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!


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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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A spoonful of transgression

I was just observing a third grade class learning/reviewing basic fraction to decimal conversion, and I overheard a great remark. A girl, reading a word problem, said to her table mate, “Jessica ate 6/10 of a cake?! She’s fat.”

There’s a part of me that hates comments like that, and a part that loves them.

I hate the comment because, you know: here’s more evidence of our appearance-obsessed culture getting into the heads of young girls, etc., etc. But I love it because this girl just showed that her relationship with this fraction goes beyond shading in the appropriate portion of the drawing. Six tenths means something to her. Maybe I’m not fully happy with what it means, but at least it’s not meaningless.

My first thought is, why don’t we have more ridiculous math story problems? People eating horrific quantities of food is funny. And what’s funny and horrible has a way of sticking in the mind. Why does Tom always eat 3/8 of a pizza?Why not 49/8? Or 149/8? Save us from the blandness of the unoffensive story problem.

(Of course, we don’t want an unsafe environment for kids. But flatlining all the content is clearly a mistake. Better to talk about issues when they come up.)

Or maybe they’re tiny pizzas, and eating 149/8 is absolutely natural, because each one has a diameter of 1 inch.  I now I don’t even now anymore: is that a lot, or a little? (A new game: I say the number of pizzas Tom ate as a fraction, and you tell me the biggest they could be without Tom suffering permanent damage.)

The point is, being able to tell when something is ridiculous or not is part of understanding math. And straying into the ridiculous is fun, and interesting.

An even more transgressive example happened to me when I was demo-ing an algebra lesson last spring in an all-girls eighth grade classroom. (Herbert Kohl, in On Teaching, remarks that those who work with middle schoolers need to have a high tolerance for the profane. I’ll find the exact quote later. Update: the quote is, “A lack of sexual prudery is almost a prerequisite for junior high school teachers.”) The lesson began with the old magic trick (try it if you haven’t seen it before):

think of a number
add 2
multiply by 2
subtract 2
divide by 2
subtract your original  number.

And then I tell you what number you’re left with, which in this case is 1. (Ta-da!) The trick to it, which we got into, is to let x represent your original number, and keep track of the algebra. In one class, though, someone asked what would happen if you could replace the twos by threes, or fours. I set the class to play around with it, and see if they could predict how changing all the twos to another number would affect the final answer. Is there a pattern?

After they’d worked on it and I was bringing them together again, one group showed me how thoroughly they understood by suggesting we try:

think of a number
add 70
multiply by 70
subtract 70
divide by 70
subtract your original  number.

I saw the answer coming halfway through, but had no choice but to complete the process on the board. It was every middle schooler’s giggly favorite, 69. And again, I hated it, and I loved it. Hated, because that was the last place I want to have anything even remotely sexual suggested to me. Loved, because these girls owned that problem, and they showed it by showing they could hit any target number they wanted. They picked the funniest one they knew.

When we’re trying to interest students, we have to respect what interest them. Reliable standbys are matters of power, death & danger, women & men. Transgressions lie perfectly at the intersection of these topics: they are instances of breaking rules with all the hilarity and danger that involves. I want a math classroom that is safe for all the students. But I don’t want one that’s sterile. A spoonful of transgression helps the math stay memorable.

Let me end with a plug for an underused but incredible educational resource that mines the ridiculous: check out Randall Munroe’s What If? Just look how he seamlessly interweaves the ridiculous and real issues of power into a readable math calculation as he investigates how many punch cards it would take to store all of Google’s data. Anyone else see the class project dying to happen here?

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Teaching Perseverance

Reading an Alfie Kohn’s article on what kids learn from failure made me think of the most common question I hear from teachers about the Common Core Practices:

How can I teach perseverance?

It’s an excellent question, and the answer isn’t necessarily obvious. As Kohn points out, experiencing failure and having a teacher prod you to keep trying isn’t—or letting you hang—isn’t necessarily helpful. He write that

studies find that when kids fail, they tend to construct an image of themselves as incompetent and even helpless, which leads to more failure… if an adult declines to step in and help when kids are frustrated, that doesn’t make them more self-sufficient or self-confident: It mostly leaves them feeling less supported, less secure about their own worthiness, and more doubtful about the extent to which the parent or teacher really cares about them.

This illuminates a real pitfall of the unsubtle approach to teaching perseverance. You can’t simply throw students into a situation where they’re likely to fail and let them founder. When I think about being productively stuck, it’s worth remembering that there’s two ways to get out of balance, and both are pretty unproductive. If there’s no challenge, there’s no real learning. If the challenge is—or feels—insurmountable, there’s no real learning either.

And the best medicine for those about to face a particularly tough challenge, and want to stave off the feelings of failure that threaten to derail them? Start with a few easy successes. Seriously. I’ve realized that I do this myself, and when I help kids do it, they can work longer and with more focus.

I start with easy cases. Ridiculously easy. Say I’m trying to figure out the problem of how many squares are on a chessboard.

There are 64 small squares. Is that they end? No, wait, the whole thing is a square. So that makes 65. But then there are middle sized ones. Suddenly, the problem seems insurmountable. Many kids, once they realize they’re facing a cacophony of counting, give up.

What I do is ask myself how I could turn this into a problem that I could have some success with. How easy could I make this problem? And here’s the interesting thing. I’ll ask kids that question (even prompting them, if necessary, that eight by eight is a pretty big situation to start with) and what they usually say is to try on a four by four board instead. Sometimes they’ll say to start on a three by three or a two by two.

Do you know where I start? Zero by zero.

It’s shameless, I know, but why should I feel shame? I’m just thinking, playing, messing around, and giving myself the gift of instant success.

Because I can see immediately that there are zero squares on a zero by zero board. There’s no board.

And then I go to a one by one board. Which has one square.

So at the point others are giving up, I’ve now experienced two successes. And so have my students, with my help. I didn’t give anything away, didn’t rob them of the challenge. Just modeled how to start simple, bolster yourself up, see what you know. And there’s a way forward, a pathway from simple to hard that seems, maybe, passable. I can see a way to slowly ratchet up the difficulty, going to two by two and then to three by three. Maybe I won’t get where I’m going—believe me, I’ve worked on some problems that go from easy cases to ridiculously hard ones quick enough to give you whiplash—but this is part of what learning perseverance looks like. It’s knowing how to reframe things, make things simpler, change focus to see if you can get new ideas. And the taste of success keeps me going.

Thinking over this, there’s something really interesting happening in the pacing. I could have said

0 squares on a zero by zero chessboard.
1 square on a one by one chessboard.
How many on an eight by eight chessboard?


The information and the problem are identical. But the experience is profoundly different. That’s because the act of making a problem simpler is empowering. It shifts the situation from one of impending failure to one of success, even if the success is paltry. We go from hanging from a cliff face to standing calmly at the bottom of the mountain, considering our path forward. And no matter how you slice it, falling from a mountain hurts, sometimes in ways that you don’t recover from. Flailing shouldn’t become failing.

In other words, the actual skill we want to teach is how to shift from the despair of feeling like you’re about to fall off the cliff into ignominious ignorance to being back at the start, thinking about your options, considering your possibilities, and know that you can take at least one step without falling down in your journey of however many miles it will be.

Want to teach perseverance? Teach how to start a hard problem with simple case.

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