A dollar that costs a dollar

I had one of those awesome experiences this week where a student thinks of a better question.

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:

CoinsPenny: 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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Who is the most famous?

One fun thing math lets us do is measure difficult-to-measure things. Like fame.

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.

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A Math Menu for Fraction Division

I’m not sure who came up with the idea of “Menus” as a math teaching device, though I first saw them at a workshop from the folks at MEC. Menus are essentially modified stations, designed to be a several-day structure that puts kids at the center of their own learning process. After a brief launch from the teacher, the students have options for challenge and engagement that will last days. It takes more work on the front end, but they are a great teaching structure.

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

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1-2 Nim Write Up

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!

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The Mathematically Inclined Shall Inherit the Earth

“… 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?

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Deeper questions with percents

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.

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Harmonic Puzzle

One of the beautiful results in mathematics is the proof of the divergence of the harmonic series. What it tells us is that the infinite series of fractions

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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Seeds and Stalks

The grails of math activities, for me, are those that involve almost no special knowledge to get into, but have near-infinite depth. (Like this one.) We sometimes describe them as having a short barrier to entry, and no ceiling. (A common suggestion when we work with teachers is to “remove the ceiling,” that is, find ways to change the problem so that the learning doesn’t end when you get the answer…)

Here’s a spanking new puzzle I’ve been playing with, and it feels like a perfect example of a problem with virtually no barrier to entry, and no ceiling either. I call it Seeds and Stalks.

Here’s how it works. We’ll generate sequences (in a kind of Fibonacci-like way) by choosing a number and adding it’s digits to itself to get the next number. For example:

16 goes to 16 + 1 + 6 = 23.
23 goes to 23 + 2 + 3 = 28.
28 goes to 28 + 2 + 8 = 38. And so on.

So we have a sequence that goes 16, 23, 28, 38, … I call this Seeds and Stalks because there are two pieces here… the seed that starts the sequence (the seed) and the sequence that grows out of it (the stalk).

Of course, there might be a seed that leads to 16 in the stalk. And indeed, 8 leads to 16. The most primal seed we could pick for this stalk is 1, since having 1 as the seed leads to the stalk:

1, 2, 4, 8, 16, 23, 28, 38, …

All well and good. But as soon as I thought of the mechanism, I was besieged by questions. The first was:

What’s the smallest collection of seeds that you need to include every number in a stalk?

I can see I’ll need 3 as a seed, since 3 isn’t in the stalk.

3, 6, 12, 15, 21, 24, …

Now 5 isn’t in either stalk, so I’ll need that too. How many seeds do I need to get every number? Or will I need infinitely many seeds?

For me, this is a perfect storm. All I need to start this problem is addition. And yet, I have no idea what will happen. I can feel that there are all kinds of patterns to find. My instinct now is to turn it over to students and see what they can find.

But I’ll look to the internet first.

What questions can we ask about Seeds and Stalks?

What answers can we find?

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Quadrilateral Puzzle

I’ve been immersed in puzzle and lesson creation lately, and I thought I should take advantage and throw some of them out here on the blog. Please take, solve, use in your classrooms or at home, and let me know what you think. If people like the puzzles, I’ll make a point of putting them out here more often.

A Quadrilateral Question for today. This sub-questions goes from easier to harder.

The Big Question: Start with any quadrilateral (Quad 1), label its midpoints, and connect them to form another quadrilateral (Quad 2). When will Quad 2 take up exactly half the area of Quad 1?

Will it happen if Quad 1 is…

1. a square?

2. a rectangle?

3. a parallelogram?

4. a trapezoid?

5. a kite?

6. Nonconvex?

7. Can you find an example when Quad 1’s area isn’t double Quad 2’s? Or will it happen all the time?

You can post in the comments if you’ve got an argument to share…

(Another question is to show that Quad 2 is always a parallelogram. Here’s my proof of that if you get frustrated.)

 

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Twin Prime Hero

I just read this wonderful interview with Tom Zhang, who made recent, important progress on the Twin Prime conjecture.

It’s a strange, quiet interview, and a lovely departure from the world of the fame-obsessed. Another thing I like: he emphasizes the love and the persistence. Here’s how the interview ends:

What would you say to a young student who wants to solve a problem?

Keep going. Do not easily give up.

Where do you suggest they find the motivation?

The most important motivation is to really love mathematics.

Is this a person who would be a hero to young people? I don’t know. Should he be? I think I can say with confidence that he doesn’t need the adulation or admiration of anyone. Isn’t that something we should look for in a role model?

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