Math Hardware versus software – Similarities & Differences with Casio

Students using technology as part of learning math is important because of the extension of learning that is possible, the visual connections, and explorations that become possible as a result of technology. The most common technology students use these days are their phones, tablets, computers, and of course, hand-held devices such as calculators. It all depends where you live, what schools you attend, what’s allowed or not allowed, and also what resources are actually available and understood by both teachers and students. From my own research, some schools/teachers have a multitude of resources, but most schools have limited options. And – even if there are many technology tools available, teachers tend to utilize the tool (s) they are most comfortable with, and that the majority of students have access to. Basically, it comes down to choosing a technology that is going to support the learning and that students and teachers can use relatively efficiently, so that time is not lost to ‘tool logistics’. Often times, again, based on my own research (dissertation), teachers choose tools that may NOT be the best choice for learning because they know how to use it over a much better, more appropriate tool, that they are unfamiliar with or uncomfortable with, so many times better technology tools go unused because of the ‘learning curve’.

What I wanted to use this post for today was to show how Casio has really recognized the ‘learning curve’ issue and tried to keep functionality consistent across handheld models and even in their software, providing intuitive steps and menu options right within the graphing menu itself that alleviate some of that ‘learning new tool functionality’ concerns that teachers and students often face when using technology. Our graphing calculators basically use the same steps, buttons, layout, even from the very basic ones (fx9750) (fx9860), to the more advanced ones (CG50), so if you know one, you know them all. And, even the new software, ClassPad.net, is built along the same lines, though obviously with more features and capabilities.  But there is no ‘searching for menus’ – relatively intuitive no matter the tool. Obviously, as you get into the newer models and then into the software, the functionality and options increase – we go from black-and-white displays to color, we go from intersection points on the graphing calculators to union/intersections on the software. But knowing how to use one tool makes transitioning easy, and if you had students with several different models of the handhelds, you could still be talking about the same steps and keystrokes.

The best way to compare and demo is to show you how to do the same thing on the different models. I’ve chosen to show graphing two inequalities, so that you can see, even on the older models, that shading and intersections occur. But also to show that as you progress into the newer and more powerful tools (i.e. memory capacity, color, larger screens, resolution, etc), allowing for more options and learning extensions.

Here are the two inequalities that are being graphed in each of these short GIF’s:

Each GIF below graphs the two inequalities and finds intersection points of the two graphs. The software extends that to allow for finding the Union and the Intersection of all points.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Be sure to check out the free software that does calculating, graphing, statistics and geometry: ClassPad.net.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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New Year’s Resolutions – A Chance to Explore Some Statistics

As I was at the gym this morning, noticing the increase in people that were there, I got to thinking about New Year’s Resolutions. I personally dread the month of January at the gym because inevitably, it is a lot more crowded with all the ‘new memberships’ given as gifts over the holidays, and full of new people who have decided losing weight and getting in shape are on their to-do list for this new year. As someone who hits the gym regularly, this month at the beginning of the year is a bit frustrating because machines are taken, the parking lot is crowded, and my regular routine is often interrupted due to the increase in the number of people. I admire everyone’s new-found commitment and applaud the goal of getting in shape and being healthier – however, my anecdotal evidence over the past several years is that this commitment is short-lived for many.  By February, things tend to get back to normal because, sadly, many of our ‘new years resolution’ folks lose the commitment and stop showing up, allowing the rest of us to get back to our routines.

Which brings me back to my thoughts about New Year’s Resolutions (NYR).

From my own very unscientific observations at the gym, those that made NYR to get in shape, lose weight, etc. usually last about a month – and this is based solely on the increase in people during January, and then the slow decrease in people as the month progresses, to the return to the regular crowd by February (with, granted, a few new ‘regulars’ who stick it out). I wondered, as I was cycling, are there any statistics out there that actually show the follow-through on New Year’s Resolutions – i.e. what were the resolutions made at the beginning of the year, and what was the actual end result at the end of the year?

I was able to find statistics on the most popular NYR made last year (2018)  However, I couldn’t find any follow-up statistics to see how many people in the survey actually stuck to their resolutions, which is what I think would be interesting to explore.

 

 

 

 

 

 

 

 

 

 

 

 

 

I then found another source that listed the 10 most popular NYR’s made for this year (2019).  A lot of the same resolutions, though maybe different priority. Some different ones as well, which could be a factor of many things – i.e. the economy, the political climate, the source of the survey, who was surveyed, etc.

I am curious why there is no follow-up from those that conducted the surveys at the end of the year. It would be fascinating to see what the graphs look like at the end of the year compared to the beginning and why or why not some people dropped off their NYR and some stayed true.  I couldn’t find any ‘proof’ for claims such as “80% of all NYR’s fail by February“, though again, going back to my personal observations, I would agree with this claim. There are definitely a lot of articles about how to ‘keep’ your resolutions, and plenty on why people don’t stick to their resolutions, but no statistics that actually support this claim that I could find. But it would be nice to have some data or evidence that supports observations – which leads me to my final thought on a fun ‘real world’ statistical study that teachers might explore with their students for the remainder of this school year.

During this short week, where school has started up again but students tend to still be in vacation-mode, why not start a long-term study to see if we can get some statistical data about NYR’s? Have students in your class make a list of 3 NYR’s – so some goals they really plan/want to accomplish by the end of the school year. Better yet, pick a specific month and/or date (so May 30 for example). Then, compile the class data to create categories and percentages, similar to the charts above. (My guess is students will have some different things on their top 10 list, which would be interesting in itself). Have students keep a record of their progress towards their goals, and maybe on a monthly basis, do a quick survey on students progress/commitment to their NYR’s.  Then at the proposed deadline, do another survey on the success/failure to see who is still working on their goals and who is not. Obviously it is going to be self-reporting, but it would be interesting, as time goes on, to see who is staying committed, who is not, and more importantly, WHY they are not staying committed if that is the case. Do the class results verify that 80% drop off by February? Is there a common theme for those that do not follow-through on their NYR’s?

I wanted to share this as an idea for teachers who might have made their own NYR to be more creative in their math class. The only NYR I ever made each year was to try at least one new thing in my math classes every month – for me a pretty easy resolution to stick to. I would imagine many teachers do something similar. For those of you who have made NYR, good luck and Happy New Year!

 

Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of Classpad.net (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in ClassPad.net called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation.

ClassPad.net – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links

 

Quadratic Functions – Sample Lessons and Resources

I am starting a monthly feature where I will be focusing on some specific math content areas and providing some resources, in the form of how-to videos (both calculator and Classpad.net) and some ready-to-use math lessons (either PDF or links, depending on the tool used). I know math teachers are always searching for resources that will help them provide more open-ended math activities, where students are collecting and using data, using multiple representations to analyze and solve problems, and where students have to make decisions and support their decisions with mathematics. And integrate technology as well! So, at least once a month I am going to be picking a math content to focus on and provide some technology options as well, sometimes both calculator and online, and sometimes one or the other, depending on content.

This week I would like to focus on quadratic functions and helping students use a real-world context to work with quadratics. I am going to utilize Classpad.net, which is FREE web-based dynamic math software where I can do statistics, graphing, and calculations in one place (geometry as well, but for this activity, our focus does not include geometry). I am using this technology for a few reasons:

  1. It’s free, so all of you should be able to access the created activity, including your students, as long as you have a mobile device with internet access.
  2. I am able to create a complete activity (i.e. directions, tables, graphs, and place for students to show work) in one place and then share it easily via URL.
  3. Everyone who opens the activity can create their own copy of it (as long as you have a FREE account on Classpad.net) by duplicating into their account. Then you can modify, answer the questions, etc. and create it’s new URL to share with others (or for students to share with you). To learn more about duplicating activities, click here.

The Problem

You are fencing in a rectangular area of your yard to create a garden. You have 36 ft. of fencing, of which you plan to use all. You can cut the fencing into whatever lengths are needed, as long as you use all 36 feet. 

What dimensions should you use for your garden?

The Lesson

I have created a shared paper on Classpad.net called Quadratic Functions – Area of a Garden which you can access by clicking on the title. The idea behind this problem is that there are actually multiple solutions since the question is rather vague. I did NOT ask what is the largest garden, so students can work on collecting and analyzing the data and come to different conclusions depending on what they think is important. Some might choose largest area for the garden, some might choose largest perimeter, some might only want a rectangle some only a square, etc. By leaving the question a little more open, you are giving students a chance to explain their reasoning and come to multiple solutions based on this reasoning.

In looking at the activity (click the link above), you will note as part of the lesson, students use multiple representations. They first use their prior knowledge about dimensions of a rectangle, perimeter, area, and an understanding of feet and inches to record different dimensions for the garden. In the directions, students are asked to create at least 10 different rectangular gardens that use all 36 feet of fencing, where some of the width and length dimensions are fractional/decimal numbers and where width is sometimes larger than length. They record their dimensions in a table to start with, and then use those table values to calculate area (and perimeter if they choose to do the Extra Challenge), and use those table values to create statistical plots (scatter plots), and from the scatter plots and tables, create functions and graph those functions to fit their data. At different points along the way (after the table and scatter plots, and then after plotting their functions), students are asked to answer the question about what the dimensions they would choose for their garden and back up their reasoning using the information at that time. The idea here is to help them see that each representation provides insight into the dimensions, and some representations help you be a bit more precise or see the relationships between the quantities a little better. And also, depending on your goal for the garden, your reason for choosing certain dimensions may differ from others. There is also an extra challenge at the end (this is a way to support students who finish early, don’t need as much teacher guidance, and/or want to explore more), where students explore how the problem might differ if there was a fixed perimeter.

ClassPad.net – Lesson In Action

This is a video that shows using the activity and parts of doing the activity to get a feel for how this looks with students. I would recommend students working in pairs or small groups (3-4). All students can be recording on their mobile devices, or if you have one per group, choose a recorder.

Other Quadratic Activities and or video links. 

Here are a few more links that are focused on quadratic functions and also utilize ClassPad.net

 

Complex Numbers – Support for Calculations

I received a question on one of my Youtube video posts on the Casio Fx991 scientific calculator asking if it was possible to do complex number calculations on this calculator. The answer is of course yes – which then prompted me to make a quick video today on exactly how to do that with the fx991. See the video below:

This of course then made me think of our other technologies and that perhaps I should show how to do complex numbers with these tools as well.

Here’s the steps on the graphing calculators (any of the Casio models, since they all basically work similarly – the beauty of Casio, the buttons are relatively consistent). This example uses the CG50, but see fx-9750, fx-9860, etc).

And finally, on ClassPad.net, the FREE online math software that does it all – statistics, geometry, graphing, and of course calculations. (You can sign up for a free account (ALWAYS free) – here’s a quick how-to).

The question of course arises, when are we even using complex numbers? Or why do we need them? As I never really taught math content that required students to utilize complex numbers, I don’t feel I am able to answer these questions with authority, so I did a bit of research. For one, if we just go from a ‘content/standards’ perspective, if you are in states that incorporate The Common Core Math Standards (or a version of, whether renamed or not), then it is actually part of the High School: Number and Quantity standards which state, “Students will…”:

  • Perform arithmetic operations with complex numbers
  • Represent complex numbers and their operations on the complex plane
  • Use complex numbers in polynomial identities and equations

But, that of course doesn’t really get at why do we need them. So here are some things I found in my search for this answer. I admit I can’t explain these any more than just listing them, but it at least points to places where complex numbers are in fact important and needed.

  • Complex numbers are used in electronics to describe the circuit elements (voltage across the current) with a single complex number z=V+iI
  • Electromagnetic fields are best described by a single complex number
  • People who use complex numbers in their daily work are electrical engineers, electronic circuit designers, and anyone who needs to solve differential equations.

Hopefully this is helpful to those of you who are in fact doing complex calculations for whatever reason!

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

#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.