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

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.

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:

4. Our website – subscribe so you can start saving and sharing your work with others! Classpad.net

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.