Month: September 2014

Dave Radcliffe’s amazing Fibonacci / GCD post

This morning I received a neat heads up on a really cool post from Dave Radcliffe on twitter:

I’ve been covering the concept of greatest common divisor with my younger son this week, so Dave’s post had some great timing.  It is such a neat formula that I thought I’d use it for a little talk with my kids tonight instead of doing our usual weekend Family Math.

We started by talking through Dave’s formula and writing down the first 15 Fibonacci numbers.  I then worked through a simple example to make sure that the kids understood the language of the formula:

next I had my younger son see if he could work through the formula using the 8th and 12th Fibonacci numbers.  He’s just learned the Euclidean algorithm, so I was assuming that he’d use that approach to find the greatest common divisor.  This formula turns out to be a great way to practice the Euclidean algorithm.

I wanted my older son to work through an example, too, and picked one with slightly larger numbers – the 10th and 15th Fibonacci numbers.  I have covered a bit more number theory with him and suspected that he would remember that you could find the greatest common divisor of two numbers by factoring.  I was glad that he did since it was a nice example for my younger son to see:

Finally, to do a few more complicated examples we went to the computer.  Since Mathematica has built in functions to calculate the GCD of any two numbers as well as to calculate any Fibonacci number, we could easily work with much larger numbers.  We did a couple of examples, including one where we forced a large greatest common divisor.

I’ve given no thought at all to how to prove Dave’s formula, but I do have a little bit of driving to do this weekend so at least I’ll have something to think about!  The boys seem to enjoy seeing fun little math facts like these, so despite not having a lot of context around this amazing formula I’m happy with how this project went.  Definitely not your standard greatest common divisor exercise!

Jordan Ellenberg’s “Algebraic Intimidation”

One of the ideas that seems to have stuck in my mind from reading “How not to be Wrong” a couple of times is the concept of “algebraic intimidation.”   Ellenberg uses this phrase to describe one of the standard ways to “prove”  that 0.9999….. = 1.   I go through the proof that he’s talking about in the first video below if you’ve not seen it before.

The idea of algebraic intimidation is, I suppose, pretty simple:  the math all looks right, therefore the result must be right because you **better** believe the math!

This concept obviously generalizes to all sorts of situations.   As the maybe useful / maybe harmful (depending on what year it is) quantatitive ideas seem to be creeping back into the financial markets, I feel like I’m seeing the old algebraic intimidation hammer at work a lot more frequently these days.  But, hey, we all miss 2008, right?

While a post about martingales might more more relevant to the attempts at using math to intimidate in the financial markets, I think Ellenberg’s example is infinitely more interesting.  Particularly for students, and I’d love to use the examples below in a room full of kids who are interested in math.

The idea of talking about algebraic intimidation once again came up this past weekend in our Family Math project.  I asked the boys what they wanted to talk about  and they gave me a surprising answer – “Infinite Series.”   The entire set of talks from this weekend is here:

The two conversations relevant to algebraic intimidation  are below and came when one of the examples that they wanted to talk about was “the -1/12 series.”  Say what you want about that old Numberphile video, but the ideas in it sure stuck with my kids!

I led off this part of our project with the standard proof of why 0.999…. = 1 and then, following some examples in Ellenberg’s book, extended the ideas in that proof to a few other areas where you get some rather odd results.  We then moved on to the “-1/2 series” and followed the ideas in the original Numberphile video.

You’ll see that both kids are quite skeptical of the results.  My younger son in particular is almost physically upset.  That’s good.  I want them to learn to question results rather than just blindly trusting the math, and I especially want them to feel free to question results that seem odd.  You certainly won’t find many results that seem more goofy than the ones below 🙂

Talking about “Infinite Series”

[Note:  I’m in a little rush to get up to Boston for Brute Squad practice today, so I’m just getting the videos up now and will expand this blog entry either later today after practice or later in the week.  I haven’t even proof read this, but it was so fun I just wanted to get it out there!]

Today I told the boys that we could cover whatever they wanted for today’s Family Math project and they chose infinite series as the topic.  In particular they wanted to talk about

(1) Fibonacci Numbers,

(2) Pascal’s triangle,

(3) the sum 1/2 + 1/4 + 1/8 + 1/16 + . . . ., and

(4) “the -1/12 series”

I’m not sure that I could have been more excited about this list of topics!

