Phi is the new root 2

My knowledge about the foundation history of irrational numbers was challenged today, and I’m pretty happy about it.

I had recently tweeted a Vi Hart video that gave a fun, geometric proof of the classic first proof of irrationality: \sqrt{2} is irrational. If it weren’t, that would mean you could build a square that had integer sides and an integer diagonal, and that would allow you to build a smaller square with the same process. To get a contradiction, repeat until you run out of integers.

After I tweeted the video, I got a response claiming that… Tweets

I was somewhat taken aback. In source after source, I’ve seen \sqrt{2} named as the first number ever proved irrational. Variations on the same proof abound. And here was a claim that \phi, the golden ratio, actually holds the rightful place in history as humankind’s first brush with “the unnameable.” There seems to be a historical argument; but how complicated is the proof?

In fact, it’s so wonderfully simple that there’s a pedagogical argument to be made for teaching that \phi is irrational before we even mention the Pythagorean Theorem or square roots. You need to know how to find angles in regular polygons and chase them around diagrams, and know how Isosceles triangles work. The thrust of the proof is the same as for \sqrt{2}, but it sidesteps the parity argument that can sometimes feel less tangible.

Let’s imagine, as the Pythagoreans might have, that every number is rational. An equivalent way to state this is there is always some scaling of any pair of lengths that allows them both to be positive integers. (To the Pythagoreans, the relationship between any two lengths was identical to the relationship between two whole numbers, axiomatically.) So suppose we have a regular pentagon with integer side length a and integer diagonal b.pentagon
The ratio of b to a, is precisely the golden ratio, by the way. But we don’t even need to know what it is. We’ll just try to show that a and b can’t both be integers.

First off, chase some angles around and you’ll see pretty much every angle is either 36, 72, or 108 degrees. This gives a bunch of Isosceles triangles. It follows quickly that

x = 2a - b y = b - a [I’ll leave that piece as an exercise. It’s pretty satisfying to chase angles around and have everything come out nicely.] This implies that x and y are positive integers. But they are the side and diagonal of a regular pentagon again, so the argument repeats! And this is the crux of the problem: positive integers can get smaller for only so long before they run less than 1. (Just like the infinite regress that was hinted at in that error-laden but inspiring work, Donald in Mathemagic Land.)

Conclusion? The original pentagon couldn’t have been drawn with integer sides to start with. And that means it couldn’t have been drawn with two rational sides, or else we would have scaled them up to whole numbers. And that means the relationship between the side and diagonal of the regular pentagon is irrational.

And there we have it. Irrational numbers without actually dealing with numbers at all. Or evenness and oddness of numerators and denominators.

A delightful discovery. We’ll likely never know for sure what length the Pythagoreans proved irrational first, but that’s a strong claim for /phi over /sqrt{2}.

Especially because, as Donald found out, the Pythagoreans were all over pentagons and the golden ratio.

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Aggression: the amazing arithmetic game no one knows about

Aggression

I first played the game Aggression about five years ago. I had recently read Eric Solomon’s Games with Pencil and Paper, and tried the game out with a student I was working with. It was a hit. I heard later that the game was in regular rotation at his house, and had become a family favorite. “I should use this game more in schools,” I thought. And then I forgot about it.

Now, Gord Hamilton of Math Pickle has a beautiful explanation of the game up at his website, Math Pickle. Seeing it again, I’m reminded of just how good the game is. And happily, I don’t need to write it up anymore, because it’s hard to imagine improving on Gord’s description.

If you’re looking for a classroom game for 2nd or 3rd graders (or, honestly 1st or 4th or 5th), or just a great pencil and paper game to play at home, I highly recommend you check out A Little Bit of Aggression. Check out the PDF with different maps at the end of the slide deck. Also, check out challenges! Gord’s put some money on the line, and you might be able to claim it!

Lessons on Volume

tiny cubes
I recently collected a series of our favorite lessons for fifth graders on the topic of volume into one tidy booklet. I like these lessons a lot. They start with tangible activities involving squares and cubes, and build up from concrete to abstract until students are able to use an understanding of volume to solve truly complex problems.

I’d love some feedback, especially from 5th grade teachers. The lessons are available here.

In another experiment, the booklet of volume lessons is available by donation. This will hopefully allow everyone who wants it to get it for free, and for us to keep track of how many times it’s downloaded. At the same time, if people want to chip in for them, now there’s a way.

