Do You Hear What I Hear? Amplitude and Intensity – Trigonometric Modeling of Sound with

I am ending this week with one more music-related activity. This one is focused on sound and the ability of the human ear to ‘hear’ loudness and intensity, as measured in decibels. This, like in yesterday’s lesson, relates to the sine wave and transformations on the sine wave. I’ve adapted yet another activity from the Fostering Mathematical Thinking with Music, Casio 2015, where we look at the amplitude of a sine wave and compare it to the intensity of music. This should resonate with students since they listen to so much music and often times very loudly, so thinking about what their ears are doing to detect that increase in loudness, or intensity, hopefully is pretty engaging.

The activity begins with some interesting facts about the human ear, which I am including here – a little science along with the math:

“The human ear is truly an amazing piece of scientific measurement equipment. Besides helping to regulate the body’s balance and temperature, ears are designed to detect microscopic differences in clusters of air molecules.These are mostly pairs of nitrogen and oxygen atoms that are, on average, 3.7 Angstroms wide – 1/100,000th of the width of a human hair!  

The ear measures whether pockets of these tiny particles have gotten bunched slightly more closely together, or spread slightly further apart. It can discern these differences to within one-BILLIONTH of a centimeter. Even more remarkable is that the ear is making these determinations as the pockets of air are arriving at somewhere between 20 and 20,000 times per second. Amazing! makes you want to stop and just listen for a moment, doesn’t it?” “Intensity and the Decibel Scale.”, Physics Tutorial, n.d. Web. 3 Apr. 2014.

The activity, Amp Up the Intensity, is originally a graphing calculator activity that I converted into a version. Students look at the sine wave and a transformation of the sine wave and compare these first. They then relate the sine wave to a sound wave and use a slider to see how amplitude and intensity are impacted by the coefficient A in y=Asin(x).  They are then given examples of sound intensity from various objects (rustling leaves, vacuum cleaner, a jet taking off, etc.) measured in decibels, and compare these visually on a scatter plot and determine a regression that will model the relationship. The whole activity really focuses on how amplitude impacts the loudness.

Below are links to the activity, a PDF of the original activity, which includes teacher notes and possible solutions to the questions, and a video overview of the activity. Now, depending on which tool you have, you can work with either version of the activity.


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Trigonometric Music – Representing Audible Notes with Sine Waves

My focus this week has been on music and math. Today’s lesson we are going to look at representing musical notes using sine waves. I’ve adapted an activity from Fostering Mathematical Thinking with Music, Casio 2015, that relates frequency (Hertz) of to the sine wave. Also it connects function transformations to the pitch of the frequency and creating a model for the tuning note of A (above the middle C) at 440 Hz using the general form of the sine curve, y=sin(Bx), where x represents time in seconds.

The lesson starts with an interesting little paragraph about oscilloscopes, which I have copied in it’s entirety here, since it relates to the wave pattern of music:Trigonometric Music – Fostering Mathematical Thinking through Music

“An oscilloscope is a piece of scientific lab equipment that draws a graph of an electrical signal. One basic way to use this device is to plug a microphone into the oscilloscope, then produce a sound. The oscilloscope will display a graph of the sound wave on its screen. The investigation (lesson) introduces you to one of the most common wave forms an oscilloscope displays: a sine wave. If you can get your hands on an oscilloscope, try producing many different kinds of sounds, using tuning forks, piano keyboards, various musical instruments, and even the human voice. Even if you produce the same pitch each time, you will fin that the oscilloscope can tell the difference between sounds!”

The Oscilloscope.” University of Evansville, n.d. Web. 10 Mar. 2014.

The lesson is a really nice connection between trigonometric functions, function transformations and real-world application of mathematics to sound and pitch frequencies. I think students will find all of this fascinating, especially those who play musical instruments. Thinking about music in terms of mathematical sine waves and visualizing how the pitch of the music alters the visualization would be a really great connection, and if you could actually get an oscillator, testing out different sounds would have even more of an impact.

