Another really neat problem from James Tanton

Last week James Tanton posted this interesting problem on Twitter:

It was especially interesting to me because it is closely related to one of my earliest experiences with “advanced” math.

My high school in Omaha had an incredible math department led by an amazingly gifted math teacher, Mr. Waterman. He taught a special class over the lunch hour – Enrichment Math – where kids from ranging from freshmen to seniors would study math that is normally not part of high school curriculum. The year before I came to high school the unquestioned leader of that class, Anita Barnes, had graduated and became the 3rd student from our school in 5 years to receive a full ride math scholarship to Washington University.

During one of the college breaks she came back to give a lecture on recurrence relations in which she showed how to write down an formula for the nth Fibonacci number. It may as well have been magic, but that was one of the first times I saw the incredible power of math first hand.

Tanton’s problem was a happy reminder of the lecture that Anita all those years ago (1987 if you must know!), and I thought it would be really fun to walk through the problem with the boys.

We started by just talking through the problem to make sure that they both understood what was being asked. They both recognized that (1 + \sqrt{2}) was irrational and thought that the irrationality of the number would come into play somehow in this problem.

The next step was to head to the computer to see what powers of (1 + \sqrt{2}) look like when you write them as decimals. It takes a little while for the pattern to emerge, but the boys are able to pick up on the pattern after looking at the first 10 powers.

After getting an idea that we may indeed eventually run into one million 9’s after the decimal point, we go back to the board to talk about why. In this video I introduce the boys to the surprising role that the number (1 - \sqrt{2}) plays in our problem. This video has a little unexpected detour where we talk about multiplying two negative numbers, so that’s why it is a little long.

Now that we’ve understood a little bit about (1 - \sqrt{2}) we head back to the computer to first understand this number a little better and then to see an amazing pattern formed by combining powers of (1 + \sqrt{2}) and (1 - \sqrt{2}). Understanding this pattern provides a neat way to solve Tanton’s problem:

Now that we’ve calculated the first few numbers in the list of (1 + \sqrt{2})^n + (1 - \sqrt{2})^n we head back to the board to see if we can see any patterns at all in this new list. It was fun to see that they did manage to find the pattern by working together.

Having found the pattern, I show them how to write a formula that describes that pattern. If we can prove that our formula is correct (which we do not do, btw) we can understand why our list will only ever have integers, and then why powers of (1 + \sqrt{2} will eventually get as close to an integer as we like. This is the math that Anita explained to our Enrichment Math class in her 1987 guest lecture:

Finally, it would be a shame to have gone through all of this math without mentioning the Fibonacci numbers. We take a few moments to calculate the first few Fibonacci numbers and then I show the boys the formula involving \sqrt{5} that describes this sequence.

All in all a fun way for the boys to see (and participate in) some neat math. Also a nice walk down memory lane for me.

A list Ed Frenkel will love

Came home from work tonight to see this incredible post by Dan Anderson:

link in case the link in the tweet doesn’t work:

Since reviewing Dan Meyer (and colleague’s) new piece Central Park  yesterday, I’ve had my mind in a bit of a muddle thinking about the difference between the kind of math that I think kids will like to see and the kind of math that other folks think that kids would like to see.

Dan’s post went in a much better and much more productive direction – what kinds of math do kids find exciting when you ask them?  Well timed, Dan, well timed, and what an absolutely awesome list.  It is so amazing to see the mix of math theory and math that connects with the lives of the kids: topics ranging from Brower’s Fixed Point theorem, Symmetry, and Graham’s number to golf handicaps, diving scoring, and formatting yearbooks (which I bet looks a lot like the “Central Park” task). I’ve got no idea at all what “Mole Train Woot Woot” is, but I want to know. Badly.

Earlier this year Ed Frenkel did a nice interview with Numberphile:

He has several interesting things to say about math education:

“So how do we make people realize that mathematics is this incredible archipelago of knowledge?”

I think that Dan’s project is an tremendously exciting and inspiring way to approach Frenkel’s question.

“If you want your kids to understand and appreciate the beauty and power of mathematics, we have to connect it to our daily lives”

You definitely see this connection in some of the projects, but it is also nice that many projects involve interesting ideas from pure math. Maybe the students are not able to fully understand all of these ideas, but these ideas from theory are still somehow capturing their imagination, and that is so great to see.

“In the case of mathematics [students] are not even aware of the masterpieces of great mathematicians the way that they are are of the existence of the masterpieces of the great artists.”

Well, Ed Frenkel, on this point I think that Dan’s post will really make you happy.