Saying Yes: Joi Ito’s 9 Principles

Recently, Dan gave a TEDx talk based on the blog post 5 Principles of Extraordinary Math Teaching. In the conversations we had with each other and with other educators in the run up to the talk, one principle came up repeatedly as the most nuanced of the five: Say Yes to Your Students’ Ideas. Perhaps the most challenging principle to implement but also the most rewarding, we’ve been thinking about this one a lot lately.

The ultimate outcomes of deciding to say yes to your students are rooted in the principles of creativity. We are living in a world of increasing complexity and innovation, the best of which requires us to offer our most dynamic and creative selves, and the worst of which hazards alienation and isolation. I recently encountered some work from Joi Ito, the director of MIT’s Media Lab, in which he offered nine variables of innovation and creativity (and, frankly, survival), principles which I thought perfectly captured the spirit of what excellent education can offer.

principles

Joi Ito’s Nine Principles of Creativity

  1. resilience over strength
  2. pull over push
  3. risk over safety
  4. systems over objects
  5. compass over maps
  6. practice over theory
  7. disobedience over compliance
  8. emergence over authority
  9. learning over education

In education we can think of these as follows:

  1. We want students who can adapt their understanding to new contexts. Resilience is adaptive; strength is resistant.
  2. We want students to find their own motivation to engage in work. The question should be designed to pull them in. Work should mostly come from inner drive (pull) rather than external coercion (push).
  3. Staying in safety stifles intellectual curiosity. We want students to take creative risks. They will practice leaping, failing, and leaping again.
  4. Objects exist in isolation; systems are objects in relationship. With the proliferation of objects in the world, the future likely belongs to those who understand the web of relationships, ie those with a systems perspective. This means developing an understanding of context: the reasons why something is true, and why it makes sense with everything else you know.
  5. A map is a fixed picture of a landscape, but it becomes useless if the landscape is shifting under your feet. A map is difficult to use in a rapidly changing world. A compass gives direction even as the landscape is altered. It can be used to find the way through an uncharted (unmapped, unexplored) territory. It is the intuition of intellect, evolved through repeated exposure to rich, novel, changing contexts. It grows through mistakes, surprises, and direct experience with complexity. We want students to develop the abilities of intellectual intuition, this ‘compass’, so that whatever map they step on to, they are prepared to start navigating.
  6. When a student has series of facts out of context and divorced from motivating reason, a student has theory. Theory tends to be tidy. Practice is messy because the work of learning is about breaking an accumulated body knowledge. The facts are associated, stress tested, fail, and the process is repeated with a yet stronger configuration. To stay in the realm of theory is to build a structure, adding yet more layers and stories, without ever testing it. It becomes precarious indeed. Better to build, test, break off the weak parts, rebuild, iterate. The best versions of practice aren’t necessarily ‘real world’ problems, but they are complex ones.
  7. Disobedience breaks into the new. Compliance sustains the old. Disobedience explores the possible, which lies outside the known. Students who are intellectually disobedient are prepared to discover what we may not have imagined yet.
  8. What emerges from a group of students working together is a reflection of what they are ready to know, what they are curious to know, and what they already know. This is the perfect medium of learning. If they are driven by the authority of the teacher, this ideal medium is trampled and it’s potential gains are lost. Teachers can meld their agendas with the students’ emergent understanding.
  9. Learning is student-centered. Education is system-centered. Actually, in education, we want a blend of the two. As above, the position of greatest leverage brings together both the student and the system, a learning-education blend that meets the student’s own innate curiosities and interests with the goals of the education system at large. This is the domain of the principle of Saying Yes: by saying yes, we engineer this meeting of the student and the system, of the individual learning and the larger interests of education.

Saying Yes is a kind of guidance that uses the student’s inner world, their own mind and curiosities, to bring them through the educational goals we desire for them. It lies in the most productive intersection of child-centered learning and teacher-driven education. It takes the best of the Unschooling movement, the recognition that a student’s desires and interests are the best motivators for their learning, and the best of traditional education, which acknowledges that the outcomes of education prepare students for the future world that they (and we) may not have anticipated yet.

