# Keith Devlin’s comments on Dan Meyer’s circle problem

Earlier in the week I wrote about a neat geometry / algebra problem Dan Meyer had posted on his blog.  That post (including the link to Dan’s blog) is here:

https://mikesmathpage.wordpress.com/2014/02/25/dan-meyers-geometry-problem/

It had been both fun and interesting to see all of the comments that have been posted to Dan’s blog about the problem during the week.   This evening Dan posted some updates to his original post including the following commentary from Keith Devlin:

I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.

So in the end I just computed.  I got an answer but no insight.  I guess that reveals something of a mathematician’s meta cognitive arsenal.  You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.

Having personally found some of the math hiding in the problem to be pretty neat, and in particular fun to talk through with a kid learning math , I wanted to take an extra few minutes to take sort of the opposite side of Devlin’s argument and advocate for something neat relating to  math I saw in the problem.

First, a quick (50 second) review of the problem:

Next, a quick solution (again, under a minute).  This isn’t meant to be instructive, but rather just a quick review of one possible solution where something interesting (I think anyway) comes up:

Finally, about 3 minutes of advocating for what I thought was particularly interesting – how does the second solution we got from the quadratic equation enter into this problem?

So that’s why I wanted to advocate a little for this problem.  Thinking through some of the geometry that comes up from the two solutions to the quadratic equations can be an important (and fun) exercise for students.  The significance of the first solution is clear – it is right on the line after all.   The meaning of the second solution is less clear and easy to ignore.  However, extending some of the geometry videos / apps posted in the comments on Dan’s blog would show that locating the point P at this second location also produces a square and circle having the same area.  This second solution naturally leads to asking what other solutions might exist with P not on the original line segment.  It turns out there’s a really cool geometric surprise hiding in those solutions.  How fun!

All in all, a really neat problem.

# Dan Meyer’s Geometry Problem

NOTE – 7:50 am on Tuesday February 25.  I’ll be editing this quite a bit over the next day or so, but just wanted to get the videos up before running off to work.  Sorry for the likely high amount of typos in this first draft . . . .

NOTE – 7:50 pm on Tuesday – a little more explanation behind each of the videos.

I woke up this morning to see a pretty interesting geometry problem for kids posted on Dan Meyer’s blog.  The article is here:

http://blog.mrmeyer.com/2014/confab-circle-square/

and the problem is this:

“Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.”

I start with my kids every day at 6:15 am so that I can try to get two hours in with them before heading off to work.  I liked this problem when I saw it and asked my older son if he’d be interested in working through it.  He was, so off we went.

(1) The first part was just talking through the problem to see what his initial reactions would be.  I was happy that he was able to understand the setup for the problem right away.  In the post about the problem Dan Meyer was particularly concerned about students being able to  “mak[e] sense of precise mathematical language.”   I wasn’t sure about the wording either and am interested to see if there are better suggestions posted on his blog, but we were lucky this time and the problem’s wording wasn’t a stumbling block.

What did prove to be an interesting stumbling block was dealing with a $\pi$ as the coefficient of a polynomial.  After the fact I realized that I should have expected this, but while we were talking about the problem I was caught a little off guard.  A short diversion switching out $\pi$ from the equation and replacing it with 25 seemed to help my son understand how to get past this problem.

(2) In the second 5 minutes we focus in on his moving from the geometric equations he’s written down to finding the point on the line segment.  We build on the formulas that we found in the last video and ended up finding a formula for the length of the line in terms of the radius of the circle.  I was really happy that he was able to get to this stage of the problem without much help from me, since the rest of the solution is just calculating.

(3) Because all of the $\pi's$ were giving him problems, I went and got a calculator so that we could see the actual numbers.  I wish I hadn’t done that without asking him to approximate the values, but these are the decisions you make at 6:00 am . . . .  Looking back on this about 12 hours later, I have no idea why I decided to go get a calculator.  Ugh.  Oh well, we recovered a little during this segment anyway, and I thought finding an approximation for $\sqrt{\pi}$ was actually a fun little diversion even if I’d already sort of given the answer away.   At least get to do a little estimating . . . .

