When will she pass me for the first time? [solution]

Recently, my dad posed the following question here:

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

If you haven’t solved it yet, give it a crack. It’s a fun problem that has surprising depth.

Here’s my solution (in it, I refer to “mom” rather than “my wife” for obvious reasons!):

Since mom’s lap rate is m laps per unit time, and dad’s lap rate is n laps per unit time, in time t, mom goes mt laps and dad goes nt laps.

They meet whenever their distance (measured in laps) is separated by an integer number of laps k. That is, mom and dad meet when

mt=nt+k, k\in\mathbb{Z}.

This happens at time


Mom will have gone


laps and dad will have gone


laps when they meet for the kth time.

And that’s it! That’s the general solution. This means that:

  • At time t=0, dad and mom “meet” because they haven’t even started walking at all (they are k=0 laps apart).
  • At time t=\frac{1}{m-n}, dad and mom meet for their first time after having started walking (they are k=1 lap apart). This is the answer to the problem as it was originally stated. Mom will have gone mt=\frac{m}{m-n} laps and dad will have gone nt=\frac{n}{m-n} laps when they meet for the first time.
  • At time t=\frac{2}{m-n}, dad and mom meet for their second time (now k=2 laps apart).
  • At time t=\frac{k}{m-n}, dad and mom meet for their kth time.

Here are two examples:

  • If mom walks 15 laps in the time it takes dad to walk 10 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=3 laps and dad will have gone \frac{n}{m-n}=2 laps.
  • If mom walks 12 laps in the time it takes dad to walk 5 laps, when they meet up for the first time, mom will have gone \frac{m}{m-n}=1\frac{5}{7} laps and dad will have gone \frac{n}{m-n}=\frac{5}{7} laps.

Boom! Problem solved! :-)

Random Walks Mural

I’ve been meaning to give the back wall of my classroom a makeover for a while. This summer I finally found some time to tackle the big project. I took down all the decorations and posters. I fixed up the wall and painted it a nice tan color. Then, I let loose the randomness!

and some added, inspirational, text :-)I struggled with what the new mural would be–I’ve thought about it over the last few years. I considered doing some kind of fractal like the Mandelbrot Set. But it should have been obvious, given the name of my blog!! What you see in the picture above is three two-dimensional random walks in green, blue, and red. In the limiting case, one gets Brownian motion:

Brownian motion of a yellow particle in a gas. (CCL)

I honestly didn’t know what it was going to look like until I did it. I generated it as I went, rolling a die to determine the direction I would go each time. I weighted the left and right directions because of the shape of the wall (1,2=right; 3,4=left; 5=up; 6=down). For more details about the process of making it, here’s a documentary-style youtube video that explains all:

Actually, I lied–it doesn’t tell “all.” If you really want to know more of my thought process and some of the math behind what I did, watch the Extended Edition video which has way more mathematical commentary from me. I’ve also posted the time lapse footage of the individual green, blue, and red. Just for fun, here’s an animated random walk with 25,000 iterations:

Wikipedia, Creative Commons License

A two-dimensional random walk with 25,000 iterations. Click the image for an animated version! (CCL)

I think the mural turned out pretty well! It was scary to be permanently marking my walls, not knowing where each path would take me, or how it would end up looking. At first I thought I would only do ONE random walk. However, the first random walk (in blue) went off the ceiling so I stopped. And then I decided to add two more random walks.

In retrospect, it actually makes complete sense. I teach three different courses (Algebra 2, Precalculus, and Calculus) and I’ve always associated with each of theses courses a “class color”–green, blue, and red, respectively. I use the class color to label their bins, to write their objective and homework on the board, and many other things.

The phrase “Where will mathematics take you?” was also a last-minute addition, if you can believe it. There just happened to be a big space between the blue and red random walks and it was begging for attention.

good question!What a good question for our students. The random walks provide an interesting analogy for the classroom. I’d like to say I’m always organized in my teaching. But some of the richest conversations come from a “random walk” into unexpected territory when interesting questions are raised.

Speaking of interesting questions that are raised, here are a few:

  • Can you figure out how many iterations occurred after looking at a “finished” random walk? Or perhaps a better question: What’s the probability that there were more than n iterations if we see m line segments in the random walk?
  • Given probabilities p_1, p_2, p_3, p_4 of going in the four cardinal directions, can we predict how wide and how high the random walk will grow after n iterations? Can we provide confidence intervals? (might be nice to share this info with the mural creator!)
  • After looking at a few random walks, can we detect any bias in a die? How many random walks would want to see in order to confidently claim that a die is biased in favor of “up” or “left”…etc?

Some of the questions are easy, some are hard. If you love this stuff, you might be interested in taking a few courses in Stochastic Processes. Any other questions you can think of?

Where will math take you this coming academic year? Welcome back everyone!

MAA Distinguished Lecture Series

If you live in the DC area and you like math, you have no excuse! Come to the MAA Distinguished Lecture Series.

These are one-hour talks, complete with refreshments, all for free due to the generous sponsorship of the NSA. The talks are at the Carriage House, at the MAA headquarters near Dupont Circle.

Here are some of the great talks that are on the schedule in the next few months (I’m especially excited to hear Francis Su on May 14th).

I’ve been to many of these lectures and always enjoyed them. Robert Ghrist‘s lecture was out of this world (here’s the recap, but no video, audio, or slides yet) and was so very accessible and entertaining, despite the abstract nature of his expertise–algebraic topology.

And that’s the wonderful thing about all these talks: Even though these are very bright mathematicians, they go out of their way to give lectures that engage a broad audience.

Here’s another great one from William Dunham, who spoke about Newton (Dunham is probably the world’s leading expert on Newton’s letters). Recap here, and a short youtube clip here:

(full  talk also available)

So, if you’re a DC mathophile, stop by sometime. I’ll see you there!

Math on Quora

quora iconI may not have been very active on my blog recently (sorry for the three-month hiatus), but it’s not because I haven’t been actively doing math. And in fact, I’ve also found other outlets to share about math.

Have you used Quora yet?

Quora, at least in principle, is a grown-up version of yahoo answers. It’s like stackoverflow, but more philosophical and less technical. You’ll (usually) find thoughtful questions and thoughtful answers. Like most question-answer sites, you can ‘up-vote’ an answer, so the best answers generally appear at the top of the feed.

The best part about Quora is that it somehow attracts really high quality respondents, including: Ashton Kutcher, Jimmy Wales, Jermey Lin, and even Barack Obama. Many other mayors, famous athletes, CEOs, and the like, seem to darken the halls of Quora. For a list of famous folks on Quora, check out this Quora question (how meta!).

Also contributing quality answers is none other than me. It’s still a new space for me, but I’ve made my foray into Quora in a few small ways. Check out the following questions for which I’ve contributed answers, and give me some up-votes, or start a comment battle with me or something :-).

And here are a few posts where my comments appear:

USA Science and Engineering Festival

If you’re local, you should go check out the USA Science and Engineering Festival this weekend. It’s on the mall in DC and everything is free.

USA-Science-and Engineering-Festival LogoThey will have tons of booths, free stuff, demonstrations, presentations, and performances. Go check it out!

For my report on the fest from two years ago, see this post. The USA Science and Engineering Festival is also responsible for bringing to our school, free of charge, the amazing James Tanton!

One thing that makes my class unique

Photo from Flickr.com, credit Alan Cleaver, under Creative Commons License.

What’s one thing that makes my class unique?

We play Two Truths and a Lie.

Let me explain. I teach 150+ kids each semester (which means I get new ones in January). I used to think that my job was to teach the material, and the kids didn’t need to like me for that mission to be accomplished. It doesn’t matter what they think of me. That’s not my job, so I reasoned. But thanks to reading awesome books like The Essential 55, The Excellent 11 (both by Ron Clark), and most important, Teaching with Love and Logic (Jim Fay and David Funk), I now know that’s completely and totally false. Here’s the truth: You can’t teach students until they like you.

Getting to know my students has become a major part of what teaching means to me now. The Mr. Chase of eight years ago would never have done a get-to-know you activity at all, since it takes valuable instructional time.

The trouble is, it’s super hard to get to know 150 students in one semester. Even learning their names is a monumental task. The cursory get-to-know-you activity on the first day is cool, and better than nothing, but can you really get to know 150 students in ONE DAY? I still do a little mini, fun first-day activity. But here’s an additional, deeper activity that I’ve come to love.

On the first day of class I hand out index cards. I don’t ask students for their information anymore. I can get their parents names, email addresses, phone numbers, address, and more, through our school’s database, just as you probably can. So asking for that information is a waste of time as far as I’m concerned–it’s just busy work for them. Instead, on their index card, I ask them to write their name and Two Truths and a Lie. They can give it to me after the 45 minute period is over. I tell them they can work on it while I’m going over the syllabus, if they find me boring :-). They can even turn it in the next day if they really want to craft an excellent set of statements that will fool their classmates.

Have you ever played this game? Here’s how it works: You write down three statements about yourself, two of which are true and one of which is false. Then people try to guess which statement is the false statement. Students share things that are interesting and unusual–things their closest friends in the class might not even know.

“I speak four languages”

“I have two dogs and a turtle.”

“My grandmother lives in Portugal.”

“I’ve never broken a bone.”

“I’ve been to five continents.”

“I’m a black-belt in Jujitsu.”

“I don’t like chocolate.”

“My dog’s name is Bubbles.”

When you play this at parties, it takes a while–a minute or two for each person. And of course you want to discuss the results afterward. “What languages do you speak??” “Okay, your dog’s name isn’t Bubbles. But do you have a dog? What kind is it? What is its name?”

So if it takes a while, and you want to take your time, how do you fit it into class time? Well, I have a stack of them at the front of the room and whenever we have extra time, throughout the first month or two of school, we pull a random card (or a few) and meet that student. I say “Today we’re going to meet Robert…everyone say hi Robert!” and everyone says “HI ROBERT!!” (way less corny when it actually happens; don’t worry they love it!). Then we read Robert’s card, and on the second reading everyone is required to raise their hand upon hearing the statement they think is false. Great fun. And afterward we ask Robert some follow-up questions.

It’s a fun activity and lets us genuinely get to know one another and learn very unique things about each other. I give them my own Two Truths and a Lie on the first day of class as an example:

1. I’ve done tricks on a flying trapeze.

2. I lived in Peru for a year.

3. My parents have chickens in their backyard.

(Feel free to make guesses as to which of my statements is a lie.)

This was a unique idea to my class, but some of my other teacher friends have adopted it now, so perhaps it doesn’t qualify anymore :-).

This blog post was in response to the prompt, “What is one thing that happens in your classroom that makes it distinctly yours?” which I was encouraged to answer as I participate in the Exploring the MathTwitterBlogosphere challenge. More challenges to come! (And more blog posts, I’m sure!)

Happy Metric Day, by the way!

Mindset List for incoming High School class of 2017

Flickr, Creative Commons License

Happy first day of school! For us, today marks the first day of the school year and we’re welcoming students into our midst. What are the new kids (the freshmen) going to be like?

Each year Beloit College describes the incoming college freshmen class with its now famous “Mindset List.” I looked around and couldn’t find a high-school equivalent. So here’s one I came up with. This is a description of the incoming high school freshmen class (class of 2017). Note that all of descriptions on Beloit’s college freshmen mindset list apply also to high school freshmen. So here’s my own “high school class of 2017 mindset list.” Enjoy!

  1. The Euro has always existed. So has Sponge Bob Square Pants. And Google, Inc. And the iMac. And Viagra.
  2. Bill Gates has always been worth over $100 billion.
  3. You can talk to them about the Sandy Hook shooting or the Virginia Tech massacre, but they won’t remember anything about Columbine, which happened the year they were born.
  4. Star Wars Episode 1: The Phantom Menace is as much of an ‘old-school’ classic as any of the original Star Wars movies. The movies Fight ClubThe MatrixAmerican PieSaving Private RyanArmageddon, and The Sixth Sense also came out in the years they were born.
  5. East Timor has always been a sovereign nation.
  6. George W. Bush and Barack Obama are the only presidents that they really know. Clinton left office when they were just 2.
  7. Exxon and Mobil have always been the same company.
  8. Movies have always been reviewed by Ebert & Roeper .(Gene Siskel died the year they were born; Roger Ebert just died this past April.)
  9. They won’t have any memories of John F. Kennedy Jr, Dr. Spock, Frank Sinatra, Roy Rogers, or Alan Shepherd, all of whom died just as they were being born.
  10. They have always had their music in mp3 format and used mp3 players (invented in 1998). CD players? SO passé.
  11. Seinfeld closed up shop before they were born.


For more events that happened in 1998 and 1999, visit the wikipedia articles. Please feel free to correct any of my above information or suggest additions!