I was recently asked on my YouTube video channel if Casio’s graphing calculators also have QR code capabilities like the Casio FX991 ClassWiz Scientific Calculator. It was a great question – and my response was the graphing calculators don’t need that QR code because they already have the power of visualization. The purpose of a QR (Quick Response) code is to get information quickly, whether that’s an audio or a visual or data (usually on your mobile device). With graphing calculators, that is part of the calculator – we can enter data in many forms and see multiple representations of that data very quickly – a graph, a table, a function, specific points, etc.within the graphing calculator itself, making a QR code unnecessary. And, if you are using the graphing software/emulators, you can put these graphs and multiple representations up very quickly.

Why does the Classwiz then have a QR code? This is a scientific calculator, which is incredibly inexpensive (from $15-19), so what’s the reasoning behind including QR code capabilities? The answer – to add the power of visualization and make this calculator have ‘graphing’ capabilities at a fraction of the cost. You can enter data in the form of functions, tables, spreadsheets, and then have the ability to see graphical representations of this data with the QR code.

Here’s a short video that talks about the differences in the graphing calculator versus the scientific calculator and demonstrates the QR code. You will also see a comparison of the tables and graphs represented on both calculators.

I was straightening up my office – something I realized I do not do enough. I found a file of student projects from when I was teaching Geometry over 15 years ago. We had done some geometry poems for Valentines day – i.e. write a poem that utilizes mathematics vocabulary (getting that ELA and creativity flowing in my students), and I had clearly saved a few of my favorites. There were other files of student projects – scale drawings of bedrooms and furniture (so students could ‘rearrange’ their rooms using a scale model), dilation pictures, transformation sketches from Sketchpad, problem-solving portfolios, and designing an aerial view of a city using geometric shapes and properties. As I walked through memory lane, looking at student work from years ago and remembering specific students, it really made me miss those classroom experiences. And what I had forgotten is how incredibly creative and thoughtful students are when given the chance to express themselves – you learn so much about them if you let them, what they know about mathematics, what they think, and what they don’t know if you provide opportunities to approach mathematics creatively.

I’d completely forgotten about the problem-solving portfolios I did with both middle and high school students in all my courses. They were given a choice of problems connected in some way to the math content we were learning or applications of prior knowledge, etc., and they were to choose from several. They had to complete one per unit and put it in their portfolio as examples of their problem-solving and learning/application of mathematics. This was way before the ‘Common Core’, but as I look at my expectations, it was very Common Core like. The idea behind was really very much centered around helping students to persevere and think critically about problems, use problem-solving strategies, and explain their interpretation of a problem, plan out a solution path, justifying their thinking, and showing multiple ways to approach a problem, and analyze their solutions to see if they made sense. Here are the ‘steps’ they needed to go through and demonstrate in their problem-solving:

Restate the problem in your own words, writing out any questions or wondering you have about the problem.

Create a solution plan – what do you think about the problem and why (is it hard, easy, does it seem similar to something you have seen or done before), what math might be needed, what problem-solving approach will you start with and why do you think this might be a good approach? What do you think might be the solution, before you begin?

Work through the problem – include everything, especially if you changed your original plan and why. Write down everything that comes to mind and what you did to think through things.

What is your solution and why do you think this is a reasonable solution?

Analysis of your problem solving – What did you think of the problem after working through it? What did you learn from doing the problem, either about yourself or about math, or both!?

In reading through some of these (I’ve posted some samples below from several different portfolios), you can ‘hear’ students personalities coming out, you can immediately see if they might have a misconception about what the problem is asking or an interesting approach to a solution, or identify those who really needed some extra support because their art work was more substantial then their mathematical work! It gives great insight into who might need some extra support or who might warrant some extra challenges. But mostly – the freedom to choose, think on their own and be creative and work through their problems provided students and ability to express their learning in a different way than an answer on a test. I remember at the time I was considered a rather eccentric MS/HS teacher because I did all these ‘strange’ things like keep math portfolios and journals, use manipulatives, used technology (Sketchpad) and projects instead of tests to demonstrate learning. But – in looking back on the past, and looking at what we want from students today in mathematics, with College and Career Ready Standards and Mathematical Practices, I think it’s the right path. Provide students opportunities to think, choose, be creative, find multiple solutions, justify their answers and question their results. It brings out their creativity and they learn to express themselves as mathematicians.

The holidays are upon us, so of course it makes complete sense to look for geometrical connections. Or maybe that’s just me?

As a geometry teacher (just finishing up a Geometry & Spatial Reasoning course), I am seeing geometry connections everywhere. From the wrapped presents, to the origami ornaments, to the snowflake patterns, I am constantly looking for those real-world connections and easy (and cheap), ways to get students working hands-on with math.

We are all familiar with ‘holiday math’ problems that connect to wrapping presents – i.e. how much wrapping paper do you need, how much ribbon, etc. Area, surface area, linear length connections all very obvious. But, as a geometry teacher, I am also curious about the gift boxes themselves. I know it is often difficult to find 3D models for learning, so boxes provide a cheap way to provide students hands-on explorations of nets, area, surface area, volume. So – teachers – get your students to bring in boxes after the holidays – so much you can do with these!!

Another thought – origami. This time of year, teachers often create holiday decorations with their students with paper-folding, which is fun, obviously, but can also be a great way to apply many math concepts. Shapes, fractions, and transformations for example. Take the following two origami designs – a star and a tree. As you are folding, you could be having students think about the individual shapes, but also the dimensions, the fractional parts after making a fold, what types of transformation have occurred – even congruence and corresponding parts.

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For example, in the star above, after folds #1, what fraction of the square does each smaller square represent? When we fold that triangle in #2, what type of triangle is it? What fraction of the original square is represented in that yellow triangle? What type of transformation does each fold represent? Are the triangles in #3 and #4 congruent? How do you know?

Again, looking at the tree folding above, what shapes do you see in #1? What fraction of the whole paper is each shape (so squares and triangles)? How about in #2? And which shapes are congruent? How do you know? Lots of great math, that you could really explore with students while they are also doing a fun hands-on activity.

Hopefully you can use some of these ideas with your students. Have a wonderful holiday season!!

With the holiday season upon us, and people often spending beyond their means, it seems appropriate to continue the CG50 (and all Casio Graphing calculators) app exploration with the Financial App.

One thing we do not spend enough time on in K-12 education is financial literacy. I know there are some states that are trying to address this, but it is not enough. This lack of understanding about money, savings, taxes, interest, debt, etc. is a huge contributor to our enormous debt crisis. Take our current political focus on the ‘tax reform’ bill that’s up for a vote soon – most people do not understand the ramifications of this because they don’t really understand anything about finances and how taxes work. We do not in this country teach the basics of financial literacy, which is why we have so many people drowning in debt, losing their homes, barely surviving month-to-month on what they make, and forget about having the ability to save for the future. How many students really understand about saving money? Or how taxes impact their hourly wages (i.e. $10/hour is not that great when you factor in all the taxes taken out)? Or how not paying of your credit card monthly can make that $300 dollar purchase become a $400 or $500 dollar purchase?

When I taught in Virginia, they started a Personal Finance course ‘elective’ (only for those students technically not on the college prep track – which was silly, as ALL students should take a course on Personal Finance). I was lucky enough to be the pilot teacher in my school, so I could pretty much create the course. My goal was to help students understand the importance of financial planning so they could survive and thrive in the world, no matter where their path took them. We started with learning about different career options they were thinking of, and what a typical annual salary might be (so plumber, electrician, hair dresser, doctor, lawyer, teacher, etc). They learned to fill out job applications, and write resumes, and then we ‘pretended’ they had been hired and were receiving biweekly payments (I actually gave them ‘checks’). We learned about payments, investing, taxes, rent, credit cards, insurance, amortization,balancing a check book (the class had a ‘bank’), etc. They had to determine where they would live, whether they would get a car, how much they could spend on food, entertainment, etc. based on their salary. What they quickly learned is that their wages, after taxes, were often NOT enough to do much else – no fancy apartment and having to make tough choices (i.e. gas or food, no car, no expensive smartphone, taking bus, walking, no movies every week, no fast food, etc.) When a student comes to you all excited about their $9/hr job and all the things they will buy, and then realize after their first paycheck that it’s going to take months to have enough, it’s eye opening. And scary.

What I learned is that we do not talk to students about real-world, practical mathematics enough – simple things like saving money, calculating tips, balancing a checkbook, interest, credit card debt, etc. This is math they need in their everyday life. This is math that has purpose. This is math that will help them make smarter decisions about their future. Maybe if we did, we wouldn’t have so many people struggling to survive or believing every unrealistic promise they hear in the news..

My message – let’s get some Financial Literacy into K-12 mathematics programs!

With that said, here is a quick video on the Financial App that is available on the Casio graphing calculators. This video uses the CG50.

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x. What would they have to do to the other numbers in order to isolate that 18? This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it. Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’. The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

I am about to start teaching an online geometry course, and it has me missing some of the things I use to do with my students to help them discover relationships, and work with angles and symmetry, which was origami. Origami is the art of paper-folding – and using it in geometry is a great hands-on and visual tool to help students discover angle relationships, symmetry, linear relationships.

Origami is something I am sure most of you are familiar with and maybe have even attempted to create some origami art yourself. I have two friends who are origami wizards and often post their creations on FB – and it’s pretty amazing the shapes they create. When I recently went to the Museum of Math in NYC there was a whole exhibit devoted to Origami.

In my class, obviously, we did relatively simple constructs – folding one piece of paper into things like cubes, birds, shapes. The focus being on the folding and shapes created from each fold and looking at the angles and relationships that developed after each fold. But – as I have discovered, there is some really complex math behind origami, and really complex shapes that are created all from one sheet of paper that are simply astounding. I just found this Ted Talk from 2008 by Robert Langdon that discusses the mathematics behind Origami and how because of mathematics, folds that before were impossible are now possible, allowing for origami constructions that are astounding. Those of you who teach geometry, I think this will be very interesting to you, though I think other math subjects as well will find some applications. At the end of the video there is also a link to some templates for folding some more intricate origami constructs.

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.