Curve Fitting with Prizm Pictures

I’ve been thinking a lot about the upcoming NCTM conference in April, the theme of which is “Building a Bridge to Student Success”.  I am excited to be heading back to NCSM & NCTM this year after having a years hiatus from math conferences.  Can’t wait to meet up with old friends and colleagues, check out what’s happening in math and math technology, and be a part of a vendor booth again. Believe it or not, I actually like being in the Exhibit Hall – it’s very invigorating and I get to connect with math teachers from all over and find out where the “points of pain” are, to use the words of my friend Stephen Reinhart.

I’ve been involved in the Casio planning for NCTM, so bridges have been a big part of my thinking these last few weeks. With that in mind, I have been looking at the Casio Prizm calculator and the built-in picture resources, and found one that is reminiscent, if not actually, the Golden Gate Bridge in San Francisco. Which led me to playing around with curve fitting and looking for applicable lessons.  There is a whole lesson sampler for the Prizm available free that provides several curve-fitting to picture lessons. You will find not only key strokes for creating the curves and using the pictures, but also questions and resources for the students.

What I love about curve fitting and using real-world pictures is that students are able to see how the math they are learning is actually used and apparent in the world around them. For example – no bridge is built without a lot of math! Prizm has an amazing number of pictures built in that would fit any level of student working with equations and curve fitting – i.e. linear to trigonometric. You can have them plot points and determine their own regression or have the calculator do it, or a combination of both. Lots of options. The point here is that the pictures and line fitting capabilities allow students to problem-solve in a real-world context.  Always a goal in any math class!

I encourage you to check out the Prizm Lesson Sampler yourself. If you don’t have a Casio Prizm of your own, you can test out the emulator free here (fx-CG Manager Plus).  I’ve included a short video showing the basics of accessing the pictures, plotting points, and fitting a regression line.

If you live in Virginia, you can actually attend a free dinner/Prizm workshop in the next two weeks and experience it for yourself. Should be a lot of fun.  Here are the Virginia workshop/dinner dates and links to register:

  1. February 1: Washington County, VA – click here to register
  2. February 2: Roanoke/Salem, VA – click here to register
  3. February 3: Danville County, VA – click here to register
  4. February 8: Rockingham County, VA – click here to register
  5. February 9: Dinwiddie County, VA – click here to register
  6. February 10: Fairfax County, VA – click here to register

Have fun playing!



The snow piled high up the back door….

For those of you in the east coast, particularly from Virginia up through New York, you probably are still digging out from the crazy blizzard that was Jonas. I had about 30 inches at my house, which was wild.  My poor dog took one look and refused to go out – considering the snow is higher than him, you can’t blame him!  I had to dig a tunnel in the back yard, which was no easy task.

Naturally, during the storm there was a lot of news-watching to see what the snow accumulation predictions would be. Also, on Facebook, there were a lot of people posting time-lapsed videos of the snow accumulation from various parts of the country.  My favorite one is posted below – it was posted by Ed Piotrowski of WPDE and shows 40 seconds of snow accumulation of 40″ taken over 27 hours with pictures shot every 2 minutes from a guy named Wayne Bennett’s camera.  Here is the clip:

Of course my first thoughts – wouldn’t this make a great math investigation for students.  How much snow is falling each minute? How does12540761_10208896719685904_6604353696648335492_n it change over time? There are a lot of these time-lapse videos out there, some with actual rulers, where students could actually collect numerical data.  And, now that the storm is over, how long is it going to take for this much snow to melt? Have students look at weather temperatures over the next few days and try to determine melting rates and how long this much snow will take to get rid of. How does rain (predicted in my area tomorrow) impact this? If the snow were rain, how much water is that?  There are a lot of interesting questions and predictions that could be made. Heck – just calculating how many frames were needed to capture the time-lapse would be an interesting math problem.

As I continue to dig out from the storm, I just wanted to share my mathematical thoughts. It’s pretty simple to find real-world math and that sure does make learning math a lot more fun.d


Math in the Movies – Let’s go to Mars

Last night I watched the movie The Martianstarring Matt Damon and directed by Ridley Scott. I really enjoyed it because I am a sci-fi fan, but even more so when I realized how much math and science was in the movie. I mean – it’s all about the math really. (I would like to say, for the record, that it is NOT a comedy, even though Matt Damon won best actor in a comedy at the Golden Globes.  Funny moments, yes, comedy?No)

If you have not seen the movie, do so. I am going to be sharing some links that will definitely have spoiler alerts in them and I will probably even share some spoiler alerts myself, so I suggest not reading if you have not seen the movie.  

