Math In Motion – Creating Simulations with ClassPad.net

The beauty of dynamic software is the ability for objects to move in real-time and measures and other objects connected and/or controlled by those also move. Basically seeing change over time happen. This allows for the ability to create some interesting simulations – such as simulating cars moving at different speeds and directions to explore rate of change, or objects turning to explore rotational symmetry and angles of rotation. Many possibilities.

Obviously thinking of ways to incorporate simulations into the teaching of more abstract concepts can be time consuming. This post, I am sharing a How-to created by Ismael Zamora, where he shows how to create moving images (using cars) and also provides a couple related ClassPad.net papers if you are interested in the activity he created.

The idea of the activity, Math In Motion, is to have students first Notice & Wonder about the movement of the cars and how they are related, what the sliders control, and is it possible to answer the question of when they will meet?

Here are the links to the publicly shared papers:

Below is a How-to video that explains how Ish created the motion of the cars, involving images, and sliders.

 

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Teachers Rock! Show Your Appreciation in a More Personal Way – Tell Them

It’s National Teacher Appreciation Week, for those of you not in the know. In schools everywhere, teachers are probably getting nice little ‘treats’ from parents and students, or having special lunches or breakfasts brought in, or being treated to free ice cream or nice messages or pep rally’s – lots of things to show how much everyone appreciates the work they do. Obviously these celebrations and expressions of gratitude vary around the country, but there is usually, based on my own personal experiences in middle and high school, some recognition for teachers at some point during this week.  Which is great. Teachers deserve to be told how wonderful they are and what a difference they make in students lives, because they do. They do every day, whether they or you realize it.

It’s the little things that teachers do every day, which often go unrecognized, that really make a difference in students lives and learning. That extra time put in to make a lesson really engaging, that eating in the classroom during lunch to spend time with students who just want to talk or get some help, the personal money spent on supplies and classroom decoration so all students have what they need and to make the classroom a welcoming place, the smile at the door as students enter, the late hours grading, the phone calls to parents to share good news about students (yes, teachers do that!)….there are too many to list here, but every day teachers are providing not only learning experiences, but emotional and physical experiences that help to mold and build students confidence and understanding. This is what I don’t think people who have never been teachers understand – teaching is unlike any other job. You can’t just come in, do the same thing every day, and go home at the end of the work day and forget about it. Teaching is more than teaching content. There is a lot of emotion and dealing with students on so many levels, and navigating that, along with teaching content, makes teaching one of the most difficult jobs out there.

Unlike many other jobs, teachers often never know the impact they had on their students. Sure, we can see grades and scores on tests, but that is a moment in time in a students life, and we don’t often ever know if what we did as teachers has long-term impact (which we hope) as students grow and move on. We think it did. We hope it did. But often, we never know. Unless a student comes back and visits, (or, we are now friends on FB, years later!) – we never really know if the things we thought would make a difference did in fact make a difference. Which makes teaching different from many other professions, who can usually see immediate results or impact of their job. Teaching is a profession of faith – where we believe our efforts are the best we can provide and are something powerful that contributes to our students potential future selves. And though we often never know, we do believe.

What I think would be a really powerful way to show appreciation during this week is for students, current and past, to let a teacher know what it is they are doing or have done that has an impact on them or helped them. Reach out to that Spanish teacher who made class funny, and embraced your obnoxious sarcasm, and influenced your decision to become a teacher yourself, or write that math teacher who helped you survive Calculus and helped you become an engineer, or that teacher who smiled at you every day and gave you a hug so that you loved coming to school. Get your kids to write a note to a teacher (now or in the past) that made school exciting or turned them on to reading or helped them perfect their dunking. It’s those little recognitions’, those personal recollections that really make a teacher feel appreciated and know that what they do is making a difference to someone. Those of you who have been out of school for a while, it’s pretty easy to locate a former teacher via FB or LinkedIn. Those of you still in school, write a note, even if anonymously – it will brighten that teachers day and reaffirm their commitment to teaching.

The U.S. Department of Education has shared some really great videos of teachers sharing what makes them feel appreciated, so I am providing links to those here:

  1. https://youtu.be/dLZXKu8fxnc
  2. https://youtu.be/eqi_kE31tZU

My favorite is what students say about their teachers though, so I am sharing that video here:

 

The Soggy GrassHopper – A Twist on Zeno’s Paradox

This month we are going to highlight an activity called “The Soggy Grasshopper”, which comes from Fostering Mathematical Thinking in the Middle Grades with Casio Technology Ish Zamora (@seemathrun), one of Casio’s ACE math teachers has created a whole lesson package centered around this activity, which includes a ClassPad.net paper, a YouTube video on using the activity, and I am attaching the PDF of the activity for those of you who want things written out and might be using a hand-held device. Though don’t forget, ClassPad.net is a free web-based software where you can do all the math – i.e. calculations, graphing, statistics, writing out explanations….all of which are needed as part of this lesson.