A review of Central Park by Dan Meyer, Chris Danielson, and Desmos

This afternoon Dan Meyer announced a new project that he and several of his colleagues put together:

A direct link to the project on the Desmos site is here: Central Park on Desmos

By chance I had some unexpected free time this afternoon, so I played around with it a little. Not with the idea of thinking critically about it, but rather just to see what it was. At the end I wasn’t sure what to think and thought I’d give it a go with the boys when I got home. They enjoyed it.

Though the videos are long, if you are interested in seeing kids reacting to this project while working through it (and struggling a little with the directions), they are probably worthwhile. The short story – both boys were engaged all the way through (~15 min for my 10 year old and 27 min for my 8 year old). My older son found the estimating hard, but did not really struggle with the math, though not all of his answers were correct. His reflections at the end were not really math related.

My younger son did struggle with the math a little bit, but stayed with it and I think learned a little bit about variables by the end. His reflections were about the math, which was good.

I’ll admit to not generally being a fan of this type of computer based lesson. If there was a way to measure the tendency of someone to like lessons such as this one, I’d guess that 80 out of 100 people in math would like them more than me. That said, though my own reaction was lukewarm, my kids did like it and after seeing them work through it, I would happily use it with other kids.

Here is my 10 year old working through it. Variables are not new to him, so that part of the lesson (which, I think, is supposed to be the main part) is not difficult for him. He does struggle a little in the beginning with the estimates, and I was happy to watch that struggle. In retrospect, I wish I’d shot this movie a bit more zoomed in on the computer screen.

and his reflections (about 1 min):

Here is my 8 year old working through it. Variables are also not new to him, but he’s obviously not spent as much time working with them as his older brother has. He didn’t seem to have as much difficulty with the estimation as I was expecting, but understanding some of the more math-related questions was difficult for him. By the end, though, he did seem to get how to use the variables. It was nice to see him stay with the lesson and to get something out of it. Though this video is long (~30 min), if you are a fan of talking math with your kids, there’s some pretty good “kid talking about math” in this video.

and his reflection on the project (about 1 min):

I was impressed that this lesson could keep my younger son engaged for nearly half an hour, and even more impressed that at the end the things he found exciting about it was the math. For that reason alone, I’d be happy to use this lesson with other kids. My older son’s reaction leaves me a little worried that students who are not struggling to understand variables will not get as much out of this lesson. The creators of the lesson will probably be surprised to hear what his favorite part of the lesson was, but I’ll let you see that for yourself in his reflections.

Mr. Honner does it again

Patrick Honner has several blog articles about poor problems (sometimes outright mistakes) on some of the exams that New York state requires for students. Since I don’t live in NY I’m not super familiar with these exams, though Mr. Honner’s posts actually make me happy that I’m not. I get the feeling that the more I knew about them the more they’d drive me crazy.

In any case, his latest article is here:

Regents Recap — June 2014: When Good Math Becomes Bad Tests

and the problem he discusses is:

“The medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians?

(a) 2 and 3

(b) 3 and 6

(c) 3 and 4.5

(d) 3 and 9”

Unlike some of the prior problems that he has written about the problem itself doesn’t have any mathematical flaws. Instead this problem is simply testing if you know a single math fact. Really no deep understanding of geometry is required to solve it at all if you know this fact. I think that Mr. Honner’s concern – which is essentially that if state exam questions end up looking similar to this one, math education is just going to turn into something equivalent to prepping for a night on Jeopardy – is spot on.

This critique hit me for another reason, though. Three years ago when I decided that I wanted to start making fun little math videos for kids, I thought that I should practice a little and see if I actually had any ability to explain math. At that point it had been more than 10 years since I’d been in front of a classroom. Oh, and 10 years in finance doesn’t exactly sharpen your explanation skills.

What I decided to do was grab my copy of Geometry Revisited off the shelf and pretend I was doing a few lectures from the book. I went through the first two chapters or so, but two of the early sections are relevant here. The second “lecture” was about Ceva’s theorem, which is a beautiful theorem with a fascinating and incredibly instructive proof (keep in mind that I’d not talked about math for a long time in this video, so it isn’t the best. Also my older son is watching for some reason I don’t remember):

So we get a beautiful theorem with a really instructive proof right in the second section of Geometry Revisited! Also, I saw that Steve Leinwand gave a lecture at a conference for teachers last week and said that ratios were one of the most important pieces of early math. Ratios play a surprising role in this proof of Ceva’s theorem, so it may have even more educational value than I realized the first time around. Finally, the result we can see pretty easily from Ceva’s theorem that is relevant to the question on the NY state exam is that the medians of a triangle intersect in a single point.