Below is the link to the version of the activity, along with the original PDF of the activity, which is geared towards the use of a graphing calculator. Now you have both versions and can utilize whichever one makes most sense for your students and resources. The PDF includes sample answers and teaching guidance that can be utilized no matter which version you are using. I’ve also included a video overview that walks through the version of the activity.

The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Musical Frequencies and Pitch Relationships – Data Tables & Regressions with

Musical pitch is defined as follows:

“Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as “higher” and “lower” in the sense associated with musical melodies”

Pitch is a perception of sounds as being higher and lower, or better put, the ‘human’ perception of frequency. The pitch of a note is how high or low the note is based on a specific frequency. Today’s mini-math lesson explores the relationships between the pitch of a note and the frequency of the note, based on the notes relationship to middle C on a piano. There are 12 notes in music, played with 7 white and 5 black keys (see image).  The black keys show two different notes, but because they are the same sound/note they only count once. On a regular piano, with 88 keys, the same 12 notes repeat but at different pitch. Guitars also have the same 12 notes, as do most musical instruments, and understanding their relationship helps understand how to create different pitch (or play in a different key, as another way to think about this).


In the activity I explore today, Pitching the Right Note (adapted from Fostering Mathematical Thinking with Music, Casio 2015), the 12 notes are looked at by octaves (group of 12), with three octaves explored (the octave below middle C, the octave at middle C and then octave above middle C).  Students are given the frequency, in Hertz (Hz), of each note, and then explore through looking at graphs and numerical calculations, the relationships between the same notes at different octaves. They discover how the frequencies change (increase, decrease) over the octaves, and proportional relationships between the octaves and corresponding notes. Through this exploration, they learn how mathematics plays a huge part in musical pitch.

This activity is a graphing calculator activity that I created a version of so that now you have options no matter which technology tool you might have access to. Below I have provided the activity link for use online (great for remote learning!) and also the PDF version (for use with a hand held graphing calculator). I have also included a video overview that walks through the version of the activity. The PDF contains possible solutions and some teacher guidance that could be used for both the or calculator, again, depending on your resources.


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Scatter Plots and Regressions to Understand Musical Frequency and Octaves (Mini-Math Lesson)

Being forced to stay home for the past few months has apparently led many to take up new hobbies and learn new things. Some are reading books they always meant to read, others are learning a new language, and many are learning to play instruments or learning new songs on instruments they haven’t touched for a while. Some are learning to cook and try new recipes. The list goes on and on. I myself am trying to learn some new guitar songs and reacquaint myself with the piano, which I played for years as a child. Trust me…not as easy as you would think after all this time, however I am happy to say reading notes and some familiar tunes like Scott Joplin’s The Entertainer, are coming back, albeit slowly.

Naturally, this renewed and/or new interest in the arts has caused an uptick in some company’s sales despite the shut down, according to an article I read. As Casio is ALSO a seller of musical instruments (keyboards, electric pianos, synthesizers, etc.), as well as calculators, watches, and software, I thought it would be fun to focus on some connections to music, math and math software   in this weeks mini-lessons. Casio Education actually has an entire workbook resource on music and mathematics (Fostering Mathematical Thinking Through Music with Casio Education, 2015) which is a nice tie in to Casio music and Casio hardware (calculators). The resource is geared towards graphing calculators, but I thought I would convert some of the activities into activities this week. I provide both the free math lesson I created/adapted, but also the PDF of the graphing-calculator lesson, so you could use this with whatever technology you and/or your students have available. I am also doing a quick video overview to walk through the activity as well.

The lesson I chose for today is one related to frequency of radio waves (VHF and UHF) and how it relates to the pitch of a musical note. The intro to the lesson has a brief history of German physicist Heinrich Hertz, who is credited with making many key discoveries related to electromagnetic waves (see image/intro that comes from the activity). It’s the reason the standard unit of a wave’s frequency (number of cycles a wave completes per second) is called “Hertz”.