We started with the Fibonacci numbers.  The idea here was to review the idea of how you create the list of Fibonacci numbers, see what the boys remembered about this sequence, and show them how the Fibonacci numbers arise in a simple continued fraction.  The boys remembered that you could use the numbers to make a spiral, so we spent a little bit of time talking about the spiral, too.

I wanted to show the continued fraction example because the Fibonacci numbers occur in both the numerator and the denominator of the continued fraction convergents, but the numbers are shifted over 1 in the numerators.  That shift of an infinite sequence will come into play in our last videos when we discuss “the -1/12 series”

The next topic was Pascal’s Triangle, which turns out to be an absolutely perfect next step by luck.  We started by reviewing how you create the triangle and then moved on to looking at some other sequences that are hiding in the triangle. We found several fun patterns hiding in the triangle including some patterns that describe some fun geometry. At the end I showed them that even the Fibonacci numbers are hiding in the triangle in sort of a sneaky way. I wanted to talk more about this but a bee flew into the room, oh well . . . :

The third topic was the sum 1/2 + 1/4 + 1/8 + . . . and why this series sums up to be 1.    This was also really fun and I got a nice surprise as each kid had a slightly different geometric way of showing why this series summed to 1.  I showed them a 3rd slightly different idea and then showed them a second neat series that also sums to 1 ->  1/4 + 2/8 + 3/16 + 4/32 + . . . Patrick Honner gave a really cool visual proof of this fact here  and show them how his visual proof works.

The last topic is the “-1/12 series” made famous by this Numberphile video:

After an introductory talk about this series and the seemingly (or perhaps, “actually”) crazy sum, I backed up a little by talking about a question that seems to be a tiny bit easier -> does 0.999…. = 1?  Following the line of reasoning in Jordan Ellenberg’s “How Not to be Wrong” I showed that the standard way of proving this also can produce some strange results.  I really like Ellenberg’s description of this standard proof as “algebraic intimidation” and you can see how that algebraic intimidation plays out in the next two videos as both kids really don’t believe that the original sum is -1/12, but also seem to be convinced by the math that it does.

Finally, I followed the ideas in the Numberphile video above and showed how you get the result that the sum of 1 + 2 + 3 + 4 + . . . . = -1/12.  I love that this result seems to actually physically bother my younger son.

This was a super fun project.  Shows the fun you can have when you let the kids pick the topics 🙂

Not quite a “day in the life” call it a “morning in the life”

Had a great morning working with the boys today that included a particularly fun exchange with my older son that I wanted to write about.

It started with a continuation of a brief conversation on Twitter about Khan Academy.  I’ve been quite happy to use their problem bank for review exercises with my kids.  I’m glad that the resource is around, and I’m glad that it is free.  No need to post the whole conversation, but you can get to it from here if you enjoy reading twitter conversations:

This school year I’m walking through Art of Problem Solving’s Geometry book with my older son and  today we were covering the section on exterior angles in triangles.  I’ve never taught geometry before and I find thinking about how to talk to my son about basic concepts in geometry to be both fun and challenging. Exploring the beginnings of mathematical proof from a few basic axioms has been particularly fun.

The talk about exterior angles got pushed back a tad because of a challenge problem that gave my son a bit of trouble.  The problem is from an old AMC 8 exam and is here:

http://www.artofproblemsolving.com/Wiki/index.php/2003_AMC_8_Problems/Problem_22

Two interesting things came up talking about this problem.  The first was showing how you could find the area of a square if you knew the length of one of the diagonals.  Maybe not the most interesting piece of mathl, but still pretty cool when you see it for the first time.

More interesting from a math point of view was the idea that 4 - \pi could never be equal to an expression like \pi - 2, or even something like \pi - \sqrt{2}.   I explained to my son that \pi was “more than irrational” and couldn’t be written as the sum of a rational number or square (or other) roots of rational numbers.  His response was great – “So it is super irrational?”  Well, sort of yes, actually!

These are the conversations that I love having.  Going back to the Khan Academy point, I’d much rather have the kids spend time doing a few Khan Academy review problems on  fractions, or basic facts about prime numbers (to name two things that I’ve used it for so far this year)  if it means I can have these conversations with them.

Not wanting to pass up the opportunity to explore this particular conversation a little more, we went to the whiteboard and I explained the difference between algebraic numbers and transcendental numbers.  I even wrote down the series for Sin(x) and Cos(x) to show him that \pi did satisfy some polynomial-like equations, just not ones with finitely many terms.  Next, almost if the morning was a set up, he asked if e satisfied any special equations.

