Probability Simulation with the fx-9750GIII Graphing Calculator – There’s an App for That!!

In a previous post, I talked about ways to use random number generators on a calculator to simulate rolling dice, or flipping a coin, using the fx-991EX scientific calculator. Probability is such a fun way to explore mathematics, especially when you can collect the data and really relate the concepts of sample size, theoretical probability, experimental probability and measures of center to a real-world application. When I was teaching in the middle and high schools, my students and I had so much fun doing math – i.e. tossing coins, rolling dice, picking cards, using candy to explore sampling, creating surveys and collecting data, etc. Unfortunately, especially in this strange new world of distance learning, doing probability experiments with real objects is a bit harder, since everyone is remote. Having technology and being able to do probability simulations virtually is a great advantage, so today I am going to revisit probability simulations, looking at Casio’s newest graphing calculator, the fx-9750GIII, which has an add-in app specifically for Probability Simulations.

The emulator software for the fx-9750GIII comes with the Probability Simulation menu already installed, which is fantastic, since as a teacher, you could have this on your screen while teaching virtually with students, and run the simulations and students collect and record the data. It would be ideal for students to have the hand-held version as well, in which case the Probability Simulator menu needs to be added to the hand-held menu via computer download.  I have provided a how-to download the app video below, along with an overview of how to use the Probability Simulator.

The Probability Simulator is really powerful because it allows you to quickly collect lots of data, see the changes as data is collected both in a table and graphically. When working with students doing the data collection by hand, the fewer samples they collect, the less likely they are to ‘match’ the theoretical probability that is predicted for the outcome. So, tossing a coin, theoretically should yield 50/50 heads/tails. But, if students only toss the coin 10 times, they are more likely to not match the theoretical, and maybe heads seems to be more likely. The idea behind the Probability Simulator is that you can start with a small number of samples, and then build, and go up to 999 data samples of the experiment (which in a classroom situation would be unrealistic time-wise). What students can see as you increase the number of samples is that the theoretical probability becomes more likely, reinforcing the idea that sample size/number of trials has an impact. There are 6 different simulations that you can run with the Probability Simulator – coin toss, dice roll, spinner (spinning to land on 4 possible numbers), marble grab (five different types of marbles), a card draw, and then random numbers. You can set the number of trials. What I love is there is a visual of the ‘trial’ (dice rolling, spinner spinning, etc), as well as then a graphical display of the outcome and a table display. Students are provided with multiple representations of the situation which really helps them make connections. The last two options, card draw and random number, don’t show a graphical display. Instead, after collecting your data, you have the option to store the data to Statistical Lists. When you then go into the statistical menu, your lists are populated and you can then decide which lists and what types of graphical displays make sense for the data collected. You can actually do that with all the different simulations – store the data and go into the statistical menu and look at different plots, such as box-and-whisker, pie, scatter….whatever might make sense. But the first four auto-show a bar graph along with the table as the simulation is running, which is a great visual.

Here are two videos related to the Probability Simulator – the first one is how to download the add-on app to the hand-held fx-9750GIII calculator and the second one is an overview of the Probability Simulator app in action.

  1. If you have the fx-9750GIII hand-held graphing calculator and want to add in the Probability Simulator app to the menu (and also the Physium), here is the link to where those are located:
  2.  Here is the overview of how to add in the app (or additional apps as well): fx-9750GIII – Downloading and Adding Add-in Apps
  3.  This video is an overview for the Probability Simulator app: fx-9750GIII Probability Simulator App Overview


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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:

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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Probability Simulation with the fx-991 Scientific Calculator and Random Integers

When I was teaching middle school, I always loved the probability section because we would do some fun activities with dice, cards, coins, candy, etc. It was a fun way to compare theoretical probability to experimental probability and get students collecting data and doing mathematics.

An obvious experiment that many start with is tossing a coin and determining the probability that it will be heads or the probability it will be tails. The theoretical probability is that there are two outcomes, 1 heads, 1 tails, so probability for both is 1/2.  However, if you have students actually experiment and record how many heads/tails out of a specific number of tosses, it is more often than not 1/2 (50%), unless you do a large number of flips (i.e. collect a larger sample). I use to have pairs of students flip a coin 10 times, then we would compare and see how many came up with the theoretical probability. There was usually a couple, but more did not.  Then we would do another 10 tosses but combine groups data and compare, and finally combine all the data from the class (so a much larger sample) and compare. The idea being that the more samples, the closer you get to the theoretical probability.

