# CG50 – What Are All Those Apps?

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.

# Fractions with a Calculator – Looking for Patterns

I have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. DynamicNumber.org for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.

# Math Magic or Calculators?

I was perusing my news feed trying to find something of interest to write about, and came across an article entitled “The Common High School Tool That is Banned in College” i.e. the calculator. It’s an interesting article, worth a read,  basically comparing the high school perspective on the use of calculators to the college perspective or non-use of calculators. There is no right or wrong answer – I think it depends on the math content, what you want students to do (i.e. basic algorithms to solve problems or using mathematics to solve deeper problems).  Depending on your goals, the use of calculators and technology differs. As with any technology, calculators are a resource that needs to be used appropriately, and we need to be teaching that.  Common Core Mathematical Practice #5 – Using Appropriate Tools Strategically is all about this. Calculators have their place and are important to help explore and expand mathematical understanding, but we have to help students understand when their use is necessary and not a ‘crutch’, as stated in the article.

This was on my mind obviously, when I then ran across a tweet post by Go!Math Videos @gomathvideos that shared a TedX talk by Arthur Benjamin entitled “Faster than a Calculator”, which naturally sparked my interest and seemed related to the question of should we be using calculators. In the video, Arthur Benjamin has members of the audience use calculators while he does calculations in his head. He then goes on to wow everyone with his math ‘tricks’ (what he calls mathemagics). He ends by doing a 5-digit square calculation by thinking out loud as he ‘solves’ a problem. It’s fascinating – he changes numbers to words to help him solve – he is definitely using his own ‘algorithm’. The video does not answer the question should we be using calculators – but it definitely shows that calculators are just one way to get a solution and it may not always be the fastest. Anyway – just some fun for this last post of 2016. Enjoy!

Wishing everyone a Happy and Safe New Years!

# Solving Equations with A Scientific Calculator

Solving  equations is a skill that students are expected to be able to do in pre-algebra and beyond. If we look at the Common Core State Standards, these skills actually come into play starting as early as 6th grade, with students expected to solve one-step equations and progressing to systems of equations by 8th grade. An important aspect of solving equations is connecting a real-world context to these and understanding what the ‘solution (s)’ mean in terms of that context.

The use of calculators or technology to help students solve equations is a controversial one at best, and as a math teacher, I do believe that students need to know the processes to solving equations without the use of technology first. But – when we get down to real-world application and problem-solving, the technology becomes a tool that allows students to go beyond just “getting the solution” and to making meaning out of those solutions, and using their solutions to make decisions – which is the ultimate purpose of finding those solutions, right? In these cases, I firmly believe that the use of technology, (more often than not a calculator), is a necessary tool so that students deepen their understanding and are not bogged down in the process of the calculation. Part of the practices – “use appropriate tools strategically”.

As an example, let’s consider a simple real-world context that involves solving a system of equations, something required by the time students reach 8th grade (see Common Core Standards). Let’s say a scientist is mixing a saline solution and has one solutions that is 10% saline and the other 25%. He needs to make a 85 ml bottle that is 15% saline. How much of each of the two solutions should he mix to create the 85 ml bottle of 15% saline? This requires our two equations, with x = the amount of 10% solution and y= the amount of 25% solution.

• x + y = 90 ml
• .1x + .25y = 12.75 (15% of the 85 mL saline)

Perhaps students are actually in science class doing a lab and creating this new solution. While it would be reasonable to do this by hand using substitution, if this is part of an experiment, then using a calculator to get the answer quickly and therefore get on with the experiment might be a more logical step, especially when time is of the essence in classes. I am going to demonstrate on the fx-991Ex how to solve this problem.  I am using a scientific calculator because in middle school, students are more than likely going to have access to these versus a graphing calculator. This video shows how you can quickly solve the simultaneous equations, and also, with the QR code capabilities, also see a graphical representation of the solution.

If a scientific calculator is all your students have access to, remember that they can do a lot more than you might think.  I will explore more features of the ClassWiz in later posts as we continue to explore mathematics and using technology to support learning.

# 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:

# Online Training for Casio Prizm – Get Free Emulator Software!

One of the most frequently asked questions I remember at the NCTM Regionals and NCTM Annual Conventions from teachers was “do you have any training to help me learn to use the the Casio calculators because I have so many students in my math classroom using them and I want to be able to support them?”  The short answer is yes, we do!

There are a couple of free options available on-demand now.  One is the Quick-Start guide that will help both teachers and students navigate and learn some of the basic functionality. We have quick-start guides for several of our calculators which you can find at this link: Quick-Start Calculator Guides. The other option, specific to the Prizm, is a free, self-paced, online course for teachers that let’s you learn about the calculator and specific features by working through modules. The great thing about this course, besides the fact that it’s free, is when you complete it you get the emulator software free for use on your computer for use with your classroom instruction. Can’t beat that deal!  There are of course several other free resources, but these two are a terrific way to get started.

Additionally, we will be offering some upcoming, regionally based workshops where you spend a few hours doing content-specific math activities while familiarizing yourself with the calculator. You leave with a free calculator and read-to-use lessons. All for only \$35, so that’s a pretty amazing deal as well, plus you get some professional development hours and collaboration with other mathematics educators.  Stay tuned for those.

Just like with students, starting the new year for teachers means learning new things and finding supports for your own learning and teaching. Take advantage of both free and inexpensive ways to develop new skills and get some great hardware & software to enhance your classroom instruction. Don’t forget we have a Youtube channel of free videos as well.

# Start the Year with Math Tool Explorations

In my last post, I talked about creating a classroom culture that promotes classroom discourse. The time to begin this is the start of the school year, when students are getting to know each other, getting to know you, and getting use to the ‘learning’ environment. My suggestion was to start these first couple of weeks with activities that are not necessarily content related, but more ‘interest’ related, because students are much more comfortable talking to others, i.e. strangers, about summer movies or vacations than they are about math. There’s no worry about not knowing something or saying the wrong thing – remember the focus in these first two days is to help students learn to communicate with each other, work together, and understand that your classroom is about learning together and everyone has a voice. Those first few times of group work or collaboration, keep it simple – remember the focus is on creating a ‘safe’ place for discussion – listening and sharing with others.  After they’ve had some time, then you begin to introduce content to the mix.

One of the things I liked to do during those first few days of school as part of this classroom culture, was to introduce the technology and other tools we would be using during the year. This allowed students to get familiar with the tools in a non-threatening, simple math focused way, ask questions, get help, and basically become comfortable using the tools. It provided a lot of engagement and interaction as well.

For example, since I used Sketchpad weekly, I made sure that first week of school classes were in the computer lab a couple times and playing around with Sketchpad (I did not have laptops or iPads…just a lab that I signed up for 2 times a week).  It was very informal these first few times, focused on getting familiar with the program and the tools of the software so that when we did eventually use it as part of our content, they’d have had their playtime already and we could focus on using it to learn math. Same thing with protractors, compass, rulers, tangrams, calculators and whatever other tools I might be using for the subjects I was teaching (usually Geometry & Algebra). We would have “explore time”, where students would be given some simple math objects or activities – i.e. angles, sides, shapes, treasure hunts, and work together to use the tools to measure angles, sides, etc. They had to confer with each other if they got different answers, help each other if they weren’t sure how to read/use a tool, and I would also do some review of how the tools were used as well (particularly the protractor).

For the calculators, they would have a scavenger hunt – things like find the on button, graph a line, solve an equation, etc. and they would have a quick-start guide to help them. Even if students had their own calculators that were different, the scavenger hunt still worked and helped everyone learn to use their own calculator.  Just letting them explore on their own and teach each other helped foster collaboration, brought out some class leadership skills, and let provided some valuable time learning tools that saved time in the long run when those tools would be needed for more difficult learning of math concepts.

Today’s classrooms I realize might be a bit more diverse with technology tools – especially if students are bringing their own devices.  But – it’s still important to provide time for students to explore and play with tools. Providing some focused math explorations (i.e. measuring objects or creating a scavenger hunt), even if students have different devices, still allows for them to get comfortable with tools they will be using for learning. It also

allows you to understand the different devices and tools students have and get a handle on where you might need to make some adjustments.

It’s pretty easy to create simple math tool explorations – think of the tool, what will students need to use the tool for, and then create a couple of ‘to-do’ activities. Provide Cheat-sheets for the more advanced tools (i.e. calculators or Sketchpad software for example) that students can reference, but really rely on students figuring it out on their own or working together to figure out how to use a tool. That’s part of the challenge, and part of the classroom culture creation – depending on each other when you are stuck or want to verify your solution.

Examples:

Protractor:

• Create a sheet with several different angles and have students measure
• Give them some specific angle measures that they then need to create the angles
• Give them shapes (polygons) and ask them to measure the angles

• Have them do basic constructions – i.e. draw a circle, make a segment, construct a line, make two lines meet at a 90 degree angle, construct a triangle
• Measure things they constructed
• Play with all the tools and describe what each one does

Calculators (graphing for this example) (provide a reference/cheat sheet, for example: http://www.casioeducation.com/resource/pdfs/PRIZM_quick_start_guide.pdf) (Some very basic sample questions below just to give an idea)

• Where is the on button (describe location)
• How do you turn off the calculator?
• What happens when you push the Graph Menu?
• How do you enter a y= equation?
• Enter 3 and 3/4 into your calculator – change it to a decimal – how did you do it?