Complex Numbers – Support for Calculations

I received a question on one of my Youtube video posts on the Casio Fx991 scientific calculator asking if it was possible to do complex number calculations on this calculator. The answer is of course yes – which then prompted me to make a quick video today on exactly how to do that with the fx991. See the video below:

This of course then made me think of our other technologies and that perhaps I should show how to do complex numbers with these tools as well.

Here’s the steps on the graphing calculators (any of the Casio models, since they all basically work similarly – the beauty of Casio, the buttons are relatively consistent). This example uses the CG50, but see fx-9750, fx-9860, etc).

And finally, on ClassPad.net, the FREE online math software that does it all – statistics, geometry, graphing, and of course calculations. (You can sign up for a free account (ALWAYS free) – here’s a quick how-to).

The question of course arises, when are we even using complex numbers? Or why do we need them? As I never really taught math content that required students to utilize complex numbers, I don’t feel I am able to answer these questions with authority, so I did a bit of research. For one, if we just go from a ‘content/standards’ perspective, if you are in states that incorporate The Common Core Math Standards (or a version of, whether renamed or not), then it is actually part of the High School: Number and Quantity standards which state, “Students will…”:

  • Perform arithmetic operations with complex numbers
  • Represent complex numbers and their operations on the complex plane
  • Use complex numbers in polynomial identities and equations

But, that of course doesn’t really get at why do we need them. So here are some things I found in my search for this answer. I admit I can’t explain these any more than just listing them, but it at least points to places where complex numbers are in fact important and needed.

  • Complex numbers are used in electronics to describe the circuit elements (voltage across the current) with a single complex number z=V+iI
  • Electromagnetic fields are best described by a single complex number
  • People who use complex numbers in their daily work are electrical engineers, electronic circuit designers, and anyone who needs to solve differential equations.

Hopefully this is helpful to those of you who are in fact doing complex calculations for whatever reason!

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CG50 – What Are All Those Apps?

As many of you know, I post quick videos in the blog to show different things about the Casio calculators or math or teaching. Many of these are posted on my YouTube Channel. I will occasionally get comments from viewers asking questions, and I do my best to answer them. If I can’t answer the question, I find someone who can, or research until I do have a response. Just the other day, when I was asked “how do you use the constants on the CG-50 calculator”, I was not quite sure what was being asked, since I tend to use the calculator from a mathematics teaching perspective, and hadn’t explored using constants (from a science perspective) and wasn’t even sure what was meant by the ‘constants’ in this particular question (as it could mean the constants in a given equation).  Turns out the viewer was asking about the Physium Menu/App on the calculator, and how to get the constants from these tables and values into calculations. This is something I have honestly never used because I am not a science teacher and therefore rarely, if ever, have need for this app. But – it got me curious and seeking out an answer (which I did find and explore so I could give a reasonable answer).

In my ignorance, I realized that there are many apps on the CG50 (and other Casio graphing calculators) that I have never really explored, not just the Physium App. Mostly I focus on the most-used menu items – Run Matrix (to do calculations), Graph (to work with functions and graphs), Table (functions using table representations), Equation (solving equations), and Picture Plot. But there are a lot of other menu items that I need to explore and learn to utilize since they all are useful for different contexts and applications. This is now a goal of mine – to try to learn and explore the basics of the other menu items (apps) of the CG50 (and other) graphing calculator, starting with the Physium Menu/app. Here’s what I have discovered:

The Physium application has the following capabilities (so science teachers, take note!!)

Periodic Table of Elements

  • You can display the periodic table of elements
  • The table shows the elements atomic number, atomic symbol, atomic weight and other info
  • Elements can be searched for by element name, atomic symbol, atomic number or atomic weight

Fundamental Physical Constants

  • You can display fundamental physical constants, grouped by category to make it easier
  • You can edit the physical constants and save them as required
  • You can store physical constants in the Alpha memory and use these saved constants in calculations in the RUN-MAT menu/application

Now, I am still not a science teacher, so this would not be a menu item I will use often, but I wanted to do a quick video of what I discovered in my own exploration.  And – there is a link to the how-to guide for the Physium Menu/App for those of you interested in exploring more. If you have a CG10 or other graphing calculator from Casio and don’t have the Physium menu/app, you can download it here.

