Math and Sound Teacher Notes

Grade 12

Trigonometric Identities

Paul, Stanley, Erwin, and Gigi

Purpose: To show the trigonometric identity:
sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2)
in an auditory manner.

Students will hypothesis, predict and test which part of the wave is associated with the pitch and which part is associated with the beat or the volume.

Description of Activities: Students will be using Audacity (a free program) to generate sinusoidal sound waves. Playing two waves together is like playing the sum of the two waves. (This shows the additive property of waves.) Students will be investigating the beat phenomenon and the trigonometric identity: sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2) . They will be following their worksheet to guide them through the activity.

Sources: http://en.wikipedia.org/wiki/Beat_(acoustics)

Estimated time: 1-1.5 Classes or 1 Class and the rest is due for homework

Students are required to produce a page of answers and supporting graphs. Students will conclude cos(2π (f1-f2)t/2) is the part responsible for the amplitude (and thereby the beating) of the wave. The pitch of the wave is the part sin(2π (f1-f2)t/2) .

Marking Criteria: Students will be marked on the correctness of their answers. Students will also be given a mark on their participation in class out of a 4 point scale worth 25% of their total mark.

1- Did not participate

2- Participated 20% of the time

3- Barely meets expectation for participation

4- Meets expectation for participation

5- Exceeds expectation for participation

Math and Sound Investigation

1) Open up Audacity. Go to Generate : Tone to create a tone also known as a sound wave. You can delete a sound wave by highlighting it and pressing delete. It is not recommended to have more than two sounds waves in your workspace at once. Try generating sound waves with different amplitudes, waveform and frequency.

2) At a fixed point in space, a basic sound wave (pure tone) can be represented by a sinusoidal function. One of the functional representation is Asin(kt).

Fill in the blanks with the following words: amplitude, frequency, volume, pitch

The _________________ of the function represents the loudness or _________________ of the sound.

The _________________ of the wave function represents the _________________ or how high or low the note is.

Copy and paste the above two statements with their answers onto separate file. You will print your answers at the end of class to hand in.

2) Investigate:

Generate a sine wave tone. Choose a frequency (call this n)
Generate another sine wave with frequency n+1.
Play the result.
What does it sound like? Record your answer.

You may hear a pitch change and a beating frequency.
Delete the wave with frequency n+1
Repeat the above and investigate with frequency n+2, n+3, n+10, n+20, n+100, etc
How does the pitch change as the second frequency increases?
How does the rate / frequency of the beat change?
Record your answers.

3) On graph paper, hypothesis what the wave would look like.

4) When you are listening to the waves together, it is like adding two sin waves.
sin2πf1+sin(2πf2) =
What is the trigonometric identity associated with this sum?

5) Hypothesis which part of the wave is associated with the amplitude (the beating) of the wave and which part is associated with the pitch.

6) Clear your workspace. Make two sound waves with different frequencies. Record this frequency. Using your hypothesis in question 5 predict the frequency of the beat and the frequency of the pitch. Use appropriate frequencies in which you can hear the beats.

7) Test your prediction by listening to the waves. Record how many beats you hear in 10 seconds. What are the experimental values of the beats? Is your hypothesis in 5 valid?

8) With the trigonometric identity found in 4, label which part is responsible for the beats and which part is responsible for the pitch of the waveform.

9) Plot out the sum in a graphing program or graphing calculator. You may need to plot twice: choose an appropriate scale for the beat and another for the pitch. Include your plots with your answers to hand in at the end of class.

10) Find two frequencies f1, f2, in which a period of the beat is 10 times larger than the period of the oscillations (part responsible for the pitch). Show your work. Graph the sum of sin2πf1+sin(2πf2) .

Extension Questions:

Go to http://www.mindspring.com/~j.blackstone/dist101.htm. Scroll down to the applet. Play around with the applet and record how the sound changes. Does the perceived pitch change? In other words, if you tried to hum the note does it change as you change the wave?

Division by Zero Reflection

As I am not much of a poet, this exercise was quite difficult for me. It seemed unnecessary to teach mathematics. In particular, I even had the feeling that the audience that I can reach through this approach could be an audience I never wanted to reach in the first place. Of course, this type of thinking (that there are 'undesirable' students) is very detrimental for an educator, I still see little value in using poetry in mathematics. This is in part due to my own belief that poetry and mathematics cannot comfortably co-exist. To quote Paul Dirac: "The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible." Of course this is an extreme view, but I suppose my philistine understanding of the art of poetry makes it difficult for me to appreciate its beauty and worth, much less advocate mathematics through this medium.

Assignment 3: Analyze a Project Idea and Develop/Extend

Group: Gigi, Paul, Stanley, Erwin

Handouts and Sample Tilings:

Reflection for advantages, disadvantages and modifications from each group member:

Gigi:
Teachers and students will need to know how to separate a line segment into different equal number of parts. I had to look this up in Google but the instructions are not always intuitive. I had to recall how to copy an angle. Not everyone has learned how to copy angels, create a perpendicular bisector to a line.
Another problem we had was with creating the star of David. Students may place the triangle intersections in different spots. Students may have to draw on the copy of the tessellation analyze the different shapes that are used to construct the larger shape. Students may need to draw on the copy of the tessellation to see the symmetry or see how the shape is constructed.