 

The Problem

A grasshopper is on the ground and notices that it is beginning to rain. It wishes to hop to a spot beneath a tree to get out of the rain. It aims for the tree, but finds it can go only halfway on its first hop. As the grasshopper gets wetter, it finds that it can only hop half the remaining distance each time.

In this investigation, students will explore the grasshopper’s journey.

The Math

This problem involves collecting data, so setting up a table, to collect the numbers of hops and distance traveled and the distance left. Students will look for patterns and write formulas for the fraction of the distance remaining. Students will create scatterplots of their data and use the table and graphs to answer questions about the grasshoppers journey.

The Resources

  1. You can find the PDF of the complete activity here: The Soggy GrassHopper
  2. You can find the ClassPad.net paper here: The Soggy Grasshopper (created by Ish Zamora)
  3. You can find the YouTube video on the activity, narrated by Ish Zamora, here: ClassPad.net How-To Activity: The Soggy Grasshopper

Women in STEM – Celebrating Women’s History Month

Yesterday it was announced that mathematician Karen Uhlenbeck . from the University of Texas at Austin, had been awarded the Abel Prize 2019 “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Impressive in itself, but more impressive because she is the first woman ever to be awarded the prize (The Abel Prize was established on 1 January 2002. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003).  A fitting tribute and accomplishment during this month, which happens to be Women’s History Month, which celebrates women’s’ contributions to society and history.

Seems only appropriate to dedicate this post to other significant women and their contributions to STEM, especially as there is still such a need for more women in the STEM fields of science, technology, engineering and mathematics. The more young girls and women see what others have done, the more they are inspired to pursue futures in these fields. I’ve done a little research and pulled together a few names to share in this post. By no means is this an exhaustive list, rather a list of women that sparked my interest, particularly in mathematics, since this has been my personal passion for most of my life. There are many more out there, but the idea of celebrating Women’s History Month is to realize how important, and often unknown/hidden, women have been in many of our STEM advances and historical events.

  1. Marie Curie the only woman to have received TWO Nobel Prizes (one for Physics and one for Chemistry).
  2. Gertrude B. Elion another Nobel Prize winner in Physiology, whose work contributed to many new drugs, including AZT, the aides drug
  3. Augusta Ada King-Noel, Countess of Lovelace – credited with being the first computer programmer!!  Very cool.
  4. Barbara McClintock – Nobel Prize winner in Physiology, credited with showing that genes turn certain physical attributes on and off.
  5. Rachel Carson – credited with creating the EPA (Environmental Protection Agency) as a result of her writings and work.
  6. Radia Perlman – commonly referred to as ‘the Mother of the Internet” for her algorithm (STP) that basically allows the Ethernet to handle massive networks
  7. Rear Admiral Grace Hopper – credited with creating the programming language C.O.B.A.L
  8. Lisa Meitner – part of the duo that discovered nuclear fission (fascinating history here about her being ignored in the awarding of the Nobel Prize)
  9. Katherine Johnson – her mathematical computations influenced every major NASA space project – wow!! (See the movie Hidden Figures)
  10. Florence Nightingale – helped pioneer the field of applied statistics and created a version of a pie chart called the ‘coxcomb‘. Totally new information for me!!

I could go on and on – it is amazing once you start looking, how many women have been pioneers, ‘firsts’, and influencers/contributors to math, science, engineering and technology. It’s exciting that so many are finally being recognized. Inspirational. There are lots of interesting articles and synopses out there that can spark student interest and maybe inspire some of our youth as well. Maybe spend some of this Women’s History Month exploring with your students or just on your own. I know I have been really surprised and amazed and plan to keep researching.

President’s Day – Sample Lesson (Statistics and Measures of Center)

In honor of President’s Day this coming Monday, it seems appropriate for this months’ lesson feature to be center around information about presidents. There is a lot of data that you could explore with students about presidents – age, number years in office, age when they died, number of children, etc. But, I simply chose one statistic – i.e. age at inauguration, to make this really a focused lesson on using a set of data to explore data representations and measures of center.

The Lesson

The idea of this lesson is to help students look at different ways of representing data (table, dot plot, histogram and box-and-whisker plot) in order to make some conjectures and observations. And depending on the representation, really thinking about the information you are able to glean. Can you estimate measures of center from visual representations of data? Is one representation better than another for a given set of data.

ClassPad.net is a great tool for a lesson like this because it allows for students to see all representations in one spot,  continually add on representations, and also provides a place to write down observations and conclusions. You will see that the shared paper (the complete lesson) has explanations, questions, directions all in the one place, and students are doing the math and making their conjectures after each step. It’s an all-in-one, multiple representation activity. Here is the link to the shared paper, that you can use freely with students. If you want to create a duplicate copy to save, you must create a free Classpad.net account and ‘duplicate’ the paper there once you open it. This will create your own copy that you can modify and then share with your own students. Either way, the idea is that students can do multiple representations and measures of central tendency very quickly and make some mathematical connections and conclusions about the given presidential data.

ClassPad.net Lesson In Action

I’ve made a short video that explains the lesson a bit more and walks through the how-to’s of creating the different plots and shows the different aspects of the lesson (i.e. the text stickies, tables, plots, etc.). As usual, you will notice a lot of questions built into the activity, where students are asked to observe, notice, compare and write down their observations. My suggestion for this activity is to do each step and really pause after to allow students individual think time, but then also bring them back together (pair up first, then whole class) to make sure everyone is on the same page before moving to the next step.

Again – here is the shared paper link: ES/MS President’s Day Lesson

Weather and Integers – The Importance of Real World Connections

A lot of my math teacher friends have been posted images from weather reports on FaceBook and Twitter, like this one to the left from @seemathrun, showing the real-world application of integers due to the extreme weather conditions that are happening across the country right now. It really is a perfect opportunity to show a true application of mathematics that students can definitely relate to, especially if they are in those freezing climates. Add in the wind chill, and you have some interesting data and comparisons and a chance to talk about the relevance of math and understanding numbers.  Here’s an image to the left showing wind chill, temperature, and frost bite times someone else shared that could help explain why so many schools are closed, even though there may not be any snow on the ground, (which is usually the reason behind winter closures). I know one of my colleagues and friends, @ClassPadnut, was sharing with me yesterday that with the wind chill, it was -60 where he lives.  Yikes!!!

There is obviously a lot of different math concepts you could explore with students, dependent on grade level and questions asked. I find the wind chill graph the most interesting. Looking at the wind chill chart, the drop in temperature is almost, but not quite, constant, like you would think – i.e. You will note that there is an equation for the calculation of wind chill at the bottom of the image. I was  curious about whether students could find that connection from the data alone -something to challenge students with. How would they graph this data? Could they? Thinking of statistical tables, what would they enter and what statistical plots would be appropriate? If students are in areas where schools actually closed, you could talk about how the data supports the decisions, and what is the ‘cut-off’ temperature/wind speed that might influence the decision? Lots of things to explore.

I found another image that showed the lowest temperatures reported in each state, so you could do a comparison across states. Even Hawaii is cold!!!  Crazy.  Below is the image, which I then used to enter the data in a table in ClassPad.net, and then make two different plots to represent the data – a histogram and a box-plot. You can see from the box plot five-number summary that the median temperature in the U.S. for this day in January is -40.  Wow!!! (And boy, don’t want to be in Alaska at -80!) Again – think of the interesting class discussions about integers, about how these temperatures will impact things such as the orange crops in Florida or the tourism in Hawaii or California. (Here’s a link to the Classpad.net paper that has the image, table, and graphs shown below: https://classpad.net/classpad/papers/share/b61b70a0-0eed-47da-947a-580e1d835f8d.

As you can see, using what is actually happening right now in our country, i.e. REAL world connections of weather (temperature, wind speed, wind chill), is an amazing opportunity to help students see the relevance of integers and statistics and how this data is being used to make important decisions, such as do we close schools? Who should not venture outside? How long before you get frostbite? The visuals help students ‘see’ mathematics in action, and particularly if we focus on the integer aspect, provide a clear connection to integer addition (and subtraction, depending on the questions asked), something many students struggle with.

Whenever possible, we should be trying to connect the math concepts students are learning and using to a real-world application. Here’s a perfect opportunity, no matter the grade level, to have some great class discussions about the impact of weather on our world, about the relevance of integers, and about how statistical information is important to decision making.

 

Math Hardware versus software – Similarities & Differences with Casio

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

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

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

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

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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