The main point of this section was to try to remove some of the confusion that was coming from all of the $\pi's$.  As I mentioned above, during the talk I was taken a little off guard by this problem and it seemed like the best way around this problem was to switch over to decimal numbers.  Unfortunately, though, making this switch only helped a little, and it wasn’t until we did an example with integers that we got around the problem.   One thing I learned from working through this problem was that I need to incorporate a few more problems where some of the numbers aren’t integers.

(4) Having finished up with his solution, I wanted to show him a different way to solve the problem (I especially wanted to do this because we just finished a section on quadratic equations).   This was actually the approach I was expecting him to take, so I’m actually pretty happy that he went in a different direction at the start.  It is always nice to be able to see multiple solutions to the same problem, even if I’m mostly leading the charge on the second solution.  The two things I was trying to accomplish in this part were (i)  working through a little arithmetic with $\pi$ because of the confusion in the first part, and (ii) introducing the idea that there is a second solution that comes in from solving the quadratic equation:

(6) Now for some extra fun – let’s do the same problem in 2 dimensions rather than in just one!!  This is the part I was really looking forward to when I read Dan’s blog this morning.  I actually considered doing this part as a separate excercise, but we were on a roll so we kept going.  It would have been fun to try to spend a little time thinking through what shape we’d expect to see before diving into the calculation, but now we only had about 5 minutes left at this point, so we just jumped right into the calculation.

(7) Finally, since the 2D equation looks like it is a little rough, let’s just end the morning on Wolfram Alpha to see the nice surprise:

All in all, I really liked this problem.  Love finding nice math surprises on the internet!

# The Prince Rupert Problem

Since 2008 I’ve spent part of my free time coaching a couple of really great ultimate frisbee teams:  Boston’s Brute Squad and Seattle’s Riot.    Other than one player on Brute Squad who was getting her PhD in math from Cornell, there hasn’t been a lot of overlap between my math activities and my ultimate frisbee activities.

BUT a few days ago Callie Mah from Riot sent me a link to a math book that she thought I’d be interested in:

I bought it yesterday up in Boston, and Callie was  absolutely right – what a wonderful book for showing fun math to a general audience!  This morning I decided to open it up to a random page and do a fun little project with the boys on whatever topic was being discussed on that page.  We landed on the “Prince Rupert Problem.”

I do not remember hearing about this problem previously, but luckily it is pretty easy one to understand:  What is the largest cube that can pass through a cube of a given size?

The problem has a neat history.  It came out of some sort of bet that Prince Rupert made,   and took about 100 years to solve.   Surprisingly, there wasn’t a lot written about the problem on the internet, and the easiest reference for the history of the problem seems to be the Wikipedia page:

http://en.wikipedia.org/wiki/Prince_Rupert%27s_cube

The approach I took with my kids for the project this morning was to use our Zometool kit to look at a similar problem for a square first. – what is the largest square that can pass through a square of a given size?  The Zometool squares worked really well in helping us compare the relative size of the two squares:

Next we moved to the slightly more complicated problem of finding the largest square that could pass through a given cube.  This problem is a great introduction to visualizing 3D geometry, and once again the Zometool kit helped tremendously with the visualization:

Finally we moved on to the actual Prince Rupert Problem – what is the largest cube that can fit through a cube of a given size?  The solution – a cube with sides of length $\frac{3 \sqrt{2}}{4}$ times the side length of the other cube (or about 1.06 times the side length) – was a little too difficult to cover in a short movie so I just mentioned the answer and showed how to think about it.  By coincidence we had two Rubik’s cubes with sizes in almost exactly the right proportion, so we had a nice visualization here, too

All in all, a really fun project.  Thanks Callie!!

