- Prove that

It feels, as Prof. S says in the video, like a beautiful conjecture.

I highly recommend trying to come up with a proof. There are many (54!), and I’ve come up with about seven since I saw the video. More intriguing, though, is the question of whether this type of thing happens in any other cases. Let’s look at a picture that summarizes the most surprising part of this problem.

It is not particularly surprising that . The shock is that as well. Are there other right triangles we can draw on a grid that have this property? The answer, it turns out, is yes.

*Prove that .*

Let’s go into even greater generality. Suppose we have two rectangles with integer side lengths. (Everything that comes later will refer to the picture below.)

**Big Question 1**: For what collections A, B, C, and D will the ?**Big Question 2**: Given any A and B, does there always exist a C and D so that ?

Here are some of the answers I’ve discovered so far:

A = 2, B = 3, C = 5, D = 1.

A = 3, B = 4, C = 7, D = 1.

A = 4, B = 5, C = 9, D = 1.

*Prove the examples above all satisfy .**Define a pattern in the numbers above. Will always work?*

Here’s another observation about the list of numbers above. ! What could that have to do with things?

**Wild Conjecture**: If , then .

*Prove or disprove the wild conjecture.*

Here’s another sequence of solutions where :

A = 3, B = 5, C = 4, D = 1

A = 5, B = 7, C = 6, D = 1

*Prove that these sets of A,B,C,D satisfy .*

What happens if we sum the squares? Once again, is double . This is evidence in favor of the Wild Conjecture. Not nearly a proof.

Notice what’s happened here. We began with an isolated question about a cool relationship between specific angles. By asking the next natural questions, we have very surprising variations of that original relationship, and an entirely different pattern emerging in the squares of the sides of the rectangles. True understanding comes not from solving one problem, but in solving families of problems in multiple ways, and following the natural questions as far as we can.

I’ll leave you with a suggestive diagram for a proof of the original three square problem.Here we have the original 1 by 2 right triangle (AEB) next to a scaled up 1 by 3 right triangle (CDB). Can you see why their two small angles (the ones at B) sum to 45 degrees? Do you see how this picture could generalize?

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Part 1 ends with a mystery, including a $100 bounty for anyone who can find a counterexample to the trick. If you know a student (or are a student) who thinks you can find a way to break this trick, let me know if you do.

]]>This Saturday session runs for six sessions, from October 18 – November 22. This session’s topic: Games, Logic and Arithmetic.

**Kindergarten & 1st grade
Section 1 **12:05pm – 12:55pm Sign up now!

**2nd & 3rd grade**

12:05pm – 12:55 Sign up now!

**4th & 5th grade**

1:05pm – 1:55pm Sign up now!

**6th, 7th & 8th grade**

2:05 – 2:55pm Sign up now!

This new Sunday session runs for eight sessions, from October 5 – November 23. Games, Logic and Arithmetic will be the topic for these new Bellevue classes too.

**2nd & 3rd grade**

10:00 – 10:55 Sign up now!

**4th & 5th grade**

11:00 – 11:55 Sign up now!

**6th, 7th & 8th grade**

12:00 – 12:55 Sign up now!

Math Circles meet before or after school, and highlight our favorite games, puzzles, and mathematical ideas. Dates and times are variable. Click the signup link to learn more, or check with your school.

APP at Lincoln — Sign up now!

Catharine Blaine K-8 – Sign up now!

McGilvra Elementary School – Sign up through the school

Queen Anne Elementary – Sign up now!

Thurgood Marshall – Sign up through the school.

View Ridge Elementary 2nd & 3rd grades – Sign up now!

View Ridge Elementary 4th & 5th grades- Sign up now!

West Woodland Elementary – sign up through the school

You may know the game 21, aka blackjack. In classrooms, I like to play with a deck that only includes numbers from 1 to 10.

*Twenty-one*. Each player gets two cards (face up). They can “hit” to take another card, or “stay” to stick with what they have. Whoever gets as close to 21 without going over wins. (Traditionally this game is played against the dealer in casinos. It’s fine to play it that way as well.)

Here’s how 21 becomes 500:

*Five Hundred*. Each player gets two cards. As in 21, they can “hit” or “stay.” The difference in 500 is that you multiply the numbers on your cards together. The goal is to get as close to 500 as possible without going over.

