Having the kids play with the tiles from Cherry Arbor Design

This week I’m going to share math projects based on items you can purchase from small businesses (in the US) who make amazing math-related products. The first project is based on the tiles from Cherry Arbor Design:

Cherry Arbor Design’s website

Please take a look at their site and their beautiful products.

My younger son chose to make a design using their Penrose tiles, which led to a great discussion:

My older son used their dragon tiles which let us talk about the symmetry he expected and then what he actually found:

Everything that Cherry Arbor Design makes is stunning. Check them out and consider supporting them if you can.

Steven Strogatz’s circle area project – part 2

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

Here’s tweet:

Here’s the project:

Steven Strogatz’s circle-area exercise

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 \pi r^2, 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!

John Golden’s visual pattern problem

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.

David Coffey’s visual pattern question

This morning David Coffey asked (on twitter) for thoughts on a visual pattern problem:

I didn’t like the problem that much. My main objection, I suppose, is that the first three choices are so easy to eliminate that a correct answer to this question isn’t that informative.

Not that multiple choice questions are going to be super informative anyway, but I suggestion a revision to make the question more like they type of question that Fawn Nguyen asks on her site visualpatterns.org

My question was: How many blocks are in row 42 and my choices were 82, 84, 87, and 89.

Tonight I had my younger son work through both questions. Here’s how it went:

Here is his work on the original question:


Here’s the work on the revised question:


Patterns of the Universe Part 2

Yesterday I found Patterns of the Universe at the MIT book store:


and did a little intro project with the boys last night:

Patterns of the Universe Part 1

This morning we went back to the book to have each of the boys look at a new pattern in more detail. Here’s what my younger son picked:


and here’s how he colored it in and some thoughts he had about the shape:


My older son picked the Harriss spiral – here are his initial thoughts. I was excited to hear that he wanted to learn more about this shape:


Here’s how he colored it in and also his description of the pattern – “it sort of reminds me of how a tree behaves”:


I’m really happy to have this book. It is a great way to get kids thinking (and talking) about both patterns and math that they might not be seeing in school. The process of coloring in the patterns gives them lots of time to think about the math, too, which hadn’t occurred to me before watching them work in their patterns this morning.

Patterns of the Universe Part 1

Was really excited to find this pile of books today!

Tonight I gave each of my kids a copy and asked for their reactions. I also had my 4th grader color in one section to see what he would do (it wasn’t what I was expecting!). My older son had an 11 hour day at school because of a club he’s in, so I decided to wait until tomorrow for his coloring.

Here’s my younger son’s initial reaction to the book:


Here’s how he colored in the Sunflower and why he did it that way:


Here’s my older son’s reaction to the book:


I’m super excited to see what patterns they pick for coloring tomorrow and then hearing what they have to say about those patterns 🙂

Volume, Scaling, and a surprising relationship to the Fibonacci numbers.

We used our Zome Geometry book for another Family Math project today. The project we stumbled on used the blue struts to talk about volume and scaling. It starts by asking a seemingly simple question about the volume of two boxes you can make with the blue struts. Here’s the problem:


and the constructed boxes – I liked hearing their ideas about the volume of the larger box:


Next they re-built the “golden box” using only medium and short blue struts. Building the box this way gave them a little bit of extra insight into the volume. Their first guess about the volume was that it would be the same as a cube with side length equal to 2x the small blue struts.


We took a few minutes to build the cube they wondered about in the last video and looked to see if the volume of that cube matched the volume of the golden box:


Once we saw that the 2x2x2 cube was too large, the kids thought to build a cube out of the medium struts. It seems possible that these two structures have the same volume – but we need to find a relationship between the mediums and the small blue struts in order to calculate the volume.


While we had the camera off, the boys played with the medium and small struts to find an aproximate relationship. They used this approximation to estimate the volume of the golden box. It was really interesting to me to hear and see their approach here:


After we finished the project, my younger son was playing around with the struts a little more. I thought it would be fun to see if they could see the interesting relationship between the medium and small blue struts. We did a little postscript. Here they see the numbers 5, 8, 13 for the long struts and 3, 5, 8 for the short ones and guess that the next numbers will be 13 and 20.


We checked if 20 shorts matched 13 mediums and found a fun surprise!


So, a fun project with a surprise postscript. Who would have thought you’d see the Fibonacci numbers pop up just by building Zometool boxes 🙂

Pythagorean triangles and differences of squares

This is the second short (in words) post for today.

Problem #7 from the 2008 AMC 10 gave my son a little trouble this morning, but talking about it led to a fun conversation about geometry and algebra. Here’s the problem:

Problem #7 from the 2008 AMC 10 A

After talking about the solution to this problem, I scrapped the actual lesson for today and talked a bit about some special Pythagorean triangles. Then I gave my son this problem as a challenge:


He was able to solve the problem in the last video by finding a pattern in the side lengths. Next I challenged him to find a different pattern. This part was a little bit of a struggle, but he did eventually find a different pattern that connects the side lengths:


Finally, I wanted to use the pattern that we found in the second video to find some new triangles. We found the next couple of triangles in the pattern and that showed him, I think, that this new pattern could be pretty useful.


This was an interesting little project for me. I guess there’s no way to know what patterns that people will find easy to see and what ones that they will find hard to see, but I do think looking for patterns that you don’t see initially is an important skill in problem solving. I’ll be on the lookout for similar project connecting geometry, algebra, and patterns in the future.

From the Virgin Atlantic lounge at JFK

About a month ago a couple of projects at work got really busy and it became clear that the week of Thanksgiving was going to be spent overseas.  This morning I dropped my wife and kids off at one airport so they could spend the week in Omaha with my family,  then headed off to the office and then to JFK.  Here’s the view from the Virgin Atlantic lounge right now:


Before leaving this morning  I spent a little time with the boys working through this fun problem:

This is yet another super project for kids from Fawn Nguyen’s visual pattern site: visualpatterns.org.   I love these projects because they are activities that I never would have thought of on my own.  Not in a million years.  The patterns are pretty simple (at least the ones that we’ve done so far), but both boys find them fun and engaging.  Last week’s purchase of a big box of snap cubes has made the projects even more exciting for them.   I see on the gallery tab on Fawn’s site that we aren’t the only people using snap cubes to play around  with her projects.  I wonder if we are the only people with Minecraft villages made out of snap cubes . . . .

This is sort of a somber trip.  One of the projects is coming back up to speed on a company I used to look after – Global Aerospace.  In 2002, Forrest Krutter and I helped to negotiate the purchase of Global Aerospace.  He and I were on the board from 2003 to 2007 when I resigned to look after some other duties in the US.  Forrest and I had a history that went quite a bit beyond Global Aerospace.  He did the local interviews in Omaha for MIT and interviewed me for college in the fall of 1988.  He then hired me into Berkshire in 2000 and we worked together on lots of projects until this past fall when  he died after a long battle with cancer.    By terrible coincidence, he died on the same day as my high school math teacher Mr. Waterman.

So, part of this trip is coming up to speed on Global Aerospace.  As much as I like the people there, and as much as I do truly believe that they are the best in a really tough industry, the circumstances will make that bit of the trip tough.
Unfortunately this isn’t the first time I’ve had to deal with the death of someone who I worked with closely.  In 2011 my partner of 10 years, Bob Bennett, died after his fight with cancer.  I spent all of 2012 coming up to speed on the parts of his work at Berkshire that I wasn’t involved in.  In fact, part of the reason I was in the office today was something that I forgot to take care of last week.

It is interesting to me how quickly I’ve grown to not like traveling for work.  At first – in my early 30s with no kids – flying all over the world was both fun and fascinating.  The thrill faded fast, though.  Since stepping down from the Global board in 2007 I’ve hardly traveled at all.  Of course, that’s made home schooling a lot easier.  My guess is that I’ll be travelling at least 6 weeks a year for work with the new responsibilities I have now.  That’s a big change.  I probably wasn’t traveling 6 weeks total from 2008 through 2012.    It seems likely that coaching ultimate is going to be very hard in 2014.  Hoping beyond hope that I’m wrong about that, though.

So, the flight leaves in about an hour.  I’m having breakfast with my best friend from college who is overlapping with my in London for about 2 hours tomorrow morning.  The rest of Sunday should be pretty uneventful – probably just reading in the hotel and hopefully hitting the gym.    The week is filled with meetings rather than math videos.  Flying back on Thursday evening and then picking Allie and the boys up from the airport on Friday evening.    Hoping Friday gets here fast . . . .

A neat Fawn Nguyen problem

One of the fun things for me about home schooling has been learning to teach elementary school math.  Maybe “learning” is the wrong word since I’m not really studying anything, but through trial and error I feel that I’ve improved my teaching a lot in the last few years.

It is always difficult teaching a new subject for the first time because you really have a good feel for how other people think about the material.  Even though the elementary school math is quite basic, not totally understanding what the kids aren’t understanding makes teaching the material tough for me.

In the last year I began to follow a bunch of math teachers on twitter.  It has been an incredible experience learning from all of the ideas that they share.  One of the most prolific writers is a middle school math teacher in California named Fawn Nguyen.  How she has the time in the day to do what she does is beyond me (as are her blogging skills), but I use so much of her stuff that it is as if my kids are being taught by an east coast amateur impersonator of her.    Seems like everything she posts turns into some sort of homework for the boys.

Today she posted a series of problems here:


The first is a problem that I wouldn’t have appreciated before I started working with my kids.  After seeing it today, though, I couldn’t wait to go home and try it out on them.

I didn’t let either of the kids see the problem before turning on the camera – the goal was really to see how they’d approach the problem rather than if they’d be able to do the multiplication correctly.

First up was my older son.  He’s always seems to want to charge right into a problem with the first idea he sees, so I was curious if he’d simply multiply out the two expressions.  He did, but in a way I wasn’t completely expecting:

I should note that he taught himself how to do basic arithmetic, so his way of adding, subtracting, and multiplying is definitely not standard.  As his approach to arithmetic was perfectly fine,  I never bothered to teach him now to carry, borrow, or do long multiplication the “normal” way.  I’ve always wondered if I’ll regret that decision later.

Next up was my younger son.  I also wasn’t sure what he was going to do.  He’s only learned the basics of multiplication, so it didn’t seem likely to me that he’d want to multiply everything out.  In the last couple of weeks we’ve been studying primes and factoring, and that was the approach he took to the problem.  I’m sorry that so much of this video is him trying to figure out if 79 was prime, but I thought the process was fascinating.  Even if I had to help him get over the last little hump.

This was a really fun problem to work through –  Thanks Fawn!!