With a few fun ideas about cevians in hand, you might be interested in learning even more, and Geometry Revisited doesn’t disappoint. The next section shows a couple of nice results about the medians that also have incredibly instructive proofs (the part about the medians starts at 3:20):

I remember vividly how going through these early sections in Geometry Revisited reminded me how much I loved math. With just two short sections on Ceva’s theorem and medians we have a couple of beautiful results and several really instructive proofs for students to see. I understand that no all kids are going to find these ideas to be as fascinating as I do, but I think that lots of kids will. It seems like such a shame to me to reduce it all to the question posed above, and frankly even worse to essentially reduce it all to something like –

Answer: They trisect each other.

Question: What do the medians in a triangle do?

What do mathematicians do

Lots of interesting math floating around the internet this week:

(1) Numberphile had an incredibly cool set of videos featuring Ron Graham talking about Graham’s Number,

(2) The NY Times had two articles on math education:  Why do Americans Stink at Math by Elizabeth Green and Don’t Teach Math, Coach It by Jordan Ellenberg,

(3) Two really interesting blog articles:  Jordan Ellenberg describes progress on understanding the rank of elliptic curves:  Are Ranks Unbounded? and Cathy O’Neil produced a neat little python notebook to walk people through RSA’s encription algorithm:  Nerding out: RSA on an iPython Notebook, and

(4) The “Twitter Math Camp 2014” teacher conference was happening in Oklahoma, which make for 100’s (of not 1000’s) of interesting discussions on twitter about teaching math.

All of of the fun math plus all of the ideas about teaching math made me want to step back and talk to the boys about what mathematicians do.    The math theme of the week seemed to be the difference between bounded and unbounded sets, so I tried to let that idea shape the discussion today.

We began by talking about Platonic solids.    Before turning on the camera we built a few of the Platonic solids out of our Zometool set for props.  Then we talked about what these shapes are and if there are infinitely many of them:

Next we talked about the prime numbers.  Ellenberg’s book How not to be Wrong has a wonderful discussion for a general audience about the prime numbers and I’ve been meaning to use some of his ideas to talk about the primes with the boys.  Luckily for me, right off the bat the boys were asking some questions about primes that Ellenberg answers. The main topic in this part of the talk is about the of primes, though my younger son wonders about the gaps between primes that will discuss in the next video:

Next, gaps between the primes. The boys seemed pretty interested in how the primes spread out. Ellengerg’s idea of using the even numbers and powers of 2 as an example turns out to be a really nice hook, and provides a great framework for talking about the new bounded gaps result:

After spending 10 minutes talking about some fun results about prime numbers, I wanted to spend the last few minutes talking about one way that prime numbers come into play in our daily lives. This part was inspired by Cathy O’Neil’s piece this week. I sort of daydreamed for a bit about an “rank of elliptic curves for kids” talk, but, um . . . , no.

What I focused on instead was the idea from O’Neil’s python notebook that it is easy to multiply two numbers and not so easy to factor. This idea forms the basis of encryption algorithms. Elliptic curves come into play, too, and Ed Frenkel discusses that a little bit in this fascinating video: Elliptic Curves and Cryptography.  But again, that’s for another day.

Definitely a fun week. Neat to see some new and exciting ideas from math in some blogs, and fun to see so much spirited discussion about math education. I think that many of the ideas in theoretical math will appeal to kids – Graham’s number and cryptography are just the two that emerged this week – and it is fun to be able to talk about these ideas and why mathematicians find these ideas interesting with my own kids.

Now, in the spirit of teaching and coaching from Ellenberg’s NYT article, I’m off to Boston to coach Brute Squad.

A small mistake in Numberphile’s videos about Graham’s number

[ not about math with my kids, but about some cool Numberphile videos that came out yesterday, and sorry this one was a little rushed]

Yesterday (July 20, 2014) I saw two absolutely incredible Numberphile videos about Graham’s number.  Part of what makes them so amazing is that the explanation of the number comes from Graham himself!   I love Numberphile’s work in bringing math to the masses.

There is, unfortunately, a little mistake in the text overlay in one of the videos that I wanted to point out.  The somewhat humorous result of this mistake is that Graham’s number is actually larger – vastly larger, in fact – than what the video indicates.