# Ben Vitale’s Triangle puzzle

I’m on vacation right now or I’d write something much more thorough longer, but I wanted to point out a really interesting puzzle about triangles posed by Ben Vitale on twitter yesterday:

Since I never know if the link’s from twitter will be preserved in the embedding, here’s a direct link to his blog:

He gives a couple of examples and challenges the reader to find some more.  One of his examples solutions is the  triangle with sides $\sqrt{61}$, $\sqrt{153}$, and $\sqrt{388}$ which has an area of 21.

Following the hint on Ben’s blog, I played around a little bit and found that the triangle with side lengths of $\sqrt{801}$, $\sqrt{932}$, and $\sqrt{3461}$ has an area of 3.   Here’s my work, where you’ll see the Fibonacci numbers hiding in the middle of the page!

I really like this problem, and as I mentioned to Ben via Twitter last night if we weren’t on vacation I’d be working through this problem straight away with my kids.  This problem shows a beautiful connection between geometry and arithmetic, and the Fibonacci numbers come up in my example from a neat connection to continued fractions.  What a wonderful problem.  Thanks Ben!

# Steven Strogatz and math writing

Today Steven Strogatz posted a link on twitter to a new article he’s written in the Notices of the AMS – “Writing about Math for the Perplexed and the Traumatized.”  It is a a great and instructive read.  Here’s the link:

Click to access rnoti-p286.pdf

This part of the piece, in particular, caught my eye:

“For any would-be pop math writer, here are a few surefire techniques.

(1) Illuminate.  Give the reader a shiver of pleasure by providing an “Aha!” experience.

(2) Make connections.  Tie the math to something the reader already enjoys.

(3) Treat the reader like a friend of yours – a nonmathematical friend.  Then you’ll instinctively do everything right.”

I think it is an interesting set of ideas that applies beyond pop math writing, and it made me reflect on the math I’ve been doing with my kids.   A few of the projects we’ve done in the last year of so fit nicely into Strogatz’s categories.  Here are a few examples:

(1) Provide an “Aha!” experience.

(a) The Chaos game.  Probably my favorite math surprise with the boys involved a basic fractal example – the Chaos game. We made a short program on Khan Academy to illustrate the math (I don’t know how else to share code).   The “Aha!” moment starts around 3:00 in the video:

The code is here for anyone who wants to play around with it:

(b)  Multiplying negative numbers.  This one turned out to be our most viewed video from 2013.  I was at work one day and someone on Twitter asked if there was an intuitive way to see that a negative number times a negative number was positive.  I spent the afternoon sort of daydreaming about ways to illustrate multiplication of two negative numbers and tried out my idea with my older son when I got home.  It was a fun watching him see the arithmetic right along side the geometry:

(2) Make Connections

(a) The Higgs Boson.  Last fall Peter Higgs won the Nobel Prize in Physics following the discovery of the Higgs boson at CERN in 2012.  My kids have always been interested in reading about physics and I pointed out a couple of articles about the new Nobel prize awards to them.  By coincidence, a few days later I saw a great article by Frank Wilczek about the Higgs boson in an MIT alumni physics magazine.  Wilczek’s article mentioned that one of the properties that made the particle so hard to detect was that it only lives for about $10^{-22}$ seconds!   I thought it would be fun to try to put $10^{-22}$ in context for the boys since you don’t encounter that number too much in daily life!

The Wilczek article about the Higgs Boston is here, if you are interested:

Click to access wilczek_physicsatmit_2013.pdf

(b) Really basic group theory.  My kids love playing with Rubik’s cubes and I’ve been able to use the cubes to illustrate a few different math ideas.  Maybe the most fun math idea I used the cubes for was illustrating a little group theory for my 7 year-old.   It was fun to see his enthusiasm for the new concepts he was seeing for the first time.

