Understanding Periodic Relations
In this activity students will create models of real-world objects that generate periodic functions. Students will determine the graphs of periodic functions and explore some of the properties of this type of relationship.
Materials Needed:
Push Pins, Fun Foam, Card Board or Construction Paper, empty toilet paper rolls, scissors, rulers, compasses, protractors copied onto acetate (see appendix)
Activities:
A Trip to the Fair
This activity is designed to introduce students to periodic relationships. They will build a model of a Ferris wheel using a 1 cm: 2 m ratio. Students will use either fun foam, card board or construction paper to cut out the circle for the Ferris Wheel. They should measure carefully to ensure that the wheel is the right size. Students should use a push pin for the axle, locating the center of the circle and inserting the push pin, then inserting the push pin into the empty toilet paper holder at the appropriate height. Again attention to detail in measurement is important.
To complete question 2, students will have to recognize that if the diameter is 24m, then the radius will be 12m. Given that the Axle is 14m high, the maximum height of the Ferris wheel is 26m and the minimum height is 2m.
For question 3, one full turn takes 36 seconds and results in a 360 angle of rotation. A half turn takes 18 seconds and is 180, and a quarter turn takes 9 seconds and is a 90angle.
For question 4, students should start measuring from the bottom when the rider gets on the Ferris Wheel so a 270 turn makes you the same height as the axle so you are 14m above ground.
In question 5, the acetate protractors can be helpful as they can easily be affixed to the circle with the pushpin. Students must be reminded that the height is to be reported in m so they will have to remember the scale they used. in building the model. In question 6, students should make observations about the repeating nature of the heights and the fact that these values are bound between 2 and 26. In question 7 students should use the data from the table to draw a graph. Question8 asks students to notice the connection between the patterns in the table and the patterns in the graph. It would be helpful to discuss the domain and range of this graph at this point. The data will generate a cosine curve. This data can easily be switched to generate a sine curve by changing the starting point - you would start at the same height as the axle on the way up. This would be an interesting extension for early finishers.
A Trip to the Fair
A very popular ride at any fair is always the Ferris Wheel. The first Ferris Wheel was built by George Ferris for the 1893 World’s Fair in Chicago. Since that time Ferris Wheels have exploded in popularity and now take on many different sizes and formations. In this activity you will build your own Ferris Wheel and use it to solve problems about periodic relations.
1. You and your friends go to the Fair where you see a Ferris Wheel and decide to take a ride. While you are waiting in line you read some information about the Ferris Wheel you are about to ride. Using a scale of 1 cm: 2 m use the materials you have been given to build a model of this Ferris Wheel. Use your model to help you answer the questions below.
2. a) What is the maximum height of the Ferris Wheel?______
b) What is the minimum height of the Ferris Wheel? ______
c) What is the height of the center of rotation?______
3. a) How long does it take for the Ferris wheel to complete one full turn? ______
What is the corresponding angle of rotation for this turn? ______
b) How long does it take for the Ferris wheel to complete one half turn? ______
What is the corresponding angle of rotation for this turn? ______
c) How long does it take for the Ferris wheel to complete one quarter turn? ______
What is the corresponding angle of rotation for this turn? ______
4. Suppose you get on the Ferris Wheel and make a 270º turn, at what height above ground will you be? Explain how you know.
5. After everyone gets on the ride and you have made one full rotation you notice that you are at the bottom again on your way up to the top. If you call this 0 of rotation and time 0 seconds, complete the chart below. Use your model.
Angle of Rotation ( º ) / Time Elapsed (seconds) / Height Above Ground (m)0 / 0
45
90
135
180
225
270
315
360
405
450
495
540
585
630
675
720
6. Explain any patterns you notice in the chart above.
7. Plot a graph of height vs. time for the Ferris Wheel from the data you collected in the table below.
8. Explain any patterns that show up in the graph. Do they connect to the patterns you observed in the table? Explain any connections you observe.
9. From examining the graph, identify the following:
a) The amplitude of the graph is ______.
b) What does this represent?
c) The equation of the sinusoidal axis of the graph is ______.
d) What does this represent?
e) The period of the graph is ______.
f) What does this represent?
g) The equation of the frequency of the graph is ______.
h) What does this represent?
10. Use the information above and the graph to determine an equation for the function. Challenge yourself by trying to find at least one cos equation and one sin equation.
Appendix