Since the formatting a blog post with the arrow notation and power towers was going to take more time than I had this morning, I decided to just go to the whiteboard. The mistake, which I explain in the first video below, is easy to make since all of the numbers are so large.  It involves confusing the number 3 ↑↑↑ 3 with 3^3^3^3.

After my video are the links to new Numberphile videos (that you might want to watch first if you aren’t familiar with Graham’s number) and two a few other fun Graham number articles, including a Family Math project about Graham’s number I did with my kids last year (which is the only reason I noticed the mistake in the new Numberphile videos).

Here are the new Numberphile videos (the text overlay error is in the 2nd video):

and here’s Evelyn Lamb’s piece on Graham’s number for Scientific American:

Finally, here’s our old Family Math project on Graham’s number, which is a really fun project to work through with kids:

A bit of a struggle with estimation

My wife was running a few errands with my older son this morning, so I decided to do today’s Family Math with just my younger son. The topic today was estimation and the specific problem we were thinking about was this:

If we drive from the Museum of Science in Boston to the Exploratorium in San Francisco, how many time will the wheels in our car turn around?

Fun question, and we began by talking about the different pieces of the puzzle we’d need to solve this problem (with a little help from the math cat):

After talking through the problem, we went outside to see if we could measure the circumference of our car wheel. We were unable to find a wheel of the same size (we tried a garbage can, a christmas tree stand, and a bike wheel) so we had to measure the diameter. I wanted to also make a chalk mark on the tire and measure the distance along the ground when it made one rotation, but I accidentally left my keys in my wife’s car. Oh well . . .

Next we came back inside to tackle the problem. In this segment my son was having trouble estimating some of the numbers we’d encounter in the problem, so instead of proceeding straight to the solution we went back outside.

We went back outside to take a closer look at some of the numbers in our problem. I thought it would be helpful for him to see what 88 inches looked like in relation to the length of our driveway, for example. That seemed to help him get a better estimate of how many times one of the car wheels would turn leaving our driveway.

The second trip outside seemed to help him get a better understanding of some of the distances involved in the problem, but he still had a tough time getting a good estimate of the number of times the wheel would turn going from Boston to California. He now understood that his estimate of 500 from before was too low, but he wasn’t sure how much to increase that estimate. We did a series of simple approximations and arrived at an estimate of 2 million.

Finally we get to the point where we plug in some numbers. We haven’t done much with unit conversions, so I had to help him through that a little bit. Eventually we arrived at a number of roughly 2.2 million for the number of times the wheel would turn.

So, a little bit more of a struggle with this one than I’d intended, but still a lot of fun. The lesson for me on this one is that I need to do a few more exercises that involve estimation and unit conversion. Hopefully I’ll remember to incorporate an exercise with those characteristics as least once per month.

A great counting problem for kids involving binary

This summer I’m slowly working through Art of Problem Solving’s Introduction to Counting and Probability with the boys. I though it would be fun to see them working together and since I haven’t covered this subject with either of them previously I was hoping the age difference wouldn’t be that big of a deal. So far so good.

Flipping through the challenge problem section in Chapter 2 earlier this week I ran across this wonderful problem from an old ARML:

If you write the integers from 1 to 256 in binary, how many zeroes do you write?

Definitely too difficult for a homework problem for either of them right now, but it struck me as a great problem to use for one of our weekend Family Math projects. Some parts were more difficult than I expected, but the kids remained engaged all the way through. It was really fun to see them talking through some of the more challenging details as well as listing out binary numbers with the snap cubes. Although this one is a little more challenging and a little longer than a normal Family Math project, I’m really happy with how this project turned out. I think it would be incredibly fun to work through this problem with a large group of kids.

We began by simply discussing the problem which, not surprisingly, meant a short review of writing numbers in binary. My younger son understood that 256 was a power of two and would be written as 1 followed by a bunch of zeroes. How many zeroes exactly was the starting point to today’s discussion:


Next was a neat idea from my older son – to solve the problem we want to break it down into easier cases. His first guess at a way to break the problem down was to organize the numbers by the number of 1’s they have. An interesting idea for sure, and one that works well for numbers that only have one 1. Unfortunately breaking the problem into these cases gets complicated fast. After we do a bit more work with permutations and combinations it might be pretty fun to return to this idea, but for today we walked down this road a little bit and discovered it is a pretty tough way to go.

Since this video is a little long and has nothing to do with how we eventually solve the problem, it is ok to skip it. However, I think it is really important to try out ideas like my son had here. Learning how to identify when an idea is working out and when it isn’t is a really important lesson:

Next we moved to our kitchen to look a little more closely at what binary numbers look like. We used snap cubes to “write” a few numbers in binary. Looking at the list we decided to group the numbers by number of digits and see if that helped us count the zeroes.

