Mathematical Precision – Can We Please Stop The Debate of 6/2(1+2)? (Or Use the fx-9750GIII!)

I would like to preface this post with my personal opinion – this is a silly argument that has gone viral. If teachers are teaching mathematical understanding and mathematical precision, and not a reliance on ‘calculators’, this ‘problem’ is not really a problem. It’s all about representation and being clear in what you want students to do, even in bare-naked math problems like this one. It shouldn’t be this hot-button topic if you are helping students understand mathematical operations and numbers. If students get different answers, it’s because the problem wasn’t clear….which actually, should lead to some great student discussion about mathematical precision.

The mathematical situation: 6/2(1+2).

Part of the problem here is that writing the problem in-line text, as the problem is represented above, does not allow for rational number representation in standard form (like ½), and many calculators have linear entry that looks like the above written problem. Some calculators use implied grouping, where anything under the division is considered a ‘group’ (so (2(1+2)), and some do not (only the first quantity, 2,  is considered the denominator).  If you were writing this problem on the board for students to solve, you would hopefully be more accurate and precise and use parentheses or the standard form of a rational number to avoid confusion. In a textbook, or on an assessment, same thing. But, as a result of the linear written form of this problem, i.e. 6/2(1+2), the ‘argument’ that people are touting is that the problem can be interpreted two ways:

  1. Either as 6 divided by 2, then multiplied by (1+2) or
  2. 6 divided by the grouping of  “2 multiplied by (1+2)”.

My thoughts – 1) you shouldn’t be relying on the calculator to be the answer-giver (meaning students should notice/wonder and question if the division is the whole quantity, including the 2, or if its just the 2 as the divisor) before calculating; 2) If they get different answers, that should become a class discussion about precision and representation; 3) IF you are being a good teacher and focused on mathematical precision, you would not write the problem this way in either case, because you would and should use parentheses if you wanted either situation. So, what you SHOULD write, is:

  1. (6/2)(1+2) in which case, the parentheses are done first, so 3*3 = 9 solution, OR if you wanted the other situation, then you write..
  2. 6/(2(1+2) in which case, parentheses done first, so 6/(2(3))=6/6 = 1

So – it’s provider error (teacher, textbook, assessment) for not being precise in what you expect students to do. If you follow PEMDAS, with no parentheses, since it is linearly written, it can be interpreted as 6 divided by 2 times (1+2), so parentheses first giving 6 divided by 2 times 3.  THEN, Divide and Multiply, as you see it, from left to right – so 6 divided by 2 is 3 and 3 x 3 is 9. But – with a calculator that uses implied grouping under a division symbol, the 2(1+2) is grouped.  Conclusion – KNOW what you want students to do and write it that way (standard notation OR with parentheses), OR have a great discussion  when they get different results that leads to the importance of precision in mathematics.

The real debate is over what calculators do when you enter the problem – which on most calculators, is linear entry, so as mentioned above, some calculators are doing implied grouping, where the 2 outside of the parentheses is considered ‘grouped’ with the parentheses under the division symbol, so it is interpreted as version 2 above (i.e. the entire quantity 2(1+2) is seen as one number). Others are not. And therein lies the argument – which calculator is correct…..which is not really the argument, since students should have clarity on that before they even get to the calculator (which, let’s be honest, you shouldn’t be using a calculator for this problem to begin with!!)

What’s the answer? Obvious one – use parentheses to clearly state what you want ‘grouped’ if writing/representing a problem linearly. Use standard notation of rational numbers so that fractions look like fractions and you can clearly see whether you are just dividing by the 2 or whether you are dividing by the entire quantity 2(1+2). If using a calculator, know what quantities are grouped, so teachers/texts/assessments need to clearly group or use standard notation to show what is in the divisor.

This brings me to the new fx-9750GIII graphing calculator, which has basically done the work for those who aren’t being precise in the way they present problems to students. The fx-950GIII has new order-of-operations functionality that basically puts the parentheses in for you if you don’t write the problem using standard notation. Basically, it is showing you how it is interpreting that division, so you can then analyze if that is truly what/ how the problem was meant to be solved. (Which again – should have been determined to begin with so this issue doesn’t even arise, but I am repeating myself……..)

Here is a short video showing you how the fx-9750GIII is helping teachers and students really analyze the order of operations if they enter a problem such as 6/2(1+2) linearly (instead of using the nice rational template or parentheses).

Video: fx-9750GIII: Order of Operations and Interpretation (No More 6/2(1+2))


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Dynamic Graphing on the fx-9750GIII – Helping Students Discover and See Relationships

We talk a lot about how learning math should NOT be focused on memorizing steps and formulas, but instead on doing and discovering the relationships and building understanding. Ideally, students should be visually seeing multiple representations of concepts (tables, graphs, words, pictures, models, manipulatives, equations, etc), comparing and analyzing these representations, and making their own conjectures and ‘rules’. Instead of telling them what they should see or know or do, they are able to figure it out themselves by focusing on relationships and connections (obviously with good structured investigations and questions and activities). This is the ideal.