In response to that question I down the series for e^x and showed how these three equations led to the relation that e^{\pi i} = -1.

With this little side track into transcendental numbers behind us, we finally got around to talking about exterior angles in triangles.  We started with a couple of simple examples and then moved on to proving the theorem that the measure of the exterior angle in a triangle is equal to sum of the other two angles in the triangle.  The proof we did is probably the standard one and uses the fact that the angles in a triangle add up to 180 degrees.  After we finished this proof my son turned to me and said that he thought there was another way to prove the same theorem.   That was a nice little surprise and I asked him if he wanted to do this new proof for his morning movie.  He agreed and I turned on the camera with no idea of what he was going to say.  Turned out that he had a really good idea:

Fun morning.  Really love having conversations like these.

A neat probability problem from James Tanton

Last month I bought a copy of James Tanton’s “Solve This:  Math Activities for Students and Clubs” on the recommendation of Fawn Nguyen.   As is true with all of Fawn’s recommendations (that aren’t related to college football), it has been a joy to go through.

I picked out a pretty challenging problem to try out for our Family Math today.  My goal was not to give the boys a complete understanding of the solution to the problem, but rather to show them a situation that they could understand and speculate about a little.    Although walking them through the clever solution at the end proved to be a little difficult, I’m really happy with how this problem engaged them and look forward to doing more problems from this book later in the year.

We started with a simple explanation of the problem and used several lego figures to help with the illustration.  Before diving in to the solution, we spent a few minutes just talking about what they thought the answer would be.

Next we tried a few examples.   Almost comically, every time we flipped the coin we got heads, so Unikitty kept falling off the cliff at the first step.  Finally we got a long sequence where we had more tails than heads for a long time – actually a really long time.   This long sequence was a lucky illustration of just how complicated this problem can get if you try to look at it case by case.   Sorry that it is hard to see heads / tails in the flips on the camera – I probably should have used something that had different colored sides rather than a coin.

Next we went to our whiteboard to try to work out the math.   It was interesting to me that both boys thought that the series 1/2 + 1 /4 + 1 / 8 + 1 / 16 + . . . . would show up in the solution.  I wanted to approach the problem with binary trees to show them that the series they were looking for doesn’t show up quite as easily as they thought it would.    I also wanted to illustrate this approach because we’d looked at binary trees last week:  https://mikesmathpage.wordpress.com/2014/09/07/binary-trees-and-pascals-triangle/

Though this part of walking through the problem wasn’t as clear as it could have been, I’m happy for the kids to see that you don’t always march right to the solution of a problem in a straight line.  I’m also happy for them to see that problems that are relatively easy to illustrate can sometimes be a little more complicated than they seemed when you start thinking about them more carefully.

Finally the clever mathematical solution.  The idea used in solving this problem is a little bit over their heads, but it is a great mathematical idea for them to see nonetheless.  The fun, and actually pretty amazing, part of this solution is the idea of finding a clever piece of symmetry in the problem.  That symmetry allows you to write down an equation whose solution is the probability that you are looking for.  Quite a remarkable idea – I don’t know what the probability is, but I know a quadratic equation that it has to solve!

As it took a while to walk through this solution, I chose to talk pretty informally.  At the end, I let my younger son see that his guess at the solution – i.e. that the probability we are looking for is 1 – does indeed solve the equation.  With my older son, I let him play with the quadratic equation a little and see that p = 1 was the only root.

Tough stuff, but hopefully still a fun example:

I’m really happy that Fawn recommend James Tanton’s book to me.   Despite the difficulty of this first project, the boys were engaged all the way through.  Can’t wait to try out a few more.

Chess vs. Golf

Rory McIlroy cause a few probably unwanted headlines earlier this week when he said that both Tiger Woods and Phil Michelson were in the “last few holes” of their careers.  Link here if you like blown-out-of-proportion golf gossip:

http://espn.go.com/golf/story/_/id/11509989/rory-mcilroy-defends-comments-tiger-woods-phil-mickelson-says-was-being-complimentary

It made me wonder, though, how old is old when it comes to pro golf?

Then, in an “oh shiny” moment, I was looking at the latest (unofficial) world chess ratings.  It made me wonder which group is older – the current top 20 male chess players or the current top 20 male golfers?  Well, wonder no more:

Age Range    Group 1     Group 2

20 – 24              6                   2

25 – 29              4                   4

30 – 34              4                   5

35 – 39              3                  7

40+                   3                   2

Before you look to see which one is which, have a good think about it.  Current rates as of September 13, 2014 are here:

http://www.2700chess.com/%20

and here:

http://www.owgr.com/ranking%20

Irrationality of the Square Root of 2

During the week I saw this neat post on Twitter from Evelyn Lamb:

The 2nd link in the tweet is to an amazing list of 27 different proofs of the irrationality of the square root of 2 on Cut the Knot’s website.

