc. A figure of the set up is shown. NOTE: The length of the pendulum should be measured to the center of mass. Students generally find it easier to measure from the knot that is formed where the string is tied to the washer. After those values are gathered it is an easy matter to measure the length from the knot to the center of the washers and add that value to each length. Analysis: 1) Make a scatter plot of (length of string, length of time to complete one swing). The length of time of one complete swing is called a period. Label the axes with the variables and a scale. Sample data are provided in the table below: AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Length in cm 7 12 17 22 32 42 52 62 Time in seconds .5 .714 .852 .962 1.162 1.342 1.5 1.614 A calculator-generated scatter plot is provided. 2) Examine the scatter plot. What type of function might be a good model for your data? Since the data are increasing and concave down, it would be reasonable for the student to conjecture the square root function as a model. Some students might propose a logarithmic function as a model. However, the log function has a vertical asymptote and we would doubt that is a good choice since nothing in the phenomena argues that the values will approach negative infinity for small lengths of the pendulum. As a matter of fact, we expect the ordered pair (0, 0) to be in the set. 3) Find a function to model the data. Using the power regression option on the calculator, we obtain T 0.186 L.528 as a function model. Looking at the scatter plot of the data and the function on the same axes, this seems to be a good model. 4) The theoretical relationship between the length of the string and the period of the pendulum is 2 T L , where T represents the period, L represents the length of the string, and g is a g constant that is the acceleration due to gravity, which is 980 cm/sec2. How does your function compare with the theoretical model? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Our function model compares well with the theoretical model. Our power is 0.528 which is quite close to 0.5 and the coefficient in the theoretical model is 0.201 compared with our coefficient of 0.186. Errors in timing the period could contribute to the differences. 5) If it takes 2 seconds to complete a swing of the pendulum, what is the length of the pendulum? Show work to support your answer. We need to solve 2 0.186L0.528 10.753 L0.528 1 (10.573) 0.528 L 89.87cm L 6) What is the period of a 72 cm pendulum, according to your model? T .0186(72)0.528 1.78 sec. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Pendulum Activity Student Handout Materials Needed to Collect Data: Pendulum Meter stick or tape measure Stopwatch If you swing a long pendulum, it takes more time to complete one swing than if you swing a short pendulum. This experiment allows you to investigate the relationship between the length of a pendulum and the length of time it takes to complete one swing. The ordered pairs will be (length of pendulum, length of time to complete one swing). Data Collection Instructions: Vary the length of the string by about 5 to 10 centimeters from one trial to the next, and measure how the period of the pendulum (time to complete one swing across and back) changes. To measure the period, students should hold one end of the string stable, pull the weight slightly (about 10 20 ) to one side, let the weight make 10 complete swings, record the time, and then divide the time by 10. Collect 8 to 10 data points. Be sure to include some long lengths as well as short lengths. It is particularly important to collect data for small values of the length of the pendulum so you might want to collect times for lengths of the string such as 5 cm, 10 cm, 15 cm, 20 cm, 30 cm then 40 cm, 50 cm, etc. NOTE: The length of the pendulum should be measured to the center of mass. Generally it is easier to measure from the knot that is formed where the string is tied to the washer. After those values are gathered, you can measure the length from the knot to the center of the washers and add that value to each length. Analysis: 1) Make a scatter plot of (length of string, length of time to complete one swing). The length of time of one complete swing is called a period. Label the axes with the variables and a scale. 2) Examine the scatter plot. What type of function might be a good model for your data? 3) Find a function to model the data. 4) The theoretical relationship between the length of the string and the period of the pendulum is 2 T L , where T represents the period, L represents the length of the string, and g is a g constant that is the acceleration due to gravity, which is 980 cm/sec2. How does your function compare with the theoretical model? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program 5) If it takes 2 seconds to complete a swing of the pendulum, what is the length of the pendulum? Show work to support your answer. 6) What is the period of a 72 cm pendulum, according to your model? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Area Under a Curve Teacher Notes Materials Needed Graph paper A carefully drawn graph of y x 2 Graphing calculator We know how to find area of a number of geometric shapes such as rectangles, circles, and triangles. Areas of some shapes, however, are less convenient. Is there a formula for finding these areas? We will explore a formula for one such unusual shape. Data Collection We want to know the area of the region bounded by the x-axis, the parabola y x 2 , and the vertical line x a for varying values of a. (See figures below for some sample regions. You should label both axes in units of 1. The vertical axis in the figures below are labeled in units of three so that the figures would fit on the page reasonably.) You will find area estimates by counting boxes in the graph paper. The ordered pairs you will gather are (x-value, area of the region under the curve and above the x-axis). For example, we know the pair (0, 0) should be in the set. We might estimate the ordered pair (1, 0.4) as another value. Repeat this process for xvalues from 2 through 7. Regions under the curve y x 2 for x = 2, 4, and 6 Teacher Note: As x-values increase, it becomes more and more challenging to estimate the area under the curve. You may prefer to estimate the area under y x . If so, the theoretical model 2 will be y x3/ 2 . 3 AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Analysis 1. Make a scatter plot of the data. What type of function do you think might model this data? A sample data set and a graph of the data are shown below. Students may have trouble guessing a function model. They might, however, guess a power function. x-value 0 1 2 3 4 5 6 7 Area 0 0.4 2.7 8.9 21 40.1 69.4 110.2 2. Use your calculator to generate a function model for this data set. (Note: you might need to delete the pair (0, 0), depending on the function-type that you choose.) Did you anticipate the correct function-type in part 1? The calculator-generated function model for our sample data is y 0.381x 2.9 . This is a power function. 3. Does your model fit the data well? Use a residual plot for your function and the data to make this decision. Examining a graph of the function and the data there appears to be a good fit. Examining the residual plot there seems to be a pattern. (See figures below.) However, all of the residuals are small compared with the y-values in the data. Furthermore, as x got bigger, it became increasingly more difficult to estimate the value of the area, thus contributing to the greater absolute error for x-values of 6 and 7. The percent error for each of these values, however, is small. 4. Using your model, for what value of x is the area 200? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Solving 200 0.381x 2.9 we have 524.93 x 2.9 (524.93) 8.67 x 1 2.9 x 5. Theoretically, the area of the region bounded by the x-axis, the line x a , and the curve 1 y x 2 is A x 3 . (We learn this in calculus.) How does your function compare with the 3 theoretical model? The calculator-generated function compares well with the theoretical model. Our coefficient is 0.381 compared with 1/3 and our exponent is 2.9 rather than 3. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Area Under a Curve Student Handout Materials Needed Graph paper A carefully drawn graph of y x 2 We know how to find the area of a number of geometric shapes such as rectangles, circles, and triangles. Some shapes, however, are less convenient. Is there a formula for finding these areas? We will explore a formula for one such unusual shape. Data Collection We want to know the area of the region bounded by the x-axis, the parabola y x 2 , and the vertical line x a for varying values of a. (See figure.) You will find some area estimates by counting boxes in the graph paper. The ordered pairs you will gather are (x-value, area of the region under the curve and above the x-axis). For example, we know the pair (0, 0) should be in the set. We might estimate the ordered pair (1, 0.4) as another value. Repeat this process for xvalues from 2 through 7. Analysis 1.) Make a scatter plot of the data. What function do you think might model this data? 2.) Use your calculator to generate a function model for this data set. (Note; you might need to delete the pair (0, 0), depending on the function-type that you choose.) Did you anticipate the correct function-type in part 1? 3.) Does your model fit the data well? Use a residual plot for your function and the data to make this decision. 4.) Using your model, for what value of x is the area 200? 5.) Theoretically, the area of the region bounded by the x-axis, the line x a , and the curve 1 y x 2 is A x 3 . (We learn this in calculus.) How does your function compare with the 3 theoretical model? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Balls in a Trough Teacher handout Materials Needed A box lid Several sets of Styrofoam balls of varying diameters. (Actually, set of balls will do.) diameter you will need enough balls to fill the length of the lid. Tape measure Graphing calculator For each Is there a relationship between the number of balls you can fit into a box lid and the diameter of the ball? If the diameter is small you should be able to fit a greater number of balls than if the diameter is larger. Data Collection Determine the diameter of the ball. Fill the lid with as many same sized balls as possible. If it is not possible to completely fill the lid with balls, approximate the fractional part of the last ball that is required. For example, in the figure shown to the right, we would say approximately 6.5 balls fit in the trough. Therefore, if the diameter of the ball is 1.75 inches we would record the ordered pair (1.75, 6.5) in our list of the data. Repeat the process for each set of different sized balls. Analysis 1) Make a scatter plot of the ordered pairs (diameter of the ball, number of balls that fill the trough). Sample data are provided in the table below. Diameters were determined by using a tape measure to find the circumference and then dividing the result by pi. Diameter in inches 2.03 2.586 1.512 .876 1.452 2.466 .676 # of balls 6 4.5 7.75 13.25 8 4.75 17.75 A calculator-generated scatter plot is provided below. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program 2) Examine the scatter plot; find a function to model the data. The data appear to have the shape of a reciprocal function. This seems reasonable since as the diameter (x-values) gets smaller the number of balls increases whereas the number of balls decreases as the diameter increases. Since a reciprocal function is a power function we can use the power regression option on the calculator to obtain B 11.8D 1.004 . k 3) The theoretical model for this data is B where B is the number of balls in the trough, D D is the diameter of the ball, and k is a constant. How does your function compare with the theoretical model? What value of k makes sense with your data? (Think about the length of the box lid.) In this case, the length of the inside of the lid was 11.25 inches. The number of balls that should fit inside the trough made by the lid is 11.25 divided by the diameter. Therefore, the theoretical 11.25 model is B . Our constant of 11.8 compares well with 11.25 and our exponent of -1.004 D compares very well with the theoretical model whose exponent is -1. 4) What is a reasonable domain for the function in the context of the problem? Theoretically the domain in the context is D 0 , but practically, in the classroom, it is unlikely that we would use a ball with diameter greater than 12 inches so 0 D 12 might be more reasonable. 5) Use your model to predict the number of balls in the trough for a diameter not in your data set. For example if the diameter is 5 inches, then we would have B 11.8(51.004 ) 2.34 balls. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Balls in a Trough Student handout Materials Needed A box lid Several sets of Styrofoam balls of varying diameters. (Actually, any set of balls will do.) For each diameter you will need enough balls to fill the length of the lid. Tape measure Graphing calculator Is there a relationship between the number of balls you can fit into a box lid and the diameter of the ball? If the diameter is small you should be able to fit a greater number of balls than if the diameter is larger. Data Collection Determine the diameter of the ball. Fill the lid with as many same sized balls as possible. If it is not possible to completely fill the lid with balls, approximate the fractional part of the last ball that is required. For example, in the figure shown to the right, we would say approximately 6.5 balls fit in the trough. Therefore, if the diameter of the ball is 1.75 inches we would record the ordered pair (1.75, 6.5) in our list of the data. Repeat the process for each set of different sized balls. Analysis 1) Make a scatter plot of the ordered pairs (diameter of the ball, number of balls that fill the trough). 2) Examine the scatter plot; find a function to model the data. k where B is the number of balls in the trough, D D is the diameter of the ball, and k is a constant. How does your function compare with the theoretical model? What value of k makes sense with your data (Think about the length of the box lid.)? 3) The theoretical model for this data is B 4) What is a reasonable domain for the function in the context of the problem? 5) Use your model to predict the number of balls in the trough for a diameter not in your data set. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Light Intensity Lesson Teacher Notes Materials Needed to Collect Data: CBL2 DataMate App Light Probe Small desk lamp Meter stick There is a relationship between the intensity of a light and the distance from the light source. You will use the CBL2 to collect data of the form (distance from lamp in cm, light in mw/cm2 ) . Directions for the data collection activity are given below. You should collect 8-10 data points. CBL2 Light Data Collection Instructions. Instructions for collecting CBL2 data. Plug the light sensor into CH 1. APPS DataMate o The CBL2 will search for the probe that has been connected. If it does not find the light sensor, press 1: SETUP. With the cursor pointing at CH 1: press ENTER. You will then have the option of selecting 5: Light (on page 2). 1. SetUp o Scroll to MODE and press ENTER 3. events with entry 1. ok Make the room dark. You will need a small wattage light bulb, like a desk lamp, and a meter stick to collect the data. Place the meter stick near the base of the lamp and turn the lamp on. 2. start (The screen will show “Press [Enter] to Collect or [Sto] to Stop. It will also show the number of the data point you are about to collect, 1 data point to begin, and the intensity, which will vary depending on the light in your room. The intensity needs to be slightly less than 1 when you begin to collect data. You will need to have the light sensor several inches from the light source.) Once the light intensity has settled somewhat, press ENTER. It will then ask you to ENTER VALUE? You should enter the distance the light sensor is from the lamp and press ENTER again. The data points will be plotted on the screen as you collect them. Move the light sensor slightly away from the lamp. Wait for the light intensity to settle again and press ENTER. Again, it will ask you to ENTER VALUE? Enter the distance the light sensor is from the lamp and press ENTER again. Continue this process until you have 8-10 data points. Once you have collected enough data, press STO to stop. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program When you are ready to leave the DataMate APP (there should be a scatter plot on your screen) press ENTER 6. quit press ENTER The data are stored in L1 (distance) and L2 (light intensity). *The DataMate App is available free of charge at the Texas Instruments web site. The address is: http://education.ti.com/us/product/apps/83p/datamate.html. If you don’t have the time or equipment to have students collect their own data, a sample set is provided below. It will be used as a guide to giving possible answers to the question on the student handout. distance (cm) 18 22 26 30 34 37 41 45 49 51 55 59 63 65 70 intensity (mw/cm2) 0.717255 0.526816 0.371356 0.285853 0.210066 0.181889 0.156626 0.133307 0.112903 0.104158 0.0905557 0.0759813 0.0730664 0.0643218 0.0565487 1. a. Before collecting the data, what do you expect the scatter plot to look like? Sketch a graph in the space below. Students should sketch a function that is decreasing. They may not know whether it is linear or curved. If they do believe it is curved, they may not know whether it should be concave up or concave down. Encourage them to label the axes with the variable names. b. Examine the scatter plot of the data. Sketch a graph of the data. Label variables on the axes, include a scale. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Here is calculator generated graph of the data using the sample data set above. Student graphs should resemble this one. 2. Describe the relationship between the distance from the lamp and the light intensity. What characteristics do the data have? Some of the descriptive words students may use are decreasing, concave up, strong, and asymptotic. 3. Based on your answers to questions 1 and 2, what type of function do you think would best fit your data? Students may be reminded of a reciprocal function. They may also think of exponential because of the shape. Encourage them to think about what would happen to the values of the light intensity beyond the values in the data set. 4. a. The theoretical relationship between the distance from the lamp and the light intensity is I A D 2 , where I represents light intensity, D represents distance, and A is a constant that depends on the light bulb. Use your graphing calculator to find the power function that models your data. f ( x) 169.27 1.88 x b. What is a reasonable domain for this function in the context of the problem? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Students should recognize that values less than zero are not possible. The maximum value of the domain is dependent on the equipment. 2 meters seems a reasonable distance to stop therefore a reasonable domain is 0 x 200. c. If your function isn’t very close to the theoretical model, what in your data collection could have caused any discrepancies? Errors in measuring distances. Too much light in the room. Not aligning the probe and the light properly. 5. a. Is your model a good fit? Use residuals to help answer this question. Sketch a graph of the residual plot. The residual plot implies the fit is a good one. The data points are scattered and the sizes of the residuals are small compared to the light intensity values in the data set. b. What is the residual value for the data point (34, 0.210066) ? Show work to support your answer. What does this value mean in the context of the problem? f (34) 169.27 B 0.2226 is the value predicted by the model. The residual value is 1.88 34 0.210066 0.2236 0.0135 . This means that the actual light intensity when the probe was centimeters from the light was approximately 0.0135 mw/cm2 less than the model would 34 predict. 