Saying Yes means teaching that the world of knowledge is rich, complex, unknown, and constantly changing. Saying Yes means teaching resilience (that the material can adapt to the student and the student can adapt to the material); it means working with the student’s internal motivation, constructing problems that pull them in and using where they are already pulled to direct teaching; it means practicing failure, courage, and risk-taking; it encourages a system’s perspective by taking advantage of the human impulse to make connections, to ask why, to look for meaning and make sense of things; it develops intellectual intuition, the compass of learning,  by making exploration of the unknown a primary goal of the classroom; it emphasizes practice because getting our knowledge is inevitable when we follow our impulses; it means valuing intellectual disobedience; it makes use of the emergent knowledge of the classroom; it strikes the perfect balance between learning and education, between the individual and the system.

The Power of 37

Our new lesson plan library is up in beta form. We’re not sharing it widely yet, but it’s getting close. Our goal is to have a collection of great problems for K-6th grade, easily filterable by topic, grade, Common Core, and keyword. We know that many teachers would love an easy place to find high-quality, easy-to-use complex problems and games. We’re hoping that Math for Love can be such a place, where each question, game, or lesson plan is a pleasure to use in the classroom and a hit with the students, as well as being rigorous and dynamic.

To that end, we want your feedback! Try these lessons out in your classroom or with your kids. Let us know what works and what doesn’t.

We’ll plan to highlight lessons from the new library weekly (or from time to time). This week: The Power of 37.37

This lesson uses a surprising multiplication pattern to motivate multiplication practice for 4th and 5th graders. It’s highly arithmetic, and fits right in with the major work of 4th and 5th grade. It begins with a mystery.

Is it just a coincidence that (7 + 7 + 7) x 37 = 777?

If not, why does 37 have this power?

Find the full lesson plan here.

The Rearrangement Puzzle

On the topic of puzzles, my puzzle in in the NYTimes Numberplay column this week. It’s built to look hard, but come apart easily if you attack it from the right direction.

The Rearrangement Puzzle

The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

I have mixed feelings about this puzzle. On the one hand, I love that it looks so hard, but can be unraveled so easily once you get past that, and attack it with the right tools (hints at Numberplay). On the other hand, I’m dissatisfied with its inability to generalize to other puzzles, or begin a larger investigation.

Perhaps it does, and I’ve just lost the scent.

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This Week’s NPR Puzzle: Extra Challenge

This week’s Sunday puzzle on NPR is a classic from Sam Loyd. Here’s Will Shortz:

This is one of the “lost” puzzles of Sam Loyd, the great American puzzlemaker from the 19th and early 20th centuries. It’s from an old magazine with a Sam Loyd puzzle column. The object is to arrange three 9s to make 20. There is no trick involved. Simply arrange three 9s, using any standard arithmetic signs and symbols, to total 20. How can it be done?

I played around with this puzzle this morning, and indeed, there is a way to solve it using only standard arithmetic (the four operations, parentheses, decimal points). However, as I was playing around, I found a second answer as well, using square roots and factorials.

Can you find both answers?

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Goodbye hexagon, hello 6-gon!

A colleague of mine once remarked how strange it is that while the Greeks talked about 6-cornered shapes and 4-sided shapes, we talk about hexagons and quadrilaterals. Why is it, aside from the historical accident that it is, that we persist in making people learn Greek to talk about shapes they see everyday?

And quadrilaterals and hexagons are the easy ones. What’s a 7-sided polygon called? It’s either a heptagon or a septagon—I never remember (do you?). What about a 12-sided polygon? That’s a dodecagon. Thirteen sided? No idea—no one ever taught me that one.

PolygonsOur vernacular around polygons is tied to an ancient system of numeration that not even experts know. We’ve created a system where we can speak properly about only a select subset of polygons: triangle, quadrilateral, pentagon, hexagon, octagon. Those prefixes denote the numbers 3, 4, 5, 6, and 8. And while there are certainly some of you who know more, I don’t think we ever bother teaching more than this. It’s like teaching inches and feet, and not bothering with miles.

We couldn’t convert to metric in the US, but we can do something even easier when it comes to polygons: name them by number. Forget the name of the icosikaitetragon? Just call it a 24-gon. Heptagons and decagons are 7-gons and 10-gons. We could even call hexagons 6-gons.

The advantages are immediate and enormous. First, every polygon now has an easy, instantly recognizable name. We’ve removed the barrier of Greek between ourselves and shapes. Second, we’re reminded of the defining trait of the thing when we name it. It’s why we call it a red-breasted robin instead of a Turdus migratorius.

