The Numbers Behind Fireworks – Math Could Save A Finger or Two

I certainly hope everyone enjoyed their 4th of July celebrations. I know I had a lovely time at the beach with my husband and friends. And, as we were at the shore, naturally we, along with hundreds of our ‘closest’ beach-going celebrants, headed down to the oceanfront with our chairs to enjoy a multitude of firework displays put on by five different beach cities. It was actually really nice because you can see all these shows, with some closer than others depending on where you are, and they are timed so you can see the end of one as another is beginning – about an hours worth of city-sponsored fireworks. At one point in time, I saw our cities show and in the background, with 3 other shows at varying distances away (due to the curve of the shore). I did try to capture it on film, but it was night – with a phone – so not the best of pictures!

While we were waiting, again with hundreds of others on the beach for a good many miles, there were those folks who brought along their own fireworks – sparklers for the children, high-grade fireworks firing off – all in all, very impressive and very scary. Especially as the bangs went off, and the ones on the ground smoked away with children running all around – and then there was the falling ‘sparks’ and debris from those larger ones set off by the water landing on folks all around (setting off some screams). The city firework displays are all set off on barges out on the water, done by professionals. Not so much the ones being set off on the shore – right around hundreds of people. While it was all good and fun, and everyone was celebrating the birth of our nation, it was actually a little frightening as well – considering how many of the ‘fireworks’ almost exploded right by us or went towards the houses instead of out to the water….

Naturally, as is my way, I felt the need to look up some numbers. The National Fire Protection Agency has research numbers specific to fireworks. And it’s kind of frightening really. Perhaps the most frightening one is the sparklers, which all the children were running around with and what I believe most people feel are fairly harmless. This little temperature graph might make you feel a bit differently. We are afraid to let children near pots of boiling water or get too close to a fire, yet we let them run around with sparklers in their hands that are burning at 1200 degrees, almost 6 times hotter than boiling water.  WOW!  That’s an eye opener.  And, as a result, according to the NFPA, sparklers account for more than 1/4 of emergency room fireworks injuries – and who is it that is usually walking around with those sparklers – young children. Just to frighten you a little more with the numbers, the circle graph to the right shows the types of injuries that occur – notice, hand & fingers have the highest chance of injuries, with head and eyes tied for second. Again – think of those kids running around with the sparklers……

If we explore the data a little more, we find some interesting statistics:

So – makes sense, if we look at the graph on the right and the graph in the middle, that because sparkler related injuries are the most prevalent, that kids 5-9 have the highest risk for injury since they tend to be the ones running around with those sparklers. But notice in the circle graph to the left that ages 25-44 actually had more reported injuries, which, based on my own experiences and observations, also makes sense when you look at the type of fireworks that are causing the injuries (graph to the right) after sparklers – illegal firecrackers, small firecrackers, those with re-loadable shells. In other words, this is what ‘the dads’ are doing or the ‘adults’ or, as evidenced last night, the large group of college-age kids. They are the ones setting off the big, scary fireworks on the beach – and getting injured more.

Obviously, not many people think about statistics when planning for some fun on the 4th of July (or New Years or other firework-worthy celebrations). It’s about the fun. But – my guess, especially with parents of younger kids who don’t see the harm of those little sparklers – if you showed them some statistics, especially that temperature graph with sparklers at the top, there might be some reconsidering of the ‘playing with fireworks’ mentality. Math could save a couple of fingers…..

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!

Math and The Electoral College

With the election looming, and yet another Presidential Debate this evening (anyone else dreading it?) and more polls than you can shake a stick at, it seems appropriate to think about the math behind the Electoral College. I admit to really not understanding this whole system – and I know I am not alone. With the rampant conspiracy theories about the November 8 election, and a ‘rigged election’ and cries to eliminate the Electoral College and go to a popular vote only, it had me diving into “what does the Electoral College mean, why do we vote this way, and is it fair?” I think this is a GREAT conversation and critical thinking activity to have with students, especially in classes like statistics where you can actually study and do ‘mock votes’ and see what the outcomes are with or without the Electoral College.

A quick summary of what the Electoral College is – and please note, I still am a little iffy about whether I truly get it. In 1787 the delegates of the Constitutional Congress made the decision to do this indirect way of voting for the President of the USA. It was a compromise between those who wanted a) individual citizens to vote for President (1 person, 1 vote, majority wins); b) letting State legislators choose the President; or c) letting congress choose the next President. The idea was to create a method where the best candidate was chosen. Individuals in a state vote for President – the winner in that state gets all the states electoral votes (though some split the electoral votes now), and the electors (who are elected by voters),  put in the final vote for President. The person who gets the majority of Electoral Votes (270 or more) wins. Still confused? How is this fair?  Bear with me….I am hoping I can figure that out myself!

