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 🙂

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!).

Pascal’s triangle and counting part 2

I’ve recently finished both Ed Frenkel’s “Love and Math” and Jordan Ellenberg’s  “How not to be Wrong.”  Both books are excellent and have given me lots of things to think about when it comes to communicating math ideas.  One theme that plays an important role in both books is that many different areas of math are tied together in unexpected and really interesting ways and these connections between seemingly unrelated areas of math can provide important insights in problems solving.

Pretty sure the first time I ran across this idea was reading Coxeter and Greitzer’s “Geometry Revisited” back in high school.  The section on projective geometry has a chart that helps you translate between “regular” geometry theorems and dual theorems in projective geometry.  Can’t say that these abstract connections made sense to me in high school, but with the nudging from Frenkel and Ellenberg I think I’m starting to get a better appreciation for the importance of communicating connections between different areas of math.

And that brings me to our little Family Math talk from yesterday about Pascal’s triangle:

Though I wasn’t really thinking about connecting different areas of math in this talk, we did talk about a lot of seemingly different ideas more or less by accident – Pascal’s triangle, counting blocks, binary numbers, and polynomials.  After finishing Ellenberg’s book last night I was thinking about how I could do a better job talking about relationships between different areas of math and realized I had a great example that was right in front of me and ready to go.

So, today I wanted to show how all of the different ideas in yesterday’s talk could be come together to help us think about a counting problem that seems to be much more complicated.   The problem involves counting blocks again, but this time we have two copies of each block.   Here’s the introduction:

After the quick little introduction we dive right into the problem and try to count the number of ways we can pick some small sets of blocks.  Hopefully these relatively simple examples help illustrate the new problem as well as some of the complexity that arises when you have multiple copies of the same block.   From some of the simple examples we construct a new Pascal-like triangle that is a little bit different from what we saw yesterday:

Next we started looking for patterns.  The first one that the kids saw was that the rows in our new triangle add up to be powers of 3 rather than the powers of 2 we saw in Pascal’s triangle.  That pattern presents a nice opportunity to show how each of the subsets correspond to a number in base 3 (though I give this point about 1 second of screen time . . . . whoops!).    The next pattern we talk about is the relationship between one row and the next one in this triangle.  This relationship allows us to speculate on the numbers in the next row rather than trying to find all of these numbers by counting blocks:

We left off in the last video thinking that we found a relationship between rows in our new triangle.   The next thing I wanted to show was how the connection between rows of Pascal’s triangle and powers of the polynomial (1 + x) could help us understand how the rows in our new triangle were related to each other.  This part is surely the most difficult part of kids, but I thought it would be fun to talk about anyway.   I only saw this connection between polynomials and block counting in college, so I’m hoping that introducing connections like this really early on will help the boys think of polynomials as more than just abstract math symbols on the whiteboard!

Finally, we end up on Wolfram Alpha multiplying out polynomials.  Sure enough, we are able to replicate the numbers that we found in our triangle.  We even find the number of 5 block subsets from 5 pairs of 2 blocks – something that would have been super hard for us to count directly.  Sorry the computer screen is hard to see on this one – I published this video in HD to hopefully make the small numbers on the screen easier to see.

So that’s my first try at presenting a problem that has a fun and surprising solution that draws from several different areas of math.  Having seen the emphasis that both Frenkel and Ellenberg put on connections in mathematics, I’m going to try a little harder to emphasize these connections in these little math projects with the kids.


Pascal’s triangle and some fun counting for kids

Last night I asked my younger son what he’d like to learn about in today’s Family Math and his answer was Pascal’s triangle.  Since we just started our little summer project on counting and probability, this was a timely suggestion.

We started with a simple review of how you create the triangle and also talked about some simple patterns.  This video went a little over 5 minutes because my younger son noticed an interesting pattern with prime numbers that took an extra minute to explain:

The next part of today’s talk was relating the numbers in Pascal’s triangle to ways we can pick groups of objects.  We illustrated our groups with snap cubes.  After a little introduction to ways to choose groups from sets of two and three objects, we show that the main identity in Pascal’s triangle – that two adjacent numbers in a row add together to get the numbers in the next row – can be understood in terms of selecting groups.   I’m not sure how clear the explanation was, but I hope it made sense:

The next step was to show one way that picking groups of blocks can help us understand why the rows in Pascal’s triangle always add up to be a power of 2.   This fact is a little easier to understand that the example in the last video (as long as you know binary).  In retrospect, I should have done this identity first.

Finally, we wrap up back at the whiteboard talking about why we see powers of 11 in Pascal’s triangle.  We actually did an entire Family Math about this fact a while ago:

I was pretty happy that my son remembered the powers of 11 in the first video, so I was really happy to be able to do go over this idea again.  We revisited why those powers show up (which involves a short discussion of polynomials) and then use the same idea to compute a few other similar computations.  (and ugh!  sorry we went off of the bottom of the screen at the end 😦  )

All in all a fun morning talking about Pascal’s triangle.  It is always fun to revisit old topics and dive a little deeper than we did previously!