Similar things can be done with cards – i.e. what is the probability of drawing a face card? Or a specific number card?  Tossing dice is another great one – you can do one dice and the probability of getting a specific number, or a really fun one is rolling two dice and summing their face value and determining the probability of getting a 7. Students have to first think about all the different combinations of sums, how many of each, etc. The possible combinations of sums is 36 (2,1) and (1,2) would be count as two separate ways to get a sum of 3). The number of different ways (works best if you use two different colored die) to get a 7 is 6, so theoretical probability is 6/36=1/6.  It’s then a nice connection to way craps uses the 7 as the ‘crap out’ roll – it is theoretically more likely for someone to roll a 7 and lose for the whole table than any other sum.

There are obviously many things you can do with basic probability. I had a bucket of pennies, a bucket of different colored dice, many decks of cards, would bring in candy (Skittles and M&Ms work great). You can use lots of hands-on things to make probability experiments fun and engaging. But….what if you don’t have the means for these manipulatives? This is where you can use technology to ‘simulate’ things such as rolling dice or flipping coins, even picking a card from a deck.

Today’s shared lesson is a brief one on probability using the Casio fx-991EX scientific calculator. Using the table menu and the RandInt# function, you can simulate tossing coins, rolling dice, and more. The activity specifically mentions coins and random number probability, but the video link shows you how to do coins, dice, and cards as examples.

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Explore, Create, and Plan for Technology Use NOW for School Year 2020-2021

With the majority of schools finishing up the school year ‘officially’, I realize many teachers and students just want to take a break. Especially since the end of the 2019-2020 school year was so different and strange, having to both learn and teach remotely, not having a graduation or prom or sports, not seeing your classmates or students, and having to recreate school in a virtual way.

For many, ‘doing school’ these last few months was a struggle, since they had never taught or learned from a distance, and had never used virtual classes or video recording before, or at least not to the extent that it was expected. Finding ways to navigate this new paradigm of teaching has been tough for many teachers. What might traditionally be done in a classroom is much harder to replicate in a virtual environment and you should be doing things differently in an online environment, so looking for new tools and resources and more importantly, strategies for teaching online, became an additional part of the teach-from-home stress.

I know everyone is ready for a break and a rest from thinking about teaching (or learning) and what tools to use and how to create learning experiences in an online environment. Definitely – take a rest! Enjoy the beach! Enjoy your family! Enjoy doing nothing! BUT – it’s during these times of no expectations (i.e. no deadlines, no students, no classes) that you should also take some time, in a relaxed way, to explore some new tools and start thinking about lessons and strategies that incorporate technology in preparation for the 2020-2021 school year. I don’t think this virtual teaching, or distance learning, is going to disappear. I think there will be a continued need in the fall, and if not full-time, definitely a hybrid approach – part in person, part virtual. Use your down time to start exploring, planning, and revising what you use to do in a way that it can be done either face-to-face or virtually. Start thinking outside of the box now so that when you actually have to start focusing again on school year 2020-2021, you are already ahead of the curve.

This week I am devoting my blog posts to ‘planning ahead’ and providing some videos and ideas for making mathematics teaching more engaging, more dynamic, and flexible – possible to teach both in-person and face-to-face. I am focusing this week on, a math software tool that can be used both in a virtual learning environment and in a face-to-face environment. I am going to share ways you can incorporate this tool as part of a regular support for teaching and learning mathematics, no matter what school ‘looks like’ this fall.

Today I am going to just suggest that you sign up for a free account and start playing around. This is the easiest way to get a feel for what this tool can do. It is a diverse, free math software that allows you to be really flexible in how you use it:

  • In a classroom – as whole class demonstration (with good questioning and ability to test conjectures immediately)
  • In a classroom – one-to-one or shared devices – students can work and explore and answer questions right on the activity and have discussions, that can then be shared whole class. A great discovery tool
  • In a virtual classroom – again, as a demonstration tool – you can show students visualizations, you can collect data from students and immediately graph, or calculate with all of them watching.
  • In virtual classroom –  You could have one paper open and share the mouse and have students move things and explore questions
  • In a virtual classroom – send papers/assignments and have them do the work on their own and send back to you, and then you can display different results
  • In a virtual classroom – demonstrate something, ask a question, have students go into ‘group rooms’ and then come back to discuss their findings.

Lots and lots of ways to use the tool. And – you only need ONE tool to do all the math since can do writing, calculations, graphing, statistics, geometry, financial calculations…..

So – for today, I just want you to explore. Check it out. Try something that you teach – i.e. if you teach algebra, make a graph. If you teach statistics, do a statistical plot and make a regression.

Below are two videos that will help you get started. Tomorrow, I am going to share some strategies for sharing papers with students. For today – explore, play, discover.

  1. Start Exploring – What to Do When You First Try

      2. How to sign up for a free account