The unit, Sounding Off About Trig, intends to take the overarching theme of sound and applies it to various aspects of algebra and trigonometry. While the unit is designed to supplement a Pre-Calculus textbook, there are parts that may be adaptable to an Algebra II/Trigonometry class. I have chosen not to center the unit on one specific aspect but instead will create short vignettes that can be applied in different places of the curriculum. This will enable teachers to do one of two things. They may either use individual lessons of this unit to enrich (or replace) a given trigonometric or algebraic concept or they can choose to continually revisit the overarching theme by using all parts of this unit throughout the school year.
In addition, these short pieces will be as “hands-on” as possible. Most students learn best when they can put their hands around what they are learning. Students will learn how trigonometry applies to science through the classroom activities. Often math is taught as a skill without being given a place to apply those skills. The activities will ask students to integrate their knowledge of mathematical skills with their understanding of sound. By having the unit’s activities focus on sound most will require students to use musical instruments as well as an oscilloscope. An oscilloscope is a piece of scientific equipment that records sound waves on a screen from a played musical note. The graphs produced will provide students with a visible depiction of an invisible concept: sound waves.
Because the unit integrates sound and mathematics, there are several specific “sound” objectives as well as “math” objectives. In a given class activity many of the sound and math objectives will both be addressed. Occasionally a few objectives will need to be introduced independently before the class activity can begin. Prerequisite skills will be clearly indicated in each activity immediately following the activity’s objective. Background information that a teacher may need to use the activity will be presented in the appropriate places.
Sound Objectives
As mentioned above there are many objectives to this unit. Some are science related and others are mathematically related. The objectives listed in the next few paragraphs are followed by background information. The objectives are offset for quick reference and the material is grouped such that a teacher with knowledge of the material could skim or skip each idea. For the novice teacher or one unfamiliar with the sound concepts there should be enough substance in the paragraphs to provide at least a working knowledge of the topics.
Objective 1
At the end of the unit students should be able to state and discuss general concepts about sound waves. These topics should include how sound waves travel, and how their wavelength is determined.
Sound waves need a medium through which to travel. For the purposes of the experiments included in this unit, we will only deal with sound waves traveling through air. The sound wave travels through air by compressing air particles as it goes. It is often reflected off of objects that are bigger than the wavelength and “washes” over objects that are smaller than the wavelength. The wavelength is calculated by the formula, c=(f, where ( stands for the length of the wave, c stand for the speed of sound, and f stands for the frequency. The speed of sound in air is 1100 ft/sec of 343 m/sec. Therefore, if we know that the frequency is 440 Hz then we can find the wavelength by dividing 1100 ft/sec by 440 Hz. This gives us a wavelength of approximately, 2.5 feet. Thus anything much larger than 2.5 feet will reflect the sound and anything much smaller than 2.5 feet will be “washed” over by the sound. Should the wavelength be approximately the same size, the sound wave will be scattered in all directions. This knowledge becomes very important in acoustics as buildings are designed for different purposes.
Objective 2
At the end of the unit students should be able to stipulate how the frequency and period of a sound wave relate to one another.
The frequency, f, of a sound is defined as the number of cycles per unit of time. Sound waves that humans can hear range between 20 Hz and 20,000 Hz with the human voice ranging in the area of roughly 100 Hz and 5100 Hz. The period of the graph of a sound wave is inversely proportional to the frequency, giving us the equation, f = 1/p, where p stands for the period. Thus, if we know that the frequency is 440 Hz, then the period is 1/440 or .00227 seconds. Similarly, if we know the period is 3.14 seconds, then the frequency is .31846 Hz. If the period is .0314, then the frequency would be 31.846 Hz and in our hearing range. You can see that as the period gets smaller, the frequency rises. Because these go in opposite directions they are said to be inversely proportional.
Objective 3
At the end of the unit students should be able to discuss what a Helmholtz resonator is and how it works.
A Helmholtz resonator is “another type of air vibrator.” It can be used to analyze musical sounds. A common type would be a bottle. Air blown across a bottle’s opening acts as a spring compressing the air in the bottle beneath the neck. There are many instruments that act as Helmholtz resonators, such as a guitar. The vibrating strings force the air inside the guitar to act as a spring as the back and front boards of the guitar vibrate. Air is then pushed through the opening on the guitar and a sound is heard.
The formula to find the frequency of a note produced from a bottle is f = (c/2()( sq. root of a/Vl) where c equals the speed of sound in air (1100 ft/sec or 343 m/s), a is the area of the neck, l is the length of the neck, V is the air volume of the resonator. We can determine the frequency of a produced note from a by finding the individual values and then substitute them into the formula. So for a bottle with a neck of .25ft, a volume of air of .097ft3, the area of the neck is .0314in3, then we get the following equation: f = [1100/(2*3.14)][sq. rt. (.0314/(.097*.25))] = 199.32Hz. Then using appendix A we find that the sound produced was approximately the G note below middle C. The Helmholtz resonator provides a wonderful opportunity for students to work with formulas, musical notes, and understand frequency better. It also gives the teacher the opportunity to talk about conversions from centimeters to inches and inches to feet, bringing in multiple levels of mathematics.
Trigonometric Objectives
As with the Sound objectives there are specific math objectives covered in this unit. The introduction mentioned that the math material is intended to be used with a Pre-calculus or an Algebra II/Trigonometry class. Many of the current Algebra II texts address trig issues in the later chapters of the book. Some even begin to focus on trigonometric functions as does the book I currently use, Advanced Algebra by Prentice Hall. Advanced Algebra has students graph y = sin x and y = cos x in the middle of the ninth chapter. Thus this unit could be used to help ground the math concepts in applications.
Objective 1
At the end of the unit students should be able to use formulas such as volume, area, and others to find the missing variable.
In mathematics it is very important that students can evaluate mathematical formulas when given the values of the variables used in the formula. It is equally important that they are able to solve equations for any one of the variables used. For example, if a student is given the formula Area = (r2, and they are told the area is 36( and the radius is 6, then they need to able to verify that this is correct. In addition, they need to be able to find the radius if only given the area of 36( or the area if only given the radius is 6. These skills are so important that they are introduced in middle school mathematics and continually revisited through high school.
Objective 2
At the end of the unit students should be able to graph simple sine functions.
Given a circle with radius of one and centered at (0,0) there are specific values for x and y that can be found as the point travels around the circle. The points at the four axes, starting on the positive x-axis are (1,0), (0,1), (-1, 0) and (0, -1). Other specific points can be found by using the ratios of special right triangles as an angle is formed using the positive x-axis as one side of the angle. ( *add a source here for teachers to refer to) The x values of these points can also be found by using trigonometric functions. As cosine is known as the adjacent side over the hypotenuse, it quickly becomes apparent that we have the x value over one. This is because the hypotenuse is also the radius of the circle, which equals one. So cosine quickly becomes defined as the x value or x = cosine ( where ( is the angle formed. Along similar lines we have y = sin (. Each of these functions can then be graphed on a new set of axes where the horizontal axis is the angle measure, and the vertical axis is either the value of x or the value of y depending on which function one chooses to graph.
By first developing the points on the circle for given angle measures, a chart of values containing the angle measures and the height of the circle (y) can be created. Then students can use this data to plot the sine curve. Some values can be found by using the special right triangles. Other values can be generated by using calculators to ensure a smooth, correct curve.
Depending on how you choose to use this unit, you may wish to develop the points and graph first. Thus, when students begin to use the oscilloscope, they have a mathematical equation in their head that defines it. I would suggest this approach if you were spreading out the unit across the curriculum. However, if you are using this unit as a separate “chapter,” then I suggest that you first model the data with the oscilloscope and then try to find the equation that accurately models it. In my classes, I plan to use the lessons to supplement and enrich the data in the chapters; so I would first have students understand y = sin x and then move to the experiments.
Objective 3
At the end of the unit students should be able to determine the period, phase shift, and amplitudes of functions when given an equation or a graph.
The period of a function is a numerical value that states how long it takes for the height of the function to repeat. If you had a fence surrounding your yard and every six feet there was a post, then the period of the fence would be six feet. A sine or cosine function repeats every 360 degrees. If you imagine the unit circle again, you can see how the points on the circle would begin to repeat after you have rotated around from the positive x-axis back to the positive x-axis. The period is represented on the function y = sin ( or y = cos ( by a value multiplied to (. In their basic forms there is an understood one in front of the (, so the period of the function is 360/1 or 360. If the function was y = sin 6( then the period is inversely shortened. Thus, the period of this new function is 360/6 or 60. The height of this function y = sin 6( at 60, 120, 180, etc should be the same because every 60 degrees the function begins a new period.