(c) Fractions with snap cubes.  My kids love building with legos and snap cubes and every now and then I get a chance to  use some of these little building blocks into the math we are doing.  After introducing fraction division using some basic arithmetic rules (defining division as multiplication by the reciprocal), I thought it would be useful to show him a few not-so-technical examples.  So we did a little geometry with snap cubes to help us understand fraction division:

(3) Be their friend.

Well, this one isn’t too hard to do when you are working with your kids.  I’ve really enjoyed covering a few advanced topics in math with the boys. The main point of covering these advanced topics wasn’t necessarily the details of the math, but rather to show that some beautiful, and even unsolved math can be accessible to (and fun for) young kids.

(a) The Collatz Conjecture.  Here all you need is to be able to divide by 2 and multiply by 3.  How fun to be able to show kids an unsolved problem that relies only on basic arithmetic!

(b) Pentagons.  With my older son I’ve been following Art of Problem Solving’s Algebra book this year.   The book spends quite a bit of time on quadratic equations.  Certainly understanding quadratic equations is great for helping kids develop a good base in algebra, but it is a lot of work.  At one point when we needed a little break from the topic, I thought it would be fun to show him an advanced topic where quadratic equations play an important role – pentagons.  It was a fun to show him how several different pieces of basic math that we’d already studied (plus one or two things we hadn’t studied) came together in a regular pentagon.

Not everyone has the time, the interest, or the ability to communicate mathematical ideas to a general audience.   I’m happy that Stephen Strogatz able to spend some of his time acting as sort of a public face for math in the US.  Hopefully this article he’s published today will  give a few other people some ideas about how they can join him in improving the public perception of math in the US.

# I’ve learned a lot about teaching math from Fawn Nguyen

We started home schooling about four years ago.  The biggest struggle for me over that four years has been trying to understand how my kids think about elementary school math.  Take fractions, for example.   Before we started home schooling, I’d guess that I’d not given 10 minutes of thought to how to understand fraction math in 30 years.   The lack of any detailed thought about how kids think about fractions means that there’s a really steep learning curve for me when we started covering that topic.  Well . . . assuming I’m aiming higher than “how can you possibly not understand this?”

In an effort to learn about teaching young kids I started reading a bunch of teacher blogs and following lots of teachers on twitter.  There’s an incredible amount of information out there ranging from detailed writing that focuses on education theory to much more conversational writing about day to day life teaching math.   One of the teachers whose writing has influenced me the most is Fawn Nguyen.

Fawn has an amazing gift for writing.  She can take the terribly difficult topic of teaching math to kids and break it down in to bite size pieces that seem easy to understand.  Her writing is also filled with example after example after example about how her students have approached problems in the classroom.   Many of these examples are cataloged on her “Math Talks” site:

http://mathtalks.fawnnguyen.com/2014/02/15/week-11.aspx

On this site you get to see first hand different ways that kids approach problems from pre-high school math.  It is an outstanding resource for me, and if you want to do some fun math with young kids I’m sure you’ll find it to be really useful, too.

I’ve linked her set of problems from Febuary 15th, 2014 above and went through these four problems with both of my kids this morning.  A few are a little above the level of math that I’ve covered with my younger son, but I still thought it would be fun to see how he would approach them. Here are our attempts at each of the problems:

(1) What is 0.48 x 650?

I would have guessed that my oldest son would have just jumped right into the multiplication, and was surprised to see that he converted 0.48 to a fraction first.  His first step helps him avoid some of the pitfalls that come with decimals, and having converted to a fraction he then charges ahead with the multiplication.  At the end of the video we talk through one step he could have done before multiplying.

I have not covered decimals with my younger son, though he’s probably seen them in a few problems here and there.  I’d hoped that he would be able to recognize that 0.48 was close to 1/2, which he did.  That was all I was trying to get out of this problem with him.

