Math Fun in San Antonio – NCTM 2017 Annual

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

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

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

Workshops:

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

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

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

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

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

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

Math and Science Discover the Unseen Planets of Trappist-1 – Now That’s Cool!

I am sure by now you have heard about NASA’s discovery of 7 – earth-like planets orbiting the star called Trappist-1 by using the Spitzer Space Telescope. And, apparently 3 of them could possibly be habitable for life. All of this is amazing in itself, but, what is even more amazing is they discovered these planets without really even seeing them.

What?!  How is this possible? How do they know then that there are even planets if they can’t see them? It all comes down to some amazing technology, some data collection, a lot of math, science and analysis. If you are looking for ways to get your students excited about math and science and real-world applications to answer questions, you need look no further.

While listening to a story on NPR, as usual, an astronomer came on to discuss how these planets were in fact discovered. In his discussion, I was just floored by all the applications of geometry and statistics used in this discovery. When he said they couldn’t actually ‘see’ any of the planets, but instead, used the dimming light of the star, Trappist-1, that these planets orbit around as an indication that there were in facts objects/planets orbiting about the star. So – basically,  looking at the stars brightness from the Earth, the amount of starlight that is blocked as each planet passes across the view of the star was used to calculate the size of each planet.  Based on the amount of dimming, they were able to determine the size of the planets, relative to Earth, with the dips in the stars light indicating how fast the individual planets were orbitting the star. This video below explains the process really well:

The star, Trappist-1, is what they call an ultra-cool dwarf star, which is about the size of Jupiter and significantly cooler than our own sun, and is only 39 light years away from us.  That seems far to me, but apparently in ‘space units’ that’s really close! (Here’s a great problem for students – how many miles would 39 light years represent?) Each planets mass was determined by the amount of tug of each planet on the other. Then, using the size and mass calculations, they estimated each planets density, which then allowed them to extrapolate that six of the planets are probably rocky. Another really interesting thing about all the planets is they appear to be tidally locked, which means the same side of the planet always faces its sun, so one half of the planet is always dark, the other always light. This is based on the length of each planets day, or its spin on its axis (determined by watching the planets for a period of days and seeing how often they crossed the star). The shortest day (compared to an earth day) is 1.5 days, the longest is about 20 days (they still have to collect more data for this last one). I found this great chart on the NASA Jet Propulsion site that compares each of the seven planets (with an artist’s rendering of what they might look like…remember, no one can actually ‘see’ these planets yet)

This infographic displays some artist's illustrations of how the seven planets orbiting TRAPPIST-1 might appear — including the possible presence of water oceans — alongside some images of the rocky planets in our Solar System. Information about the size and orbital periods of all the planets is also provided for comparison; the TRAPPIST-1 planets are all approximately Earth-sized.

This infographic displays some artist’s illustrations of how the seven planets orbiting TRAPPIST-1 might appear — including the possible presence of water oceans — alongside some images of the rocky planets in our Solar System. Information about the size and orbital periods of all the planets is also provided for comparison; the TRAPPIST-1 planets are all approximately Earth-sized.

I find the whole process exciting, interesting, and fascinating. I think students would too and there is so much application of mathematics and science going on here. And, as a certified sci-fi geek, just thinking of the possibilities of other life on those ‘M’ class planets (shout out to my fellow Star Trek groupies) is sparking my imagination. Right now, we don’t have the technology to see these seven planets, but who knows? Maybe a student who explores the math and science behind these now might create that next telescope that lets us see the planets, or the space ship that allows us to travel there? Fun to imagine, and fun for students to explore these ‘brave new worlds where no man has gone before….”.

If you are interested in finding out more about this Trappist-1 discovery, here are some more links:

  1. http://www.vox.com/2017/2/22/14698030/nasa-seven-exoplanet-discovery-trappist-1
  2. http://www.csmonitor.com/Science/Spacebound/2017/0222/Exoplanet-update-Discovery-of-seven-Earth-like-planets-heats-up-search-for-life-video
  3. https://www.washingtonpost.com/news/speaking-of-science/wp/2017/02/22/scientists-discover-seven-earthlike-planets-orbiting-a-nearby-star/?utm_term=.801bc7e5c159
  4. http://www.foxnews.com/science/2017/02/23/keys-to-life-scientists-explain-how-newly-discovered-exoplanets-could-be-habitable.html
  5. https://www.sciencenewsforstudents.org/article/new-solar-system-found-have-7-earth-size-planets
  6. http://www.spitzer.caltech.edu/images/6286-ssc2017-01f-TRAPPIST-1-Statistics-Table
  7. https://exoplanets.nasa.gov/news/1419/nasa-telescope-reveals-largest-batch-of-earth-size-habitable-zone-planets-around-single-star/

A Math Nerd’s Dream Museum

img_3760I went to the National Museum of Mathematics (MoMath) today – what else would I do while in NYC?!!  If you were unaware, this is yet another img_3766great attraction to add to your to-do list next time you are in New York City. I was lucky enough to have a few hours today to myself and thoroughly enjoyed my hands-on experiences – me and several hundred school-age children.