 

Learning from Webinars

Online learning takes many forms, of which webinars are one. For teachers, webinars are nice because they are usually content or instructional strategy focused and they are relatively short in length, so you can fit them into your busy schedule. Webinars are often live – meaning happening in real time, and you register and sign on at the designated date & time and can interact with the presenter (usually via a chat forum).  This is nice because if you have questions, you can ask them right away. But – the disadvantage here is you have to have the time to sit in on the webinar, which is often not the case, considering our crowded school and personal schedules. If you can participate ‘live’, I highly recommend it. When I worked at Key Curriculum and hosted our weekly webinars, I know the live interaction was a very positive aspect of the learning.

Let’s face it – the reality is it is not always easy to get to a scheduled ‘webinar’, even if it is the most interesting topic in the world. Which is where the beauty of technology helps out – because most webinars are recorded and archived for on-demand viewing (much like our on-demand television binge watching craze!). Most education companies or organizations that host webinars will have them archived somewhere because they WANT you to log on and watch – it’s good for business. There are a couple of good sites listed below that have some educational archived webinars that might spark your interest:

There are more out there, but this should be a great start. And – not to let Casio be outdone, we have many archived webinars as well that focus on integrating Casio technology and math content. These are free and accessible on our Casio YouTube site  We have short how-to videos on this site as well, so the way to determine a longer, content focused (and/or technology focused) video is to look at the time stamps – those that are 20 minutes or longer tend to be the webinars. I’ve included one below on Proportional Reasoning, since this is such a huge issue with students of all ages, and the presenter, Jennifer N. Morris is one of my favorite people and math teachers. Enjoy!!

Exam Mode on the Prizm CG50 Graphing Calculator

As my last post stated, it’s that time of year for standardized testing. As part of this, certain states require that students use calculators that have been set to exam mode. This means that certain features of the calculator have been ‘turned off’ or are inaccessible to students while the exam is going on.  I remember spending hours setting all my calculators to exam mode for students and then having to spend hours undoing that once exams were over – quite a pain.

The beautiful thing about the CG50 Prizm graphing calculator is that you never have to undo the exam mode – it will automatically turn off exam mode after 12 hours. Which means, you can set it, students can take their test, and then next day, the calculator is ready to go again with full functionality restored.  Another nice feature is that when the calculator is in exam mode, you can actually see it on the screen – there is a green highlighted border when in exam mode. This makes it easy to walk around and visually check that the calculators are indeed still in exam mode (or were set to exam mode to begin with, if you have your students do the process for you).

I made a quick video on how to put a CG50 Prizm into exam mode. I apologize for the lighting – very hard to film the actual calculator (vs. emulator) while holding my computer video camera…and those shadows?!!  But – hopefully you can get the gist of things!!

Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out:

Conics – Casio Prizm vs. TI-84+CE

I am currently teaching a course at Drexel University and we are starting a unit on circles. I loved using Sketchpad when teaching because it allowed for dynamic manipulation of objects (shapes, functions) so that students could visually see the impact of variables to the shape, size, position of the object. Unfortunately, my students (math teachers in a Masters Math Teaching Program) do not have access to Sketchpad, though one does use Geogebra, and as this is a course focused on teaching, they need to use what they have access to in their own classrooms with their students. For many of them this does not involve any technology at all, which is sad, but for some, they do have access to graphing calculators.