6. According to your model, if the probe is 67 cm from the light source, what do you expect the light intensity to be? Show work to support your answer. f (67) 169.27 1.88 B 0.0625 If the probe is 67 cm from the light source, I would expect the light 67 intensity to be approximately 0.0625 mw/cm2. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program 7. If the light intensity is measured to be 0.25 mw/cm2, how far is the light source from the probe? Show work to support your answer. 169.27 0.25 1.88 x 1.88 0.25 x 1.88 x 169.27 169.27 0.25 169.27 x 0.25 1 1.88 B 32.04 If the light intensity is measured to be 0.25 mw/cm2, the light source is approximately 32.04 cm from the probe. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program Advanced Functions and Modeling Light Intensity Lesson Student Handout Materials Needed to Collect Data: CBL2 DataMate App Light Probe Small desk lamp Meter stick There is a relationship between the intensity of a light and the distance from the light source. You will use the CBL2 to collect data of the form (distance from lamp in cm, light intensity in mw/cm2 ) . Directions for the data collection activity are given below. You should collect 8-10 data points. CBL2 Light Data Collection Instructions. Instructions for collecting CBL2 data. Plug the light sensor into CH 1. APPS DataMate o The CBL2 will search for the probe that has been connected. If it does not find the light sensor, press 1: SETUP. With the cursor pointing at CH 1: press ENTER. You will then have the option of selecting 5: Light (on page 2). 1. SetUp o Scroll to MODE and press ENTER 3. events with entry 1. ok Make the room dark. You will need a small wattage light bulb, like a desk lamp, and a meter stick to collect the data. Place the meter stick near the base of the lamp and turn the lamp on. 2. start (The screen will show “Press [Enter] to Collect or [Sto] to Stop. It will also show the number of the data point you are about to collect, 1 data point to begin, and the intensity, which will vary depending on the light in your room. The intensity needs to be slightly less than 1 when you begin to collect data. You will need to have the light sensor several inches from the light source.) Once the light intensity has settled somewhat, press ENTER. It will then ask you to ENTER VALUE? You should enter the distance the light sensor is from the lamp and press ENTER again. The data points will be plotted on the screen as you collect them. Move the light sensor slightly away from the lamp. Wait for the light intensity to settle again and press ENTER. Again, it will ask you to ENTER VALUE? Enter the distance the light sensor is from the lamp and press ENTER again. Continue this process until you have 8-10 data points. Once you have collected enough data, press STO to stop. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program When you are ready to leave the DataMate APP (there should be a scatter plot on your screen) press ENTER 6. quit press ENTER The data are stored in L1 (distance) and L2 (light intensity). *The DataMate App is available free of charge at the Texas Instruments web site. The address is: http://education.ti.com/us/product/apps/83p/datamate.html. 1. a. Before collecting the data, what do you expect the scatter plot to look like? Sketch a graph in the space below. b. Examine the scatter plot of the data. Sketch a graph of the data. Label variables on the axes, include a scale. 2. Describe the relationship between the distance from the lamp and the light intensity. What characteristics do the data have? 3. Based on your answers to questions 1 and 2, what type of function do you think would best fit your data? 4. a. The theoretical relationship between the distance from the lamp and the light intensity is I A D 2 , where I represents light intensity, D represents distance, and A is a constant that depends on the light bulb. Use your graphing calculator to find the power function that models your data. b. What is a reasonable domain for this function in the context of the problem? c. If your function isn’t very close to the theoretical model, what in your data collection could have caused any discrepancies? AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program 5. a. Is your model a good fit? Use residuals to help answer this question. Sketch a graph of the residual plot. b. What is the residual value for the data point (34, 0.210066) ? Show work to support your answer. What does this value mean in the context of the problem? 6. According to your model, if the probe is 67 cm from the light source, what do you expect the light intensity to be? Show work to support your answer. 7. If the light intensity is measured to be 0.25 mw/cm2, how far is the light source from the probe? Show work to support your answer. AFM Power Functions NCSSM 2004-05 DPI Educator-On-Loan Program