I thought it would be fun to review of few of these proofs with the boys this weekend.  Though obviously some of the details will be over their heads, I think it is still interesting to share ideas like these with younger kids every now and then.  It helps communicate some of the beauty of math and also shows that problems can have many different and surprising solutions.

We started with the “standard” proof.  In this proof we assume that \sqrt{2} can be written in the form A / B where A and B are integers and the fraction is in lowest terms, and then arrive at the contradiction that both A and B have to be even.

Both boys have some familiarity with this proof since we talked about it before (probably more than once, actually). Before jumping into the review of this proof, though, we talked about what the boys knew about \sqrt{2} and what they knew about irrational numbers.

The next proof is similar to proofs 3 and 3′ on Cut the Knot’s page. This proof uses a little bit of number theory and a little bit of combinatorics. Since my younger son is studying basic number theory right now (from this book: http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt&%20) and since we studied some basic combinatorics over the summer, I thought this proof would be fun for them to see.

The idea in this proof is once again to assume that \sqrt{2} is equal to A / B where A and B are integers.  This time we count the number of factors in A^2 and B^2 to arrive at a contradiction:

[Post publication note – a former student of mine, Andrew Gacek, has point out that the video below has an error.  I’ve corrected the argument in the video below this one, but will leave the original video – see if you can spot the error, too! ]

Here’s the correction:

Next up is a neat geometric proof attributed to Tom Apostol and is proof #7 on the Cut the Knot list (though the comments on the Cut the Knot page indicate that this proof was known prior to Apostol talking about it).  This proof has a neat geometric idea – if \sqrt{2} is rational, say A / B,  we can find a right triangle with legs of length A and hypotenuse of length B.  Among all possible such triangles there will be a smallest one  (this statement is essentially the same as the idea that we can write the fraction A / B in lowest terms).   Given this smallest triangle, we show that there’s actually a smaller one which contradicts the idea that the first one was the smallest.  Thus, the idea that \sqrt{2} is irrational!

Finally, proof #21 on the Cut the Knot list – using continued fractions to show that \sqrt{2} is irrational.  I’ve loved continued fractions since learning about them for the first time in high school.  My high school math teacher was also really fond of them and I’ll always remember him standing in front of our class telling us to “split, flip, and rat”!

The idea in this proof is that the continued fraction for \sqrt{2} goes on forever, but continued fractions for rational numbers have only finitely many terms.  The other neat thing in this proof is that we get to see a surprisingly simple representation of \sqrt{2} which is a nice contrast to the idea that irrational numbers are simply infinite, non-repeating decimals.  I love the part in this video where my youngest son suddenly recognizes the pattern!

So, though some of the details in these proofs are a little difficult for kids to follow completely, it was still a lot of fun to go through this exercise this morning.  We got a little practice with logical reasoning, basic number theory, basic geometry, and even a little bit of work in with fractions!  We also saw some nice examples showing that there are many different ways to look at a problem, even a problem involving only a single number.  Definitely a fun morning.

Binary Trees and Pascal’s Triangle

We tried a new Italian restaurant last night.  It used paper to cover the tables and they let kids draw on the table covers with crayons which was a nice way to pass the time.  I was surprised to see that my younger son was drawing binary trees.  He said that he remembered them from an old Vi Hart video, which was a little strange since it is a Thangsgiving video.  Oh well, no telling what kids will remember:

I had a different project on tap for today’s Family Math, but when your kid is drawing binary trees on the table it is probably a sign, so plans changed!  We started our talk this morning with a quick review of what binary trees are, and then talked about a few simple properties that they have:

Next we build on the topic that we touched at the end of the last video – representing coin flips in a binary tree.  If we want to keep track of only the number of heads and tails that we’ve seen, some sequences that we’ve seen before make a surprising appearance in our little tree:

Next we moved on to showing a picture of how the binary tree can merge into Pascal’s triangle.  It was neat to see that the kids had seen how the “diamonds” would appear.  We also talked a little informally about why the pattern here is indeed the same as in Pascal’s triangle.  One of the other fun things we look at in this video is how the row sums that were easy to see in the binary tree carry over to this setting:

Finally, I wanted to show how this idea could help us solve a problem that they’d not seen before (though a pretty standard Pascal’s triangle problem).  The problem ask about counting different paths in a lattice.  We can think of the go right / go up choice as similar to the heads / tails coince from the binary tree example:

So,  a little doodling on our restaurant table cloth last night turned into a fun little Family Math talk.  Always fun to see what kids remember from things that they’ve seen (and when they remember them, too, I suppose!).

