Last night Fawn Nguyen posted a neat problem for kids:

How would you solve this? There are 75 olives, 40% of which are green. I eat some of the green olives until 10% of the olives that remain are green. How many green olives did I eat?

I thought it would be fun to try out the problem with the boys this morning. My younger son went first (while my older son was practicing viola in the background). He described his approach as “guess and check”:

My older son went next. I think we wasn’t super focused at first because Fawn’s problem about 75 olives became a problem about 40 apples, but once he got back on track he found a nice solution. His approach looked at the number of non-green olives since that number stayed constant:

It was fun to see the two different approaches, and also interesting to see that my two kids approach percentages very differently. This was a nice problem to start the day.

Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.

You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:

Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.

At the end I asked them when they thought a shape would require 1x1x1 cubes.

After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:

Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.

Yesterday we did a really fun project inspired by a tweet from Steven Strogatz:

Here’s tweet:

Unusual intuitive argument for why A= pi r^2 for a circle, found by one of the tables in our #math exploration class. I love these surprises pic.twitter.com/dch9PfmynZ

During the 3rd part of our project yesterday the boys wondered how the triangle from Strogatz’s tweet would change if you had more pieces. They had a few ideas, but couldn’t really land on a final answer.

While we punted on the question yesterday, as I sort of daydreamed about it today I realized that it made a great project all by itself. Unlike the case of the pieces converging to the same rectangle, the triangle shape appears to converge to a “line” with an area of , and a lot of the math that describes what’s going on is really neat. Also, since my kids always want to make Fawn Nguyen happy – some visual patterns make a surprise appearance 🙂

So, we started with a quick review of yesterday’s project:

The first thing we did was explore how we could arrange the pieces if we cut the circle into 4 pieces.

After that we looked for patterns. We found a few and my younger son found one (around 4:09) that I totally was not expecting – his pattern completely changed the direction of today’s project:

In this section of the project we explored the pattern that my son found as we move from step to step in our triangles. After understanding that pattern a bit more we found an answer to the question from yesterday about how the shape of the triangle changes as we add more pieces.

Both kids thought it was strange that the shape became very much like a line with a finite area.

The last thing that we did was investigate why the odd integers from 1 to N add up to be $late N^2$. My older son found an algebraic solution (which, just for time purposes I worked through for him) and then we talked about the usual geometric interpretation.

So, a great two day project with lots of fun twists and turns. So glad I saw Strogatz’s tweet on Friday!

We seem to always start our year off with a Fawn Nguyen-like problem. Today it happened by accident when I saw this visual pattern problem from John Golden:

We tried out the problem for a little after dinner math challenge tonight. Here’s what the boys thought initially – I was happy to see that they noticed that they could look at the pattern going backwards as well as forwards:

At the end of the last movie the boys wanted to make the base for the next tower. We did that with the camera off and then started looking at the pattern again.

I was a little surprised that they wanted to make this next piece rather than just talk about it, but making it did seem to help them see what the pattern was. In fact, their initial guess at the pattern was totally different from what I saw 🙂

So, although we didn’t get all the way to the formula for the nth step, we did find a way to determine (in theory) the number of blocks on any of the steps. I remember playing around with these difference tables in high school and being absolutely amazed – it is fun to be able to play around with them with the boys now.

and decided to give the problem a try with the kids this morning.

Here’s the problem: In a town, 3/7 of the men are married to 2/3 of the women. What fraction of the people in the town are married?

I did the problem with pairs of snap cubes instead of marriage. Here’s now it went:

/

After they finished their solution, I showed them Fawn’s solution. My younger son was a little confused about adding fractions versus what’s going on in this problem. Hopefully that confusion was straightened out by the end of the video.

/

So, another nice problem from Fawn. It is fun to be able to talk through a non-standard fraction problem like this one with the boys.