If you look at the image above, which outlines the number of Electoral votes per state, you can see a huge difference – some states have an enormous number of Electoral Votes, and some very few. As you can see – size might matter (CA, TX, FL), but not always – VA is relatively small and yet has 13 Electoral Votes compared to say Montana, a larger state with only 3 Electoral Votes. So – how is the number of Electoral Votes determined? Hawaii gets 4 vs the very large state of Alaska only 3. So – it must have something to do with population numbers, which in fact is the case. The number of Electoral Votes is the number of state representatives in Congress (both Senate and House of Representatives), which are based on the population of the state. Every state will have at least 3 Electoral Votes (2 Senators, 1 Congressman). Obviously you now see why winning states like CA, TX, FL, PA, and NY are so crucial because of their large populations and large number of Electoral Votes.

I have been reading a lot and searching for good websites that might be helpful for teachers wanting to figure this out with their students. There are several sites that talk about the Electoral College and what it is – I didn’t find these too helpful from a teaching perspective, but they may be of interest to some of you from a historical, “why do we do this” perspective.

  1. Nice interactive map – http://www.270towin.com/
  2. Article about the ‘fair or unfair’ aspect of Electoral College and the funding – not sure it answers the questions but makes you think: https://blogs.scientificamerican.com/guest-blog/the-funky-math-of-the-electoral-college/
  3. This is a nice site with lots of historical perspective and answers to questions, like does my vote count? http://www.learnnc.org/lp/media/lessons/davidwalbert7232004-02/electoralcollege.html
  4. This article makes a case for the Electoral College system being fair: http://www.politico.com/story/2012/12/keep-electoral-college-for-fair-presidential-votes-084651

Here are some sites I found that would be helpful for doing some simulations and having interesting conversations with students. Many of these are interactive, with the ability to create election results (or simulated) to get a better understanding of how the Electoral College system works, and hopefully make a determination about whether it is fair or not.

  1. This was my favorite – be sure to check out the “Play Presidential Politics” link, as it has a simulation vote where you can create your own populations for states.  Would be great for students. http://www.sciencebuzz.org/topics/electoral-college-math
  2. Lots of information (some of which I used) in kid-friendly language: http://www.congressforkids.net/Elections_electoralcollege.htm
  3. Nice lesson here – some interaction/lesson plan info as well: http://www.scholastic.com/teachers/article/math-majority-rules
  4. Yummy Math – nice lesson here (using the picture above!) http://www.yummymath.com/2016/electoral-college-vs-the-popular-vote/
  5. From NCTM – a lesson on the “fairness” of the elections – Love Illuminations! https://illuminations.nctm.org/lesson.aspx?id=2825

So – is it fair or not fair? Does your vote count? I am not sure I can give you a definitive answer. It probably depends who you are, who you want to win, and where you live. But, in my readings looking at several charts that compared the ‘weights’ of individual votes toward the outcome (i.e. Does your vote count?), I think my personal opinion is yes, your vote does count, and yes, it is fair.

Notice Alaska, with a smaller population, has a much more weighted vote compared to CA. This may not seem fair – but, Alaska, a huge state with many diverse needs and interests but with a small population, deserves an equal representation in the government, which may not happen in a one-vote majority rule election, if we look at populations sizes of CA or NY, with their enormous populations. This is why the Electoral College was created in the first place – so every state gets a fair share of representation for their interests in the outcome, no matter their populations size, and those ‘states’ with larger populations don’t end up  deciding everything. NY’s interests are vastly different than Alaska’s, after all. A popular vote would be unfair because those larger states, who lean more one way or another, would control the outcome, leaving those states with fewer people, left out of the equation, and their interests not accounted for or lost in the process. My personal conclusion – I am actually for the Electoral College, after all my reading and my still foggy, but much better, understanding of the system.

One final note – look at the overall United States (in chart above) – the total population, the total Electoral votes, and the weight of each individual vote.  It’s 1.

So YES – your vote counts – get out and VOTE!  (And make it an educated vote, based on candidates proposed policies and plans – not based on emotion).

 

Statistics – Understanding vs. Deceiving

Someone posted a Facebook link to a blog post w/pictures from Ben Orlin‘s blog, Math with Bad Drawings, entitled “Why Not to Trust Statistics”. If you click the link you will see the post is several drawings with stick figures quoting statistics, and then a mathematical representation of those statistics showing how deceiving statistics can be if you don’t in fact have all the data yourself. The point being, if you are given statistical comparisons without the data, the ‘words’ you hear or the graphs/pictures you see may in fact be incredibly deceiving and not accurate.

