# The Power of Math Exploration

If I had a dollar for every time I hear “I would do more hands-on, inquiry, problem-solving, collaborative learning, in math class if I ________________________ (insert any one of the following):

• didn’t have as many students
• didn’t have to get through the ‘curriculum’
• had students who would actually talk
• if I didn’t have to make sure they were ready for the test
• if I didn’t have to review all the things they didn’t learn from last year…..
• ….the list goes on…….

I would be a very wealthy woman. What is mind boggling to me is there is so much research out there that shows students do better when they learn for understanding and not for memorization, which means learning through context, through inquiry, through problem-solving, through struggle. Time is one of the biggest ‘road-blocks’ teachers throw out there, and granted, there definitely is a time crunch to get all the content in before those dreaded assessments. What I try so hard to get across to the teachers I work with, is that you can  save time by taking time – you actually can ‘cover’ more ground by teaching from a more contextual, experiential, problem-solving way. As students make connections and problem-solve, they are able to learn more efficiently and more than one concept at a time because they are working from a connected-math view point instead of the single-skill/concept at a time approach we traditionally provide.

An example from Geometry: (this is using Classpad.net, free math software)

Concept – identifying polygons, and then what’s the difference between congruent-sided polygons versus regular polygons (identifying what a regular polygon is).

Activity: Using the drawing tool, have students draw examples of 3-side, 4-sided, 5-sided (and more….) polygons.  At least 2 of each kind that look ‘different’. Can be convex or concave

• Have students compare their shapes noting similarities and differences and coming up with definitions – attaching specific words to their definitions like convex, concave, closed, etc.
• Now have students use the arrow tool, and select one of their triangles, and the Adjustment menu to make all sides congruent. Then, choose a second triangle and Adjustment and make the shape a ‘regular’ polygon. What do they notice? Have them measure sides and angles and compare to others.
• Do the same for two different 4-sided figures (so Adjust congruent, then adjust regular), the 5-sided, etc.  Each time compare the two on their paper, and then compare to others, and try to come up with what the difference is between congruent-sided polygons and regular-polygons.
• Come to group consensus, and by the end of class students have manipulated, explored, collaborated and defined several things: polygons, convex polygons vs. concave, triangle, quadrilateral, pentagon,….regular polygon, congruent sides, etc.

An example from Algebra: (this is using CG50 Graphing Calculator (CG10 is similar):

Concept: Parent Function and Vertex From of a Parabola

Activity: Students graph the parent function of a Parabola (y=x^2) and then graph another in standard form using variables for coefficients.

• Have students use the modify feature of the graphing calculator to animate the different coefficients (one at a time)
• Observe what changes in that coefficient does to the parabola by comparing the modified to the parent
• Make conjectures and compare with other students till consensus is reached.
• Do this with all the coefficients.
• Have students then test out their conjectures by providing them several equations of different parabolas and, based on their conjectures, determine the shape, direction and location of the parabola BEFORE they do anything, and then test their guesses by entering in the calculator.
• Time saver: Doing this activity with linear equations first will then give students a general understanding of transformations of functions which they then extend and solidify with quadratics, which then can be easily extended into other equations, like the absolute value function. Time saver!

Obviously I am using technology here, because technology allows for conjectures to be made and tested very quickly. But technology is just a tool that is appropriate in some instances, but there’s so much that can be done without technology as well. You can make math much more of an exploration just through your own questioning (i.e. why do you think? can you explain that more? Are there other ways to do this?) and by providing students a chance to puzzle things out on their own, ask questions, use tools (so objects, paper, pencil, etc).

One of my favorite things to do is to provide them with a situation that has lots of information, but no question (basically, find a rich math task, but don’t give students the question(s)). Students then write down all the things they notice, such as quantities, relationships, etc. and then come up with their own wondering’s and questions. Then you let them choose a path they want to explore (this works well with small groups or partners). Usually it ends up that there are several different questions and solutions generated and explored using the same information. When students then share their findings, you find that there is a lot of math going on, which leads to some really interesting class discussions – some you yourself might not have thought of. You can then maybe even give them the question that might have been given in the problem – by that time students may have already explored it and if not, by now they have a real sense of what information in the problem will help them and they are more willing to actually solve the problem.

The key here – students only become problem-solvers if they are given the opportunities to explore math, make their own connections, and collaborate with others to verify their thinking. The more you give them opportunities and provide tools and resources and challenging problems, the more efficient they become at using math, connecting math concepts, and viewing math as a connected whole instead of isolated skills and facts. Take the time….it’ll come back in the end.

# 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.

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

# 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.

# Free Online Casio How-to’s & Content Focused lessons – Great Personal Learning Resources

I am clearly on a ‘what should you do with your summer’ kick, if you look at my previous two posts. But – my belief is that summer, while a time for fun and relaxation, is also a time to brush up on some skills you may be lacking or things you want to learn, find new ideas for the classroom….basically, use the time to foster your own personal learning.

This learning doesn’t need to be expensive, it doesn’t need to be long – it’s all about improving skills or learning new ones. With that said, I thought I would remind all of you, whether you are a teacher, a student, or a parent of a student – if you want to brush up on your Casio calculator skills, we have a lot of free online tutorials and how-to’s that might fit the bill. In my interactions with teachers, I am often asked if we have ‘tutorials’ so that the teachers can support all those students coming to class with Casio calculators (because they are more affordable and much more intuitive to use).  The answer is yes!