As I said, the movie is all about math and science – to survive, to communicate, to pull of a dramatic rescue. It reminded me at times of another great sci-fi space movie, Apollo 13, which also involved a space rescue. As in Apollo 13, there are scenes in the Martian where scientists/mathematicians on earth are testing out theories and calculations on replicas on earth so Matt Damon’s character, Mark Watney, can then perform the same things on the real equipment on Mars. There are also numerous mathematical and science calculations and experiments that Mark Watney does during his long time alone on Mars to try to survive – many of which go horribly wrong, but that’s what makes it so realistic and such a great example for students, because if you mess up, you try again and try a new approach.  And sometimes the obvious solutions aren’t the best solutions.

I am not going to go into all the math and science in the movie, because other people have done a great job of that already. What I am going to encourage is that math and science teachers use the math and science from The Martian in your classroom. Bring the movies into your classroom and see the fun students will have applying mathematics and science in an engaging way with something that is of interest to students. When Apollo 13 came out in 1995, I was teaching 6th grade mathematics on a collaborative team. We did a ‘field trip’, taking students to see Apollo 13 and then doing a lot of math/science/english/history activities to bring what they saw back to the classroom to connect to what they learned. One activity I specifically remember is having students work with partners. One partner had to build a structure (we used Legos) and write down what they did, and then read the directions to the other person and see if the other person could make a replica of what they’d built. The idea here was to emphasize the importance of good instructions, logical sequence, etc. But – it simulated the scene in the movie where the scientists on earth had to basically make a square peg fit into a round hole and create a structure that would clean the air in the module so the astronauts would live.  And they had to tell the astronauts what to do clearly and quickly or they would die, and only through their voice – so good descriptions and logical steps. We did a lot of other activities, but what I remember most is how engaged and excited students were to be using their writing, their math, their science, their history, to expand on and understand the movie.

Here are several of the articles and links I have found related to the math/science in The Martian.  Choose a couple and explore with your students and see how much fun you and they have!

  1. Math of “The Martian”: How It Adds Up, Sarah Lewin
  2. “The Martian” is Full of Math Word Problems, Says Author Andy Weir, by Liana Heitin
  3. Do the Math: How to Survive in “The Martian”, by Kari Tate
  4. The Science of “The Martian”: 5 TED-Ed Lessons to Help You Understand the Film, by Laura McClure


Modeling with Mathematics – Math Practice #4

Whether or not you teach in a state that has adopted the Common Core State Standards for Mathematics (or a modified version of them), the Common Core 8 Standards for Mathematical Practice should be something every math teacher fosters in their instructional practice. These practices are based on NCTM’s processing standards and NRC’s standards for mathematical proficiency in the Add It Up report. They are about helping students become problem-solvers, creative thinkers, communicators, users of multiple resources, and most importantly, able to apply what they know in multiple ways. That’s what teaching math should be about – helping students use what they learn in the world around them, now and in the future.

I am NOT going to get into a debate with anyone on the pros or cons of the CC Math Standards themselves.  That is a politically charged hot mess. Whether you are for or against these standards is irrelevant. How you teach and support student understanding – i.e. your TEACHING PRACTICES, is what makes the difference, NOT the standards you follow in your curriculum. No matter the content standards, they way you help students learn, understand and apply those standards is important and vital, and is what the practices are all about. Lack of how to incorporate effective practices is what I have found, from years of working with teachers, is one of the biggest deterrents in student learning. And, as evidenced in many articles and classrooms I have observed, there is a great deal of misunderstanding of the Standards for Mathematical Practice and how to incorporate them effectively.

In a previous post, I highlighted Mathematical Practice #5, Use Appropriate Tools Strategicallywhere I tried to explain what the practice meant and provide some examples. Today I’d like to do the same thing with Mathematical Practice #4, Model with Mathematics as I think this is one of the most misunderstood, or ‘misused’ practices. I myself, before doing an in-depth study of the practices, interpreted this practice wrong. In my mind, I thought it meant that I should be using manipulatives and ‘models’ (i.e. technology simulations, physical models, etc.) while teaching and I would therefore be modeling with mathematics.  That is part of the standard, but NOT the true purpose.  Remember, these standards for practice are what we, as teachers, are trying to foster in our students – meaning, we are trying to help our students model problem situations with mathematics to help them better understand it and/or solve it.