At the end of the last video we formed a conjecture that the number of zeroes in our 4 digit binary numbers would be 9. In this next video we write out the numbers using our blocks and discover that we have 12 zeroes in the 4 digit binary numbers rather than 9. We then talked through how we could see the 12 zeroes as 3 groups of 4. The boys struggled a little to see the three groups of 4 that made these 12 zeroes, but eventually saw it and understood it. That led to another conjecture:

After the last video they were really engaged with this problem. We were guessing that we’d find 32 zeroes when we wrote out the 5 digit binary numbers. The boys spent about 10 minutes off camera building these numbers out of snap cubes, and we did indeed find 32 zeroes. We then talked for a bit about why the pattern we found makes sense. After the video was over I mentioned one extra reason that we could see why half of the digits (not counting the left most digit) were zero – the last 4 digits are the same forwards and backwards if you reverse the colors. That helps us see that for every yellow block we have a black block that can be paired with it.

Finally we went back to our big whiteboard to add up the results. Nothing super special going on here – we write down all of the cases and add up the numbers. The kids thought we’d need a calculator, but we somehow managed to add up the numbers without one!

As I said at the beginning, I thought this problem would make a great project to work through with the kids. Although it was a little long, I’m really happy that we worked through this one. Certainly this was one of the most challenging problems that we’ve gone through together, but since the kid remained engaged all the way through, I’m super happy that we gave this one a shot. Again, I’d love to go through this with a large group of kids.

What a fun start to the day today!

How would you teach this problem?

I’m doing a slow walk through Art of Problem Solving’s Introduction to Counting & Probability with the boys this summer. Mainly just for fun, but also because I thought that it would be neat to see them working together.

Peeking a little bit ahead in the book last night I found this problem in the Challenge Problem section at the end of chapter 2 (problem 2.33 on page 47). It is from an old ARML:

How many zeroes do we write when we write all the integers from 1 to 256 in binary?

I like this problem a lot and am planning on using it for one of our Family Math activities this weekend. Before we dive into the Family Math project, though, I thought it would be fun to ask around – how would you approach going through this problem with kids?

Any and all ideas are welcome!

Just for fun – some infinite sums

I was listening to Jordan Ellenberg’s book “How not to be Wrong” on the way back from Cape Cod yesterday. This time through his short discussion on infinite series caught my attention. Since in heading to DC for an ultimate frisbee tournament this weekend, I thought I’d do our weekend Family Math a day early and talk a little bit about infinite series.

We’ve talked a little bit about infinite series before – motivated mainly by Vi Hart’s videos about why .9999…. = 1 and the Numberphile video about the sum 1 + 2 + 3 + 4 + . . . – and although this talk goes a tiny bit deeper the goal isn’t rigor, just fun. I’ll link the Vi Hart and Numberphile video at the end of this post.

I started off asking the boys about infinite series and they mentioned the two examples that they’d seen before. Neither of them seems to believe the Numberphile video which was nice to hear – at least they are thinking about why the result in the video seems strange. Next we talked about why 0.999… = 1 and a few of the common “proofs” including the one that Ellenberg refers to by the catchy phrase “algebraic intimidation.”

In the next video we sort of explore Ellenberg’s “algebraic intimidation” phrase by looking at another example from his book – the series 1 + 2 + 4 + 8 + 16 + . . . . Here we apply one of the techniques that we used in the last video to show that this series seems to have a value equal to -1. Wait – what??

We finished up with another series where the algebraic techniques we used to show 0.9999… = 1 produce a strange answer. The series that we consider here is 1 – 1 + 1 – 1 + 1 – 1 + 1 . . . . The boys arrive at the conclusion that the sum seems to be either 0 or 1. We then go through the algebra to show that you get the surprising answer of 1/2, but they are not convinced.

This was a fun little discussion. Obviously the details of infinite series are a little bit over their heads right now, but it is neat to see them thinking about results that make sense and results that don’t seem to make sense at all. One of the other neat ideas that I’ve taken away from Ellenberg’s book is understanding what ideas are “obvious” and what ideas are not “obvious.” Once mathematicians started asking questions about infinite sums, it took a couple of centuries to get their heads around the issues. It is a nice that Ellenberg is able to provide lots of examples of “obvious” results that are not obvious at all.

Finally, here are the Vi Hart and Numberphile videos just for completeness: