Category Archives: Recreational Math

Summer Assignment!

With the academic year winding down (and already finished for some), take a moment to think about what you will be doing this summer.  I’m sure you are planning to relax and refresh, but don’t neglect recharging your professional batteries…

READ A BOOK.  Is there one on your teaching bookshelf that you’ve been meaning to read?  Here are a few that I’ve read recently or have on my “Pedagogy and Learning” list for this summer:

  • Make It Stick: The Science of Successful Learning by Brown, Roediger & McDaniel
  • 5 Practices for Orchestrating Productive Mathematics Discussions by Smith & Stein
  • Embedding Formative Assessment by William & Leahy
  • Mathematical Mindsets by Boaler

If you have the opportunity, find a “buddy” or group and read together.  Try one chapter a week and discuss in person or via email.  Our Teachers Teaching with Technology cadre of instructors did book “discussion chats” this past year.  Here are some ideas and prompts to organize your comments on Embedding Formative Assessment and 5 Practices (thanks Jennifer!)

And since it is summer, I’m also planning to do some fun math reads.  Consider these, or maybe there are others you have your eye on (tell me in the comments!). The last two have the advantage that each chapter is a stand-alone essay, which is especially good if your attention span is shortened by summer distractions.

  • How to Bake Pi by Cheng
  • The Man Who Knew Infinity by Kanigel (now a major motion picture!)
  • Here’s Looking at Euclid by Bellos (also a math columnist for The Guardian)
  • The Joy of X by Strogatz (originally an essay series for The New York Times)

One more suggestion for your reading list is to catch up on an NCTM journal article you meant to read this year but didn’t have time to.  There are free previews of some articles on the website if you aren’t yet a member.

LEARN SOMETHING NEW (anything! Doesn’t have to be math-related.)  If you come back to school in the fall and share your experience with your students, they will see you as a learner and it may encourage their efforts.  Here are some ideas:

Watch a webinar.  Did you miss one this year you meant to join?  On-demand recordings are easily paused so you can take notes or try a problem yourself.  For example, Texas Instruments has an archive of their free webinars here.

Go to a workshop/class/conference. There are plenty of these available in person and online.  Your department or district may have local offerings.  The TI Teachers Teaching with Technology PD workshops are listed here, and include “virtual” workshops as well.  Jo Boaler’s YouCubed organization offers an online course for teachers “How to Learn Math” [info here].

For my Connecticut and New England colleagues, two great opportunities are nearby.  The T3 Northeast PD Summit is June 22 & 23 in New Britain, covering both the TI-84+ and TI-Nspire [info here and sign up here].  And the Geogebra Institute of Southern CT is holding their 4th annual conference in New Haven on August 16 [info here], including a pre-conference workshop August 15 for beginners.

MAKE A PLAN.  One of the best things for me about being a teacher is the chance to revise and improve my teaching practice on a regular basis.  Some years I taught the same course to more than one class, so each lesson got two or three tries in the same day (or week).  I reflected on how it went with the first group and made adjustments and improvements for the next class.  Remembering the details for the next school year is harder to do, so I would make quick notes right in my lesson plan to capture the changes I’d like to try in the future.  Here is the reminder I used as the last item on my “Lesson Plan Template”:border self eval

If you have notes from this year about lessons you’d like to modify, find them before leaving the building for the summer.

In 1994, Steve Leinwand wrote an article in Mathematics Teacher  “Four Teacher-Friendly Postulates for Thriving in a Sea of Change”.  One of them has resonated with me ever since then:  “It is unreasonable to ask a professional to change much more than 10% a year, but it is unprofessional to change by much less than 10% a year.”  While many of us try to change and improve our teaching each year (or are asked/mandated to implement new practices), change is challenging and daunting.  Leinwand suggests that teachers consider changing about 10% of what they do each school year, a very reasonable amount (just one new lesson every two weeks, or one unit out of the ten you teach).  What will be your 10%?  What topic/class/unit needs work?  Get a jump start this summer on something you’d like to teach differently than you’ve taught it before.

So, take this as your summer assignment: PICK SOMETHING you intend to do to learn and grow professionally this summer, along with your plans to decompress and have fun.  Let me know how it goes.  Have a wonderful summer!


NOTES & RESOURCES:

Leinwand, Steven. (1994).  Four Teacher-Friendly Postulates for Thriving in a Sea of Change. Mathematics Teacher 87(6):392–393. [Reprinted in 2007 during 100th anniversary of MT, with commentary by Cathy Seeley: Mathematics Teacher 100(9):580-583.]

More on  Recreational math books here: Math-Frolic.

Leap Years & License Plates

Thoughts on Divisibility and Counting

I am always in search of good numbers. When I park in a commercial parking lot, I look for a space with a number that is divisible by 3. I like addresses and phone numbers which involve multiples (such as xxx-1696 since 16 x 6 = 96). I enjoy mathematical dates, like 11/11/11, 10/11/12 or 3/5/15.

calendarSo the year 2016 has been a good one so far, numerically speaking. In February, I noted 2/4/16 and 2/8/16 and 2/14/16 (you can do the math). And on 2/29/16, otherwise known as “leap year day,” I started thinking about how to determine if a given year is a leap year.

Leap years occur every four years in order to synchronize the calendar with the astronomical seasons. To be precise, they occur in every year that is exactly divisible by 4, however years that are multiples of 100 are NOT leap years, unless they are multiples of 400. And how can one look at a number and determine if it is a multiple of four? Simply examine the tens and the units digits; if that two-digit number is a multiple of four, then the larger number is also a multiple of four. So 2016 is divisible by 4, but 2014 is not.

The rules for divisibility are taught at a variety of math levels. I remember learning the rules for divisibility during a seventh grade math unit on bases other than 10. My children learned them in fifth grade while doing factors, multiples and prime numbers. The Common Core State Standards doesn’t mention them by name, but begins discussion of division in grade 3, and factors, multiples, and primes are contemplated in grades 4 and 6.

In my work with middle school and high school students, I find great variation; some students are skilled at divisibility rules and others are surprised to hear about them. Everyone knows how to tell if a number is divisible by 2 or 5 or 10. Many of my students think multiples of 3 should work the same way (they should end in 3 or 6 or 9) and although they may know the “finger trick” for the first ten multiples of nine, they haven’t considered how to test larger numbers. The rules for divisibility by 3 and 9 rely on the sum of the digits (in contrast to the rules for 2, 4, 5, and 10).

My favorite divisibility rule is the one I learned most recently; while doing KenKen puzzles, I wanted a method other than long division to determine divisibility by 7. The technique is this:

  1. Chop off the units’ digit and double it.
  2. Subtract this from the remaining number.
  3. Continue this process until the result is 1 or 2 digits.
  4. If the final number is divisible by 7, then the original number was divisible by 7 [Note that 0 is divisible by 7].

For example: 3052

  1. Take the 2 and double it to get 4.
  2. 305 – 4 = 301
  3. Take the 1 and double it to get 2.
  4. 30 – 2 = 28 which is divisible by 7.

In the same manner, 2016 can be shown to be divisible by 7. So 2016 is a very good year: it is not only a leap year divisible by 4, but is also divisible by 3, 7 and 9.

Can we use technology to help students test for divisibility? On any calculator, we can simply divide each proposed factor. For more efficiency, use the TI-84+ table with the function Y1 = (number)/x and scroll through the table looking for whole number results.

 

This can be impractical for very large numbers, but on the TI-Nspire there is a factor command which comes to the rescue.

 

The question of why divisibility rules work is a fruitful exploration for students. For more on proving divisibility rules, see the resources below.

Connecticut_license_plateIn my search for multiples of 7, I have a new place to look: license plates. In Connecticut, they recently converted the license plate format to two letters followed by five numbers, such as AB-12345. I can check these 5-digit numbers for divisibility by 7 while stopped at a traffic light.

Which leads me to the question of license plate formats and the Fundamental Counting Principle. Connecticut used to have license plates with three numbers followed by three letters: 123-ABC. When those ran out, they briefly used a format with one number, four letters, and finally one number, such as 1ABCD2, at first without a hyphen and later with one inserted after the third character, like 1AG-HJ2. I disliked those plates because they were hard to remember (a 3-character chunk is more memorable to me if it is all letters or all numbers). Thankfully, the new seven-character plates appeared last year even though the sequence hadn’t been exhausted.

How many possible license plates are generated by each of the designs? Multiply the number of ways to select each character to determine the total:LLcorrect counting

Old style:         123-ABC:        10*10*10*26*26*26

Interim style:   1ABCD2:         10*26*26*26*26*10

New style:       AB-12345        26*26*10*10*10*10*10

The interim style, with 45.7 million plates, has 2.6 times as many plates as the old style.  The new style has 67.6 million possible plates which is about 3.8 times as many plates as the old style.  And with a one-in-seven probability that the number is divisible by 7, there will be about 9,657,143 multiples of seven out there.

Now that’s a lot of license plates! In the meantime, I’m looking forward to the next great date coming soon: 4/9/16.  Not only is it a progression of perfect squares (when will that happen again?) but 492016 is a multiple of 7.


NOTES & RESOURCES:

For more on why divisibility rules work, see the following:

  1. Math Forum: Explaining the Divisibility Rules [link]
  2. James Tanton: Divisibility Rules Galore! [link]

For more about using fingers to multiply and why the tricks work, see Kolpas, Sidney J., “Let Your Fingers Do the Multiplying”, Mathematics Teacher, 95(4), April 2002.