Lesson 1: Irrational numbers: Application to natural frequency and resonance

Project Unit Title: Moving and Pumping Through Fluids at Different Sizes

Unit Theme:

Fluid dynamics is a complex discipline that describes the flow of fluid in and around objects. Students will get a glimpse of fluid dynamics through three lessons related to real and irrational numbers, cubic functions, density and viscosity, and scale models. Most importantly, students will experience how mathematics can be used to model natural phenomena.

Lesson Theme:

The study of natural frequency and resonance and its effect on natural vegetation and trees, agriculture, and man-made constructions is an important and practical application of the fluid dynamics discipline. Natural frequency is the frequency at which an object vibrates once it has been set in motion. Resonance is the tendency of an object to oscillate at its maximum amplitude at certain frequencies. Resonance also has applications in acoustics.

Title:

Irrational Numbers: Application to natural frequency and resonance

Introduction/Theme:

After a facilitated discussion about the effects of natural phenomena such as wind on an object’s natural frequency, pairs or groups of students will conduct an experiment to model resonance in plant stems and an activity to measure the natural frequency of pine trees. Through these activities, the students will gain an understanding of frequency, natural frequency, resonance, oscillations and square roots.

Learning Outcomes:

The student will learn to apply real and irrational numbers in a real world setting. The student will use a geometric model to replicate the resonance in plant stems. The entire lesson promotes flexibility in problem solving by taking an issue, natural frequencies, directly related to agriculture and architecture/engineering, modeling it and drawing conclusions based on their findings.

Curriculum Alignment:

Grade 8 Mathematics

• Goal 1, Objective 1.01 – develop number sense for the real
  • Define and use irrational numbers
  • Use estimates of irrational numbers in appropriate situations
• Goal 1, Objective 1.02 – develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil.
• Goal 3, Objective 3.01 – represent problem situations with geometric models

Classroom Time Required:

Three consecutive class periods on the traditional 60-65 minutes classroom schedule or two consecutive class periods on the block 90-minute schedule.

Materials Needed:

1. Plant Resonance Model
   • Handout of the warm-up on square roots (included in the lesson).
     This warm-up includes six questions on simplifying and calculating square roots.
     This can be done individually at the beginning of class or students can complete this in their groups before beginning the activity.
2. Oak wooden dowels. Each group of students will need four wooden dowels, one of each diameter.
   

   Length of Dowel (ft)	Diameter of Dowel (in.)
   1-3 feet	⅛ in
   1-3 feet	¼ in
   1-3 feet	¾ in
   1-3 feet	½ in

3. A table clamp holder for each group. The clamp must be able to attach to the edge of a flat, horizontal surface (table) and hold the wooden dowel in place vertically. Chemistry classes and science labs typically house these clamps. They can also be purchased at a local hardware store.
4. Each group will need a ball of playdo approximately 1-4 inches in diameter.
5. Each group will need a stopwatch.
6. Each group will need a ruler.
7. Pine Tree Frequency:
   • Each group will need a stopwatch
   • Each group will need a pencil and paper.

Technology Resources:

Each group of students will need a calculator. Optional: The instructor will need a computer with an LCD projector and speakers in order to watch the video clips.

Pre-activities:

• Before the students arrive: The teacher should organize the dowels, clamps, playdo, stopwatches and rulers into the appropriate number of groups. If the teacher chooses to view the videos, the computer and LCD projector should be set up. For the pine tree activity, the teacher should select a few medium-sized pine trees that can be pushed with some effort. The top of the pine tree must be visible. The teacher should also build flexibility into the schedule due to weather. The pine tree activity will not work well on a windy day. It is preferable if the pine tree activity is conducted before the plant resonance model activity.
• During the first class: The students should complete the warm-up activity on calculating square roots. Following this review, the students need to be engaged in a discussion on the effects of an object vibrating at its natural frequency. Lead the discussion by offering examples of vibrations such as a high pitched noise breaking a glass or the resulting damage from vibrations caused by an earthquake. Natural frequency is a predictor of tree and plant stability. A natural phenomenon that may cause a tree to oscillate at its natural frequency is wind. The students should be encouraged to provide examples of how wind may affect their world, particularly regarding crops, trees, and structures. Since the lesson will most likely be taught in the fall near the end of hurricane season, the discussion may also be opened with examples from a recent storm. During the discussion, the teacher needs to guide the students to hypothesize why trees snap or uproot and why some structures may fall and others remain standing. Natural frequency is also an important characteristic of buildings and structures. Short buildings have high natural frequencies and tall buildings have low natural frequencies. The natural frequency of a building, dependent on the physical characteristics of its components and on the surface below, is an important characteristic to consider for safety during an earthquake. Buildings should be designed outside of their natural frequency. The teacher should have the students hypothesize about the movement of structures and share examples of when the students experienced the feeling of a building moving (St. Louis Arch, bridge). The teacher should highlight that structures collapse and trees uproot or snap when winds gust at their natural frequencies or when the ground creates movement in the structures at their natural frequencies. Mention that winds typically gust between 0.2-2 Hz. The teacher may choose to introduce the activity with a video of the Tacoma Narrows Bridge. The teacher needs to introduce the terms frequency, natural frequency, resonance, and oscillations.

Activities:

Pine Tree Frequency (1 Day):

1. Break the students into groups of 4.
2. Each group of students should have the required materials. Group members can be assigned tasks – timer, recorder, oscillation counter, tree shaker.
3. With the pine tree standing still, the tree shaker will start gently pushing the tree forward and then pulling it back. The tree shaker should do this gently until the tree is swaying back and forth. When the tree shaker begins, the top of the tree will most likely be moving in the opposite direction of the way that the shaker is moving the tree. When the top of the tree begins moving with the tree, the shaker may stop pushing and pulling on the tree. This is when the timer starts the stopwatch and the oscillation counter counts the number of oscillations.
4. When the tree begins to stop swaying steadily, the oscillation counter should say stop when the tree has completed an oscillation and the time should stop the stopwatch. The recorder will record the seconds and the number of oscillations.
5. This should be repeated 10 times
6. Once the students have their data, the class may return to the classroom.
7. The students need to record their data in data sheet 1. To find the period, T, for the pine tree, the students need to calculate the time it takes, in seconds, for one oscillation. This should be completed on data sheet 1.
8. The frequency of the tree should be calculated on data sheet 1.
9. Each group should write their natural frequency on the board and a discussion should compare the results. The students should be guided to answer why the data is different for the different groups. Teacher notes: The differences may be due to different shapes and sizes of the trees, data collection techniques and possible errors/differences. The students should also be asked to hypothesize how the frequency of certain hardwood trees (oak, walnut) might compare to the frequency of coniferous (pine) trees. Teacher note: The frequency would be higher since the hardwood is typically stiffer. It should also be noted in the discussion that the leaves of certain trees may affect the frequency.
10. From the discussion, the students should write a paragraph explaining their choice of trees to plant if they lived along the coastline of North Carolina. The students should support their decision using the knowledge of the natural frequency of trees and real life observations.

Plant Resonance Model (2 days):

Day 1:

1. Review the terms frequency, oscillations and resonance with the students. Explain to the students that this activity will model the resonance in plant stems.
2. Assign the students into groups of four and distribute the materials.
3. The students should measure the length and diameter of each wooden dowel and record this information on data sheet 2.
4. The students should form a ball with the playdo and place it on the end of the dowel.
5. The students should measure the diameter of the playdo and record this information on data sheet 2.
6. The dowel should be clamped to the edge of a table or desk (stable surface).
7. Beginning very gently, the students should stand to the side of the dowel and pull the dowel sideways (direction of the horizontal surface).
8. Once the dowel is released, the timer should start the stopwatch.
9. The oscillation counter will count the number of oscillations the dowel completes before stopping. When the dowel is no longer completing full oscillations, the oscillation counter should tell the time keeper to stop. The recorder should record the number of oscillations and the time in data sheet 2. Do this for five oscillations.
10. This method should be repeated for each dowel.
11. Teacher note: remind the students that they are calculating the frequency of the dowel and that this information will be used to find the mass of the playdo.

Day 2:

1. Complete data sheet 2 as a class. Before beginning the second page of the data sheet, caution the students that there are terms and variables that they have not been exposed to, but variables just represent numbers.
2. Conclude with a discussion on the extension of plant modeling to man-constructions and the importance of considering natural frequencies. Remind the students that mathematics and science allow people to design safe and durable structures as well as more sustainable forests and crops in certain environments. By establishing variables, formulas, and equations and sharing this information, scientists and mathematicians have benefitted people around the world.

Assessment:

The assessment will take place in two forms. The assessment will occur at the end of day 1. The paragraph should include information and terminology from the pine tree experiment. The students should propose a type of tree and explain, using frequency as support, why the tree was chosen. The second form of assessment is the concluding discussion on day 3. The students should be able to understand questions using the terms frequency, oscillations, resonance, etc. The student should be able to explain the importance of natural frequency and the mathematics in terms of plants and man-made structures.

Modifications:

This plan is intended for students placed in Algebra I at the eighth grade level or a math equivalent to pre-algebra. This plan could be modified by supplying some of the equations and data on the data sheets, particularly on data sheet 2. The data sheet could also be cut into separate sheets so the student only sees one part of the problem at a time and is not overwhelmed. The language used could be simplified, leaving out terms such as elastic modulus and second moment of area. This information could just be provided.

Critical Vocabulary:

1. Frequency is the number of cycles or occurrences per unit of time. Frequency is measured in Hertz (Hz).
2. Natural frequency is the frequency at which an object vibrates once it has been set in motion.
3. Period of Oscillation, denoted by T, is the time that it takes for motion to repeat itself in a regular pattern or cycle.
4. Resonance is the tendency of an object to oscillate at its maximum amplitude at certain frequencies

Websites and Resources:

Tacoma Narrows Bridge Video
This link provides an actual clip of the Tacoma Suspension Bridge disaster.
http://www.encyclomedia.com/video-tacoma_suspension_bridge_disaster.html

Supplemental Files: 

Data Sheet #1 Group Member Names

Data Sheet #2 Group Member Names

Warm Up for Lesson 1

Warm Up for Lesson 1 Answer Key