So in the image above, the 19 in Twenty-One would be a 240 in 500. Worth sticking in either case.

I only just made this game up last week, and haven’t played too much, so please experiment. Is 500 the best number to have as the bust point? Still, the game makes kids estimate, make a single strategic decision, and multiply one digit numbers and two digit numbers. And it takes almost no time to teach it.

**Announcement: Math Salon on August 16!**

For Seattleites: we’re happy to announce that we have a Math Salon on the calendar. Supported and hosted at the Greenwood Library, this event is a great opportunity for you and your kids to spend a Saturday afternoon playing with math. If you’re interested in joining us, please rsvp here.

Want to volunteer? Email dan [at] mathforlove.com to join us.

]]>There is a tension between intrinsic and extrinsic motivation in teaching mathematics. Our answer to the classic student questions *Why do I need to learn this?* is a good measure of where we look for motivation. You can appeal to the extrinsic, or instrumental, rewards: you need math to succeed in get a good grade, to succeed in middle school, high school, college math, to get a good job, and so on. And of course, that’s what a lot of people do.

On the other hand, you can take the tougher route of appealing to the intrinsic rewards. You need to learn math because it is beautiful, challenging, elegant, amazing. The reward of math is that it is engaging right now, in the present moment, and you should learn it because something in you *needs to know it*.

Any reader of this blog knows where we stand. Our name is Math for Love, after all. (Early motto: the only reason to do math is for love.)

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Sidenote: Creating the conditions that encourage the growth of intrinsic motivation is nontrivial; it defies the casual effort. It is one of the central jobs of a teacher, and the reason that teaching is a serious profession.

In a recent talk at Los Alamos, Bill Gates described the difficulty of reforming education as greater than the difficulty in curing malaria.

New technology to engage students holds some promise, but Gates says it tends to only benefit those who are motivated.”And the one thing we have a lot of in the United States is unmotivated students,” Gates said.

If we could automate what it takes to instill curiosity, passion, and love for a subject in a group of kids, then there wouldn’t be much of a reason to respect the work of teaching. But the nut of creating student motivation from tech solutions has barely begun to be cracked. In fact, it’s precisely because motivating is so deviously hard that teaching ranks as one of the most interesting and respectable professions (in my eyes at least. And in the eyes of those nations that tend to have more well-educated populations.) Not surprisingly, those who don’t understand how difficult creating motivation is are the same ones who malign teachers.

Sub-sidenote: Gates might have been in the camp that thought education reform was easy and teaching was rote before. If so, it sounds to me like he’s coming around. Nothing like working in education to see how hard it is to change it. There’s a possibly apocryphal story about some founding father—I forget which one—who tried and failed to reform schools, so went on to found the country… an easier job.)

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A natural reaction when considering intrinsic vs. instrumental motivation would be to conclude that trying to motivate students using both internal and external rewards would be the best way to go. But new research hints that two motives may not be better than one. In the study described by its authors Amy Wrzesniewski and Barry Schwartz, the researchers surveyed over 11,320 West Point cadets and found that among those with strong internal motivations, those with powerful extrinsic motives actually did worse in every capacity—graduation rates, performance in the military, etc.—than those without them.

In other words, extrinsic motivations may perhaps weaken the long term power of love the work. If this is really true and applies more broadly, it suggests that wanting the money, fame, renown, etc. from doing great work actually gets in the way of achieving it. We get external prizes when we take our eyes off them, and focus on our passion for the real work.

In other words: the only reason to do math is for love.

There’s more to say about motivation, and the depth and complexity of it will always make teaching a fascinating profession. But for now, I’ll leave you with this wonderful RSAnimate video of Dan Pink on the counterintuitive nature of motivation.

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I thought I’d give an update of some interesting mathematical things coming out right now: a board game (ours), a book, a video game, and a video series. All of them cost money, but depending on your interests, needs, and financial status, I think they all might be worth it.

Our new board game, Prime Climb, had a great showing on Kickstarter last month, and we’re working on the next step of getting the design finalized and the game put out into the world—not to mention the fascinating and challenging work of figuring out how domestic and international distribution works (we hope to be big in Japan). It’s very hard to predict what the future of Prime Climb will be, but you can ensure you get a game from the first printing if you pre-order this summer. Just click the “games” link above, or click here. You can get the game shipped to you in the US for $35.

