Exploration of Periodic Phenomena
DATE: 2/14/11 CLASS PERIOD: Precalculus UNIT: U.C. 2-5a .
LESSON OBJECTIVES:
At the end of the lesson, students will understand how certain natural phenomena are modeled by sinusoidal waves. The presentations the next day will encourage deep understanding and allow the entire class to learn about more than just one phenomenon. This lesson corresponds with Common Core State Standards-Trigonometric Functions F-TF 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Schedule / ActivitiesWarmup (10’) / Hand students a warmup sheet that says:
Describe three things in the real world that can be modeled by a sine wave or other periodic function.
Ask for responses: write them on the board. Then have them choose which they’d like to study and form groups. Topics for study include: spring oscillations; ripples on a pond; circular motion; pendulum motion; tides; daylight variation over the year; sound waves produced by musical instrument.
Ask them how they might explore these topics and record the data.
Activity (30’) / When discussion is done, ask students to select one topic to explore and find others who wish to explore the same topic. Hopefully there will be four evenly distributed groups. Provide each group with the materials they need and instructions on how to graph their data. Ask them to create an equation that models their data and to justify their mathematical decisions. Presentations will be tomorrow.
Ending (5’) / Great job! I can’t wait for tomorrow’s presentations. Please put everything away neatly.
Project / Materials
Daylight Data Tables / Instructions; Tables from http://aa.usno.navy.mil/data/docs/RS_OneYear.php.
Tide Tables / Instructions; tables from http://www.saltwatertides.com/dynamic.dir/oregonsites.html.
Ripples / Instructions; plastic tub and marbles.
Pendulum Motion / Instructions (Vernier); graphing calculator; motion detector, string and weight; timepiece.
Sound Waves / Instructions (Vernier); graphing calculator; microphone and a musical instrument.
Spring Oscillations / Instructions (CPM); Slinky Jr.; weight; timepiece.
How to print out data tables.
Daylight Data Table
Can be computed at http://aa.usno.navy.mil/data/docs/RS_OneYear.php. Scroll down to Form A, enter year, Type of table (any type should be fine), State (Oregon) and City (Portland). Then click “Compute Table.” You should create your graph based on the first of every month, or twice a month; at most, once a week, as it would take too long to graph 365 days. You have two graphs: sunrise and sunset, or you could graph the number of hours of daylight if you calculate the time between sunrise and sunset.
Tide Tables
Tide tables can be found at http://www.saltwatertides.com/dynamic.dir/oregonsites.html. Select a location, month and day to begin, and the number of days. Select a several days so you have plenty of data. The max is 14 days. Then click “Get Tides.
Exploration of Periodic Phenomena – Hours of Daylight over 12 Months
Over the course of one year, the tilt in Earth’s rotational axis creates a variation in the number of hours of daylight. Can we model this variation with a periodic or sinusoidal function? If you consider one circuit around the sun to be a period, then perhaps we can. You have chosen to explore this question with the other members of your group.
Attached is a table that shows the time of sunrise and sunset for every day of the year 2011. You will use this table to create a graph, and the graph will (hopefully) resemble the sine and cosine curves we’ve been exploring in class. Be sure to be neat and accurate with your graph, as you will present and explain it to the rest of the class (probably on Wednesday when everyone is done).
You have a number of choices on how to use these data. How many people have chosen to work in this group and their preferences will likely help you decide how you answer these questions. Discuss them and reach a unanimous decision before moving on.
· How smooth do you want your graph to be? You have data for every day of the year. A graph with 365 data points will be extremely accurate and smooth, but would take the most time. I’d suggest a graph with 104 data points (twice a week) or 91 data points (every four days), but if your group is small you may do 52 data points (once a week). Anything less, like twice a month, will not show enough of the curve to be instructive. This will make it more difficult to determine the equation of your graph (your final step).
· What graph(s) do you wish to make? The data you have creates two graphs: one for sunrise and one for sunset. It would be natural to draw both graphs on the same axes. You may instead create one graph that shows the amount of daylight hours. Simply subtract sunrise from sunset and plot that.
· Where is the origin on your graph? The horizontal axis should be time (days) and the vertical axis should be hours or daylight. It makes sense to have January 1 be zero on the horizontal axis, but there is no clear choice for zero on the vertical axis. It could be midnight, or it could be the smallest (earliest) value in the data. Alternatively, you could use the times for the spring equinox as your zero, so that your graph ends up going below the horizontal like an unshifted sine wave.