(2) Counting toothicks

This is the first of two problems in this set that focus on recognizing patterns.  Fawn has an entire site dedicated to these pattern recognition problems:

http://www.visualpatterns.org/

These problems are great for getting kids to understand how to move from the concrete (seeing the specific pattern at each step) to the abstract (writing a formula for the general pattern).   I personally think that both understanding patterns and being able to describe patterns are an important part of learning math, so I’m always excited to try out Fawn’s pattern problems with my kids.

For this particular problem, both kids approached it the same way – first counting the number of triangles and then figuring out how to write down the formula for the pattern.    Both kids were able to write down the number of triangles in each step and struggled a little going from that list of numbers to the general pattern.  That struggle is such an important part of learning.

(3) It is 2790 miles from Los Angeles to New York City – how many inches is that?

I work with a guy whose arithmetic skills are so incredible that he can do problems like this in his head.  For the rest of us, learning how to estimate is a really useful tool.  It is a useful skill way beyond arithmetic, too.  I remember being absolutely amazed at the way my undergraduate physics adviser could draw solutions to various differential equations on his chalk board.  Problems like this one help kids get some useful practice at basic estimation.

As with the first problem, I’ve not spent much time with my younger son on this topic yet, but still wanted to see how he’d approach it even if it meant I’d have to help out more than I’d normally want to:

(4) One more pattern problem

Not much more to add from what I said on probelm #2 – these pattern problems are great ones for kids:

So, I’m happy my attempts to learn more about teaching math to kids eventually led me to Fawn Nguyen’s writing.  If you are interested in learning about teaching math to young kids, I’d encourage you to follow her, too.

# A could’ve-done-better attempt at talking through the Schwarz Lantern

Last night I was watching the Olympics and  flipping through one of my favorite recreational math books – “The Penguin Dictionary of Curious and Interesting Geometry” by David Wells.

One of the sections of the book discusses the Schwartz Lantern, and that discussion gave me the idea to talk through this interesting shape with the boys today.    Unfortunately that talk didn’t go as well as I’d hoped – too many details are over their head – but, despite my failed attempt,  I do think there are some really neat things that kids could learn from the Schwarz Lantern.  Students with a little bit of background in geometry and trigonometry will, I think, have a ton of fun (and learn a lot, too!) studying this topic.

The first time I heard about the Schwartz Lantern was reading this wonderful article by Evelyn Lamb last fall:

http://blogs.scientificamerican.com/roots-of-unity/2013/11/30/counterexamples-in-origami/

Lamb has a amazing gift for explaining mathematics to broad audience and this article of hers is one of my favorites.  I wish I had the origami skills to make of the lanterns.

After reading Lamb’s piece I did a little more searching and found a nice write up on Cut the Knot:
http://www.cut-the-knot.org/Curriculum/Calculus/SchwarzLantern.shtml

as well as  a neat example from the Wolfram Demonstrations Project:

With Evelyn Lamb, Alexander Bogomolny, and Wolfram already covering the topic, only a fool would think he could add something to the discussion . . . so here’s what I did:

The thing that made me stop and think  about the Schwartz lantern last night was this curious statement from the Penguin Dictionary – when you approximate the surface of the cylinder using Schwartz’s idea “instead of approximating more and more closely to the surface of the cylinder, the triangles turn against the surface, and the total surface area tends to infinity.”

I honestly didn’t get the point the book was trying to make.  As I was sort of day dreaming about the approximation while watching the Olympic last night, I was not finding that the area of the approximation went to infinity.  Finally I took out a pencil and paper and figured out where I’d gone wrong.  Finding the error that I’d made helped me get a better understanding of why the Schwartz Lantern was so interesting.  I thought the kids would enjoy learning about this neat shape (particularly after the discussion of fractals that I had with my younger son yesterday) and decided to try to  talk through it with them this morning.