Research shows that when students discover the patterns and come up with rules/formulas/relationships themselves, they are much more likely to retain the information, or more to the point, more able to ‘recreate’ the experience they had and recall and/or rebuild the information. Not possible when memorizing isolated facts or skills. It’s why just teaching struggling students rules and steps and skills and making them repeat it over and over does not improve learning…..there are no real connections being made.

Algebra is a subject that many students struggle with because of the ‘unknowns’ or variables – it is abstract, and without exploring patterns and relationships of the variables in many forms, it is hard to help students understand and make sense of things like “equations to model real-world situations”. Sadly, there is still the tendency to just teach process – i.e. follow these steps, or “use the quadratic equation to get the solutions” without even talking about what those solutions mean, where they are located, why there are two (or one or none). If we just teach skills and if students just memorize processes/steps, when confronted with something similar, but not exactly the same, they can’t do it, and give up, because they have not had experiences that help them build understanding and see relationships and connections to prior knowledge.

This leads me to today’s focus – Dynamic Graphing.

Dynamic mathematics is when something in a given representation is changing (a measure, a value, a construct), and as it changes, you see the impact of that change. As an example, in geometry, this might mean a triangles three angles are measured, and as you move one or more of the vertices, the angle measures change as you see the triangle change. If you sum the three angle measures, you get 180, and as you change the triangle dynamically, even though the triangle is now different, with different measures that are changing as the triangle changes, the sum of those angles is still 180. Which leads to a conjecture.

Dynamic graphing is the same thing – it’s using variables (instead of static numbers), in equations and changing one or more of them, and then watching what happens to the graph as that variable changes. Dynamic graphing really emphasizes what a variable means – i.e. a quantity that varies, and as it varies, the graph/tables/equations also change. As an example, in a linear equation, defined as y=Ax+b, if we make A- a dynamic variable and change it, students will see the line change it’s ‘steepness’ and direction, and make a conjecture that A- must have something to do with a line’s steepness. If we change b- dynamically, they will see the line move up and down vertically (but not change it’s direction/steepness) – so they conjecture that b- determines where the line crosses the y-axis (because they notice the line always seems to cross at the y-axis at a point that corresponds to b-). Simple things like dynamically changing and visually seeing the impact of that change can lead students to more deeply understand what those coefficients/variables in an equation mean and do, so that when you give them y-=-3x+5, they already know the direction of the line (negative), how steep it might be, and where it’s going to cross the y- axis. You didn’t have to tell them – they figured it out by observing, looking for patterns, and making conjectures and then, most importantly, having discussion with others to confirm their findings.

I wanted to share how you can do dynamic graphing on the new fx-9750GIII graphing calculator. Not a new feature – it’s possible on all the Casio graphing calculators. However, I don’t think many people realize this functionality exists, especially on a black-and-white, inexpensive (but powerful!) graphing calculator. It’s something I think many people expect only mathematics software to be able to do, so I wanted to show you this feature. (And, you can do dynamic geometry too with the Geometry Add-in Menu!)

Video: fx-9750GIII: Dynamic Graphing and Visualizing Variable Changes

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Exam Mode On Casio’s New fx-9750GIII Graphing Calculator

Exam Mode – something that makes teachers and test providers happy. Maybe not something on everyone’s mind right now, but an important feature on Casio’s newest addition to their graphing calculator line. The fx-CG500, fx-CG50 color graphing calculators already have Exam Mode, so the fx-9750GIII now has that capability as well. In today’s post, I want to discuss briefly what Exam Mode even means, and then provide a how-to video which shows how to set the fx-9750GIII into exam mode.

Letting the calculator ‘do the work’ for you is always a big concern for teachers, especially during testing, particularly standardized testing. If students are asked to find the factors of a given equation, we want them to find them using mathematical skills and understanding, not by pushing ‘solve’ on the calculator and letting technology take over. This does not show what the students can do. Which in theory is the point of exam mode – forcing some functionality of the technology to be temporarily suspended so that the students must do and show the work. My personal opinion is that if this is your fear, then you aren’t asking very good test questions…..but….that is obviously a post for another time!!  Since a majority of ‘tests’ (particularly standardized) are more focused on process and steps, turning off the auto capabilities of the technology makes sense – it forces students to have to show and do the steps manually. This is exam mode, something required by many state and national tests in order for a calculator to be on the approved list.