Traditionally, trigonometric functions are graphed in radians as well as degrees. The more advanced the course, the more likely the function will be in radians. To convert from degrees to radians you only have to multiply the degrees by ( and then divide by 180. Therefore the period in radians in a basic y = sin ( function is 2(. And multiplying the angle can change the period by some value. So if the equation was y = sin 2( then the period would be 2( divided by 2 or just (.
As with any function in math, sine and cosine functions can be translated (moved) around in a given coordinate system. There are several ways to translate a function. To translate a function horizontally a value is added to the independent variable before any operations are performed on that variable. For example, if one has a line y = x, it can be translated right four units by subtracting four from the x. This results in the equation y = x – 4. To translate the function y = sin ( one merely has to add a value to ( before taking it’s sine. This is modeled by the equation y = sin (( + c) where c is any real number. In trig functions, a horizontal shift is known as a phase shift.
The amplitude of an equation affects how high and low the function goes on the vertical axis. It is mathematically calculated by taking half of the difference between the maximum height and the minimum height. Thus, as the sine wave hits a height of one and a low of negative one then amplitude = ½ (1 – -1) = ½ (2) = 1. The amplitude can be found in a trig function by taking the absolute value of the number multiplied by the sin (. For example, if y = - 3 sin ( then the amplitude is |- 3| = 3.
The concepts of amplitude, period, and phase shift help students understand what changes can be made to a function in order to move it in the plane. If these are approached without some context, they tend to be very abstract for students. Thus sound is incorporated into the unit. How we change sounds directly translates into a phase shift, period change, or amplitude adjustment. The incorporation gives students a chance to see that the concepts they learn have direct application.
Objective 4
At the end of the unit students should be able to translate a sine or cosine function.
Objective 2 dealt with each of the possible changes as if they occurred in a vacuum. But as we all know it is rare for just one thing to occur - many things simultaneously happen in real life. As the purpose of this curriculum unit is to show uses of trig in real situations, it is necessary to look at problems that have more than one of the transformations occurring.
A general equation for a trig function is in the form y = a sin b(( - c) where |a| is the change in amplitude, 2(/b is the change in period, and c is translation left or right. It is my experience that students have difficulty dealing with multiple changes. I often have students draw several layers of their graphs from the equations, advising them to alter the function y = sin ( one step at a time. I have my students first deal with the new amplitude, then the new period, and finally the horizontal shift. For example, given the equation y = 2 sin 4(( + (), I would have my students mark that the height of the function will go to 2 and –2 instead of 1 and –1. Then I would recommend that they mark that the period will end at 2(/4 = (/2 instead of 2(. Often I even recommend that they sketch a light graph given just these two changes. Finally I have them shift this whole sketch by the correct amount, which in this example would be ( units to the left.
As students will be exploring different graphs generated on the oscilloscope, they will need to know how to do multiple transformations at once. For a sound that is introduced a split second later than a first the sine graph has been shifted right, that is a phase shift has occurred. And since period is inversely related to frequency - as the frequency increases the period shrinks and as the frequency shrinks the period is increased.
Objective 5
At the end of the unit students should be able to graph the sum of two or more sine functions.
Two functions can be mathematically combined by using any of the basic arithmetic operations (+, -, *, or /). As stated earlier, very rarely is math represented in real situations in its basic forms. Musically pure notes from a flute are present, but notes produced from other instruments such as an oboe are not “pure” tones. Thus the graph of their sound waves would not be a simple sine function…not even a transformed sine wave. In fact the graph of their sound is a periodic function that is much more complicated. (See diagram below next paragraph)
The graph of the non-pure tones is created from adding several different sine waves together that have different periods because their frequencies are different. Thus the graph of these tones can be generated by finding the graphs of the different frequencies and then adding their y values together to form the new function. This is demonstrated mathematically by having the function y = sin 3( and y = sin 2(. If ( = (/4 (or 45 degrees) then we get y = sin 3((/4) = sin 3(/4 = .707. We also get y = sin 2((/4) = sin (/2 = 1. Thus if we add these two functions together, producing a new function y = sin 3( + sin 2( and use the same angle, then we get y = .707 + 1 which equals 1.707. Hopefully you can see that many of the y-values will change and a new graph will occur. See diagram below to see how the two functions add together to give us the third.
(figure available in print form)
If students can understand how these functions can be combined then they can comprehend why the graphs of the non-pure tunes do not look like plain sine curves.
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