Pee In the Pool and Other Summer Problems – Problem Solving Resources

As part of my daily brush-up-on education news, I read over my Twitter feed to see what interesting articles or problems the many great educators and educational resource companies I follow might have shared. I laughed so hard when I saw the Tweet from @YummyMath asking how much pee was in the water, with a picture of a large pool and many people in it. Come on – let’s admit it, we have all asked that question at one time or another (especially if you are a parent!!)  It’s a great question. And now I am curious. Where to start? My thoughts are I’d probably need to do some research on the average amount of pee found in a pool and then go from there. The great thing here – Brian Marks from @YummyMath has done that work for me, and even has an engaging ‘lesson starter’ video to go along with the lesson (link to the lesson). So – this would be a really fun problem to start out with that first day of school – funny, lots to notice and wonder about, getting ideas from students on where to begin, what information they might need, etc.

In an early post this summer, Summer Vacation – Use Your Experiences to Create Engaging Lesson Ideas, I talked about how your own summer experiences could raise questions and interesting problem-solving experiences to bring back to the classroom. But – as the tweet from Brian Marks @yummyMath reminded me, there are other amazing educators and resources out there who are already thinking of these questions and even creating the lessons for you. No need to reinvent the wheel, as they say – if there are some interesting questions and resources already being posed and shared, then use them. Saves time, maybe provides some ideas you hadn’t thought of before, or maybe it takes something you did think of and provides some questions or links that you hadn’t found yourself. As educators, we need to really learn to collaborate and share our expertise so that we are not individuals trying to support just our students, but we are educators trying to work together to improve instructional practices and student achievement. Isn’t that what we try to stress within our own classrooms – i.e. working together, communicating, and sharing ideas because this leads to better understandings and new approaches? Same goes for our teaching practices and strategies.

Here are some fun problem-solving resources, with lots of different types of problems, but definitely some ‘summer-related’ things already started for you!

  1. YummyMath – (check out the ‘costco-size’ beach towel activity….that’s funny!)
  2. Mathalicious – (Check out the ‘License to Ill’ lesson – relevant to todays’ debate on Health Care & Insurance)
  3. Tuva|Data Literacy (Check out their lessons and their technology for graphing and analyzing data, and their data sets – so much here!)
  4. RealWorldMath
  5. TheMathForum
  6. Illuminations 
  7. Center of Math
  9. MashUpMath


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.


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!

Basketball Math

It seems appropriate to focus a little bit on basketball since we are in the midst of “March Madness” and the final four looming this Saturday, April 2.  I myself 2016-03-29_14-33-42have a difficult time staying excited when my team, Virginia Tech, was not even in the tournament, though, our arch rival, Virginia, did make it to the Elite Eight. So – home-state pride and all!!  As a math teacher, events like the NCAA Basketball tournament, provide opportunities to connect mathematics to real-world happenings. Students tend to be excited to learn and use the mathematics in context because they are either watching the games or at the very least, aware of them so there is a connection. With these types of current events going on, it provides an opportunity to research and collect relevant data and explore numbers in a variety of ways.

Statistical measures and comparisons are the obvious mathematical focus here, though not the only one.  You could explore some physics, such as what’s the ideal arc and location to make a 3-pointer (or a foul shot), or some geometry, such as what’s the ideal volume of a basketball to create the perfect bounce (if there is such a thing).  Having your students come up with their own questions and then do some research and get their own data would be a fun exercise all by itself. Here are just a few sites with NCAA 2016 statistics you could use to support your students questions:

2016-03-29_15-26-59Obviously, you could use these statistics in several ways, but for me, the easiest, and most efficient way to get students asking questions and then use the data they find to help answer those questions is to a) provide them access to data; 2) provide them with technology tools to explore the data; 3) allow them to explore and make conjectures; 4) have them share out their findings and justify their conclusions.  (Very Common Core!) If you are like me, where access to technology in the classroom consisted of a projector screen and computer, and then calculators, getting the data becomes the biggest hurdle.Printing out data from sites (like those above) is one way around that, though a bit cumbersome and it does not allow for student-driven questions, since the data you print may not be what they are questioning. You could have students in groups, and give each group time to formulate some questions first, and then provide each group some time on the computer to search for data that will help support their questions. Some of you may be lucky enough to have a few tablets or computers in your classroom for students, or allow students to access their smartphones/internet to do searches.  In that case, each group can do their own searches on devices within their group, which will make the search process easier.  No matter how students search for and gather data, having calculators for each student would be important, since they can then enter the data and explore quickly and make conjectures.  Yes, yes, you are right – you could just have them do everything by hand. But – what is the point of all this research and data collection? Is it just to find measures of central tendency?  Is it just to plot a histogram? No – it’s to use the data to answer interesting questions, make comparisons, and explore. The use of the calculator (or other technology if you have it) opens the door for students to explore.  They can compare multiple teams or players or test out ideas, or plot points to come up with equations, show multiple graphs from different teams on one grid. Lots of things that ‘by hand’ become cumbersome and tedious and defeat the purpose of this type of mathematical activity. I personally loved using a graphing calculator mainly because every student had access to one. This allowed everyone to be involved and they could work on separate, but related questions or they could work together to verify their conclusions.

Hopefully this gives you some things to think about and maybe try with your students while the NCAA is still going on. Happy calculating and if you have entered one of those work-pools for predicting the winner, hope you are beating the odds and your brackets are paying off!


Histograms with ClassWiz & QR Codes

Students should engage and be hands-on with mathematics as much as possible. One of the activities I loved was the Sum of Two Dice, whether in my middle school classes or in my Algebra classes. That’s the great thing about mathematics – you can take an activity/concept and make it more or less rigorous depending on the questions you ask.

I am sure many of you have done this activity – I am choosing it for this post because it’s a nice way to 2015-12-10_11-23-28demonstrate how the Casio ClassWiz (fx-991EX) scientific calculator allows you to create frequency tables and with the QR code, see an online visualization of the data.

First thing – have students roll two dice and collect some data – i.e., the number on each die and then the sum of the two together. If you don’t have die (or don’t want to hear all the noise!) you can utilize the random integer option on the calculator to simulate rolling die. I usually had my students in pairs to do this – one record, one ‘roll’. Then have them tabulate the frequency of each sum and create a new table with possible sums & frequency.

2015-12-10_12-32-42The next step is to have them make a histogram of their own frequency table and then compare to other students.  In my example, students only rolled 24 times each, so everyone’s graph will look different and not be what we expected (more 7’s). Great discussions can come from these observations.

Hopefully, discussions lead to the idea that each group o2015-12-10_12-17-22nly collected a small sample of rolls, and that if we had more samples,
perhaps the distribution of data would be more what we expected – i.e. more sums of 7 appearing. Here’s where having a class emulator is a great tool – you can display one frequency table and collect the class data.  So – same sums, but combine each groups frequency to get a total frequency for the whole class.

Once you collect the class data, you can then create a new histogram. With the 2015-12-10_12-18-07ClassWiz you can easily do so just by creating a QR code of the table data, and, with the emulator, go directly to the visualization. You can also do this with the SmartPhone App Edu+ if you have that option. The nice thing about the emulator is you can immediately pop the visualization up and begin discussions and comparisons of whole class histogram versus individual groups. Being able to immediately see the visualization with only a scientific calculator, is powerful, especially as you can quickly compare between previous “one-group” histogram and current “whole-class”.

There are so many ways to use this activity – I use to use it with TinkerPlots, graphing calculators, students hand-drawing the graphs. The ClassWiz and its ability to create QR codes and online visualizations is another way to help students make meaning out of the math they are doing, especially when they only have a scientific calculator to use, as most middle school students do. Hopefully this gives you some more options.  I have included a short video clip on how to actually create the frequency table, QR code, and online graph using the ClassWiz fx-991EX. Try it!