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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CG50 – What Are All Those Apps?

As many of you know, I post quick videos in the blog to show different things about the Casio calculators or math or teaching. Many of these are posted on my YouTube Channel. I will occasionally get comments from viewers asking questions, and I do my best to answer them. If I can’t answer the question, I find someone who can, or research until I do have a response. Just the other day, when I was asked “how do you use the constants on the CG-50 calculator”, I was not quite sure what was being asked, since I tend to use the calculator from a mathematics teaching perspective, and hadn’t explored using constants (from a science perspective) and wasn’t even sure what was meant by the ‘constants’ in this particular question (as it could mean the constants in a given equation).  Turns out the viewer was asking about the Physium Menu/App on the calculator, and how to get the constants from these tables and values into calculations. This is something I have honestly never used because I am not a science teacher and therefore rarely, if ever, have need for this app. But – it got me curious and seeking out an answer (which I did find and explore so I could give a reasonable answer).

In my ignorance, I realized that there are many apps on the CG50 (and other Casio graphing calculators) that I have never really explored, not just the Physium App. Mostly I focus on the most-used menu items – Run Matrix (to do calculations), Graph (to work with functions and graphs), Table (functions using table representations), Equation (solving equations), and Picture Plot. But there are a lot of other menu items that I need to explore and learn to utilize since they all are useful for different contexts and applications. This is now a goal of mine – to try to learn and explore the basics of the other menu items (apps) of the CG50 (and other) graphing calculator, starting with the Physium Menu/app. Here’s what I have discovered:

The Physium application has the following capabilities (so science teachers, take note!!)

Periodic Table of Elements

  • You can display the periodic table of elements
  • The table shows the elements atomic number, atomic symbol, atomic weight and other info
  • Elements can be searched for by element name, atomic symbol, atomic number or atomic weight

Fundamental Physical Constants

  • You can display fundamental physical constants, grouped by category to make it easier
  • You can edit the physical constants and save them as required
  • You can store physical constants in the Alpha memory and use these saved constants in calculations in the RUN-MAT menu/application

Now, I am still not a science teacher, so this would not be a menu item I will use often, but I wanted to do a quick video of what I discovered in my own exploration.  And – there is a link to the how-to guide for the Physium Menu/App for those of you interested in exploring more. If you have a CG10 or other graphing calculator from Casio and don’t have the Physium menu/app, you can download it here.

 

Access & Equity in the Classroom – A Teachers Role (Equity, Equality, and Access to Quality Education -Part 3)

This is the 3rd installment in my 3-part series on equity, equality and access to quality education. Here are links to Part-1 and Part-2, where I first define these terms and then I talk about funding issues that impact access and equity. As noted in Part 2, funding is a huge component of why schools and districts don’t provide equitable access to support student needs, and why low-economic areas tend to have inequitable education experiences and poor access to the supports and resources needed to help all students learn and achieve, based on their individual needs.

As a teacher, school funding is out of our hands for the most part (except for the personal funds we all spend to make sure the students in our classroom have resources and support). Parents and community leaders need to take a really close look at the money teachers spend out of their own pockets to address some of the inequities within their own classroom and school – it’s not right, it’s not fair and there needs to be more push-back on education policy and more support from local businesses, community advocates, and state and local school boards to ensure that schools that need funding and resources are getting those in an equitable fashion (remember, not equal, but equitable – all schools do not need the same). Teachers will spend their own money, even when they have very little, because they care about their students and what happens in their classroom, but they shouldn’t have to.

But, I digress.

What I want to talk about in this post is what teachers can do in their classrooms to address equity and access to quality education. Teachers, even without adequate funding, resources and support, are the most able to provide equity and access for the students in their classroom because that is where the learning happens. And it’s the learning, it’s the teaching strategies, it’s those interactions and learning experiences that can provide equity and access for all students. Let’s remind ourselves about what equity and access means – it means each student getting what THEY need to learn, meaning they have access to rich learning experiences and teaching that provides them with the support they need to understand the content, to think, to make connections, to apply that learning, and to achieve to their potential. To learn, despite their gender, their race, their socio-economic status, or their disabilities.