Do you agree? Before you answer, let me add one more point: mathematicians use this nomenclature already. We even say n-gon instead of polygon, just so we can decide what n is later, or use the variable in equations.

We could have gone further. The prefix -gon is just Greek for -angle, as in triangle (3-gon). And while 5-angle and 9-angle have a certain poetry, I like the staccato of 5-gon and 9-gon. And it’s not so bad to keep a little Greek in there.

Sure, it’s nice for students to know the word triangle. And you can argue that learning vocabulary for the polygons is fun. But do we really need to add barriers around mathematical objects when we could just call them what they’re called? Do kids need to know quadrilateral or heptagon to relate to 4-gons and 7-gons?? If you’re a math teacher, you can start calling pentagons 5-gons and decagons 10-gons tomorrow.

And the beauty of it is, your students will know exactly what you mean.

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The Four Questions

Math Play Venn Diagram

We’ve argued for a long time that the real experience of mathematics is inextricably tied to play. But if you’re a parent or teacher, you’ve seen kids play in mathematically irrelevant ways. How do we hit that sweet spot of mathematical play?

One way is to recognize mathematical questions and ideas when they arise from the play itself. Another is to subtly introduce them, without sacrificing the play when you do. So it isn’t, “Stop playing so we can do these flashcards.” Instead, it’s “I wonder how many Jenga blocks you could stack on a single block without it falling.”

We’ve found that there are four central questions that can help uncover or motivate mathematical play.

1. How many?
2. How much?
3. What kind?
4. What if?

Each of these questions highlights a topic from mathematics, and arguably these same questions are the ones that continue to motivate modern mathematical research. Roughly, the correlation goes:

1. How many = questions of number
2. How much = questions of measurement
3. What kind = questions of classification & geometry
4. What if = questions of logic & imagination

For younger kids, the first two questions can often be simplified into a “which is bigger” or “which is more” kind of question, which can avoid counting if they’re not ready to do it.

Once you have these four questions at your disposal, opportunities for mathematical play start popping up everywhere. A walk around the block can be a chance to play the game of estimating steps and then checking whether you were right. (“How many steps do you think it will take to reach that tree?”)

Or an argument over who has more juice can become an experiment to see which cup actually holds more liquid. (How much juice can it hold? Which cup holds more? How can we figure that out?”)

Little kids will automatically classify and sort. What’s fascinating to me is how central this question is throughout mathematics. It seems like every field begins with a question of what kinds of objects are possible. What kinds of pentagons tessellate? What kinds of of polygons can you make by putting squares next to each other?

As for What If questions, cartoonist Randall Munroe has practically built a career out of them. The freedom to assume that the rules or the setup is different is one of the keys to owning your mathematical experience.

If you’ve got a favorite question you ask to find that sweet spot on the Venn diagram, let us know!

A last thought: perhaps the Venn diagram above isn’t to scale. Maybe it should be more like this:
Math Play 2

The Problem We All Live With

The problem we all live with

Back in 2006, I had the chance to see Jonathan Kozol when he visited Seattle touring his new book, The Shame of the Nation. The country, he said, had more educational racial segregation in 2006 than it did in 1968, when his first book, Death at an Early Age came out.

While the US rejected the doctrine of “separate but equal” in the landmark 1954 Brown vs. Board of Education Supreme Court decision, racial segregation in schools continues to be a fact of life. It almost never comes up. That’s the reason that the This American Life episodes, linked below, are so relevant. Integrating schools has been shown to be one of the most effective way to close the educational outcome gap between students of different races; it is also one of the only options for school reform not on the table.

As the beginning of school rises up on the horizon, these radio shows couldn’t be more timely. They are required listening for anyone who cares about race, class, education, and the future of the American democratic experiment.
The Problem We All Live With – Episode 1
The Problem We All Live With – Episode 2

After that somber note, here’s news on Math for Love in September. Our Sunday class registration is open for K-8th grade; Prime Climb has won another award (it’s fourth!); we’ve wrapped up our work on the Seattle Summer Staircase—a fantastic project once again—and got a brief vacation before this year kicks off. We’ll be running professional development with more schools than ever this year, as well as expanding our partnerships with libraries, pre-Ks, and possibly embarking on some new, exciting partnerships (more on these later).

We’ll have lots more news to share as the year progresses. Here’s a little inspiration for the beginning of the school year that made me smile.