Pascal’s and Sierpinski’s triangle

We’ve really been enjoying “The Math Book” by Clifford Pickover (sorry, I don’t know Latex well enough to embed the \alpha and \beta into the book title).  We started reading it last weekend and did a project on the Prince Rupert problem:

During the week I’ve been having the kids read a section of their choice and write a little one page report.  They’ve written on the Menger sponge, the Klein bottle, the Hilbert hotel, slide rules, the 15 puzzle, and Pascal’s triangle.  The short (one page, mostly) sections in the book allow the kids to read and the write about interesting math, so these short projects have been a lot of fun.
Yesterday my youngest son wrote about Pascal’s triangle, and my older son had an interesting comment on the pictures in the book – why was there a picture of Sierpinski’s triangle in the section about Pascal’s triangle?    Good question, and one that we attempted to tackle this morning in our weekend Family Math series.

The first step was a short talk about the basics of Pascal’s triangle.  It is a nice little arithmetic review for younger kids, and there are so many fun identities hiding in the triangle that you could talk about Pascal’s triangle many times without worrying about running out of material.  In fact, just this week Alexander Bogomolny at Cut the Knot posted this neat set of identities that I don’t remember ever seeing previously:

For now, the infinite series math is a little over our heads, but there is still plenty of interesting math for kids in Pascal’s triangle:

After this short little discussion of Pascal’s triangle and how it worked, I showed them how you could simplify the triangle and get something that starts to look like Sierpinski’s triangle:

Always fun to play around with Pascal’s triangle, and if you are looking for a book that can help kids see some really fun math, get your hands on Pickover’s new math book as fast as you can!

Pascal’s Triangle and Powers of 11

The most difficult thing about teaching my kids, by far, has been that I have no experience at all teaching elementary math.  When a concept is difficult for either of them to understand, quite often I struggle to work out exactly what it is that they struggling to understanding.  But we muddle along.

One obvious consequence (and sometimes it is a “feature” and other times a “bug” !!) is that I have no idea at all about the accepted ways to teach some of the most basic subjects.  Fairly often I just let them figure it out the elementary stuff on their own.  Basic arithmetic is a good example of a subject where my older son came up with his own methods (which have nothing to do with borrowing and carrying).  Since his ideas were perfectly fine mathematically, I just ran with ran with them.   Here are three examples that illustrate his method:

(1)  356 + 672 = 900 + 120 + 8 = 1028,

(2)  532 – 384 = 200 – 50 – 2 = 148

(3) 25 * 13 = 200 + 60 + 50 + 15 = 325

I think the main disadvantage of this way of doing arithmetic is that it is a little slow, but there are at least two nice advantages.  First, the approach highlights place value and therefore made it very easy to talk about arithmetic in other bases.  Second, this method looks pretty similar to the way we normally do arithmetic in algebra, so learning to multiply polynomials, for example, was not particularly difficult.

Yesterday I got a nice surprise when we stumbled on a new problem where this arithmetic method added a lot of value. We were doing some review work and one of the questions was simply to find the cube root of 1,331.  My son told me that he new the cube root of 1,331 was 11 because 1,331 was a row of Pascal’s triangle.

I love opportunities to talk math that come out of the blue, and this was as good an opportunity as any.  I’m not actually sure where the connection between powers of 11 and Pascal’s triangle came from since we haven’t talked about Pascal’s triangle in a long time.  However, as luck would have it, we’d spent the last couple of months talking about polynomials, so we were primed for a fun discussion.  It turned out to be even better than I’d hoped!

The first question I asked him was what he thought 11^4 was.  He drew out Pascal’s triangle and said he thought that 11^4 would be 14,641.  Fine, but the next row of Pascal’s triangle is 1 5 10 10 5 1, so what would 11^5 be?    Since the method of finding powers of 11 using Pascal’s triangle now appears to break down, he proceeded to calculate:

14,641 * 11 = 146410 + 14,641 = 100,000 + 50,000 + 10,000 + 1,000 + 50 + 1 = 161,051.

My plan, of course, why the connection to Pascal’s triangle has disappeared, but his unusual method of addition meant that the coefficients of Pascal’s triangle were right there on the board!  Ha ha, the joke was on me.  We revisited the calculation this morning:

Following that discussion this morning, we spent a few minutes connecting polynomials to Pascal’s triangle and showing why the powers of 11 are hiding inside the triangle.  Definitely a fun and surprising weekend of math!!

For me moments like these have always been the best part of teaching.  As I’ve said many times, I’m glad that I have the time and flexibility to teach my kids.