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:

Planes, Wind, Velocity, Vectors – (Scientific Calculator fx-991EX ClassWiz and Vectors)

Happy Friday! I wanted to end my week of fx-991EX Classwiz focus with something that I have been missing – flying. As part of the work I do, I usually travel a bit each month. Whether that is to speak at conferences, work with schools and teachers on math pedagogy, technology, or traveling for vacation, I spend a lot of time on planes. But – as of March, I haven’t traveled once, for obvious COVID19 reasons. And, since schools and teachers are going through so much uncertainty, the professional development I was scheduled to do this summer is cancelled. And who knows when live face-to-face conferences and events will happen again. I am grounded.  And I miss it!!

As I looked through the menu options for the fx-991EX Classwiz, I saw the Vector menu, which got me thinking back to high school (yes, a LONG way back!!) where we actually learned about vectors. Adding/subtracting vectors to determine the direction a plane would have to fly to get to it’s destination to account for the impact of the planes velocity and the winds velocity. This clearly stayed with me because it made so much sense and I found it fascinating, and to this day, I think about the calculations that must go on by the pilots and the air-traffic controllers to make sure planes get to their destination, counteract wind, avoid other planes, etc., much of which is connected to vectors. I don’t know why vectors was taken out of the geometry curriculum – I think it is now more in upper level courses, but to me it really should still be included in geometry, which most students still take, because it is so relatable and relevant. Students see planes (even if they don’t necessarily go on them) flying above them, and I think it would be a great way to bring in a very real answer to ‘when are we ever going to use this?” Obviously there are more applications of vectors, but planes is the one that really stuck with me.

But, I digress. Let’s get to the focus of today’s post, which is using the Vector menu on the fx-991EX Classwiz to work with vector problems. I am focusing on simple vector calculations in my video related to determining a planes path based on their velocity and direction and the impact on this from the winds velocity and direction. A vector has both magnitude (length/size) and direction (often  includes angle). If visually finding the direction the plane must fly, which is what I remember doing in school,  you place the two vectors (wind/plane)(represented by directed arrows for length and direction) head-to-tail, and then construct the resulting vector (basically making a ‘triangle’ (see image 1).  We describe each vector by it’s x-vector magnitude, and it’s y-vector magnitude (so (x,y)). We add/subtracting the vectors by adding/subtracting the (x, y) vectors magnitudes of each, which results in the the solution (x,y). To find the magnitude of the resulting vector you would use the (x,y) solution and the Pythagorean Theorem. If you look at the image 2, what you should see a connection to the Pythagorean Theorem and a hint at Trigonometric functions if given angles (if given bearing degrees vs. magnitude of vector). You might have to do some trig to figure out those x- and y- vector magnitudes or the magnitude of a given vector if only given it’s bearing degrees.

Image 1

Image 2


There are many applications for vectors (navigation, force, displacement, acceleration, etc). What I show you in this video is how to do vector calculations in 2-dimensions (you can also do vectors in 3 dimensions which would be the same process, just choosing 3 dimensions vs. 2). I show adding/subtracting, scalar multiplication, and also the Dot Product.

Video: fx-991EX ClassWiz: Vector Calculations with Scientific Calculator


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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)

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Solving and Visualizing Solutions to Simultaneous Equations/Polynomial Functions

I decided to keep focusing on the fx-991EX Classwiz Scientific Calculator this week, since it’s one of my favorite calculators, in particular because of the added capability to show a visual representation through the QR code, which is something most scientific calculators can’t provide. Multiple representations are so important to helping students see relationships, recognize patterns, understand solutions, etc. Particularly to understand what it means to ‘solve’ an equation or system of equations – i.e. what does the ‘solution’ represent? Being able to see it numerically and visually is important for overall understanding. Especially if you are not just doing ‘naked math’, but have provided a context (problem-solving) for the functions/equations you are solving, where students really need to explain the solution in relation to the context.

With that in mind, I wanted to share a system of equations activity from the open-source curriculum, Illustrative Math, which has some great problem-solving tasks that allow students to apply mathematical skills, explain their thinking, and think beyond just skill-based practice.  For this particular posting, to show how the fx-991EX can help support finding solutions and visualization, I searched the Illustrative Math tasks for simultaneous equations, and found the task Kimi and Jordan. This is a grade-8 task that provides information about two kids, Kimi and Jordan, who both earn weekly allowances and also work jobs where they get paid a certain amount per hour. The problem asks students to create a table that matches total earned to hours worked each week, and also to compare who is saving more money if they work the same number of hours. This last question really is the ‘solve the system’ question, but involves more than just a single answer, since there can be several answers dependent on the number of hours. This is where having the ability to both visualize the equations and look at the graphs and really explain what is happening in the context of the problem. What does the intersection of the graphs represent? When, if at all, would Kimi earn more? When, if at all would Jordan earn more. So, adding the context of the situation makes this a beyond skill-practice problem, since they have to explain their thinking and there isn’t just one solution (i.e. the intersection point). This a nice problem to use with a graphing calculator or dynamic math software, but also if you have the fx-991EX as your technology tool, you can do the multiple representations as well, including a visualization.  Below are some images from the fx-991EX on this specific task.

This is the two equations, in standard form, for Kimi (top) and Jordan (bottom)

This is the ‘solution’ (intersection point) for the simultaneous equations. However – what does that mean in context? Does it answer the question who will earn more for a certain number of hourse?

This is the visualization of the equations and solution (graph) – notice you see the equations in standard form (and the decimal is in standard notation). Students now can look at the graphs and talk about when Jordan might earn more, when Kimi might earn more, and when they earn exactly the same amount….all based on hours. So three answers to the question “who earns more…”















I like this problem because it goes beyond a single solution for (x, y), and instead forces students to explain the different possible solutions based on hours worked. Real applications of mathematical skills require students to analyze, justify and understand that mathematics is just a way to model situations and help you make decisions.

Here’s the link for the activity on the Illustrative Math website, which also includes a discussion of the solution, and also a link to the Youtube video that shows you how to use the fx-991EX Classwiz to solve simultaneous equations and look at the visualization with the QR code option.

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