The museum is focused on providing hands-on, interactive img_3782mathematical experiences so students can see, create, and play with mathematics.  There are games, art exhibits, bikes with square wheels to ride, cars to control around a mobius strip, img_3780angles, tessellations, fighting robots, logic puzzles….it was really fun, and there was a lot of ‘learning’ embeddedimg_3775 in all of the exhibits, though I did find I was the only one reading – the kids wanted to just ‘do’. But can you really blame them?

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One of my favorite exhibits when you walk into the museum is the wall of etchings done on metal plates. There are parabolic lights above them that move and due to the angles the metal etchings are at, it appears the whole display is moving and that the etches are 3D when in fact they are flat. The etchings themselves are beautiful – lots of mobius strips in there!!  I tried to capture it on video but it doesn’t do it justice.

Another favorite was the art exhibit showing the amazing geometric img_3770sculptures of Miguel Berrocal – famous for creating sculpture puzzles – i.e. sculptures built by pieces fitting together. There are numerous sculptures on display along with puzzle books showing the steps to build some of the sculptures. There are also two hands-on opportunities to try to build some of the sculptures. I tried my hand at the above sculpture, “portrait de Michele”, which they recreated the pieces using a 3D printer and then provide ‘directions’ to build.  My results are below….I was very proud of myself!

There was a little bit of everything – I made myself into a human fractal tree (that’s me as the trunk if you look really close). And then I made two 3D shapes (sphere and star) by putting together flat plates with 2D shapes (circles and triangles) in a layered order so that they end up looking 3D.  That was a challenge trying to piece the different sized shapes in the right order.

There was a lot more fun to be had – from the square tire bike to the shape challenges to building polyhedra. All in all, a fun-filled few hours doing some math and experiencing students enjoying doing math as well. If you ever get the chance to get to NYC, be sure to include the MoMath in your itinerary!

Numeracy – Skills for Life

I just watched this very interesting, and slightly alarming, TedX talk by Alan Smith. It drew my interest because of the title: Why You Should Love Statistics. Statistics is one of those math topics that I really believe all students in high school should take, yet it is often considered secondary in importance to pre-calculus or Algebra 2. My feelings about Statistics is that it is more important for the majority of students (and adults) because statistics are used daily and without an understanding of statistics, it is possible to be continuously deceived or misled. I think our present day political climate is a clear indication of this.  It all comes back to numeracy and understanding numbers and what those numbers, or data, are telling us about the world around us.

Smith begins his talk with some information about numeracy in the UK and then shows some data from the OECD Survey of Adult Skills (PIAACX2012) comparing the numeracy rates from 12 countries, shown below.

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There is a clear problem numeracy in several countries.

Smith goes on to talk about statistics and how statistics are about us – “the science of dealing with data about the state, or the community that we live in….it’s about us as a group, not us as individuals”. He goes on to show how the way people perceive statistics is remarkably different than the reality of those statistics. It’s much more interesting to actually listen and watch Smith as he talks and shows results, so here’s the talk:

His basic message is we need to be more excited about numbers and statistics in general because it gives information about us. And if we are excited about numbers, we become more engaged in looking at numbers and data which can only improve our understanding of numeracy rates and more importantly, an understanding of us. Something much needed in this day and age.

Global Warming? Let’s Look at Some Data

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

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

 

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

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

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

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

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

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

Fractions with a Calculator – Looking for Patterns

calculatorI have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. DynamicNumber.org for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

Let’s take fraction addition. Obviously, we are going to start with adding fractions with like denominators.  You can put several different problems into the calculator and students can observe both the added fractions and the answers. Students can talk and share what they notice about the multitude of fractions they are adding (all with like denominators). They can make up their own addition problems and see if the pattern or things they notice hold true. Fraction and answers showing up quickly help them discern patterns because they can quickly see many examples, and use ‘funky’ fractions, not just the typical ones we tend to always rely on (i.e. halves, thirds, etc.). It’s even okay that the numerator might occasionally end up larger than the denominator – the pattern still holds true (i.e. the denominator remains the same, the numerators are added together).

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.