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.

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Learning from Webinars

Online learning takes many forms, of which webinars are one. For teachers, webinars are nice because they are usually content or instructional strategy focused and they are relatively short in length, so you can fit them into your busy schedule. Webinars are often live – meaning happening in real time, and you register and sign on at the designated date & time and can interact with the presenter (usually via a chat forum).  This is nice because if you have questions, you can ask them right away. But – the disadvantage here is you have to have the time to sit in on the webinar, which is often not the case, considering our crowded school and personal schedules. If you can participate ‘live’, I highly recommend it. When I worked at Key Curriculum and hosted our weekly webinars, I know the live interaction was a very positive aspect of the learning.

Let’s face it – the reality is it is not always easy to get to a scheduled ‘webinar’, even if it is the most interesting topic in the world. Which is where the beauty of technology helps out – because most webinars are recorded and archived for on-demand viewing (much like our on-demand television binge watching craze!). Most education companies or organizations that host webinars will have them archived somewhere because they WANT you to log on and watch – it’s good for business. There are a couple of good sites listed below that have some educational archived webinars that might spark your interest:

There are more out there, but this should be a great start. And – not to let Casio be outdone, we have many archived webinars as well that focus on integrating Casio technology and math content. These are free and accessible on our Casio YouTube site  We have short how-to videos on this site as well, so the way to determine a longer, content focused (and/or technology focused) video is to look at the time stamps – those that are 20 minutes or longer tend to be the webinars. I’ve included one below on Proportional Reasoning, since this is such a huge issue with students of all ages, and the presenter, Jennifer N. Morris is one of my favorite people and math teachers. Enjoy!!

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

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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.

DSCF3005At 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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Math Fun in San Antonio – NCTM 2017 Annual

Next week is the NCTM Annual Math Conference in San Antonio, TX.  It’s a great time to go to Texas, as the weather hasn’t gotten too hot. I remember the last time NCTM was in San Antonio, when I was still teaching high school, and met up with all my teacher friends. We had such a great time, not only going to different workshops at the conference, but exploring the area (a trip to the Alamo was a must) and eating and shopping along the River Walk. I am going this year as part of the education team for Casio, where there will be a lot of fun to be had at our exhibit booth and sponsored events.  I have been going to NCTM Annuals for over 23 years (what?????), and as usual, I am looking forward to reconnecting with math educators and friends from all over the country, many of whom I only get to see this one time a year. So, it’s more than just a place to learn new ideas and collaborate with like-minded educators, it’s a time to renew friendships and share memories. I certainly am hoping to catch up with as many folks as I can, even if just to share a cup of coffee or a hug as we pass in the conference hall.

Naturally, the goal of attending a conference is to learn new things to bring back to your classroom or to the educators you work with. It’s one of the aspects of these conferences I love the most – the ‘renewed’ energy and excitement that occurs when you see a strategy that you want to take back to your class or you learn a new approach to a familiar concept that you know will resonate with your students, or you find that perfect resource for an upcoming unit. I always consider these conferences as a way to reaffirm why we teach math – seeing what others are doing, sharing stories and ideas, and leaving with at least one or two ideas that are going to spark your students creativity and understanding. For me personally, I always had a key focus (say Algebra, or Geometry or technology or manipulatives) to narrow down the workshops I went to, with the goal to find a few new resources, ideas and strategies to incorporate into my teaching over the summer so that next years classes would be even better. This type of focus helped to make ‘teaching’  a new adventure every year, even if I was teaching the same subjects, and it also made sure that as a teacher, I was always challenging myself to be better and find relevant strategies and multiple ways to help my students learn.

One aspect that I always look for is technology applications and resources. I am a firm believer in the idea that technology, whether it be a calculator, a tablet, a computer, a video, can be a valuable resource to help students both learn and develop mathematical understanding, but more importantly to visualize abstract concepts and explore ‘what if’s’.  I am sure there are many of you out there as well looking for some technology workshops as you attend NCTM this year, so I wanted to share some workshops from some of the amazing teachers that work with Casio, as these are always such great hands-on experiences.

Workshops:

  • Thursday, April 6 – 9:30 – 10:30, Room 213AB Conv Center: Exhibitor’s Workshop What’s New At Casio: Viewing Mathematics through a New Prizm (or Two) 
  • Thursday, April 6, 3:15 – 4:30, Room 217C Conv. Center: Polar, Parametric, Rectangular Graphs – Really See the Connections! with DeeDee Henderson
  • Friday, April 7, 11 – 12:00, Room Presido ABC (Grand Hyatt): Conceptualizing Polynomials with Jennifer N. Morris
  • Friday, April 7, 1:30 – 2:45, Room 224 Conv. Center: Conics – The Ugly Duckling of Algebra 2 with Denise Young & Tracey Zak Johnson.
  • Friday, April 7, 2:o0 – 3:00, Room 008AB Conv. Center: The Probabilities and Mathematics of “Wheel of Fortune” with Mike Reiners
  • Saturday, April 8, 8:00 – 9:15, Room 006D Conv. Center: Hands-on Activities + Technology = Mathematical Understanding through Authentic Modeling with Tom Beatini

We will also be having a fun time at the booth, Thursday – Saturday, playing games, having give-aways, talking and doing mathematics with our hand-held technology, so be sure to stop by and say hi (Booth #631) and come play with math. I will be there most of the time and hope to meet some new math educators and give a hug to old friends!!

The Power of the fx-991EX – It’s Not JUST Solar

I read the Casio Twitter feed and FB feed every day, just to answer questions and see what followers might be saying. Recently there have been some kudos shared about the fx-991EX solar powered scientific calculator that got me curious. In particular. that the fx-991EX does engineering problems so well and they would be lost without it (someone said he uses it in all his higher-ed courses). This was intriguing to me since I assumed engineers, with their complex calculations, would more likely use graphing calculators like the Prizm or ClassPad or even engineering software.  Naturally, I set out to explore some of the ‘engineering’ capabilities of the fx-991EX, since I hadn’t really spent too much time with this aspect of the calculator.

As I refreshed my memory of the menu and capabilities of the fx-991Ex, it kind of boggled my mind how
much this solar-powered scientific calculator can do, and with it’s QR code capabilities, it can even show graphs and printable spreadsheets and tables. (See my previous posts about Graphing & QR code capabilities). After looking a little more closely at all the menu icons and what each does, I understood why this one calculator would in fact be sufficient for engineers, or really anyone. I spent some time playing around with different features that I had not previously explored, and have shared a couple of my explorations in the video below.

For those of you who have not experienced or explored this powerful little calculator, I suggest you do. If you are at NCTM San Antonio this April, stop by the booth and get some hands-on experience, or just explore some of the videos, or download the free 90-day emulator trial and give it a go.  You can access our Quick-Start Guide to get you on your way.

Global Warming? Let’s Look at Some Data

I realize that I am most likely among the minority of folks when I say I miss snow. I have lived in the Philadelphia area going on 3 1/2 years now, and this ‘winter’ has to be one of the most disappointing ones so far.  I think we’ve seen maybe 3 days of snow – less than 3 inches, and all gone in a couple hours.  I haven’t even had to shovel or scrape the car but one time…. There has been a lot of rain. It’s raining today, and suppose to get to 60. Yep – sounds like spring to me, NOT winter! Where’s my snow? Where’s the sledding?

I grew up in Virginia and spent most of my life in Virginia, where we got a lot of snow – I remember some pretty amazing snow storms and tobogganing down the driveway with my brothers and sister. I then moved to Houston, TX for five years back in 2008 and basically lost any hope of seeing snow or even seasons. There is no real winter….no real spring…definitely no change of seasons in Houston, though it is definitely as hot as people say. When we moved back east to the Philly area 3 1/2 years ago, I was so excited to experience a fall again, and my first winter here we had so much snow, we were actually tunneling our way out.  It was great! Sledding at the castle, power outages forcing us to hunker down at the local bars – snowstorms were fun – even the shoveling brought out the neighborhood and a lot of goodwill!

 

The lack of snow this year, and the weird warm temperatures this winter, where it has felt more like spring than winter, has me thinking about whether this is a normal pattern for the area or is it ‘global warming'(which according to our illustrious leader is a hoax), or is it something else? I think it would be an interesting and relevant real-world investigation for students to look at and analyze and make some conclusions and even some predictions, no matter where they live. My guess is lots of you are experiencing some weird weather patterns this ‘winter’ – i.e. Utah & California for example.  I know the kids around here are disappointed there have been no snow days, so they’d probably love the chance to study the numbers and see if this is an expected pattern and hopefully find a chance of snow still exists.

No matter where you live, weather patterns are a great way to analyze data and apply mathematical concepts. Most countries, states, cities and town keep a historical record of weather data – by year, by month, by day.  There are lots of different measures taken into account – temperature (lows & highs), precipitation (rain and snow), barometer pressure, wind, etc. This data is relatively easy to find as well just by doing a simple internet search. Many sites provide customization, where you can specify month, year and other data that you are interested in looking at. I did a relatively simple search for Philadelphia historical data, and compared the month of January from 2013 to 2017 – here are the numbers:

Granted, a little hard to see, but just in a quick glance, students might note that this past January 2017 we had about 5.59 inches of snow fall compared to 19.41 inches in 2016 (all in one day?!!), 3.9 in 2015, 25.86 in 2014, and 3.75 in 2013. Based on this, maybe it’s every other year that we get a lot of snow? Maybe this has nothing to do with global warming? Is there enough data to make these conclusions? Should we be looking at more months or more years? What about the average high or the average lows for each month? Does that make a difference? There are so many interesting questions and comparisons that students could explore with weather data. As a teacher, you could be applying a lot of things like ratio, proportion, measures of central tendency, different types graphical displays, fractions, decimals, algebra.  It’s a font of real-world data that could be used in so many different ways and in so many different math courses. And students would be interested, especially if you are using data from where they live.  Maybe compare the data to other similar cities or other very dissimilar cities. Do a cross-curriculum investigation – i.e. science, language arts, history.

Depending where you live, you can use weather to help students relate mathematics to their own world and explore their environment while doing math. In CA, as an example, you’ve received a tremendous amount of rain this winter – is it enough to end the drought? How long would that take and how much rain? Interesting and relevant questions students would love to investigate. In Utah, how has all the snow impacted the skiing and tourist dollars coming into the state? In Louisiana, South Carolina, Georgia, Florida – how common are tornadoes in ‘winter’?

Lot’s of questions. Lot’s of data out there ready to explore.

One last question – will there be a big snow storm in the Philly area in the next few weeks? I hope the answer is yes…I need a snow day!