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.

]]>I had been playing around with this issue of what money costs to make. (Get the lesson here.) Not a pretty picture, by the way. Rounding only a little for simplicity and age appropriateness, we had this chart of costs:

Penny: 2.5 cents

Nickel: 11 cents

Dime: 6 cents

Quarter: 11 cents

My question was, **what coin makes the cheapest dollar, and what coin makes the most expensive dollar?** (For example, a dollar in dimes would cost 60 cents to make.) There was some controversy as kids staked some initial claims, and then everyone set to work.

But one girl ignored my question in favor of her own, which I have to say, is straight up better: **can you make a dollar in coins that also costs a dollar to make?** The dollar in dimes costs only 60 cents. What about eight dimes, two nickels, and ten pennies? As you can check, that costs 95 cents to make. So close!

And here’s the even better sequel to the story. This same girl worked with incredible focus for the next thirty-five minutes, and answered her own question. Can you find how she did it? Is there more than one answer?

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We all have an instinct for what fame is, and the more we put it into words, the more we’ll find we can translate fully into math. So what let’s us know if someone is famous? Well, famous people are well known. We tend to know them, and they don’t know us. In fact, we could say that people become famous when more people know them than they know.

As soon as we’re dealing with quantities, we have something that’s pretty easy to describe mathematically. Let *U = the number of people you know* and *M = the number of people who know you*. Then we could define the *fame quotient *as *U/M*, the ratio between the number of people who you know and the number of people you know.

Happily, we can get good estimates on these numbers from twitter. I’m not a huge user, with 154 followers and 93 people I’m following. But I can still calculate my fame quotient as 154/93, or about 1.66. That feels about right… I’m not particularly famous.

Let’s try some more famous people.

Obama has 41.3 million followers and 654 thousand he’s following, so his Fame Quotient is 41,300,000/654,000 = ~63

Justin Bieber has 49.4 million followers and 124 thousand he’s following, so his Fame Quotient is 49,400,000/124,000 = ~400

*Who’s more famous: President Obama or Justin Bieber?*

With a fame quotient of close to 400, Bieber is about 6.5 times as famous as the president.

**Here’s the challenge: who is the most famous living person in the world?**

Remember, I’m using our twitter-based version of fame here. It might not match other intuitions or definitions of fame. In fact, we’ll probably have to fix it as we go. But it’s a start.

]]>I put together seven worksheets designed to be “appetizers,” “main courses,” and “desserts” in a menu structure. The course refers to the level of challenge. The topic is fraction division, and this one is a little different for me, in that many of the problems are more “normal” than those I often do, and there is more time spent on routine operations and practice to attain mastery. My hope is that they are still compelling to do, and the desserts still connect to some broader concepts in pure math. I thought I would post them here to see what folks think.

Feel free to try them out, and please let me know what you think!

Download Menu for Fraction Division

You will also want to use Counterexamples, which is a fantastic logical whole class game, as an introduction.

Download Counterexamples

I’ve been taking some time to write up some lessons, and I’d love to get some feedback. You can click here for a pdf of this lesson on 1-2 Nim. It’s one of our favorites: a surefire way to get students of any age playing and thinking.

The question is: how’s the write up? We’ve been shooting for 1-2 pages maximum, so we keep it streamlined and easy to use. Teahcers, is this a usable format for you? Parents, do you feel ready to try this out with your kids?

Anyone who tries this lesson out with a child, student, or class, please let us know how it went in the comments. Any constructive feedback is welcome!

Thanks!

]]>“… at this point, it’s in the hands of people who are mathematically inclined.”

—Stephen Hsu

The January 6th New Yorker contains an article on B.G.I., a Chinese company seeking to do major work in the field of genetics. According to them, the massive amounts of genetic data they (and others) are collecting and interpolating will help “explain the origins and evolution of humanity, improve our average life span by five years, increase global food production by ten percent, decode half of all genetic diseases, understand the origins of autism, and cut birth defects by fifty percent.”

They’re also hoping to find some genetic factors that contribute to intelligence. “Probably by tweaking a certain number of variants in a positive way, you could rev up human intelligence quite a bit,” says Hsu, one of the principals on the project. Giving human intelligence a genetic nudge is one of those projects that I instinctively don’t believe will work… until I think about it. My resistance to the idea has to do with the inherent complexity of genetic expression—the path from gene to trait is a chaotic and messy one. But that’s precisely what statistics is for. “Everyone is coming around to believe that things are controlled by many genes, and there has been a tendency in the field to just throw up your hands and say, Well, this is going to lead nowhere, or this is all a boondoggle. But I actually think that, at this point, it’s in the hands of people who are mathematically inclined.”

I think Hsu is right about a lot of things, and I liked the article. What gets my ire up, though, is that the U.S. isn’t doing nearly enough to invest in its future, to maintain its position as the center of scientific inquiry. What we need seems pretty obvious: investment in education starting with pre-K, and investment in science from basic research on up.

I don’t think we’re doing particularly well, nationwide, in education, and the latest international comparisons bear that out. Meanwhile, I’m watching scientist friends apply for grants only to be told that their application is excellent, but their timing is bad. (Summarized beautifully here). The pool of money for research is so shrinkingly small that it’s starting to unnerve me.

I’ve been saying for a long time that we need to invest in the capacity of teachers and schools, and make the choice to take the long road toward maintaining what’s good in our education system, and working on what isn’t. But scientific research is something the US is already the best at. At this rate, we won’t be for much longer. To quote the article again,

… at a time when the N.I.H. is cutting back on funding scientific research, China is not. Recently, the Chinese government published an ambitious fifty-year plan to advance its technical and scientific position in the world. Few scientists would claim that they can predict that far into the future. But the fact that China would even try demonstrates how serious the country is about its technological place in the world.

You don’t stay at the top by taking it for granted. You have to care about investments in the next generations. China’s making a play. So is Estonia. What are we going to do?

]]>The good thing about teaching percents is that they connect to the real world, particularly with money. The bad thing is, it can be hard to find really dynamic problems. Too often, you’re just marking prices up or down in imaginary shops, or looking for discounts at imaginary sales. Not a bad thing to be able to do, but not exactly the beauty and depth we want in a math class.

Here are three problems appropriate for high elementary or middle school level (and, let’s be honest, probably a lot of high schoolers and adults too) that involve some deeper thinking.

**Problem 1. Which is more, 23% of 71, or 71% of 23? **

Of course, I’m curious about the general question: is x% of y bigger or smaller than y% of x. Is there are general rule that allows you to tell? Or will they be the same? Right off the bat, it’s not at all obvious which will be larger. Answering this question involves 1) doing a lot of work with specific percentages to see what’s going on (possibly simplifying, since mathematicians always avoid arithmetic they don’t have to do), 2) making a conjecture about what’s actually true, 3) understanding what taking a percentage really means and finally 4) seeing that if you really understand percents, the answer is almost breathtakingly immediate.

**Problem 2. I bought a shirt that was marked 15% off in a sale. As I was walking away, I glanced at the receipt, and noticed that the salesperson had added sales tax (9%) first, then given me the 15% discount on the total. **

**I went back and complained to the manager. After all, I got charged sales tax on the full price, and didn’t get my discount factored in till after. The manager said that I’d actually gotten a deal! Her reasoning was that my discount was greater, since it was calculated on the tax as well as the cost of the shirt. **

**Who is right? In general, is it fairer to calculate the 9% sales tax, then the 15% discount, or the discount first, then the tax?**

This is a lovely little question with that old attention-grabbing issue of fairness woven right into it. (Of course, changing the numbers to fit your state and students in encouraged.) It might seem like information is missing, since I didn’t say what the shirt cost. Students can plug in different amounts for what the shirt might have cost and see what happens. Some students will doubtless try to convince you that it comes out to a 6% discount either way. Plug in some numbers and see why this doesn’t work. In fact, the assertion leads us to our last problem.

**Problem 3. I buy a stock on a very bad day… it drops in value 80% the day I purchase it. I mention my misfortune to a friend the next day and he tells me that the stock has just increased in value by 80%! Have I made my money back?**

There are a million variations on this question, but the main ones in my mind are

-If a stock goes down x% on Tuesday and up x% on Wednesday, can you find the overall change?

-Is it better for you if it goes *up* x% on Tuesday and *down* x% on Wednesday?

I like all these problems. They force students to confront a real mystery about percents. Let me say that for those who actually know percents through and through (and possibly some algebra), these problems verge on the trivial. But I’d be willing to gamble that that isn’t how most students—child or adult—will experience them.

]]>1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …

gets infinitely large. Recently, I got to wondering which numbers it hits on the way up. In particular, if you can rearrange the fractions, can you hit any rational number?

I was thinking about this because I’ve been looking into the Egyptian fraction problem lately. The problem is a great one for students if you need to get a sense of how fractions really work. Unlike us moderns who would write 3/5 as the answer to how to divide 3 loaves of bread into five pieces, the Egyptians would first cut every loaf in half and give everyone a half, then divide the remaining half into five pieces. Their final instructions for the division would be that everyone gets 1/2 + 1/10.

Thus the problem of Egyptian fractions: given a fraction, can you always rewrite it as the sum of distinct *unitary *fractions, that is, fractions with a 1 in the numerator? (The Egyptians apparently didn’t like to repeat their fractions.) The next question is, how many unitary fractions does it take. This latter question is still unsolved in many cases. For example, it is conjectured that any fraction of the form 4/n can be written as the sum of at most three unitary fractions. But whether that’s always true is still unknown.

So here’s the harmonic puzzle: given *any* positive rational number, can you always write it as a sum of distinct unitary fractions?

When I first thought of this problem, it seemed like it would take very sophisticated tools to solve. Yesterday, I stumbled on the answer when working with a student on Egyptian fractions, and it takes nothing more sophisticated than a little algebra—and a clever idea.

Here, if you like, are the questions:

1. Can you write any rational number between 0 and 1 as the sum of distinct unitary fractions?

(Example: 4/13 = 1/5 + 1/10 + 1/130.)

2. (The Harmonic Puzzle) Can you write any positive rational number as the sum of distinct unitary fractions?

(Example: 2 = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + +1/8 + 1/12 + 1/13 + 1/20 + 1/42 + 1/43 + 1/56 + 1/132 + 1/1806

Unless I made an arithmetic error… tell me if I did.)

3. (unsolved) Can you always write 4/n as the sum of three unitary fractions?

One thing that’s supercool: if you can answer question 2 in the affirmative, you get a slick proof that the harmonic series diverges!

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