Stanley:
After attempting one tessellation I realized that the concept of 4-fold symmetry is key to the tessellation. What a fantastic way to introduce the concept of symmetry. Can easily expand the project by first asking the relatively intuitive question of whether or not any 2-D object with n-fold symmetry can be used to tessellate the plane(the answer is no when n is infinite, as the only object with infinite number of symmetries is the circle which cannot tessellate the plane). Also with this type of construction it is not so important to enforce 'straight-edge and compass' rules of construction as this type of construction is only 'sacred' because of Greek influence.

Paul:
In terms of evaluating the benefits of this project, I think this project caters to the student's artistic flare and mathematical prowess. The first part (drawing the pattern, describing the pattern in words) appeals to those people inclined towards art, not requiring any math skills (except trying to keep things in proportion). The second part (recreating the repeat shape with straightedge and compass, making minor modifications) caters to the mathematician. It requires visual analysis of the shape to look for patterns that can be recreated using the geometry knowledge that students have thus far. Teaching analytical skills will be helpful not just in math but in so many other areas of life. One weakness of the project is that it could be overwhelming for students who cannot see the pattern that can be recreated with the two simple instruments. Tessellations are covered in the Math 8 IRP. Some of these might be difficult. (I will admit: I got a bit frustrated that I could not recreate the hexagram in the middle.) Of course, this problem could be alleviated by choosing simple tiling patterns for the students. I would not modify this too much, but I would pre-select some not-so-challenging tiling patterns or at least have two sets of patterns (medium and hard). One constraint of using this project in my classroom would be time. I might not know how long to prolong this project or how many class sessions to allocate for in-class work. Overall, I could see this being used in my class to get their creative juices flowing.

Erwin:

I was excited about this project. By mixing art and mathematics this was a nice departure from the regular math problem solving typically done in a class. All throughout the assignment, I was unsure with how far I needed to go if I imagined myself as the student. Some students might scrape only the surface of the project and others may go very deep into much detail. The construction is very mathematical because it doesn't follow the usual math problem. By trying to find an object from basically experiment is more real and connects with the individual. I needed to look back at my geometry math book to remember some simple constructions.

Benefits:
-Gets students that are not particularly excited about math a bit more engaged
-Gets students to apply what they learn to out-of the box problems ike finding minimal parts of tiling
-Assessment of mathematical ability can be made

Disadvantages:
-Students will not achieve the same outcome because the instructions/expectations are not clear enough
-The project may in fact take too long for individuals and it might not be able to be applied to groups of people. The project is based heavily on the discovery of shapes not inpendent steps.
-Harder to evaluate, giving a letter grade on effort of the final presentation.

Modification:
Break up the assignment and assess or evaluate each of the steps. This gives clearer purpose to the assignment and is better for time management of the assignment.

Division By Zero Poem

Division by zero,
Deemed the greatest mortal sin,
Long upheld,
Ever since zero came into the lexicon.
Until whence,
Bernard Riemann graced Earth with his presence,
And discovered,
That there is but one infinity among a myriad,
And naturally,
Zero got paired with infinity, and became a whole.

Division, Zero free write

Zero:
- The symbol for 'nothing'
- Looks like a hole
- Under the map z -> 1/z is sent to the point infinity
- 0 times infinity = 0
- the additive identity in a ring
- obliterates everything under multiplication (even in an arbitrary ring)
- the 'center', versus infinity as 'the limit'
- 0! = 1, because there is only one way to arrange nothing
- nCk = 0 if k > n, because there is no way to select a set of k items from only n < k items

Division:
- Multiplicative inversion
- Given an arbitrary integral domain one can construct a fraction field where multiplication makes sense
- Makes sense in a division ring
- More problematic than multiplication because the integers have no multiplicative inverse in the integers
- Perhaps motivated the advent of rational numbers
- 'Division by 0' makes perfect sense in Riemann sphere

Reflection on Mathematics' Role in Citizenship

In Elaine Simmit's article she discussed the potential problems with associating mathematics as a mere collection of facts; as black and white. Surely no pure mathematician would ever contend with these notions. Mathematics is built on a foundation that EVERY statement can only take on exactly one of two values: true and false. All of mathematical logic is built around this concept and all theorems proven is based on this essential assumption. Essentially, assuming the 'universal' axiom that a statement is something that is either true or false (but not both) and a given set of axioms, all mathematical theorems may be proven. While this is essential to building rigor in mathematics it is problematic in mathematics education. The problem is the lack of emphasis that PURE MATHEMATICS need not translate or apply to REAL LIFE situation. For example, there can be no doubt that a sequence of independent and identically distributed random variables with finite mean converge in mean almost surely, it is NOT without doubt that a given sequence of random variables in real life SATISFY the conditions for the theorem. Thus it is important for us educators to identify the APPLICATION of mathematics in real life situation. This is already strongly advocated in, for example, social studies classrooms where there is a strong emphasis on analysis of current events and to unfold the intricacies of modern society and politics. This is not done in mathematics. When thinking of 'real life' problems in mathematics we usually associate extremely trivial things like finding the correct amount of change. This is tantamount to being able to read road signs as 'reading in real life'. While not a FALSE statement this obviously does not achieve the objective. Students are not challenged and don't learn anything. Instead, students need to be implored to discover more sophisticated applications of mathematics, for example how survey companies collect data, how data collected from the census is analyzed by the government and how does this influence public policy, the soundness of using 'average income' as a barometer to judge the social well-being of a community, etc. These will ultimately produce a more engaged learning environment and produce better, more quantitatively literate citizens.