To Infinity and . . . to the next infinity

I started in on a geometry course with my older son this week.  On Wednesday we were discussing some basic shapes and he asked a neat question:  If you have a circle in a plane, are there more lines in the plane passing through the circle or more lines in the plane that don’t pass through the circle?  Fun!

I told him that the answer to the question was a little more complicated than it seems, but we’d talk through it over the weekend.  Well, the weekend is here and we talked through it this morning!

We started by introducing the question and talking about some of the non-intuitive properties of infinity.  I thought the easiest place to start would be comparing the set of positive integers with the set of positive even integers since this comparison is a nice way to show that infinite sets are a little strange!  I think that kids can understand some of the basic ideas about infinite sets, even if some of the concepts my be a little over their heads:

Next we moved on to a slightly more difficult question – comparing the set of positive integers with the set of positive integers that are powers of 2. In this case it looks like the second set is much smaller than the first one, and finding a way to see that these two sets have the same size did prove to be a challenge. However, with a little nudge, they were able to find a way to map the two sets to each other and even sort of answer the question “what is the opposite of powers?”

Probably the next natural step would be to show that the rational numbers are also countable, but I decided to skip that proof because I was worried that it would be more of a distraction and wouldn’t help so much with the question about lines and circles. Instead the next thing we talked about was comparing the real numbers to the integers via Cantor’s diagonal argument. This argument shows that there are more real numbers than integers. Although I didn’t necessarily want to focus on the different infinities, I thought it was important to help them understand the idea that just because two sets are infinite, they may not be the same size. In retrospect, I wish I wouldn’t have called this the “next infinity,” I guess we’ll have to correct that little slip the next time we talk about infinity.

With all of this background behind us, we moved on to answering the original question about lines and circles. We began by looking at a problem that is a little easier – what happens if we look only at vertical lines? Restricting our attention to this slightly easier problem allows us to see a surprising result – the number of points between 0 and 1 is the same as the number of points between 1 and infinity!

Now with the discussion of the vertical lines out of the way we can solve the general problem if we can figure out how to deal with lines that aren’t vertical. As luck would have it my older son thought looking at horizontal lines would be a good way to start. That idea got the boys thinking about rotational symmetry and led them to the solution to the original problem! Unfortunately I got confused on one of the pictures, but hopefully that 30s of confusion didn’t cause too much confusion – the perils of illustrating some of these ideas early on Saturday morning!

This was a really enjoyable project and the boys seemed to have a lot of fun and stayed engaged all the way through. I’m extra happy that this project came from a question that my son asked earlier this week. It is nice to talk about some of these ideas from pure math every now and then. It helps show younger kids that math isn’t just about playing around with numbers.

Fawn Nguyen shares a really neat Math Forum problem

Well, I made it exactly one day into the school year without copying something Fawn Nguyen posted.  That probably beat the over under . . . .

Cool problem, and one that I asked the kids to work on for part of their homework today. We talked through how they approached the problem when I got home from work.  We started with just the problem statement to make sure that they understood it. As I twitter-discussed with Fawn yesterday, reading this problem carefully is really important:

Next we moved on to how they thought about the problem. The first thing they did was look at what possible sequences of numbers might be on the chips. It was interesting to me to see that they wanted to check all sums for each sequence rather than just checking the high and the low sums.

Having found the sequence, we moved on to finding the numbers that were on each chip. Their approach to finding the first number was pretty clever and pretty quick, but after the first number we hit an unexpected stumbling block. My younger son picked a sum that added up to 17, but it was inconsistent with the first number we’d found. Hmmm – I hadn’t seen this little twist coming.

My older son found a way around this first little problem and we ended up with an arrangement that seemed to work. But, having seen the problem with 17 I thought I’d ask them a few more questions. Although there was still a little bit of confusion about the numbers on the  chips, hopefully these extra questions went a little way towards helping them understand their solution a little bit better:

The boys definitely had a lot of fun working through this problem today – thanks to Fawn Nguyen and the Math Forum for sharing it.