I remember as a teacher in Virginia, when getting ready for the state Standards of Learning tests, having to put all my classroom calculators into test mode. An incredibly time consuming task (using a TI at the time), and then having to reverse the process once the testing was over, which was insanely even more cumbersome. Casio has at least made the process easy.  Getting into Exam Mode is relatively quick – following some prompts on the calculator. But getting out of Exam Mode is brilliant – just wait 12 hours and it auto-shuts off.  You don’t have to remember to do anything, or go through the process of connecting to your computer.  Obviously, if you don’t want to wait, you can do the longer process, but as a teacher – waiting the 12 hours is golden!!

What does Exam Mode prevent students from doing? Basically, all programs that might have been entered are locked out, e-Activities, vector commands, E-con mode, and storage memory access are locked out, and no add-in apps either (like Physium, for the Periodic table). This renders the calculator safe from ‘doing the work’ for the students (i.e. a program to solve for example). Below is a link to a how-to video that shows you how to put the fx-9750GIII into Exam Mode and what steps to follow to take it out of Exam Mode. The beautiful thing being you don’t have to do anything to take it out of Exam Mode – just wait!!

Video: fx-9750GIII Putting the Calculator into Exam Mode


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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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Hand-held Graphing Calculator – Newest Model from Casio fx-9750GIII

A new model of graphing calculator from Casio came out recently (see press release), the fx-9750GIII. I can hear some of you asking “why, since everyone is going to mobile devices/computers and the internet?” The reality, which this pandemic has brought more to the forefront, is there is a HUGE disparity in access to digital technology and the internet, for a variety of reasons, which has made the mandated remote learning these last few months incredibly difficult for many schools districts, parents, teachers, and students.  There have been several articles on just how glaring the inequities are, so I have listed a couple here:

For many students, having access to mobile devices, laptops, computers, and even the internet is not possible or is a huge struggle. Hand-held calculators are a technology tool that is a much easier and more affordable option to put into the hands of every students than any tablet or laptop. Hand-held graphing calculators can do amazing things and help students explore and discover a multitude of mathematics. Calculators are still a required tool for most schools because of access, affordability, and other mandates, such as most standardized testing situations (at least for the foreseeable future after the online-assessment debacles that occurred recently). Casio, being the better calculators on the market (compared to Texas Instruments and other options) both in functionality, cost, and ease-of-use, is still improving and updating their calculators, so the new fx-9750GIII is an example of supporting the current needs that still exist and providing hand-held technology that improves based on on those needs. (To be totally upfront, this is MY opinion, from over 30 years in mathematics and having been forced to use TI calculators for over 17 of those years.) TI has a monopoly due to really good marketing and/or brainwashing, depending on how you look at it, but they are NOT a better calculator – they are hard to use or remember how to use, and let’s face it….they haven’t really changed much for years. The most popular model, the TI-84 Plus, has been around since 2004, with a slight update in 2015 to a color edition with more memory. But for the most part, the same tool for over 16 years. And still expensive. Crazy.

But I digress. Clearly don’t get me started on my TI rant!! (Though it is partly why I am a consultant for Casio – to try to deprogram people!!)

Back to the new updated Casio graphing calculator, the fx-9750GIII. I am going to focus my posts this week on this calculator, sharing some of the newer features, and looking at some of the menu options that are available and sharing some lessons as well. Today I just wanted to explore some of the new features of the calculator. Here’s a short list of some of the changes/additions, compared to the fx-9750GII and fx-9860GII models. Think of this new graphing calculator as a combination of these plus more.

Here is a short list of just some of the updates/changes/additions:

  • Math Input/Output, Linear Input/Output AND Math Input/Mixed Output modes
  • Order of Operations – clarifies entries such as 6/2(1+2) or 4π/2π (I will explore this in tomorrow’s post!!)
  • New Menu Icons – e-Activity, Spreadsheet, Add-Ins: Python, Geometry, Physium, Probability Simulator (we will explore this later in the week)
  • New types of regressions
  • 9 new probability functions
  • Ability to graph x=f(y)
  • Exam Mode
  • Catalog QR code
  • Fraction Template Button change, Standard-Decimal Button Change, and scientific notation button change (x 10^x)
  • Storage Increase (Mass Storage similar to the color graphing calculator, the fx-CG50)
  • Ability to get OS updates

Its’ a crazy powerful graphing calculator. I will explore some specific functionality in later posts this week, such as probability. Today’s short video is just a quick overview of what’s there. I have also included below a link to the Quick Start Guide for those of you interested. If you want to start exploring, you can download the emulator software (which I am using in the video), which provides free access for 90-days. It’s a great way to get a feel for the tool before you decide to purchase the hand-held hardware version.

  1. fx-9750GIII_QSG Quick Start Guide
  2. Emulator Software (free download for 90-days – choose the (fx-9860GIII download, and then when opening, select the fx-9750GIII calculator)
  3. Video Overview – fx-9750GIII Newest Casio Graphing Calculator

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