Day: November 3, 2014

Fun math that I saw this week

At the beginning of 2014 Numberphile published an incredible interview with Ed Frenkel with the provocative title:  “Why do people hate mathematics?””

There is a particularly interesting exchange around the 5:00 mark:

Numberphile:  “Let’s apportion some blame.  Let’s blame someone.  Sounds to me like you are blaming high school teachers . . . . back in our school days they were making us paint fences instead of showing us Picasso.”

Frenkel:  “Well, if I really were to assign blame, I would assign the blame to myself and to my colleagues – professional mathematicians.  We don’t do nearly enough of exposing these ideas to the public in an accessible way.  Often times we aren’t willing to come up with good metaphors and analogies.

That exchange has really stayed with me.

Later in the year, actually about a month ago, David Coffey wrote a nice piece in which he answered a similar question – Whose fault is it that you aren’t good at math?:

Whose Fault is it that you aren’t good at math?

I like his answer a lot -> you aren’t good at math because you didn’t have the experiences that you needed to be good at math.  I wrote a little too long of a response to Coffey’s piece here where I suggested some places where students (and, well, everyone) might find fun math experiences:

Responding to Cafid Coffey’s Challenge

Despite writing way too much last time, I want to write more because I’ve seen so many great math experiences  just in the last few days.

(1)  Let’s start with this incredible public lecture from Terry Tao at New York’s Museum of Mathematics:

 

 

This lecture seems to be almost exactly what Frenkel was taking about in the piece of his interview that I quoted above.  Tao shows some wonderful ideas of mathematics and how those ideas helped us understand how to measure distance in the universe.  A beautiful and accessible lecture from one of the world’s top mathematicians.   It has been online since the beginning of June and as of today doesn’t even have 700 views!!   Ugh, though to be fair I try to look out for stuff like this and didn’t even see it until yesterday.   I hope more people find out about this lecture and are able to watch it.

(2) Speaking of the Museum of Math, this weekend they finished their portion of the MegaMenger project:  http://www.megamenger.com/ .  My family went down to participate in the build and it was really fun to see all of the kids helping out and playing around in the Museum.  Hopefully there will be more projects like this one that can show kids that math is more than their 20 question algebra assignment.  Here’s a picture of my son standing inside of the finished product:

(3) Fawn Nguyen’s digit puzzle.

Last week Fawn Nguyen shared a wonderful digit problem that she did with her class.  A few days later the online math world was buzzing left and right about her problem.  On Sunday morning 5 different people had shared it on my twitter feed.  So fun and I’m so happy for Fawn’s incredible work is being recognized by an ever-growing audience.  But don’t take my word that this is an “utterly kick-ass” exercise.  Try it out, too:

(4) Continued Fractions

Continued fractions have a special place in my heart because my high school math teacher, Mr. Waterman, loved them.  He taught us out of the book pictured below (sorry I only have the picture side by side with Geometry Revisited, but trust me, that’s a great book as well!):

Book Pic

Amazingly I saw two ideas in the last week where continued fractions either have or could have played a role.  It really is a beautiful subject and it is a shame that it no longer has much of a place in high school or college math programs.

In any case, here’s a really neat blog post from Sam Shah showing how he incorporated continued fractions into a lesson in his class:

Substitution and Continued Fractions

and here’s a fun tweet from Steven Strogatz from this morning showing a problem in which continued fractions could help students make a fun connection:

Since the picture from Gilbert Strang’s book doesn’t come through, let me expand a little on the continued fraction connection.  On page 36 of the book Strang mentions the pattern he’s showing comes from a connection that 2\pi has to the fractions 44/7, 25/4, and 19/3.  Those three fractions just so happen to be the first three “convergents” of the continued fraction for 2 \pi.  The next one is 333/53 which might be fun to look for in Strang’s pattern.

Anyway, it has been great to see all of this fun math online (and in person) during the last week.   Hopefully there will be many more weeks like this one to come.

A neat place value exercise: what about fifty ten?

My younger son and I started a new chapter in our Introduction to Number Theory book today – “Algebra with Integers.”  It is big step up in difficulty from the prior chapters, and even more of a step up because I have really not covered any algebra with him at all.  Even though the discussions are a little longer now, many of the problems and examples remain accessible to him.  Sadly, that’s not going to remain true for much longer, so our little tour through introductory number theory will come to an end soon.

One of the first problems that we looked at today focused on place value:

Find a two digit integer that is equal to three times the sum of its digits.

Even getting going with this problem is a challenge if you’ve not had algebra, but a few concrete examples like 34 and 87 helped get the ball rolling.  Eventually he was able to write down an equation that would help solve the problem.  If the 10’s digit of the number is A and the 1’s digit of the number is B, we know that:

(1) 10*A + B = 3*(A + B),

and so 7A must be equal to 2B.  It was really interesting for me to see the leap from “the number is AB” to “the number is equal to 10A + B” happen right in front of me.

Solving this equation is no small task  and really is only possible after you recognize that A and B have to be single digit positive integers.  Once you do recognize that important piece, though, it is not super hard to see that the solution to the original problem is A = 2 and B = 7.  Thus, the number we are looking for is 27.

With that problem as a warm up we were ready for another challenge: find all two digit positive integers that are equal to four times the sum of their digits.

If we proceed as above we find that the equation we need for this problem is:

(2) 10*A + B = 4*(A + B),

which simplifies (a little more than the first one) to 2A = B.  Instead of just one solution here, there are several:  12, 24, 36, and 48 come to mind quickly once you know that B = 2A.

At this point my son told me that he thought it was interesting that these numbers were all multiples of 12 but then dropped that thought to tell me that he thought that there were no more solutions.

“Why not?”

“Because the next one would start with 5, but double 5 is 10 and 5(10) isn’t a two digit number.”

“What would it be if it was a number?”

“510”

“But the 5 is the tens digit, not the hundreds digit.”

” . . . . fifty ten . . . ?”

“Interesting.  What do we usually call that number?”

“60.”

“What is the sum of the digits of fifty ten?”

“15”

“What is 4 times 15?”

“60 . . . hey, it works!”

“What about sixty twelve.”

“. . . . yes, 18 X 4 is 40 + 32 = 72 = sixty twelve.”

I left it there, but thought it was a fun conversation because it got him thinking about place value.  Just like our conversation last week about the relationship between the rows of Pascal’s triangle and powers of 11:

A day in the life: Building and Extending Number Sense

I really like being able to sneak in some ways to think about number sense and place value while talking about problems that (to him) aren’t obviously related to those topics.

Made for a fun morning.