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:

Data, Roller Coaster’s and Using Scientific Calculator for Statistics (fx-991EX Classwiz w/QR code)

It’s summer. As a kid living in Virginia, I looked forward to summer for many reasons, but one was the ability to go to amusement parks. King’s Dominion outside of Richmond , VA and Busch Gardens in Williamsburg, VA, were the two amusement parks that were the ‘big deal’ in my day. The Rebel Yell (renamed to The Racer now) – at the time King’s Dominion’s biggest wooden roller coaster – could scare the life out of me every time and I loved it.

I was thinking about this summer and how different it is for so many, and one of those differences is things we usually associate with summer – amusement parks, pools, beaches, etc. – are going to be vastly different experiences with the social distancing and health-risk situations. I am not even sure amusement parks are open. I know the pool near us is open with limited people, where you must make a reservation and can only stay for 3 hours at a time and only come 3 days a week.  With these thoughts of summer and roller coasters in particular, I remembered a great site I always went to to get data to use with my students when teaching in Virginia – the Eeps Data Zoo. This was a great site for real-world data, and I remembered there was a nice data set on roller coasters, so today’s post is using that roller coaster data to demonstrate how the fx-991EX Classwiz scientific calculator does statistics and allows you to also visualize the table, the statistical plot and do a regression, as well as calculate the statistics relevant to the data.

The Eeps Data Zoo  has several data sets that were used in scientific research. You can cut/paste into excel spreadsheets or data software, or if using a handheld calculator that has the ability to enter statistics, you can enter the values manually as well. The roller coaster data is data on 15 different roller coasters around the world, both steel and wood, that compares their largest drop, their top height, their total length and their top speed.  So you can do lots of comparisons – i.e. only the wood ones, or only the steel ones or speed vs. height, etc.  There are also additional links to roller coaster data if you want to research other roller coasters.

The fx-991EX scientific calculator that I use is the video below is really amazing because you can take the data, make the table, and do the statistical calculations needed for your comparisons. But – because of the QR code, you can also see the plot of your data and visually look for patterns and relationships. Additionally, the statistical graph that is created allows you to then do a regression as well, so you are getting the benefits of a graphing calculator with a simple scientific calculator, which is awesome. Especially if you are a teacher and your students have these, you can be up at the front and display the results and have small group/whole class discussion about the visualization of the data they just entered, and so students get multiple representations and discussion about the relationships they are seeing. It’s just one possible way to work with this, and the video shows using the emulator software and the internet so that the QR code quickly pops up and goes immediately to the visualization. Great as a whole-class demonstration and discussion lead-off.

Here is the link to the video that demonstrates entering data in the Statistics Menu of the fx-991EX, looking at the statistics, then using the QR code to see a visual representation of these and looking at regression. In the video, I am really only doing one comparison of the roller coaster data – there are so many more you could explore, so I encourage you to do so!

Video How-To: fx-991EX: Tables, Statistics, Regression and Visualization (QR)

Be sure to visit Casio Cares:

Here are quick links:

STEM – Newton Knew Forces (F=ma) (Mini-Math Lesson Algebra/Algebra2)

Yesterday’s lesson was related to the motion of a pendulum and it’s graph as its swing was impacted by friction. Today we are looking at force, using a pendulum as the push (force) on an object (scooter). This is a fun little experiment that you could do with students, and in this time of home-schooling, it would be a relatively easy experiment for parents to set up with their kids. You just need some rope, a bucket full of water or even a jug of water to use as your pendulum, a door frame to swing the ‘pendulum’ from, and then a scooter or something with wheels that can move when pushed (so a wagon or a skateboard). You will also need a measuring tape, some weights to add to the scooter (three different weights), and something to mark start lines and height release for pendulum.

If it’s already sounding like too much work, don’t worry!! There is sample data in the activity, so you can still explore the mathematics if you don’t have the time to set up!!

