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

 

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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.

ClassPad.net – My Math Love-Affair Continues….

I am a lucky woman.

For my almost 30 years in education, I have loved what I do. Teaching math, helping others teach math, finding amazing tools and resources that make learning math engaging and exciting – my ‘work’ is a labor of love. My love-affair with mathematics and teaching has been influenced by many experiences and people and has led me to yet a another new adventure in my quest to help others love and appreciate the beauty of mathematics – Classpad.net,  a free, web-based software that I have been directly involved in, from conception, to development, and now, to public release and hopefully, viral usage!

Some of my Key family

It’s been a weird path of growth, with connections leading to new opportunities, and more connections, and more opportunities. As a new teacher, and also working on my masters at VCU in VA, I worked under John Van De Walle, who started me on the path of making mathematics hands-on and visual and based on problem-solving. This quest led me to look for resources and share my love of math at conferences – sparking my professional development/training itch.

DG5 Groupies!

My search for visualization and hands-on resources led me to a closet in our math department, where I found Discovering Geometry and Sketchpad. And as I used these resources to present at conferences, I got to know and LOVE Key Curriculum and become, I admit, a groupie. This led to getting to know the Key sales folks and being asked to become a Key consultant. All this PD experience led to an administrator job, where, miracle of miracle, all the Discovery books from Key were just being adopted, so I was part of this implementation, which led to meeting Key’s PD trainer, Tim Pope. As a result – lo and behold, this groupie is working for Key!

It was a dream come true! The Key family, one full of former math educators all trying to share the love of mathematics and create inquiry-based, engaging math through great problem-solving and dynamic math technology tools, was amazing. Then – the dream burst, the family split up, and the books went to Kendall Hunt (with Tim), and the technology to MHE (with me).

Heartbreak.

Casio Family

Time to open a new door: I decided to finish my doctorate and branch into the unknown world of education consulting. And that Key family? They are still there – sending connections and opportunities, which is why I now teach at Drexel, work with Casio, travel the world for The Dana Center and Department of Defense Education Activities, among many other experiences.

At this moment in time, my worlds have collided. My Casio family, which is a group of math educators trying to share the love of math and teaching and learning math through dynamic visualization, is inspirational. We’ve worked as a collaborative team, with Casio‘s incredible R&D team in Japan, to create a tool that is going to revolutionize mathematics. It’s everything math teachers want on one page, and it’s just in it’s baby-phase right now with potential for growth that is exciting.

The guys behind booth magic!

Classpad.net has a partnership with Kendall Hunt just recently announced. Those very Discovering Mathematics books I so love will be adding to their power of inquiry by providing our tool as the discovery math tool embedded in the ebooks. My new family is joining with my old family….(and Tim and I are reunited) (and we have a podcast too – 180days Podcast)(shameless plug)!

Right now? It feels like I’ve connected many parts of my life – where many of my previous ‘experiences’ and worlds have joined together. Not sure if this is the circle of life, or a Mobius strip, or maybe an example of a network with many nodes. But whatever it is, it feels right, it feels exciting and it feels limitless.

So, what is Classpad.net?

It is something that makes me proud to be a part of because it is a web-based software, freely available to teachers and students, that encompasses all the things I wished for as a teacher, and it’s all in one place instead of several different tools that don’t communicate with each other. My doctorate dissertation was on edtech, and how teachers have so many technology tools forced upon them (hardware, software, apps, tablets, PC’s, interactive whiteboards, student response systems, etc) and none of them talk to each other, and each require separate training and support. Instead of using any of these tools effectively, teachers use the ones they are comfortable with, and often not the tool that makes the most sense for helping students learn. Or worse, no tools at all.

Classpad.net solves that problem by being a tool where you never have to leave the page – you can do geometry on the same page you are doing statistics. You can add a calculation, you can make a graph – all from one place. You can dynamically show mathematics and students can explore math and make their own discoveries on a table or a laptop or a phone – with the touch of a finger. There is a complete CAS (computer algebra system) engine behind this software, so it’s capabilities and functionality are incredibly robust. We are just in the ‘beta’ stage of release, which is even more exciting because we are really seeking input and feedback from users – what’s not working for you? what do you want? And, just like a start-up tech company, our team is responding quickly and changing based on what teachers and students want and need. The possibilities are endless because we have Casio’s 60 years of worldwide technology expertise and the experiences and input of math teachers building something that can be what teachers and students really need, want, and use – all in one place.

We have a Classpad.net Youtube Channel that we are just starting to build out, but here’s a quick overview of Classpad.net

It’s only the beginning – so check it out. But, as someone who has had a long-standing love affair with math and math technology, this is going to be a fun ride with so much more to come!! Join the fun and start creating with math and sharing your love of math as well on Twitter and Facebook!

Geometry and the Holidays

The holidays are upon us, so of course it makes complete sense to look for geometrical connections. Or maybe that’s just me?

As a geometry teacher (just finishing up a Geometry & Spatial Reasoning course), I am seeing geometry connections everywhere. From the wrapped presents, to the origami ornaments, to the snowflake patterns, I am constantly looking for those real-world connections and easy (and cheap), ways to get students working hands-on with math.

We are all familiar with ‘holiday math’ problems that connect to wrapping presents – i.e. how much wrapping paper do you need, how much ribbon, etc. Area, surface area, linear length connections all very obvious. But, as a geometry teacher, I am also curious about the gift boxes themselves. I know it is often difficult to find 3D models for learning, so boxes provide a cheap way to provide students hands-on explorations of nets, area, surface area, volume. So – teachers – get your students to bring in boxes after the holidays – so much you can do with these!!

Another thought – origami. This time of year, teachers often create holiday decorations with their students with paper-folding, which is fun, obviously, but can also be a great way to apply many math concepts. Shapes, fractions, and transformations for example. Take the following two origami designs – a star and a tree. As you are folding, you could be having students think about the individual shapes, but also the dimensions, the fractional parts after making a fold, what types of transformation have occurred – even congruence and corresponding parts.

money-origami-star-finishedIstep-step-instructions-how-to-make-origami-star-toy-cartoon-cute-paper-steps-84628139

For example, in the star above, after folds #1, what fraction of the square does each smaller square represent? When we fold that triangle in #2, what type of triangle is it? What fraction of the original square is represented in that yellow triangle?  What type of transformation does each fold represent? Are the triangles in #3 and #4 congruent? How do you know?

images (1)step-step-instructions-how-to-make-origami-christmas-tree-illustration-67138886

Again, looking at the tree folding above, what shapes do you see in #1? What fraction of the whole paper is each shape (so squares and triangles)? How about in #2? And which shapes are congruent? How do you know? Lots of great math, that you could really explore with students while they are also doing a fun hands-on activity.

Hopefully you can use some of these ideas with your students. Have a wonderful holiday season!!

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.