Just like last year, I made it exactly two days into the school year before working through an activity that Fawn Nguyen shared with the boys. Guess the over / under for nest year should be pretty easy to set!

Today Fawn tweeted about her new project:

Math 8 or Alg Ts: I'm doing the 2nd lesson "Algebraic Thinking" in this post if you'd like to check it out: http://t.co/Vi8kkmTiaK

Had a fun time yesterday showing up at the end of the NCTM conference in Boston. What was originally a plan to grab lunch with Fawn Nguyen and Dan Anderson turned into a full day of meeting tons of people I’d only known online.

Over drinks at one of the bars at the conference hotel, Chris Hunter and Fawn showed me a really cool problem from Chris’s blog:

The problem is pretty easy to state – you and two friends go to the store to buy shoes. You have a “buy 2 get one free” coupon that allows you to get the lowest price pair out of 3 pairs for free. The question is what is the fair way for the three of you to split the savings?

I was so excited to try out this problem with the boys today that I stole the napkin that Fawn was writing on!

It seemed like the best way to go through this problem was with each kid individually. I started with my younger son. He had a little bit of trouble understanding the problem (so this video goes about 7 min) – the cost savings combined with the free item confused him, for example. However, with a few little clarifications he was able to get to an answer that he thought was fair.

Next up we looked at a similar problem with different numbers. These numbers present a new issue to deal with if you want to split the total price equally. This second problem also served as a great way for my younger son to get a little more clarity on some of the previous parts of the problem that had confused him.

Next up was my older son. His initial focus was on everyone paying the same amount, but after thinking about it for a little bit longer the equal split idea started to bother him. He wasn’t sure what a “fair split” meant. He thought for a while about other fair ways to split the price and eventually found the idea of splitting the savings. That thought process shows what I really like about this problem – lots of opportunities for thinking here and no obvious “right” answer. At the end, though, he thought splitting the total price equally was the most fair.

The neat thing about my son’s conclusion in the first problem is that it set up the next problem perfectly. If we split the total price equally in the second problem, there’s a strange issue for one of the three people. It takes him a while to notice the problem, but when he does notice it he thinks that the “splitting the savings” here is the fair way.

So, definitely a great problem for getting kids to talk about math. I really like the idea that different people are going to have different ideas about what is fair here, and I imagine those different ideas would lead to some really fun debates in a classroom setting.

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:

(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:

I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

It was really neat to see the pictures of the work her students were doing. What also struck me is that there are so many different ways to approach this problem.

My own personal style of problem solving has always been heavy on computation. I saw Fawn’s problem as a similar triangle problem:

I was actually having a hard time understanding the approach her students were taking in some of the pictures and asked Fawn about it. It turned out that the approach was not the computationally heavy similar triangle approach, but a much more simple approach with areas:

That solution got me wondering about other geometric solutions and also how my son would solve the problem. Since we are just now finishing up a section in his geometry book about similar triangles, I guessed he would look at similar triangles. He didn’t. He actually drew in the extra piece to make the large rectangle exactly as Fawn’s student’s had done (!), but then he went in a slightly different direction from Fawn’s diagrams above which I wasn’t expecting at all (and sorry for the interruption from the cat):

Finally, I noticed a (slightly) different area solution which is, of course, is a little more computational. I wanted to go through this solution just to get in a little extra algebra practice and also to show my son a different approach to the problem that also uses areas:

I love problems that have lots of different types of solutions. The setup in Fawn’s problem seems so simple – just a rectangle and a triangle – and it is amazing to me that there are so many different kinds of solutions. Fawn’s lesson here is a great reminder to me to focus more time on the more geometric solutions since my natural instincts seem to be to focus on the computation.

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

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

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

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

That exchange has really stayed with me.

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

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

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

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

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

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

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

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

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

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

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

If you graph y = sin(n) for whole numbers n = 1, 2, …, 1,000, what do you expect to see? Surprise: http://t.co/NGP0m3MiQf

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

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