The idea is to set up the pendulum so that it hangs from a door-frame/beam at the height of the scooter (so place scooters back end directly at where the pendulums lowest point so that the pendulum will hit the scooter and push it). Measure the weight of the pendulum bob (bucket/jug of water) before you begin. Measure the weight of the empty scooter. You then decide on a release height for the pendulum (will be the same height for each release). Mark the start position of the scooter (back end), position the scooter (empty) and release the pendulum from the designated height. It should hit the scooter and push the scooter. You then measure the distance the scooter traveled after it was hit (so from the start position) to where it ends (be sure to measure at the back of the scooter, to be consistent with the start position). The next step is to add some weight to the scooter (make sure it is secured on so it doesn’t fall off when pushed!). Repeat the experiment, record the distances and weights.  Do this for at least five different weights of the scooter – empty to heavier. (Don’t worry – I have included the PDF as well with all the detailed instructions).

Once data has been collected,then you will graph the data and look at how force and mass impacted acceleration. The goal of this experiment is to show that the rate of change of momentum of an object is proportional to the resultant force acting on the object and in the same direction.  Students will explore their table of data, make scatter plots, look at the relationships. The experiment has to do with Newtons’s 2nd Law, where Force=mass x acceleration. There are a couple things to keep in mind:

  • The pendulum weight represents the force (because you don’t change the drop height)
  • The scooter weight is proportional to the mass (because gravity is constant through the experiment)
  • The average distance the scooter moves is proportional to the acceleration
  • Weight is used in the experiment instead of mass, but for this experiment it is acceptable to deal with weight because the force of acceleration is constant throughout the activity. The weight in this activity is always proportional to the mass.

This activity comes from Fostering STEM Education with Casio Technology, Casio 2013. I have converted part of the activity (the pendulum component) to a activity, which is shared in the link below and also overviewed in the video below. I have attached the complete activity, which includes a wagon-pushing activity. The PDF is the whole activity with a lot of description and calculator suggestions, as well as sample data.

  1. STEM – Newton Knew Forces (F=ma) Activity
  2. STEM Newton Knew Forces (F=ma) (PDF)
  3. Video Overview – STEM Newton Knew Forces (F=ma) Mini-Math Lesson (Data & Regression)


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:

Mini-Math Lessons – Proportional Reasoning: Comparing Rates and Looking At Scaling

What size frame do I need? Why is that candle burning faster than the other one? These are questions we are going to explore today. And it all has to do with rates, and proportions, along with other factors such as type of wax for the candles. Proportional reasoning comes into play in seemingly mundane things, like determine the size frame needed for a picture that you might be enlarging (or shrinking). Like yesterday, it’s about comparing and using mathematics to help understand and model real-world situations. What I love about these types of problems is that they can be approached several different ways, and each way can provide a different perspective and answer because you get more and/or different information. This is what modeling with mathematics is really all about.

Both activities today, as with all the activities this week, are adapted from Fostering Mathematical Thinking in the Middle Grades with Casio Technology, Casio 2011. I have made version of them, but if you have handheld calculators, these same activities are available in the free Math Activities under graphing calculators for middle school at the Casio Education Website. The first activity has to do with two candles, the same height to start, but burning at different rates due to different types of wax. Students will explore fractions by looking at the fraction of each candle that is burned. They will compare using tables and graphs and use proportional reasoning to determine things such as when is one exactly half of the other. The second activity has to do with wanting to frame an image, and depending on the room it is to go in, the image will be sized-up or sized-down, so how much framing is needed and how much glass is needed?  This is a perimeter and area ratio problem and there is some nice simulations that students use to collect data on side length, perimeter, and area as side length increases. From experience, I know students struggle with the understanding that if you double the dimensions (length/width), that perimeter also doubles but area quadruples (exponential). The data collection and looking at the tables and graphs

Here are the links to the two activities and the video overview that explores the activities and some of the skills/features:

  1. Proportional Reasoning – Burning Bright
  2. Proportional Reasoning – Stretch That Picture


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: