The Power of Visualization – Modifying Graphs with a Graphing Calculator

I have had some great discussions with teachers in my courses lately about the power of providing opportunities for students to see and manipulate mathematics as a way to test out their ideas, play with patterns, and develop their own rules and understandings. Visualization, manipulation, experimenting – all contribute to students developing deeper understanding and their own ‘algorithms’, and because of these contextual experiences, they are much more likely to recall how to do a math process than if they were just given the rules/algorithm to memorize.

In a recent final reflection, one teacher wrote, “As a high school teacher, I have always stayed away from using manipulates for fear they were “too elementary” for my classroom.”  This attitude – that older students don’t need those physical objects or need to see – that they just need to  memorize rules and practice – is sadly still prevalent today. Which is frightening really. I experienced these same attitudes and beliefs over 2 decades ago when I was teaching in  middle and high school, and bringing out my two-colored chips, algebra tiles, and Sketchpad. Allowing students to play with math, to use physical objects, and virtual objects, to represent the math and then be able to manipulate change and see what happens was always considered ‘babying’ them. Clearly that attitude is still going strong today, since as you read above,  I hear it in the courses I teach with current classroom math teachers. This despite even more tools being available to provide a way for students to experiment, play, discover, create and find the mathematical patterns and rules themselves. The tendency to just give them the rules and the process and the definitions and have them memorize and regurgitate is still very much a part of our mathematical education. What we really want to do is provide multiple ways to look at and explore math concepts, so that when students ‘forget’, they have that experience where they built the understanding to recall where they can rebuild it again. Much easier to recall something they saw or something they physically moved and connected to than an isolated, memorized fact.

In most typical high school classrooms I visit and work with these days, it is rare to find physical manipulatives (more often in Geometry, but much more rare in an Algebra 2 or Pre-calculus class for example). But – there is almost always a technology tool – whether that be the teachers projector attached to the internet, or students on tablets/laptops, or more often the case, graphing calculators of some sort. Which means there is no excuse NOT to be providing students the opportunity to visually see the mathematics, and to manipulate and explore to come up with those algorithms they are often asked to just memorize. Meaning: use the technology for more than checking answers!  Use it to help students find the patterns and connections and create their own algorithms and definitions, use it to delve deeper into the math, to gain insight, to test out conjectures and really get a sense of what all those numbers and variables mean and how they interact with each other to change the shape of a graph and what that might mean in a application of that math in the real world. Use the tools to manipulate and see the math; technology allows for students to test a conjecture quickly, make predictions and check if they are right, and explore very large and very small numbers, etc.

As an example of this, I am going to use the Graphing Calculator App (for mobile devices), since I haven’t previously used this before in any of my videos, to show the power of visualization and technology to make conjectures and immediately test them with modifying features/dynamic math capability. You can do this on our hand-held Prizm series graphing calculators  (handhelds and emulators).

 

Additional Note: Try our FREE new dynamic math software that is web-based – perfect for tablets, PC’s, mobile devices: ClassPad.net

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Classpad.net Version 1 – Just In Time for School!

Welcome back to a new ‘school year’ (for some anyway). I’ve been on a bit of a hiatus the last couple months, working hard and doing a bit of travel. But, time to get back to it and what better way to start things off but with the launching of Version 1 of Classpad.net.

I posted about Classpad.net back in May, in my post Classpad.net – My Math Love-Affair Continues, This time I want to actually delve much more into what Classpad.net is and share some activities and images to give you a sense of the power of this web-based software. We’ve been in Beta-mode, where we’ve been fixing bugs, working on functionality improvements, and other things while teachers and students have been playing around with the software. Big shout out to all of you who’ve been giving us feedback – we’ve been updating and making changes and fixing bugs in large part to your input. Today is the launch of Version 1, so no longer in ‘test-mode’. Does that mean it’s done? Absolutely not! The beautiful thing about web-based software is that we are constantly improving and updating and adding features. It’s really in its infancy, with so much more growth and functionality and improvement on the horizon, which makes it even more exciting knowing this is only the beginning.

What is Classpad.net?

Great question. At it’s heart, it’s FREE (yes…forever) web-based, dynamic, math software. We call it ‘digital-scratch-paper’ because you can pretty much do whatever you might do when you pull out a piece of paper – i.e. write some notes, do a calculation, make a graph, create a table, draw a picture, measure something. As we know, there are lots of math software and tools out there – but most have specific purposes (i.e. only do statistics, only graph, only do calculations, etc.), so we end up having to use one tool to make graphs, another tool to create geometry constructs, yet another one to do some statistical analysis. And then, if we want to create an assignment for students, we have to use yet another tool to copy-cut-paste our various tables, graphs, constructs, and directions into a usable document. Classpad.net allows you to do all of that on one ‘paper’, which can then be printed (PDF), or shared (unique URL), or saved.  You can send this to students via URL (email or post on your website), students can make their own copy and do their work and send it back to you. It’s all there on one page – and, the beauty is, you can arrange and rearrange things on that paper as you want. To the right is a snapshot of a ‘paper’ showing all the stickies – i.e. text, calculate, graph, geometry, table/statistical plot. You have unlimited scroll and vertical space, and all objects are moveable – arrange and rearrange to your hearts content. You can title the pages and change the banner color to help sort and group content areas.

What Are The Components of Classpad.net?

You can pretty much do all the mathematics you need with Classpad.net for all K-12 curriculum content areas, including Calculus and AP Stats. There are some features that as of today are behind a ‘paywall” (i.e. nominal fee for the add-on app feature), but these are features that most K-12 teachers would NOT want students to have or necessarily need (re: CAS ability, allowing for solving equations or factoring polynomials, as an example; handwriting recognition, and a few others as we add in functionality).  But, here are the general components of Classpad.net, and with each there is a quick GIF showing some aspect of each component:

TEXT – text is just that – you can pull up a text sticky to write directions (for student homework/tests) or descriptions. You can also type in mathematical expressions/equations/terms in the text. Text stickies can be moved and resized as needed, color changes, and you can set a sticky for students to respond to (or students can add their own text sticky to write in answers and reflections as they work on things.

 

 

 

 

CALCULATE – as you would expect, calculate does calculations and so much more. You can define functions and lists, and use them later in graphs and statistical tables. Due to natural display, you can get exact answers. You can use function notation and shortcuts (see the ? at top right of Classpad.net for the function list). And, as with all the stickies, you can move the calculation stickies wherever you need them to be or pull them up whenever needed – all on the same paper.

 

 

 

GRAPH – again, you can graph anything – equations, defined functions, inequalities, integrals, etc. You can create sliders to move graphs and compare functions. You can find area under the curve, click on the graph to see key points, add moveable points to a function plot, look at the table of values, or plot from a table a values, make moveable lines for lines of fit. Comparing graphs is easy too – you can put graphs together or pull them apart to look at things separately. You can have multiple graphs on your paper – either merged or separate. You can add pictures to your graphs as well.

 

 

 

GEOMETRY – Yep, you can even add geometry to your page. We are still building out the geometry component, but right now you can do what you would expect with a geometry tool – i.e. create geometric constructs and specific constraints (perpendiculars, parallels, etc.), measure (area, length, angles, etc.), transformations including dilation, with features that are also unique (so you can construct conics, you can draw free-hand and then ‘adjust’ shapes and objects to have particular constraints. There’s the ability to create a rotational slider. You can create Hide/Show buttons and functions and expressions, and of course typical things like hide objects and change size, colors, etc. I am excited about geometry because I know it’s only the beginning and there’s so much more we are going to be adding.

 

 

STATISTICS – So much to do already, and still so much more to come with statistics. But, what’s the most fantastic part is you don’t have to go get a ‘statistics’ tool for students to be able to collect data, record it in a table, and then analyze that data. This could mean measures of central tendency, or standard deviation, or making different statistical plots to represent the data. Normal distributions, many types of regressions, box-plots, dot plots, histograms…so much there already and we are adding more in the future. As you would expect, we have a spreadsheet that can do calculations or use pre-defined lists (see calculate). You can then add functions to your statistical plots – so everything is all in one place for students to explore and connect.