The first thing I did was show them some of the information from the three web pages I linked above.  Then we read about the problem in the geometry book, and even drew out an approximation on an empty  can.  Seemed like as good a way as any to get started:

Next we moved to the whiteboard to draw some pictures and get our arms around the problem. The first thing we needed to do was to count the number of triangles in the approximation.  I chose to emphasize this step because it is one of the few pieces of this project that they could do on their own.  Next I showed them how to find the surface area of a cylinder  so that we could later compare the area we get from the triangles to the actual surface area of the cylinder.

With that mistake out of the way we moved on to the correct calculation of the area of the triangle.  This is the part I really wish I’d done better.  We got caught up in too many of the details and I feel that the main point was obscured.  Nonetheless, we did arrive at the conclusion that as the number of triangles increases, the area of the approximation goes to infinity.

Despite feeling that I could have done this activity a little better, I think that there are some fantastic ideas for kids involving the Schwarz Lantern.  It really is amazing to me that you can approximate the cylinder with triangles and get (i) the circumference right, (ii) the volume right, (iii) and miss the area by quite a bit!    It shows that you can’t just blindly apply the ideas you learn in calculus to any situation.

But you don’t really need calculus to understand this example – just a little geometry and basic trig.  For kids with that background, an hour or so working through some of the geometry followed by another hour of playing around with the origami in Evelyn Lamb’s post would, I think, make for a really fun day (or two) of math.  Even though this definitely didn’t go as well as I’d hoped, it was still a fun project for us to work through.

# Neat coin flipping problem found on Twitter yesterday

An essay by Jeremy Kun, a graduate student at the University of Illinois at Chicago, published on Edsurge on Feb 12 caused quite a stir on twitter yesterday.  I’m not interested in getting drawn into the discussion about the essay, but if you haven’t see it, you can read it here:

https://www.edsurge.com/n/2014-02-12-opinion-you-never-did-math-in-high-school

One response that seemed to get a lot of traction is here:

http://christopherdanielson.wordpress.com/2014/02/13/edsurgemath/

I first saw a reference to the Edsurge piece in a tweet from Justin Lanier yesterday morning:

I didn’t know what Justin was referring to, so I checked out @mathprogramming’s blog and found a really neat problem about coin flipping.    That piece is here:

Simulating a Fair Coin with a Biased Coin

We had lots of free time yesterday because of the blizzard, so I turned the coin flipping article into a fun afternoon project with the boys.

First I walked them through the problem – if we have some coins that don’t flip heads/tails 50% of the time, how can we use these coins to make 50/50 decisions?  It is a pretty neat problem:

With that as the introduction, we moved to the whiteboard and discussed a little bit about probability.  Some of the details were a bit above what my kids have studied, but not so much that they were totally lost.  I was actually pretty happy to be able to talk through a problem that used fractions since I just finished up a unit on fractions with my younger son last week.  It is always nice when a new or advanced problem lets you sneak in a little review 🙂

Finally, having worked through the probabilities, we jumped over to the computer to write a little program to simulate the problem.  I like to use Khan Academy’s programming site with the kids because it is easy to share.  The code in the original blog entry is obviously more compact than mine (to say the least!), but I thought breaking the problem into a few different chunks would help them understand the code a little better.  The program is here if you want to play with it:

and the discussion where I walk the boys through the code is here:

As I said above, I really liked this activity.  I think the problem will be really fun for any kids who’ve played around a little with fractions and percents and also for any kids looking to learn some basics of computer modelling.

# Fawn Nguyen and Bridges

This weekend Fawn Nguyen was speaking at a conference and posted the following problem that she was using as an icebreaker:

I use so much of Fawn’s stuff with my kids that we may as well move to California and have her teach them directly.  This problem seemed like one the boys would love, so we tried it out right away:

I definitely enjoyed the discussion and it was a nice surprise to see how this problem engaged my younger son  In fact, so much so that my older son actually complained after the video that he couldn’t get a word in.  Ha!

