Grid with 100 Paths

Due to current circumstances, the National Council of Teachers of Mathematics (NCTM) had to cancel their Centennial Annual Meeting and Exposition, which was to be held April 1-4 in Chicago. As a replacement, though, they presented an amazing gift to the math education community — 100 free webinars led by selected speakers from the Chicago program. Dubbed 100 Days of Professional Learning, these webinars are to be held on select days from April through October.

As part of the 100 Days, I presented “100 Problems Involving the Number 100” on May 14. To celebrate NCTM’s 100th Anniversary, I collected or created 100 problems, each of which included the number 100. One of my favorites, Dog Days, looked like this:

From this information alone, what could you determine?

At the end of the webinar, the conversation continued “backstage” with several members of NCTM staff, NCTM President Trena Wilkerson, and me. During that conversation, Trena made the outlandish suggestion,

Now we need a collection of 100 problems for which the answer is always 100.

I had just finished preparing a webinar with 100 problems, and now she was asking for another 100 problems. But never one to shy away from a challenge, I began to think about what kinds of problems might be included in such a collection. One type that came to mind was path-counting problems, like this one:

Moving only north or east on the segments in the diagram, how many distinct paths are possible from A to B?

That particular grid, measuring just 3 × 4, has fewer than 100 distinct paths from A to B. (How many paths, exactly? That’s left as an exercise for the reader.) What got me excited, though, was wondering if there were any grids that have exactly 100 paths — and hence providing 1% of the content for the collection that Trena requested.

As it turns out, there are no unmodified m × n grids that have 100 distinct paths. But what if some segments were removed? For instance, what if one of the middle vertical segments were discarded from a 4 × 6 grid, as shown below? How many distinct paths from A to B would there be?

As it turns out, a lot more than 100. (Again, finding the exact number is an exercise I’ll leave for you. If you need help, Richard Rusczyk from Art of Problem Solving has a video showing how to count paths on a grid.)

So, this is where I leave you:

Can you create a grid with some segments removed that will have exactly 100 distinct paths?

Have fun! Good luck!

As for the webinar, which lasted only 60 minutes, there wasn’t nearly enough time to cover 100 problems, but we had fun with 5 of them.

If you missed the webinar, you can hear the discussion about the Dog Days problem and 4 others, as well as get a PDF of all 100 problems, via the links below.

• Webinar Recording
• Webinar Chat
• Webinar Slides
• Webinar Handout

Enjoy!

There Are 2 Things that Happened Yesterday…

Yesterday was a banner day.

Last night, I was finally able to carve out some time to binge-watch Season 2 of Trial & Error, and I was rewarded with a classic math joke in Episode 1. When lead investigator Dwayne Reed arrives at the house of accused murderer Lavinia Peck-Foster, he says:

There are two things that Reeds don’t trust: doctors, Pecks, and math.

I love it!

Upon realizing that I might be able to get my sitcom-writing career off the ground by reformulating stale math jokes, I promptly submitted my resume to NBC.

But, wait… there’s more!

Earlier in the day, I received NCTM‘s email newsletter Summing Up, which contained an unexpected surprise. In the section titled “NCTM Store,” there was a blurb about my most recent book, More Jokes 4 Mathy Folks, under the headline Just Published!

I had no idea that NCTM decided to sell my book, let alone that they were going to publicize it. My ignorance not withstanding, I couldn’t be more delighted!

If you’re looking for some great, light summer reading — something that can be enjoyed poolside while sipping a mojito — then pick up a copy of More Jokes 4 Mathy Folks from NCTM today! Not only will your purchase support a great organization (and my sons’ college fund), you’ll also receive a 20% discount for being an NCTM member.

Following the lead of Dwayne Reed, here are jokes that begin, “There are n kinds…,” all of which appear in More Jokes 4 Mathy Folks:

• There are only 2 kinds of math books: those you cannot read beyond the first sentence, and those you cannot read beyond the first page. (C. N. Yang, Nobel Prize in Physics, 1957)
• There are 2 kinds of people in the world: those who don’t do math, and those who take care of them.
• There are 3 kinds of people in the world: positive, negative, and relative.
• There are 2 kinds of people in the world: those who are wise, and those who are otherwise.
• There are 2 kinds of statistics: the kind you look up, and the kind you make up.
• There are 2 kinds of experienced actuaries: those who say they have made significant forecasting errors, and liars.
• There are 10 kinds of people in the world: those who understand binary, and those who don’t.
• There are 10 kinds of people in the world: those who understand binary, and 9 others.
• There are 10 kinds of people in the world: those who understand ternary; those who don’t understand ternary; and, those who mistake it for binary.
• There are 11 kinds of people: those who understand binary, and those who don’t.
• There are 8 – 3 × 2 kinds of people in the world: those who correctly apply the order of operations, and those who don’t think that 6 ÷ 2 × (1 + 2) = 9.
• There are 2 kinds of people in the world: logicians and ~logicians.
• There are 2 kinds of people in the world: those who can extrapolate from incomplete data…

Prime Time at NCTM Minneapolis

This afternoon, I’ll be presenting “Experience the Math Practices with Games and Online Tools” at the NCTM Regional Conference in Minneapolis. So if you unwittingly find yourself in the Minneapolis Convention Center at 1:30 p.m. today, please stop by.

But how cool is this? My session is #210, and today is Friday, November 13. Awesome, huh?

Wait, maybe you don’t see it:

210 = 2 × 3 × 5 × 7

Friday, November 13 = 11/13

Yeah, that’s right! The factors of my session number combined with today’s date are the first six prime numbers. You don’t have to be a math dork to appreciate that! (Though it doesn’t hurt.)

Why is 6 afraid of 7?

I assume it’s because 7 is a prime number, and prime numbers can be intimidating.

Thanks to Castiel from Supernatural for that new twist on an old classic.

 

Humor at #NCTMNOLA

Last Wednesday evening, Steven Strogatz delivered the opening session at the 2014 NCTM Annual Meeting in New Orleans.

His talk shared a title with his bestselling book, The Joy of x. During the talk, he described five keys in bringing math to the masses, including what worked — and what didn’t — when he wrote a 15-part series for the New York Times Opinionator blog. He identified the five elements as follows:

1. Humor
2. Empathy
3. Relevance
4. A-ha!
5. Listen to Your Wife (Husband, Partner, etc.)

I was ecstatic to see humor at the top of his list. As an example of humorous mathematics, he played the now infamous Verizon .002 phone call.

As it turns out, the week was full of humor. (Who’da thunk, at a math conference?) Bill Amend, author of the comic strip Foxtrot, delivered the closing session at the conference. Earlier the same day, yours truly gave my soon-to-be-famous Punz and Puzzles talk to a standing-room-only crowd.

Following the conference, Jennifer Silverman tweeted the following:

@jsilvermath Tweet

The joke I actually told was:

Why is 6 afraid of 7?
Because 7 8 9.

Why don’t jokes work in base 8?
Because 7 10 11.

But who cares? If her son is laughing, I’m smiling!

After my session, I was accosted by an overly gregarious gentleman who had written a collection of math jokes on a yellow sheet of paper in red ink. While a queue of people who wanted me to sign their copies of Math Jokes 4 Mathy Folks formed behind him, he proceeded to tell me ALL of the jokes that he had written. He shared one joke that I found funny:

Pythagorean serum

Though funny may not be the right word. Perhaps interesting is a better choice, because Pythagorean serum was the name we used for the concoction that was served at my book release party.

And while at the conference, I was told a joke that I think works better visually than verbally…

Last but not least, I was sent the following image of Newton’s Cradle by Zachary Kanin with the suggestion that maybe I use it the next time I present:

Jim Rubillo Receives Lifetime Achievement Award

Jim Rubillo has been a member of the National Council of Teachers of Mathematic (NCTM) for more than 1.4 billion seconds. For his four decades of service to improve mathematics education, he received a Lifetime Achievement Award at the 2013 NCTM Annual Meeting.

Jim was the Executive Director of NCTM from 2001 to 2009, and he was my supervisor for the last five of those years. But he was more than just my boss — he was also a mentor, friend, and problem-solving companion. So when Ann Lawrence, chair of the Mathematics Education Trust, called to ask me to prepare a tribute video for Jim’s award ceremony, I was honored by the request.

I didn’t want to prepare a talking head video — I have a face for radio — yet I don’t have access to elaborate film equipment. Consequently, I opted to create a PowerPoint presentation with narration, which I then uploaded to authorSTREAM. Here it is, for your viewing pleasure.

Rubillo Lifetime Achievement Award

Prior to its showing at the awards ceremony, Ann Lawrence mentioned that the tribute video had been created by me. Upon hearing this, Jim murmured, “Oh, no…” (Truth be told, I think I was rather kind.)

One of the many reasons that I loved working with Jim is that he always had a good math problem at the ready. He shared more problems with me than I can count, but here are two of my favorites:

• What percent of the numbers in Pascal’s Triangle are even?
• Many years ago, it was believed that the Earth was the center of the cosmos. This was a reasonable hypothesis — it appears that the Sun rotates around the Earth. But if Earth were the center of the solar system (instead of the Sun), and if Mars rotated about the Earth, what would it have appeared that the path of Mars was?

Both of these problems have non-obvious answers, which is a trademark of the problems that Jim likes to share. Jim often looks at things with a unique perspective, and he willingly talks math with anyone who’s willing to listen. Consequently, Jim was an exceptional choice for this award, and I’m proud to call this lifetime achiever my friend.

Turn the Page

After eight fantastic years as the Online Projects Manager at NCTM, it’s time for my next chapter. On Monday, I become the Director of Mathematics for Discovery Education, leading a team that will build digital math techbooks for K‑12. I’m looking forward to building something great. As I mentioned during my interview, “I’m not coming to Discovery to create a textbook; I’m coming to create a movement.”

Leaving is such sweet sorrow. I’ll miss my friends and colleagues at NCTM, and I’m sad that I’ll no longer be creating resources for Illuminations. On the upside, my departure brought three stories worth sharing.

A Day Off

My last day at NCTM was February 28. That evening, I mentioned to my sons that I would not be going to work the next day. “Do you know why not?” I asked them. Alex suggested, “Because it’s Dr. Seuss’s birthday?” I love that! Celebrating the birth of Theodore Seuss Geisel certainly seems like a great reason for a federal holiday, but the truth is that I was just taking some time off between jobs.

Lesson Learned

The east coast was hit with a snowstorm during my time off, and both the NCTM and Discovery offices were closed. Had I been employed by either organization, I would have spent a day at home with pay. Instead, I spent an upaid eight hours designing the Vennebush Family Flag and playing Uno, Swish, and Qwirkle with the boys, while my gainfully employed wife dialed in to back-to-back-to-back conference calls. Moral: Check the forecast before quitting a job prematurely.

A Parting Gift

One of my colleagues at NCTM gave me a broken calculator. (And, no, this isn’t just a cheesy, elaborate set-up for a silly math problem.) The calculator used to be a normal, fully functioning, scientific calculator, but now it can’t add, subtract, multiply or divide without making an error. The good news is that the error is very predictable. The following video shows the results when using the calculator for four basic arithmetic problems.

The following (incorrect) results are shown in the video:

• 310 + 677 = 982
• 13 × 15 = 190
• 512 ÷ 64 = 3
• 75 – 10 = 60

And after the last problem, continual presses of the equal key should repeatedly subtract 10, but instead it shows consecutive results of 45, 30, 15, and 0.

Can you discern the pattern?

Teasing Out Some Math Jokes

Let’s do a quick warm-up before jumping into this post.

A boy leaves his house headed for school, walking at a rate of 4 miles per hour. Ten minutes later, his sister leaves the school headed for home, walking at a rate of 3 miles per hour. Assuming they travel the same route (just in opposite directions), which one will be nearer the school when they meet?

This problem is based on the first puzzle that appears in Mathematical Teasers by Julio A. Mira. As best I can tell, this book is out of print, and probably for good reason. Written in 1970, it contains no semblance of political correctness. For instance, the image below appears at the beginning of Chapter 1:

Click to Enlarge Image

Hmm… a pigtailed coed in a skirt sitting on a desk, tickling the chin of a math professor? There is no doubt an editor in 2013 would prohibit such an image from appearing in a publication; quite honestly, I’m even surprised an editor allowed it 43 years ago.

My copy of this book was obtained when the National Council of Teachers of Mathematics cleaned out their staff library. Inside the front cover, an insert states that it was a review copy “sent with the compliments of Barnes and Noble, Inc.,” and a stamp on the insert reads:

RECEIVED
MAY 26 1970
NCTM

Despite its political incorrectness and outdated contexts, my five-year-old sons have been enjoying the puzzles in the book. But they aren’t just puzzles. Like the warm-up problem above, they are jokes, in the sense that the punch line (answer) is unexpected. The following are a few of my favorites.

1. A man with $50 in a bank account withdraws $20, leaving $30. He then withdraws $15, leaving $15. Then $9, leaving $6. And finally $6, leaving $0. The sum of his withdrawals is 20 + 15 + 9 + 6 = $50, as expected, but the sum of the remainders is 30 + 15 + 6 + 0 = $51. Where did the extra dollar come from?
2. If it takes 3 minutes to boil an egg, how long will take to boil a dozen eggs?
3. How many cubic inches of dirt are in a hole that measures 1 ft. × 1 ft. × 1 ft.?
4. A man purchased a pair of shoes that cost $25 and gave the shop owner a $100 bill. After the man left with the shoes and his change, the owner took the $100 bill to the bank, where he was told that it was counterfeit. What was the total loss to the owner?
5. Every day, Johnson’s cat would climb 11 feet higher in a tree that is 63 feet tall. But every night, the cat would climb back down 7 feet. How many days would it take her to reach the top of the tree?

No answers will be posted. Y’all can attempt to reach consensus in the Comments section.

Where I’ll Be in October

Seven cities. Four math conferences, a committee meeting, and the USA Ultimate Club Championships. All in 18 days. Yikes.

Yep, October’s going to be very busy for me. During the month, I’ll spend more nights in a hotel bed than in my own bed.

If you happen to find yourself in any of the same locations, be sure to introduce yourself… but please don’t leave before telling me your favorite joke.

Oct 10-12: Dallas, TX
NCTM Regional Conference

You like games? You like fractions? Come to one of my sessions in the Lone Star State.

Friday, 8:30-10:00 a.m., Room D167
Session: Calculation Nation: Game On!

Friday, 12:30-2:00 p.m., C Ballroom 4
Session: Engaging and Free Online Resources for Teaching Operations and Fractions

Oct 12-15: Austin, TX
MathCounts Question Writing Committee Meeting

No doubt, you’d enjoy attending this event — we spend two straight days working problems and talking math. But sorry, this is a closed meeting… we’ll be compiling the tests for the 2013-14 MathCounts competitions, and that’s confidential information.

Oct 15-18: Washington, DC

Home for a couple days before I fly off to…

Oct 18-21: Victoria, BC
Northwest Math Conference

I’m excited for a return trip to NWMC. In 2010 I had an SRO crowd for my math joke hour, so I’m really jazzed to be giving a keynote math joke session this year.

Friday, 12:30-1:45 p.m., Grand Pacific Vancouver Island Centre
Session: Engaging (and Free) Online Resources for the Secondary Classroom

Saturday, 8:00 – 9:30 a.m., Empress Crystal Ballroom
Breakfast Keynote: Math Jokes 4 Mathy Folks

Saturday, 10:15-11:30 a.m., Empress Downstairs Balmoral
Session: Engaging (and Free) Online Resources for the Elementary Classroom

Oct 21-22: Vancouver, BC

A few days of R+R before heading to…

Oct 24-26: Greensboro, NC
North Carolina Council of Teachers of Mathematics Conference

Two math joke keynotes in one week? Get outta town! Punz and Puzzles is my favorite talk.

Thursday, 8:30-9:15 a.m., Auditorium 1
Session: To 10 and Beyond Using Free Illuminations Resources

Thursday, 10:15-11:45 a.m., Imperial D
Keynote Presentation: Punz and Puzzles

Oct 25-28: Sarasota, FL
USA Ultimate Club Championships

I play for Chesapeaked, a team with players from Washington, DC, and Philadelphia. We’re in the masters division (age 33+), which thankfully means I won’t have to cover a 21‑year old.

A Busy Week — Fun at NCTM and USASEF

The 2012 Annual Meeting of the National Council of Teachers of Mathematics (NCTM) is happening next week, April 25‑28, in Philadelphia, PA. As it winds down, the USA Science and Engineering Festival starts in Washington, DC, and will occur April 28‑29. It will be a busy week for me — I am performing twice at each event! If you happen to be attending either event, please stop by and say hello.

At the NCTM Annual Meeting…

• To 10 and Beyond Using Free Illuminations Resources
  Friday, April 27, 8:30-10:00 a.m.
  Salon A/B (Philadelphia Marriott Downtown)
• Using Free NCTM Resources to Promote an Understanding of Proportion
  Friday, April 27, 1:00-2:30 p.m.
  Salon A/B (Philadelphia Marriott Downtown)

At the USA Science and Engineering Festival, Washington, DC…

• Puns and Puzzles
  Saturday, April 28, 2:00-2:30 p.m.
  Franklin Stage (Washington Convention Center)
• Puns and Puzzles
  Sunday, April 29, 3:00-3:30 p.m.
  Franklin Stage (Washington Convention Center)

I am expecting an engaged crowd at each event, and I am hopeful that my presentations are received better than this…

A mathematician and an engineer attend a physics lecture. The topic is Kulza-Klein theories involving physical processes that occur in 9-dimensional space. The mathematician is enjoying the lecture, but the engineer is confused and frustrated. At the end, the mathematician comments about how wonderful he thought the lecture was. The engineer asks, “How do you understand this stuff?”

The mathematician replies, “I just visualize the process.”

“But how can you possibly visualize something that occurs in 9-dimensional space?”

“Easy,” says the mathematician. “First, I visualize it in n-dimensional space, and then I let n = 9.”

Blown Out of Proportion

In preparing a workshop about proportional reasoning for the 2011 NCTM Annual Meeting, I came across the following from “Learning and Teaching Ratio and Proportion: Research Implications” by Cramer, Post, and Currier, which appears in Research Ideas for the Classroom, edited by Douglas T. Owens. The authors discuss the following problem, which they presented to a class of pre-service elementary teachers:

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

The authors claimed, “Thirty-two out of 33 pre-service elementary education teachers in a mathematics methods class solved this problem by setting up and solving a proportion: 9/3 = x/15.”

I wanted to be surprised. Sadly, I was not.

Participants in my workshop (all current middle or high school math teachers) were asked to solve the same problem. Several incorrect answers were suggested, among them 3, 15, 27, and 45. Less than 40% of the attendees obtained the correct answer, 21.

This is frustrating, as proportional reasoning is extremely useful in analyzing real-world phenomena. In fact, it’s even applicable to language arts, as evidenced by the following graphic from my presentation:

The teachers in my workshop aren’t the only ones who have difficulty with proportional reasoning. Students have endless trouble, too…

Teacher: Today, we will discuss inverse proportion. Here’s an example: “If it takes 6 days for 2 men to finish a task, how long will it take 3 men to complete the same task?” The number of men needed is inversely proportional to the number of days required. Consequently, 3 men will be able to complete the task in 6 × 2/3 = 4 days.

Student: Oh, I see! I think I’ve got a real-life application of this. If it takes 6 hours for 2 men to hike to the top of a hill, then it will only take 4 hours for 3 men to hike to the top!