Let’s look at the actual standard.  I have highlighted key phrases that help clarify this standard:

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Notice the different ways “modeling” is represented: apply the mathematics; write an addition problem; apply proportional reasoning; use geometry; make assumptions and approximations; map relationships; use tools. This standard is not talking about physical models, but rather helping students look at a real situation (so context is important) and use mathematics to help understand it, explain it, make it simpler, relate it to something else, etc. Can they use physical manipulatives? Sure – geometry, tables, graphs, manipulatives – but all of those tools are to help them make sense and model the real world situation. As a teacher, what does this mean YOU should be doing to help foster students ability to model with mathematics? Well, for one thing, give them relevant and real problems to solve that are not “naked math” (see my previous post on this!) but contextual problems that force students to think, analyze and decide what mathematics will help them solve the problem. And then, provide them opportunities to collaborate, use a variety of tools, ask questions, and approach these problems in a variety of ways. Modeling with math means students apply mathematics and tools in ways that make sense to them so they can apply their mathematical understanding. That’s how YOU know they really understand the math, and that’s how THEY know math is relevant.

Obviously, the key here is making sure you provide relevant problems that “arise in every day situations” (as the standard emphasizes). Students learn and are more engaged if what they are doing is relevant.  And there are real problems every where.

lottery-abstract-illustration-dynamically-falling-balls-39580479So – here’s an every-day situation that arose just last night. There were 3 winners in the 1.6 billion dollar Power Ball lottery. Depending on the grade of your students, you could ask them a simple question – should these 3 winners take an annuity or take a lump sum? (Naturally – expect explanations and support for their answers)! In order to answer this question,  a lot of decisions need to be made concerning how to model the situation with mathematics. What, if any, equations would be helpful? Do they need a table to organize the data? What are the taxes and how will that impact the amounts?  These are big numbers – so could they make it simpler by using smaller numbers or proportions? Think of all the great mathematics that will happen and all the modeling with mathematics students will do within the context of this real-life problem. Think of the engagement. Think of the conversations. Hard to get that with a worksheet!




Playing Around with Data

Brothers Sisters Casio ActivityI’ve been exploring the different types of graphs that can be constructed using data lists and the Casio graphing calculators. Data collecting is a powerful way to help students use mathematics in a real-world context. It provides students the opportunity to collect data that is interesting and relevant to them, and then make decisions about that data, such as what graph best supports the data, what questions can we answer from the data, what predictions, if any, can we make, etc. Students apply so many mathematical skills when working with data. What to do with the data once it is collected is obviously a major part of the process, and being able to visualize the data to help answer questions requires students to understand what the different types of graphs mean and show about the data, and, depending on the question asked, which graphical representation is best.

To help me in my exploration, I used one of the activities from our Fostering Mathematical Thinking in the Middle Grades with Casio Technology resource book (Dr. Bob Horton, 2013), as it has some great real-world activities and sample data that allowed me to explore a variety of graphs. Casio calculators can create many graphical representations from a single set of data. All the calculators function the same way, so that’s nice – if I know how to use one, I know how to use them all. Obviously, the Prizm, aside from color, also has some extra features, but no matter which graphing calculator you have (9750GII, 9860GII, Prizm), you can create all these different types of graphs and statistical representations.

The activity I chose, Brothers & Sisters, is one where the data collected from the students in the class is the number of siblings they have, and the two lists created are the numbers of siblings (0 – the highest # in class) and frequency of each.  From this data, we explore box plots, pie graphs, histograms and then measures of central tendency. I have attached a PDF of the activity at the end of this post for those of you who might be interested in trying it with your own students. It includes the keystrokes for the Prizm, but as I said before, all Casio graphing calculators use the same keystrokes, so even the $50 version can do powerful things.

I am not going to explain the whole activity, since I have attached the PDF that you can peruse at your own leisure. But, I did create a short video clip using the 9860GII version of the graphing calculator, to show the steps. I started with sample data already entered so that I could get right to the various graphs more quickly.

Start playing with data with your students, if you have not done so already. Provide students an opportunity to collect their own data, make decisions on how to represent and use the data, and see how much math happens!

PDF: Brothers Sisters Casio Activity

Math Creativity – Blast From the Past

I was clearing out some files to make room for my daughter to store her things, and ran across some saved student projects. I had completely forgotten about these samples of student work and projects from long ago.  The “mathematical poems” – combining math vocabulary and writing and definitions. The monthly problems, where students had to restate, plan, solve, and justify their solutions (very Common Core, I must say!). The Logo symmetry/4-color theorem project.

Student Project2

A Geometry Love Poem

Student Project3

(inside) A Geometry Love Poem










These artifacts from the past brought back some great memories for me and reminded me how much time and effort I put into finding activities, projects, and learning opportunities that allowed my students to be creative and apply mathematics. I remember creating learning centers for my middle school students where at least twice a week during center time, they had choices of activities (problem solving, building/making, technology applications (Sketchpad, calculators), research, tutoring) and created portfolios that demonstrated their learning. When I taught in high school,  for every unit of study my students did an application project, where they built something or wrote something or researched and created something that applied or expanded what they had learned about math that unit. They had choices, they had opportunities to explore and deepen their understanding – to connect to mathematics in fun ways, because to me, learning mathematics was joyful and applicable and I wanted my students to love math like I did.

Student Project4

Monthly Project – restate & plan

Student Project5

Work, answer











It was a lot of work – but it made teaching exciting and different on a daily basis. I will say, however, that it was a challenge to come up with ideas and activities.  Today, there are so many more resources than I had just 10 years ago, especially if you consider all the open education resources and technology resources that were not around back then.  With Common Core and the emphasis on mathematical practices (LOVE THEM!!), the maker movement, the blended-learning movement, there are so many innovative ways that a teacher can create an engaging learning environment. One where students are given opportunities to be true problem-solvers, use their creativity to make something that connects their mathematical understanding, and apply the mathematics to real situations.

Student Project

My blast-from-the-past reminds me again of how creative students can be and how important, as a teacher, it is to provide opportunities for them to explore mathematics in different ways. Go beyond the textbooks and curriculum and find ways to help students create, make, and do mathematics. Challenge yourself and challenge your students.


Real-world Math Applications vs. “Naked” Math

During my past week off celebrating the holidays, while perusing my Facebook feed, two articles that popped up that made me very dismayed about the state of mathematical understanding. One was quite old – from the 1980’s.  Here’s the picture and information that was posted:


(Fractions.  It always comes back to fractions doesn’t it?! )While this was old, the point it was making, onw that is VERY relevant today, is that there are still misconceptions and lack of knowledge about mathematics, in particular, fractions.

The second story was from an incident just last week, where a trucker destroyed a historic bridge because she drove her semi-truck onto the bridge. The bridge had a posted 6-ton limit, and her excuse was, “I didn’t know how many pounds 6-tons was”. Forget the more obvious math discrepancy of her semi-truck having a 30-ton weight and the bridge having a stated 6-ton limit (30 IS NOT LESS than 6..?!?). Clearly there is a lot of math misunderstanding going on here!


What these two stories brought up for me is the need for teachers to really provide context to mathematical problems. When I think about how most students ‘practice’ fraction skills or conversions (the trucker situation), it is still most likely through a drill-and-kill worksheet/app situation consisting of several “naked” math problems (meaning just numbers) with no context attached. In the age of Common Core, where real-world context and application and understanding of what those fraction and conversion skills actually mean, “naked” math practice is a disservice to students. When people encounter math , outside of school, it is almost always in the context of a real-world situation – i.e. is a 1/4 pounder from McDonalds a better deal than a 1/3 pounder from A&W, or will this bridge be able to support a 30-ton truck if the limit is 6-tons. (Or, as the trucker actually asked – how many pounds is 6 tons?)  Rarely is it a “naked” math problem in the real world – i.e. compare 1/4 to 1/3 – there is ALWAYS a context that helps make sense of the problem.

It is so easy to find real-world examples of those mathematical skills we want students to know and understand, so why don’t we always make sure that students are provided a context? It helps deepen their understanding, provides parameters and a need for mathematics (i.e. “When are we ever going to use this?”), and helps students be better prepared for the reality of math around us. Sure, as a teacher, I know it is significantly easier to print out a worksheet with 20 fraction problems, or 30 times tables problems. But – wouldn’t it be better to provide 5-6 real-world context problems that accomplish the exact same things and force students to connect and apply the mathematics in a way that makes sense?

I know I am beating a dead horse here.  “Real-world” context is spewed about everywhere. But come on – a few relevant, truly applicable problems where students have to think and connect math to real situations is a lot more powerful than a page of drill-and-kill, “naked” math. Not to mention more interesting. And – if students don’t get it – the context makes it a lot easier to explain and model than a “naked” math problem.

Go on – stop being naked and get real! Math is everywhere, so go find it and use it.