Once you have reached consensus as a group on these questions and have a plan in place to complete the project, go ahead and get started. Feel free to be as elaborate and colorful as you want, but try to keep the graph simple so that when you present it to the class, it is clear to the other groups what you’ve done. Once you have the data graphed, don’t forget to complete the final step. Looking at the graph you’ve created, try to determine what the amplitude, period and shifts (if any) are. In other words, find the equation of the graph you made. Be prepared to answer why you chose the values you chose.
Exploration of Periodic Phenomena – Tidal Fluctuations
The gravitational pull of the moon, combined with the earth’s rotation, causes the oceans to ebb and flow at the coast. These, of course, are the tides. Since the tide comes in and goes out once or twice a day, this phenomenon is periodic and can be modeled by a sinusoidal wave like we have been studying. You have chosen to study this more closely, along with the members of your group.
Attached is a table that shows the next week’s high and low tides at Seaside, Oregon. (This table was downloaded from a website, so please ignore the ads.) Using the data on this table, your task is to create a graph that shows the cyclical (periodic) nature of the tides. The graph will (hopefully) resemble the sine and cosine curves we’ve been exploring in class. Be sure to be neat and accurate with your graph, as you will present and explain it to the rest of the class (probably on Wednesday when everyone is done).
You have a number of choices on how to use these data. How many people have chosen to work in this group and their preferences will likely help you decide how you answer these questions. Discuss them and reach a unanimous decision before moving on.
· Where is the origin on your graph? The horizontal axis should be time (marked in two- or four-hour increments) and the vertical axis should be depth of tide (in feet). A logical intersection would be the first data point you have and zero feet. However, if your group looks at the data and comes up with a better idea, feel free to use that.
· How will you connect the data points? Your data simply provides the minima and maxima of your function. You have no indication of any of the points in between. Therefore, you may simply connect the dots and end up with a jagged, sawtooth-type picture. You will probably understand that the tides do not move at a constant rate between high and low (which is what a jagged line graph would indicate), so you can try to make the graph smoother to match your understanding of the tides.
Once you have reached consensus as a group on these questions and have a plan in place to complete the project, go ahead and get started. Feel free to be as elaborate and colorful as you want, but try to keep the graph simple so that when you present it to the class, it is clear to the other groups what you’ve done. Once you have the data graphed, don’t forget to complete the final step. Looking at the graph you’ve created, try to determine what the amplitude, period and shifts (if any) are. In other words, find the equation of the graph you made. Be prepared to answer why you chose the values you chose.
Exploration of Periodic Phenomena – Ripples in a Pond
When you toss a stone into a pond or lake, the disturbance created by the stone makes waves on the surface of the water. The ripples that move across the pond can be modeled by a sinusoidal wave like we have been studying. You have chosen to study this more closely, along with the members of your group.
You have been supplied with a tub of water with markings on the side. Using this tub, your task is to drop marbles into the water and measure how high and low the ripples fluctuate on the side of the tub. You will need to collect the data and create a graph to present to the class.
You have a number of choices on how to collect and use your data. How many people have chosen to work in this group and their preferences will likely help you decide how you answer these questions. Discuss them and reach a unanimous decision before moving on.
For data collection, each member of the group should have a specific role. One person will need to drop the marbles; one person will need to note the fluctuation of the ripples. If you have enough people, one person could note the high water mark and another could note the low point. A fourth person could record the data as it is provided. If your group has more than four people, the optional extension exercise should be included in your observations.
I would suggest you use only one size of marble so your data is consistent. Also, drop the marble from the same height every time; remove the marble from the tub after the waves have died down; and always aim for the middle of the tub on every experiment. Your group will also need to discuss how many trials to record data on (I’d suggest 15 or 25) and how to find an average set of values to graph. Don’t forget to discuss where to place the origin on your graph.
Once you have reached consensus as a group on these questions and have a plan in place to complete the project, go ahead and get started. Feel free to be as elaborate and colorful as you want, but try to keep the graph simple so that when you present it to the class, it is clear to the other groups what you’ve done. Once you have the data graphed, don’t forget to complete the final step. Looking at the graph you’ve created, try to determine what the amplitude, period and shifts (if any) are. In other words, find the equation of the graph you made. Be prepared to answer why you chose the values you chose.
Optional Extension: Place a small piece of paper in the tub to simulate a fallen leaf or small, water-skimming insect. Note how long it takes (in seconds or number of waves that pass under it) for the paper to reach the side of the tub. Experiment with different locations: close to the center, close to the edge, and somewhere in between.