In this activity, the pitch of a note is discussed, and how the common musical referent for a pitch is the note A, above the middle C on a piano keyboard. I have played the piano for years (and as mentioned above, just started up again), and never really knew that the A should be vibrating at 440 vibrations a second, giving the pitch A a frequency of 440 Hertz (Hz). The activity goes on to explore notes that are certain number of octaves above and below the given pitch, A-440. Mathematically, this means the frequency of the pitch is doubling (or halving if going below) each octave. Students have to create a table based on this information, graph the information and find a formula that relates the frequency of pitch the the number of octaves the pitch is from the given pitch of A-440.  This is also then related to finding patterns of positive and negative exponents, in the context of musical octaves. I assume this relationship between frequency of pitch and octaves is what piano tuners are doing when they ‘tune’ a piano. They use an instrument called a tuning fork to create the wave pattern (though now their are electronic tools that let you match the wave patterns and tune).  Fascinating when you really look at the mathematics of it.

Below you will find three things – the activity, the original PDF and solution examples/discussion (for a graphing calculator but really for any mathematical technology), and a video overview that walks through the version of the activity.  The rest of the week (Tuesday, Wednesday, and Thursday) I will explore three other music-related math lessons that I am adapting from this Casio Resource.

The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Found This Great Math-with-Technology Lesson – Now What?

Let’s admit it, there are a LOT of math lessons freely available out there if you are looking. But just because they are there doesn’t mean they are good, doesn’t mean they align to your learning goals, and doesn’t mean you know how to implement them effectively into your instruction. What to do with lessons once you find them is often as much work as finding them. Today I want to finish the week out with some suggestions and how-to’s for implementing technology-integrated lessons you have found and/or created yourself.

This whole week, my posts have focused on planning for implementing technology into math instruction, starting with an exploration of the technology itself and some strategies (specifically focused on, dynamic math software). This was followed by how to look at what you teach and plan and organize for where technology-integrated lessons make sense, and how to then find some ready-to-use lessons to start building up your resource bank. And yesterday, I discussed how to modify found lessons to make them fit your instructional needs – by adding questions, reordering activities, and/or deleting components that might not fit into your learning goals. Today, the idea is to discuss ways to share these activities with students, giving you some strategies and showing you how to work specifically with activities and the sharing capabilities.

Here are some ideas for working with technology-integrated math lessons, whether in a face-to-face classroom or in a remote-learning setting.

  1. Whole class demonstration. This means at the front board/projector or on a zoom meeting, where you are directing/controlling the activity and movement. The key here is questioning – moving the objects, demoing etc., and asking students to notice and wondering, make comparisons, look for patterns, make connections to prior knowledge and make conjectures. Questioning is the key. Have them write down their answers individually (think time), then share with a partner (so online maybe have assigned groups/teams), and then each group/team shares. This is great for discovery type lessons and investigations, where you are trying to have students look for patterns and relationships.
  2. Small Groups (Pairs/Threes) – I have always found small groups to be a great way to work with technology, if you have the resources (mobile devices that can be shared or in an online setting, internet access and group break-out rooms). Send the groups the activity and give them a specified amount of time to work through the questions and manipulation (maybe assign roles if needed).  Groups come back together whole class and share their findings and everyone comes to a consensus. This works great for activities that have multiple solutions paths – so maybe you assign one group to approach with using a table, another to approach using a picture, etc. (jigsaw)
  3. Individual Work – I tend to only like individual work when it is to practice something students have learned, possibly an assessment type lesson like a homework, when working with technology. Communication about what you are seeing and doing and making connections is so important, so my preference is always #1 or #2. But, assigning a technology-integrated math activity as a homework or assessment is a great way to provide students a chance to explore and share what they know, especially with a tool like, since they can do visuals, text, graphs, calculations, all on the one page. If you are using this as homework, the important part would be to discuss the problems and work as a whole class after the fact. In an online setting, is ideal as you just share the URL, students work on the activity, and send their URL to their individual work back to you (so think of it as virtual collecting of papers/tests). In a classroom setting, it would be the same – you could put the URL link to the assignment on your class page, and students would do it at home and send you their URL for their paper. Or, you could also print out the activity and give them hard copies, but that sort of defeats the point of having a technology-integrate activity……

Below are some how-to’s on using activities with students. This includes how to share activities you have created and/or found and how to set an activity as a Homework.