I’ve read about a third of *Playing with Math*, but I wanted to get this review out now since a crowd-funding campaign for the book is underway. Sue Van Hattum set out years ago to put together the book she wanted to read, and Playing with Math is the result. The organization is simple: Sue contacted dozens of the people she saw doing cutting edge work in mathematics teaching and asked them to write a brief piece about their work. Among those contributing to the book are leaders of math circles, homeschooling parents, innovative teachers, bloggers and writers, and more, each telling stories that encapsulate their perspective and the results of their experience. There are math puzzles, games, and projects peppered throughout the pages as well.

Playing with Math is an exciting book. There’s a thrill of people really experimenting here, working out the possibilities of how math can be taught. The authors share their missteps and their successes; through it all, you start to see a coherent philosophy taking shape. Learning math requires struggle and joy… it’s serious play. But over and over, the right challenges at the right time empower the student, and build momentum for their continuing journey in mathematics. It’s easy to say it–the details are what’s tricky. This book is a story of those details.

Not surprisingly, the pieces can be a bit uneven, but it’s easy to turn the page if one of the stories doesn’t speak to you. And there are some gems in this book. The Kaplans’ story of leading a prison math circle is a laugh-out-loud pleasure to read, and Colleen King’s description of turning math subjects into student-designed games is a vivid picture of a teacher discovering a new way to teach as her students discover a new way to think. This book is a snapshot of the work of some of the trailblazers of math education working now, and worth reading.

Support the Playing with Math campaign here, and get yourself a copy of the book. ($25 in the US.)

Even the best math video games tend to be about skill building. Add in good graphics, first person game play, etc., and you still tend to have a textbook approach to math underneath the play. But there’s a game out that looks like it might be different. Mathbreakers, on Kickstarter till Saturday, seems to be about creative play with mathematics in a way that other games aren’t. It looks like a Legend of Zelda with mathematics underneath it; you don’t rise in the game by answering a specific math problem, but by finding creative ways to make the necessary numbers in any way you can. It looks like a leap forward in math gaming, and I want to play it.

This campaign has three days to make its last 15%. You can pledge here and get the game for $25.

I’ve never bought a Great Courses DVD–they strike me as overly expensive and not necessarily better than what you can get for free online. That was before I saw that James Tanton has a Great Courses course on geometry. James is one of the best math teachers and curriculum designers in the country, and one of the few people who always teaches me something new. James has tons of free videos on his website, and a great monthly newsletter, but this new course looks like it’s a step up in terms of production values and pacing. If you’re looking for a video geometry program and planning to spend some money on it, you should check out James Tanton’s new geometry in the Great Courses.

Expensive at $320 – 375. But if you go for Great Courses, then this is one to get.

]]>More than a year ago we had an idea to build a game around what felt like one of the unsung ideas in math: *prime numbers make multiplication easy*.* *(Why does no one learn this in school?) We realized that with the right color-coding, it would be possible to see multiplication and division as combining or removing colors.

A year later, Primo is ready. The game plays beautifully in play test after play test. It’s one of the most mathematically rich games we have ever seen, and at the same time avoids that icky “educational game” feel. Primo is a real game and it’s worth playing because it’s fun. Really fun.

The game is a race. Arithmetic is the engine, but not the end. Players add, subtract, multiply, and divide their way to the center of the board, knocking each other back to start and collecting Primo cards as they land on primes (the red circles) along the way. It’s a very easy game to learn, and infinitely replayable. Kids (and adults) practice their arithmetic without even noticing that they’re learning, and the game makes prime numbers intuitive and multiplication legible.

We decided to launch Primo via Kickstarter, and the campaign is up and running right now. This means that you can get a first run copy of Primo by donating to our campaign right now.

But if you really want to help us get Primo into the world, tell people about it. The beauty and terror of Kickstarter is that if we don’t make our goal, we don’t get to print up the first run of 1000 copies, and Primo gets relegated to some dusty closet. Email your friends and colleagues, post on Facebook, tweet, and let others know that they can get the game here:

https://www.kickstarter.com/

We hope you love this game. Thanks for your support.

Dan & Katherine

Math for Love

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One great outcome is that the game raises tangible issues about arithmetic. One amazing question that came up with a group of teachers: would 58 x 69 be greater or less than or equal to 59 x 68?

It wasn’t until a teacher suggested the partial products method of multiplying that everyone could see why we should expect these products to be different. But the insights didn’t stop there. Someone argued that the difference between a “switched product” like 28 x 79 vs 29 x 68 will always be a multiple of 10. Then someone conjectured that you could figure out exactly what the difference is based on the differences between the pairs of digits in the tens and ones places. A very cool discussion that came from an oddly compelling game.

So here’s the game:

Horseshoes is a simple game with almost limitless potential. It’s fun, quick, and can lead to differentiated practice, creative math practice, and also seed interesting conversations about math patterns.

**How to Play**

Horseshoes can be played with the whole class, or in small groups. Remove all face cards and tens from a deck of cards, so the only cards are from 1 to 9. The teacher/leader picks out two cards, forms a two digit number with them, and writes it on the board. This is the target number. Then the leader picks four more cards and writes those digits on the board.

The goal of the game is to create an equation using only the four digits that were drawn that equals an amount as close as possible to the target. Whoever is closest to the target wins that round. It doesn’t matter whether someone goes over or under.

**Example Game**

The leader draws a 3 and a 7, and writes the target number 37 on the board. Then the leader draws the four digits 2, 4, 4, and 9. After all the digits are written on the board, there are three minutes of quiet, where everyone writes their attempts and equations down on their own paper.

When the three minutes are up, the leader calls on people who say what they got, and how they got it.

Student 1: I got 43, by taking 49 – 4 – 2.

Student 2: I got 38. I took 9 x 4 to make 36, then added 4 and subtracted 2 to get 38.

Student 3: I got 37 exactly! I did 44 – 9 +2.

**Variations**

By taking different targets, we can encourage different kinds of arithmetic practice. For example:

Four digit Horseshoes: Pick a 4-digit number as the target. This forces multiplication. Fraction Horseshoes: Arrange the target as a fraction (i.e., 3/7 instead of 37).

**List of Counterexamples
**

- Communism
- Hitler
- Klein 4 group
- Petersen graph
- Plasma
- Hawaii
- Neutrinos
- Tacoma Narrows bridge
- Rwanda
- The 1%
- Duck-billed platypus
- Seahorses

I’ve posted one of my favorite problems of all time as this week’s NYTimes Numberplay puzzle.

*Consider this simple game: flip a fair coin twice. You win if you get two heads, and lose otherwise. It’s not hard to calculate that the chances of winning are 1/4.*

*Your challenge is to design a game, using only a fair coin, that you have a 1/3 chance of winning.*

My relationship with this problem literally spanned years. It manages to be simultaneously simple, devious, and deep.

Read more and join in on the discussion here.

The Julia Robinson Math Festival is this Saturday, and we are filling up fast! If you want your kids to be there, sign up now!

This is going to be our best festival yet. We’ve got the best activities ever, including a bunch of great new problems and activities, math/art projects, and games.

We’ll be capping off the event with a talk by Kathleen Tuite. Kathleen has been involved in some cutting-edge game design at the UW that encourages scientific discovery and collaborative problem-solving.

Our theme this spring is **geometry**. We’ll explore the subject in some new and surprising ways, tackling a variety of compelling problems as we go. Potential topics include perimeter, area, maximal and minimal shapes, reflective geometry, fractals, and any other topics that I or the students want to study. In each class, we’ll explore the topic at a level appropriate to the age, tackling a variety of compelling problems along the way.

Classes run Saturdays, April 12 – May 24.

**6th, 7th & 8th grade**, 11:05 – 11:55

Sign up now!

**Kindergarten & 1st grade**, 12:05pm – 12:55pm

Sign up now! Update: Full!

Email dan@mathforlove.com to join the waiting list or be contacted if we open a second section.

**2nd & 3rd grade**, 1:05pm – 1:55pm

Sign up now!

**4th & 5th Grade**, 2:05 – 2:55pm

Sign up now!

Save the date: June 23 – 27 we’ll be offering a week-long camp/math classes. Details are still forthcoming, but if you’re interested, email me and let me know how many kids you have who would like to participate, and how old then are/what grade they will be in next year.

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