In this post I am just going to share some links to our free online resources, and highlight a couple of the videos here as well.

Casio Education has a Youtube Channel where we post previous webinars (so these are longer and actual ‘lessons’), shorter how-to’s, and some quick reference videos and overviews as well. Here’s the link to the Casio YouTube Channel.

A couple highlights here:

Here is an example of a short look at the fx-9860 Stat menu:

Here is a much longer lesson with the Casio Prizm on families of functions:

There are also Prizm specific guided tours at this link.

And I have my personal YouTube channel where I do comparisons and how-to’s on the different calculators, so there might be something of interest there as well.

Here’s a quick how-to using the fx-991 Scientific Calculator to solve systems of equations (and use the QR code):

So – if you have a spare 10 minutes or a spare hour, there’s something for you and we will continually add to these so come back often!

# Using Pictures on the Casio Prizm CG-50 Graphing Calculator

I previously wrote a post a while back about the power of using pictures to connect mathematics to the real world. In that prior post I talk about the built-in pictures that already come with the Casio Prizm Calculator (CG-50 and CG-10), and wrote down the steps. With our new model out, the CG-50, I thought I should probably revisit this but make a quick how-to video instead just to demonstrate how easy it is and show off how many pictures are there.

Currently in my online course I am teaching, we are exploring transformations, and creating some real-world dynamic math examples, so Ferris Wheels have come up. Which got me remembering the Ferris Wheel picture that is one of the many available. Keep snowballing my thoughts, and you end up with me thinking of all the possible applications you could do with the calculator just using that one picture – i.e. what is the angle of rotation for one of the cars to ‘move’ onto another? Why are there concentric ‘circles’ as part of the structure of the ferris wheel – is this a strength issue? What is the length of one radius of the Ferris wheel (in real life – how could you calculate this from the picture? Is similarity involved?) Whats the distance between each car (measuring from the point they are attached on the Ferris wheel – so, arc length?)  And this is just one picture!

There are also ‘movies’ within the Picture Plot menu that allow you to see moving objects and plot their path as well, so again, some real-life connections to mathematical concepts right at your fingertips. As the school year is drawing to an end, this is definitely a time when you want to assess if students can make those connections of mathematics to the world around them, so exploring these types of pictures is a great way to engage students and provide them a reason for why they were learning all those math concepts. (Hopefully you were doing that all along as part of the learning process, but never too late….)

Here’s a quick video on how to access the pictures and ‘videos’ on the CG-50 Prizm, though the process is the same for the CG10 Prizm as well. Have fun exploring!

# Annual ASSM, NCSM, and NCTM – A Week of Math Ed Leadership & Collaboration

Just returning from a week of fun in San Antonio where the annual math leadership and teacher conferences were held. Casio was a proud sponsor of a few events and at NCTM we had such a blast showing off our new graphing calculators (both approved by College Board for use on the PSAT, SAT, & AP exams), the CG-50 Prizm and the CG-500 Prizm CAS (3D graphing anyone?!) Not to mention the added bonus of blowing TI out of the water! (Side note: I will be doing specific posts for each of these in the next couple of weeks showing off some of the new and exciting features).

Thought it would be fun to highlight some of the moments we had sharing math education and technology with the dedicated math leaders and teachers we met throughout the week.

ASSM & NCSM

For the second year, we were honored to sponsor the opening session of ASSM (Association of State Supervisors of Mathematics). Mike Reiners, one of our amazing math teacher leaders and Casio user from Minnesota, provided some technology talking points after the main speaker and then everyone enjoyed some good food and conversation.

At NCSM (National Council of Supervisors of Mathematics) we were able to connect with many math leaders at our exhibit booth. We had a great time sharing our new calculators at our Showcase workshop and everyone walked away with a brand new CG-50 prizm to explore

Benjamin Banneker Association Reception at NCTM

It was a privilege to sponsor the BBA Reception at NCTM for the 2nd year in a row. What a great group of math educators who work so hard to ensure equity for all students. We were excited to continue our scholarship for a deserving student to support their future education endeavors.

NCTM & The Calculator Face-Off Challenge

NCTM was a big endeavor, with game-show stage and podiums, screens, lights, calculator displays. Thanks to the amazing team of Chris and Lionel from Events Special Effects and our own Casio Exhibit gurus John and Jason, the vision was made into a reality and it was a pretty beautiful booth if I do say so myself. Kudos to the team – it’s hard work designing, building and creating everything, but they did an amazing job. Some behind-the-scenes photos:

We had some crazy fun at the booth with hourly game-shows, and T-shirt spotter program where we gave away Kindle-Fire to those spotted in our t-shirts. We had G-shock watch giveaways, calculator prizes for our volunteer contestants and a magician, Mark Paskell, doing some magical give-aways and tricks. (My mind is still blown away by the reproducing bunnies….)

We loved all the connections and interactions we had with math teachers, showing offthe amazing capabilities of all our calculators, but definitely our newest CG-50 and CG-500 graphing calculators. The look on our game-show participants faces when our CG-50 just blew the TI competitor out of the water was priceless. I know I am excited by the number of converts!

Here is a slide show highlighting some great moments from the games, demonstrations, sharing and talking with math educators, winners of our T-shirt spotter program, and some magic as well. Thanks to all the great math educators who came by and participated! Big shout out to our Casio teacher contestants, Jennifer North Morris, Tom Beatini and Mike Reiners.

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