Naturally, this got me exploring what the graphing calculators could do, and surprise, surprise, I noticed quite a difference between the Casio Prizm and the TI-84+CE graphing calculators, which are the ones my class seems to have. I was investigating conics, and in particular circles, and what options the graphing calculators gave me, especially when thinking about dynamically modifying the variables to see how each impacts the graph of the circle. Here’s is a quick summary of what I found:

  1. Both TI-84 & Casio Prizm can graph conics (circles, ellipses, hyperbola, and parabola, though how to access these conic graphs is different on both.
    • It is more apparent/easy to find on the Casio (there is a Conic Graph menu).
    • TI requires knowing that there is a Conic app in the app menu, which is a button on the calculator. It is not seen from the main screen, and if you don’t know it exists, you won’t know it’s available.
  2. Both provide more than one equation form for each conic.
  3. Both show the graph of the conic, but how is very different.
    • Casio shows the graph on the coordinate grid, where you can see the whole grid, see values on each axis, and identify quadrant and key points on the graph
    • TI shows the graph in the entire window with a weird yellow frame around it. It is difficult to determine where on the coordinate grid the graph appears – there are axis marks, but no values, not origin, making it difficult for students to understand where on the coordinate grid the graph is. Very difficult to identify quadrant and key points on the graph.
  4. Both allow you to enter different values for each coefficient variable,
    • Casio has a modify feature that allows you to see the equation, graph, and coefficient variables on one screen. You can then modify one coefficient at a time and see it dynamically change on the graph, allowing students to visually see how each impacts the graph and see the conic change shape, location, and/or size.
    • TI84+CE only shows the graph or the equation/coefficients – never together.  You have to go back and forth between them when changing values. The TI does not clearly show where on the grid the graph is, does not show a size change (all conics look the same size, but the grid scale is changing). It’s actually very confusing and would be difficult to help student visually see the impact of changing coefficient variables on the size, location and shape.

Below is a video I made showing how to graph a circle and modify the coefficients on both calculators so you can see the differences I am talking about.

Hopefully you will come to the same conclusion I did – Casio Prizm is far superior when graphing conics than the TI84+CE.

Casio Graphing Calculators – Which One’s For You?

It being the start of the school year where everyone is getting their school supplies, one question that gets asked by parents and students seeking to get a graphing calculator is which one should I buy? I’ve already done several posts comparing Casio graphing calculators to TI graphing calculators, so there’s no question when comparing these – buy Casio!  So….now that you’ve made the smart choice to go with Casio, which of the models is the right one for you? What’s the difference, aside from the cost? If you go with the most affordable version, the fx-9750GII, will you be able to do all the things you need to do in your math and/or science courses? What’s the advantage of the fx-CasioPrizm model, that costs a bit more, over the other two?

Great questions – questions we get frequently, especially when we are out at workshops and conferences. The short answer is they will all do what you need in all K-12 courses and on standardized tests (ACT, SAT to name a couple), so you wouldn’t go wrong purchasing any of the three. And, they all follow the same keystrokes, so knowing one means you know the others. But, there are some differences, which might matter to you, depending on your preferences. You can see a complete comparison of all our graphing calculators to each other and to the TI graphing calculators in our program book, pg 16-17.

What I have done in this post is compile a short list of the major differences between the three Casio calculators (Casio Prizm, fx-9860GII, fx-9750GII) and made a quick video so you can see both their similarities and their differences.

Short-List Comparison  (for all the features, refer to our program book, pg 16-17):

Feature Casio Prizm fx – 9860GII Fx – 9750GII
Display 384×216 128×64 128×64
LCD Color High Color Monochrome Monochrome
Storage Memory (Flash Memory) 16MB 1.5MB
Rechargeable Battery Available Yes No No
Exam Mode Yes Yes No
Natural Textbook Display – input/output Yes Yes No
Simultaneous/Polygon Results Yes Yes No
Irrational Number Natural Display Yes Yes No
Modify Yes No No

This is just a few of the features that differ. The obvious one being color in the Prizm, the size of the display, the Flash Memory capabilities. But for the most part, if you check out the complete list of features, you will see that all three they have comparable functionality and many features/functionality that the TI calculators do not. So – if you like color, want more flash memory (for pictures, movies) and the ability to modify one variable at a time, then the Prizm is your choice. If color is not important, but you like the natural display, then go with the fx-9860GII. If your school requires exam mode capabilities for standardized testing, then the Prizm or the fx-9860GII would be your choice. But – the fx-9750GII, for its lower cost, is going to meet most of your functionality needs, so if the extra features aren’t necessary for you, go with that calculator. You won’t go wrong with any of them.

Here’s a quick video showing some of the differences: