 In this video, I will give an overview of a lesson that was created with a music and a mathematics teacher to be used in either a math class or a musical composition class. In terms of the mathematics, the lesson is appropriate for algebra, advanced functions and modeling, or pre-calculus students. There's no musical background required, but if a student does have musical background, it's nice for them to be able to see this connection between their musical background and a mathematical topic. The students will create melodies using a free online tool called NoteFlight that is very easy to use, and we have a resource to help them do that. Then they'll create a mathematical model using data, functions, transformations and a graphing tool. We like to start by just showing students some music, and even if they don't read music, they should be able to look at this particular student's melody that she created and think about what they notice when they compare the lines of music to one another. So for example, this first line of music looks very much like this second line in that they start on the same note, but instead of going up here, you'll go down. Similarly, the notes go up here, you go down, and then this one came down and now it goes up. So this is actually called what's known as inversion. It's as if we had put a horizontal axis on this first line of music and reflected the music about that axis. This third line compared to the first line looks as if we're playing this first line backwards. So the first note on this line is the last one. These four notes are these four notes reversed, and then this measure is also in reverse order here. Lastly, the fourth line looks like the second line but played in reverse order or listed in reverse order. So this note, last note here is the first note here, etc. So each of these has a musical counterpart and a mathematical counterpart to create the third line from the first one that's known as retrograde. And the fourth line from the first one is actually known as retrograde inversion where we did the reflection both across that horizontal axis and across a vertical axis if we put a vertical axis at the end of this measure. In terms of the tools that we have or the handouts, we have a note flight handout for students who have never used that tool before. We have a musical composition tools handout for students who don't have any background in music. They can make sense of both the timing and what is meant by a pitch on the musical staff. And we have a student assignment handout. If you think about the link to the mathematics, what we want to do is create some kind of mathematical representation of the music after students create their own melodies using these compositional tools. So we're going to map the rhythm or the timing of the notes to the x-axis and we're going to match the pitch or the frequency of the note to the y-axis. In this particular scheme, we'll create all of our melodies using the 4-4 time. So each quarter note is associated with an integer on the x-axis. If you have four quarter notes in the measure of your melody, the corresponding x-coordinates of the points will be 0, 1, 2, and 3. We'll use rational numbers to represent a note that's shorter than a quarter note. So if your first measure has four eighth notes and two quarter notes, then the x-coordinates of the points for the first measure will be 0, 0.5, 1, 1.5, 2, and 3. The pitch will be associated with the y-coordinate of the data points. The very first note will define what our horizontal 0 is. So the y-coordinate of the very first note will always be 0, and all of the other y-coordinates will be relative to that note. So for example, if we start on a D, which is what this note is in the treble clef, then the y-coordinate of the E note will be a 1, the y-coordinate of an F would be a 2, this is a G, so it's a 3. The half note is represented using two points with a repeated y-value. This scheme doesn't distinguish between a half note and two repeated quarter notes. That's a limitation of the model that maybe students can find a different way of working around. In this particular example, we'll continue, we have our 0, 0 for this half note, 1, 0 is the second half of that half note in a sense. This one is 2, 1, this one is 3, 3. The second measure will be 4, 2, 4.5, 3, 5, 5.5, 3, 6, 4, and 7, 4. We'll create a data set. I've used GeoGebrant to create a data set for this and plot it, but you can use a calculator or Desmos or an Excel spreadsheet. And what we'd like to do is create a functional representation of this melody so that we can then use transformations of functions. We could certainly transform the points, but because this is appropriate for algebra or pre-calculus, it makes sense to try to create some kind of a functional representation of the music. So what we're going to do is we're going to use our technology to create a polynomial and let students decide which polynomial represents their music the best. So here's an example. The orange one is a fourth degree polynomial and the black one is a fifth degree polynomial. I used GeoGebrant, but again, you can use another tool. So if you think about playing the melody that we created, we can go to the NoteFlight file, play the melody, and then we can create our mathematical representation of that melody. So if you look at this first part, that's our polynomial fit of that melody. So if you look at this first part, that's our polynomial fit. We reflected it about the x-axis and shifted it over to tack it on to the rest of our melody. This next one was in retrograde, which is a reflection about the y-axis or the vertical axis, and then again we had to shift it over to tack it on to our melody. And the last one is both a reflection across the x-axis and the y-axis, which is also known as retrograde inversion. If you look at the terms both for musical and composition tools and the mathematical transformation tools, there's a counterpart for everything in each of the lists. So we've used inversion, retrograde, retrograde inversion. There's also something called diminution, which is when you take the length of the notes and have each of the lengths of the notes. So that's equivalent to compressing horizontally. Something called augmentation where you take each of the original notes and have them played for twice as long, and that's equivalent to stretching horizontally. If you change the key of a musical piece, it's equivalent to a vertical shift, and if you offset the timing like singing around like row, row, row your boat, that would be equivalent to a horizontal shift. Some of the student reflections are interesting to look at. In this particular one it says it was interesting to see how music and math could relate to each other. I would not consider myself a musically talented person at all, but seeing how math could relate to music made it easier for me to understand and allowed me to see how math can relate to things that don't seem remotely close to math itself. It was nice that everybody got to make his or her own music because you could make it easier or harder based on your knowledge of music. In another reflection, a student said, I learned you can transform the measures of music just like you can transform graphs and that composers actually use that technique to compose music. I liked that we could use note flight and geogibra together to get a better understanding and more practice with transforming graphs. I also liked that we applied math to a real life application. This interdisciplinary activity was a great reminder on how math is applied to our everyday lives. I like how we were able to extrapolate an eight measure piece of music with two measures. I think it would be nice to go even deeper and make longer pieces or even reverse engineer it to find out what a certain graph might sound like if it were music. If you look at the resources, we have several things. We have this math and music lesson overview video which you're watching right now. We have a teacher handout. We have a note flight handout to show you the basics of note flight. We have a handout on music composition tools, a student assignment. We have a couple of videos, one of which is a note flight demo video, a geogibra demo video, and a student reflection video. Here's some more references and resources. I hope that you enjoy using this mathematical and musical activity with your students to help them see a beautiful connection between math and music.