After we finished up the video yesterday we went out to join a little neighborhood dog walking group that meets every morning.  One of the people who joins that walk on the weekends is the principal of one of the local high schools.  He and I spend a lot of time talking about fun math activities for kids and I mentioned this problem to him.  He joked (in a good-natured way) that there wasn’t much real world connection to this problem.   Connecting math problems to the “real world” isn’t something that I spend a lot of time worrying about  – we do plenty of problems with real world connections and plenty that are more theoretical.  The reason that Fawn’s problem was attractive to me is the problem solving techniques required to solve it.  The unusual circumstances described in the problem do not bother me at all.

But . . . I couldn’t get the comment out of my head and it occurred to me that there are a few other neat math problems related to bridges that do have “real world” connections.   Two of those problems made for great discussion topics this morning.    The first is a short explanation of why bridges have expansion joints.  The answer to the math problem in the example is really surprising – almost no one guesses right when they see the problem for the first time:

This one turned out to be such a neat problem that my younger son joined in about half way through because he was so interested in what we were doing.

The next problem is the Königsberg Bridge puzzle which is a famous puzzle that was solved by Euler in the early 1700s.  Unfortunately I stated the problem slightly incorrectly – you just have to go over every bridge once, you don’t have to return to where you started.   Since I wasn’t planning on going into any theory here, I decided not to go back and correct the mistake.  The boys really liked playing around with this and spent 10 minutes after the video trying to solve a modified version of the problem (add two bridges – one to each land side – to the island that started with three bridges).  Definitely a fun example for kids:

Finally, I could do a post about bridges without mentioning the book that Mr. Waterman gave to me my sophomore year of high school that was my first introduction to abstract mathematics:

So, thanks to Fawn Nguyen for inspiring one more fun weekend of math!

# *The quadratic formula and Fibonacci numbers

I knew ahead of time that I was going to have a busy week of work this week and was looking for something fun to cover with my older son in the limited amount of time that we would have.  We were supposed to be covering properties of functions so I was looking for a topic that would at least be tangentally related to that, but I also wanted to get him a little review with solving quadratic equations.  Diving into the formula for the Fibonacci numbers seemed to fit the bill quite nicely . . .

We started with a short talk about the Fibonacci numbers that focused on thinking about the usual recurrence relation definition:

I didn’t do a great job with the graph at the end, and we spent another 10 minutes after the video talking through the graph and comparing it to $y = x^2$ and $y = 2^x.$  I was pretty happy with how that talk about the graphs went after the first video and wanted to reinforce some of those ideas the next day.  With that in mind, the next talk begins by discussing how the Fibonacci numbers grow and then considers what happens if the Fibonacci numbers could be written in the form $F(n) = C * \lambda^n$.

I’ve always found this technique for finding the closed form for the Fibonacci numbers to be really beautiful.  Turns out it is a great little tool for algebra review, too!

After finding that the Fibonacci numbers somehow related to the numbers $\frac{1 \pm \sqrt{5}}{2}$ at the end of the last video, in this one we finish up the calculation and write down the closed form for each Fibonacci number.  Lots of great algebra review for kids in these calculations, too!

With the formula for the Fibonacci numbers now in hand, I wanted to play around with the formula so we jumped over to Wolfram Alpha.  The first neat thing I wanted to go through was how we could now easily calculate each Fibonacci number without reference to the prior two numbers.  Fun!  The second thing I wanted to show was that the second term in the formula doesn’t play much of a role for the larger Fibonacci numbers.  It is pretty amazing to see how well $\frac{1}{\sqrt{5}} * (\frac{ 1 + \sqrt{5}}{2})^n$ approximates the larger Fibonacci numbers.  The meat of the formula is all in the first term!  I thought this would be an especially fun fact to show him since the two terms look so similar when you write them down.

We’ve talked a little bit about the Fibonacci numbers previously, but not with this level of math.  I’d chosen this topic because I thought the math is really interesting – which it was – but all of the algebra review turned this in to a nice learning opportunity, too.  For what was supposed to be just a  little diversion way from the book during a busy week of work for me, this topic turned out to be really fun.