I can only speak from what I know, so I am going to take a mathematical approach to equity and access in the math classroom, but even if you are not a math teacher, these ideas and processes work in your classrooms as well, with the only difference being in the content.

NCTM (National Council of Teachers of Mathematics) has a position for what it means to have equity and access in the math classroom, so I am including it here (this links to the full article):

Creating, supporting, and sustaining a culture of access and equity require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics program and assessing its effectiveness. Acknowledging and addressing factors that contribute to differential outcomes among groups of students are critical to ensuring that all students routinely have opportunities to experience high-quality mathematics instruction, learn challenging mathematics content, and receive the support necessary to be successful. Addressing equity and access includes both ensuring that all students attain mathematics proficiency and increasing the numbers of students from all racial, ethnic, linguistic, gender, and socioeconomic groups who attain the highest levels of mathematics achievement.

This means that all students should be engaged in real-world learning, problem-solving kids-girl-pencil-drawing-159823experiences, and applications of the content. These types of learning experiences are not just for those ‘advanced’ students. This means providing opportunities for students to engage in collaborative learning, where they are communicating their thoughts and ideas with others, where they are taught and allowed multiple approaches and multiple solutions, where they have supports (i.e. questioning by the teacher, partnering with others, hands-on materials, technology/visuals, etc.) that might help them make connections or get to that next ‘aha’ moment.  Lower-performing students shouldn’t be relegated to doing drill & kill worksheets and ‘remedial’ math classes where the focus is on test-taking strategies and memorization, but rather should be exposed to the same challenging problem-based, inquiry approaches as the high performing students, but with different supports to help address their needs (so scaffolded questions, or suggestions on strategies, or working with a partner, etc.).

A large part of this equity and access means teachers need to BELIEVE that ALL students can achieve and learn, with the difference being that some need more supports than others. I can’t tell you how many times I hear, “well, my lower-level students can’t do that” or “my students won’t talk or show me different approaches” or “my students will just wait for the ‘smart’ ones to do all the work’ or “my students have a hard time reading so we don’t do word problems” or “my students will just give up or just ask me to show them the answer”. I could go on, but I think you get the point (and have perhaps made those same comments yourself). It becomes a self-fulfilling prophecy if you think this way, try something once and it ‘fails’, and therefore you don’t do it again – and then you and the students believe they can’t learn, or they can’t talk, or they can’t solve problems, etc. This is where inequity becomes a huge issue in classrooms – because we then resort to teaching students the ‘one way’ to do things (i.e. often the ‘way that’s on the test), and those students who need a different approach or who can’t memorize, can’t ‘perform’ or ‘achieve’ because they are NOT getting what they need to learn, and the cycle continues. To promote equity and access within your own class, you need to do some planning, some hard work up front, and be consistent – but it can change how you teach and how students learn so that all your students are getting what THEY need to learn. As a teacher, this is your responsibility within your own classroom.

cute-children-drawing-teacher-preschool-class-little-40195392Here are some suggestions:

  1. Starting day one, begin creating a classroom culture that promotes communication, collaboration, and respect. Students need to ‘learn’ how to talk with each other and listen to each other – so practice getting them in and out of groups, sharing ideas (start with non-academic sharing first, like ‘what’s the best movie you saw this summer and why”), working with partners and presenting their thoughts. Practice respectful listening. Practice and model appropriate responses when someone might make a mistake (mistakes should be accepted as part of the learning). There are several places to go to help you learn some collaborative teaching strategies – this is a nice list of articles with good tips.
  2. Learn to ask questions instead of giving answers or telling students they are right/wrong or yes/no. Simple questioning skills force students to start thinking, communicating, making connections, asking their own questions. Again, many resources out there to support questioning skills and provide some sample questions (“Why” is always a good one, or “Can you explain?”). Here’s one resource.
  3. Set high expectations and be consistent with those from day one. Expect students to not only show their work, but to explain their thinking (write out in words or draw pictures or explain verbally). Model this when you teach or show things to students (think-out-loud is a great way to model this type of behavior in mathematics class). Consistency is important!
  4. Provide problem-solving strategies from the beginning so that students realize that they have multiple ways to approach an unknown problem or situation. These are great strategies to incorporate in those first couple weeks of school and then to reference as they come up the rest of the year. And yes – even elementary students need problem solving skills.  (Notice & Wonder should become a habit of mind for all students, no matter the age because it provides that ‘think time’ and that ability to try and connect to prior knowledge and use what you know). The Math Forum is a wonderful resource for learning about the strategies and for getting problems to use in class.
  5. Expect and allow for multiple ways to approach math problems. As long as students can justify what they did and it is mathematically sound reasoning/thinking, it should be okay. This is probably the single most important piece to equity in the math classroom – allowing students to solve problems multiple ways, using the strategies and methods that work for them, and allowing for multiple solutions/solution pathways. This is the hardest thing for teachers i think because we ‘know’ the ‘right’ way – but the right way is not the only way, and some students may never get the ‘right’ way, but they have a way and it gets them there and that should be okay AS LONG AS THEY EXPLAIN THEIR THINKING (see #3). To make this work, see #4.
  6. Provide interesting learning experiences that promote thinking, multiple pathways to a solution, even multiple solutions. You will not get students working and communicating if you give them a worksheet with 30 process/skill based problems. You need to find interesting, relevant, problem-solving experiences that engage all students, that allow all students, no matter their ‘ability level’, a way to start thinking about solving. These types of problems should require previous math content knowledge and/or applications of new math content, require some analysis…..so think rich tasks.  There are many resources for interesting problems out there – content-related too – (Math Forum, Mathalicious, YummyMath, Illuminations, links to other resources)
  7. Less lecture, more inquiry, student-based learning. Hands-on, visualizations, student questioning, student explanation. This does not mean you need to have a different activity for every student – that would be exhausting. You need to find learning experiences that address your content that allow all students a way to ‘enter’ the learning from whatever level they are at.

Teaching one way and expecting the ‘same’ approach for all students, no matter the level, will always leave some students behind and others stagnating.Our teaching should always be focused on the standards and content, with the way we structure the learning and the way we allow students to demonstrate their understandings providing the differentiation that will let all students achieve – those who are ‘behind’ learning to catch up and those stagnating able to move ahead and explore. The more students can connect with, engage in, and explain mathematics using what they know  and building on this knowledge, with the teacher guiding them to deeper understanding through questioning, modeling, and supports as needed, the more equitable the learning becomes.

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!!

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!

Exam Mode on the Prizm CG50 Graphing Calculator

As my last post stated, it’s that time of year for standardized testing. As part of this, certain states require that students use calculators that have been set to exam mode. This means that certain features of the calculator have been ‘turned off’ or are inaccessible to students while the exam is going on.  I remember spending hours setting all my calculators to exam mode for students and then having to spend hours undoing that once exams were over – quite a pain.

The beautiful thing about the CG50 Prizm graphing calculator is that you never have to undo the exam mode – it will automatically turn off exam mode after 12 hours. Which means, you can set it, students can take their test, and then next day, the calculator is ready to go again with full functionality restored.  Another nice feature is that when the calculator is in exam mode, you can actually see it on the screen – there is a green highlighted border when in exam mode. This makes it easy to walk around and visually check that the calculators are indeed still in exam mode (or were set to exam mode to begin with, if you have your students do the process for you).

I made a quick video on how to put a CG50 Prizm into exam mode. I apologize for the lighting – very hard to film the actual calculator (vs. emulator) while holding my computer video camera…and those shadows?!!  But – hopefully you can get the gist of things!!