Here is one illustration from Orlin’s post to clarify this (see rest of his post for the remaining ones):

2016-07-19_11-13-06

 

The warning here – don’t trust statistics without understanding where the data comes from and what the data actually represents. Not something I think is done generally – just listen to the news or read the papers/magazines.  Heck – look at our politicians and the ‘false’ or deceptive statistics (among other things) constantly quoted or visually shown in their speeches. A lot of statistics (verbal and/or visual representations) come with none of the background data or context, so imagine how much deceit – intentional and unintentional – is occurring. (Always fun to check out the ‘fact checks’ of political speeches to realize the spin put on many statistics).

A major problem here is that many people do not understand the statistics – what’s a mean? what’s a range? what’s a mode? what’s a median? Lack of understanding, lack of data, combined with a deliberate spin on the statistics either verbally and/or visually leads to confusion, misrepresentation, bad decisions, believing something to be true when it’s not, and so much more. It’s scary. And only with education can this “lack of understanding” or maybe it’s better to say “willingness to believe what we see/hear” be combated.

Ben Orlin’s illustrations made me think about how we teach statistics – usually with a group a data points or just a list of numbers with little context, which students then calculate the statistical measures and graph the results. But do we spend enough time comparing these different measures ( I am just thinking about measures of central tendency here) or really work with outliers and how these impact the measures (see example above for the ‘mean’ salary). Do we put enough context to these numbers so that the meaning of these measures truly makes sense? Do we provide context, real data, and real opportunities to look at visuals and verbal representations of statistics and make sense of them, in order to help students make informed decisions? My personal experience is no….though with new standards such as The Common Core, I think this might be changing as there is definitely more emphasis on statistics, real-world context, and interpreting and making sense of data. That’s encouraging.

Thinking about students and teaching, here are some visuals (using the Casio Prizm) and the data from Ben Orlin’s example (i.e. 8 salaries from a ‘company’: 7 at $30,000 and 1 (CEO, of course!) at $430,000.  These can really get the conversation going on what is a measure of central tendency, how can the same data reveal different numbers of be perceived differently, and how do outliers impact data and data reporting, how do visuals distort or reveal?

Here are the 1-variable statistics – as you can see the mean (in the illustration above, and the one used to give the “average” company salary of $80,000) is a distorted statistic, since the CEO’s salary (max value) is so much larger than all the other salaries. A better ‘measure’ to use would have been the median or mode, as those are more realistic to this set of data, where all but one person makes that salary.

DispCa7DispCap8

 

 

 

 

 

Visually, if we look at two different versions of a box plot, one with $430,000 as an outlier, the other as part of all the data, you see some funky looking box plots (which would be a conversation all to itself…where’s the box? where’s the whiskers?).  But – in outlier mode, you can see the red outlier is significantly different than all the other data (which is all the same).

DispCap3   DispCap6

If we look at the data with a pie chart, bar graph, or histogram, we also see visually how the $430,000 is an extreme data point.  All of this leading to the question of how an average is sometimes NOT the best statistical measure if you know what the data is and how it is spread.

DispCap1  DispCap2  DispCap5

Statistics is so important and prevalent in so many areas of our society, so let’s make sure we are helping students not only know how to find these statistical measures, but more importantly, help them to question what they see and hear, make sense of the data and understand the potential discrepancies, distortions and misuse of statistics so that they are making informed decisions based on real data and not swayed by a pretty picture or a scary number that is meant to deceive or sway.

Power bills as sources of math questions.

I’ve been thinking a lot about graphs lately, and how in general, many people are deceived by graphs because they don’t understand numbers, scale, sampling size, etc.  In this very contentious political time, it seems many people are fooled by the statistics they “see” graphically.  In my last post, I quoted Dan Finkel’s line “when we are not comfortable with math, we don’t question the authority of numbers”, specifically referencing people’s willingness to believe statistics they see or hear because they don’t really understand where these numbers came from or what they represent.

We can help our students get a better sense of statistics and numbers by providing them as many opportunities to explore, in context, graphs and statistics and ask questions and make sense of these. That could mean exploring all the statistics and poll results currently happening with the presidential election.  Or looking at weather predictions. As I looked at my power bill yesterday, I realized how easy this type of access to real numbers can be, as I stared at the graphical representation of my gas and electric over the past 13 months. (There is also a numerical table showing daily use of kilowat hours (kWh) and 100 (C) cubic feet volume of gas (Ccf). There alone is a whole bunch of mathematical calculation/conversions/ratios).  What I love about my graphical representations is there is a 13 month trend – so I can see where my usage was last year at the same month, and then see how my usage has changed throughout the year.  Below are my December & January graphical representations for both gas and electric usage.