 

 

As you can see – there’s so much to do, all one one page and one platform (#one-stop-shopping) and it’s free! It’s designed to be usable on touch-screen devices and mobile-devices as well as laptops and PC’s. The perfect tool as you are preparing for this school year, or are just starting your school year (or maybe you are already in-deep to your school year….it’s never too late). Go explore and give it a try and make sure you are letting your student know about this tool. We are also building out our ready-to-use lessons and our video library of support, as we continue to add and improve functionality, so stay tuned. Check out our social media sites for updates and support and we would LOVE to hear from you – share what you and/or your students are creating!!

Check Us Out and Share Your Papers and Experiences:

  1. Classpad.net Youtube
  2. Twitter (@classpadnet)
  3. Facebook
  4. Our website – subscribe so you can start saving and sharing your work with others! Classpad.net

 

 

 

#TheMathContest – Supporting Student Problem-Solving

My last several years in high school, I was a ‘roving’ teacher, meaning I didn’t have a classroom of my own, but switched classes just like the students. This made for a very challenging prep experience, and required me to be super-organized and self-contained on my little rolling cart. The rooms I ‘borrowed’ for my classes did allow me to keep an area for my students (to turn in homework and pick up missing work, etc.). In each class, my students had a portfolio (i.e. file folder), where they kept their work, one of which was the daily ‘warm-up’ problems.  These were basically a set of 5-6 problem-solving activities – some applications, some skills practice, some real-world scenarios, some puzzles, etc.  Students were expected to pick this up daily and work on these in the first few minutes of class, which gave me time to: a) get there; b) check homework; and c) set up for the class/lesson, etc.  It gave everyone a chance to ‘settle’. Students had a choice – they could do some or all of the problems by the end of the week, and I just checked portfolios and work at end of week. We would always discuss possible solutions the following week (and also they earned points for their efforts).

Needless to say, since I was providing these problem-solving experiences daily, I had to find lots of different resources for these problems, especially those that were more application and thought-provoking. Can’t tell you how many problem-solving books I purchased! There were other sources, such as The Math Forum P.O.W. (now no longer in existence, though their P.O.W. “s do still live on at NCTM), and even my textbooks had some great problems if you looked for them. The point is, it took a lot of effort to provide these challenges for my students. Obviously, I could have done it once a week instead, but for me, it served that duel purpose of focusing my students every day while I was en-route. My goal, and something I think all teachers should be striving for, is to provide students some challenges and problem-solving experiences on a regular basis – ones that may utilize prior knowledge or challenges them in different ways of thinking with new skills.

For those of you looking for such challenges, there is a new resource available from Ole Miss’ School of Education called #TheMathContest. It is actually a reboot of something Ole Miss did in the past, but it’s been revamped and improved, and now is sponsored by Casio Education and encourages the use of the new, FREE, online math software, Classpad.net that I talked about in my last post. Basically, new problems are posted each Monday, and each user can submit one answer per hour. Correct solutions earn points and you can view rankings on the website. Go to the link above to get more details on the contest. There are monthly rankings and annual rankings, which you can view online. How points are awarded is explained here.

This would be a great way to engage students and get them doing some challenging math, not to mention trying out the new software as well! If I were still in the classroom, I think I might add this as extra credit for students (for trying) and then maybe have a collaborative problem-solving time where we discuss possible approaches to the solutions after the previous weeks problem has ‘expired’.  Or maybe group students in ‘teams’ where they submit as a team? In any case, it would be nice to have a problem challenge already done for me each week, that’s for sure!

One thing Classpad.net is doing is posting video solutions to past P.O.W.’s which you can find on our Youtube Channel  Here is an example from May 7, 2018’s Problem of The week:

 The Problem:  Find the 1-millionth term ins the sequence {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, ……}

 

There is still time to try this weeks #TheMathContest Problem of the Week for May 14, 2018!  And check out the rankings – you will see students from countries all over the world who are participating.

Casio Scientific Calculator QR Code – The Power of Visualization

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.

Financial App (Pt 3 in series) – Let’s Talk About Money

pexels-photo-164501With 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 stock-photo-working-coffee-phone-work-check-budget-finances-personal-finance-e841754e-765d-426e-af94-4b6a4ce9891fthose 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.

pexels-photo-164527What 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.

Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

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.

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.