ELECTRIC:

December Electric

December Electric

January electric

January Electric

 

GAS:

December Gas

December Gas

January Gas

January Gas

 

Just from these graphs, there are a lot of assumptions that can be made, and questions that could be asked, that would then lead to more exploration.  For example, December electric from 2014 and 2015 is about the same, but January 2015 is significantly less than January 2016 – why is that? (hint: my children are home for break, so we use more electricity). Gas use in December of 2015 was much lower – was this because it was warmer in December? Are we having a warmer winter than last year? The gas bills seem to show that – but, we could then go look at the weather temperatures for the same time frames in the area I live and see if there is a correlation between temperature and gas usage (i.e. heat). Why is the electric so much more in the spring/summer months and gas is lower? There are so many questions, and, if we brought in the tables of daily usage, cost of kWh and Ccf (volume) we could be doing math calculations, comparing costs, etc. Maybe compare bills from last year to this year and see if the price in oil/gas has had an impact on the overall monthly charge. I like the idea of bringing in the weather and comparing to the electric/gas usage. You can get average weather for the area you live in pretty easily, but it would be even better for students to collect actual temperatures over time and make their own graphs and comparisons.

2016-02-26_12-00-35

Average Climate Chart

The point I am making here is that a simple thing like a power bill can be a powerful tool for visualizing math, doing math, making connections, and asking questions. Or try looking at some statistics from car sales or stocks or polls on the presidential election. It leads kids to ask interesting questions, explore mathematics they care about, and opens them to the real-world aspect of mathematics and how numbers can be used to inform, deceive, and help make decisions.  These types of explorations are interesting and help students become involved in the world around them as well and better prepared for the realities of things like gas bills! Anyway, just another suggestion on how to bring some context into your math instruction in a relatively easy way.

 

 

 

 

 

 

 

 

 

 

The Election Process – Confusing process but applies very real-world math!

I will admit that I find the U.S. election process very confusing – primaries, caucus’, delegates, voting. With the Iowa Caucus’ last night I actually learned something I didn’t know before – that Democrats and Republicans hold their caucus’ in very different ways. Democrats actually physically move around the room until there are viable candidates (those with over 15% of the vote) where as Republicans do secret-balloting and the percentage of votes for each candidate determines how many delegates they receive. I found this great article that helped me understand things a little better – “Caucus math: An NBC Primer” by Carrie Dan and Mark Murray.

Basically, in a nutshell, here’s what happened last night at the Iowa Caucus’.  At least I think….I could still have some misunderstanding, I admit.

Democrats: At each caucus site, Democratic attendees physically went to their candidates ‘corner’ to show their support – i.e. 2016-02-02_11-50-14O’Malley corner, Clinton corner, Sanders corner and an ‘uncommitted’ corner. Attendees can speak out and try to convince others to come to their side, and after all the debate and moving about, a final count is taken for each candidate. Based on the total number of attendees, a percentage for each corner was calculated, and if any candidate did not have at least 15% (making them a viable candidate), then that candidate was out, and their supporters could move to either of the other candidates or go to the uncommitted corner, and new percentages calculated. Based on all the caucus sites, the number of delegates awarded to the candidates is their percentage times the number of available delegates.  So, according to the results last night, Clinton and Sanders had a virtual tie, with Clinton getting 49.9% and Sanders 49.6%, and O’Malley only .6%.  This means Clinton gets 22 of Iowa’s 44 delegates, Sanders gets 21, and apparently the last delegate is “uncommitted”.

2016-02-02_11-49-29Republicans: At each caucus site, there is a set number of delegates being fought over that adds up to the total Republican delegates for the state (there are 30 Iowan Republican delegates). A secret ballot is cast, and a simple formula is used to determine how the delegates are divided up for that site/district.  Basically, the formula is: #of votes for each candidate at the site x ratio of delegates for site/30.  There is rounding done and each candidate ends up with a portion of the available delegates, which then, when all sites report, give a grand total for each candidate.  Last nights results for the Republicans are as follows: Cruz, 8 delegates; Rubio, 7 delegates; Trump, 7 delegates; Carson, 3 delegates; Bush, 1 delegate; Paul, 1 delegate; and everyone else, 0.

 

Here are some graphs that are more visual represented:

2016-02-02_11-50-48                   2016-02-02_11-51-12

 

But this is just a small blip in the overall race to get delegates, which then determines who eventually becomes the official candidates running for President for each party. It is  a long, confusing process but interesting! And then we have the race for presidential votes and the electoral college after that….. My point here is look at all the math!?!  And it’s real-world, and it’s happening right now and has an impact on students lives, so it’s relevant and interesting and can be a great source of real-world application and learning in your math class. Why not today in your classes (or sometime this week), hold a mock class-caucus? One the Republican way, and one the Democratic way.  See who the winners are based on your class results.  Discuss the process – is it fair, unfair? What’s the math? And as each primary/caucus comes along, involve your students and do the math.

I just want to plant a seed. Explore with your students some of the math involved in the election process, no matter what age/grade you teach. Getting students involved in our country’s electoral process in a way that lets them apply what they are learning and feel a part of the world around them can only be a good thing. Hopefully, down the road, it will help them become interested, informed and voting citizens.

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