Product Placement

The discussion from two days ago led to another nice observation from Marilyn Burns.

Michael Jerrell suggesting thinking about it as an array. That makes it pretty clear that the products can only be different by 11 if the factors add up to 10.

I had never thought of that before... but that meant that if you decomposed the other way, then you could turn the product into a series.

So 4x6 = 3x5+3+5+1 = 3x5+9 = 2x4+7+9 = 1x3+5+7+9 = 3+5+7+9.

Hmmm, so 5x8 = 4x7+12 = 3x6+10+12 = ... = 1x4 +6+8+10+12.

Which means
MxN = (M-1)x(N-1) +M+N+1 = 1x(difference of M &N+1)+ (that +2)+...+(M+N-1)

But that's an arithmetic sequence.

So something like 23x28=(28-23+1=6)+8+10+...+46+48+50. These are easy to sum - ask any 5 year old Gauss. 23x28 = (50+6)x(# of terms)/2. So how many terms? Whichever is smaller determines, since you're going from 28-23+1 to 28+23-1. So 28-22 to 28+22, difference of 44 by 2s. 22 times up plus the first term... that's ... 23 terms. Hmm. Of course, since we're saying 23x28 = (56)x(something)/2.

Of course! First + last = (N-M+1)+(N+M-1) = 2N. So (first+last)*(# of terms)/2 = (2 x larger) (smaller)/2.

No new product, which I was hoping for when I started, but a nice connection from products to series.

Note if M=N, this is the classic squares are the sum of odds. Since N-N+1=1, N^2= 1+3+5+...+(2N+1), making this a good generalization, too.

Thanks, Marilyn and Michael!

Quick Story over a Long Time

I'm at a church meeting for a couple of days. If you can't find teachers to hang out with, church meeting people are a good substitute. Selfless, friendly and some pretty good stories.

But what if you meet a teacher at one of these meetings? Boom!

John had many stories, most of which are not mine to tell. But he did tell about teaching in England in the 60s on a Fulbright year. On the ship over, they showed To Sir With Love. (One of my all time favorites. Must see teacher movie.) When he gets to East London, that's the school he's teaching in! Were they really that cockney? I ask. "Oh, yes."

At some point he became the first dean of students for a nearby high school. Since one of our greatest challenges with student teachers is building classroom culture, I asked what he advised. He said there's no short answer, but let him tell a story.

He's teaching chemistry, and a student came in to find out if he really needed the class. "What do you want to do?" "I'm going into biological research." "Well, you're almost certainly going to need chemistry." So he takes the class, and does brilliantly. One day they're working with condensors, and John warns the student that this is expensive glassware. The student accidentally lets it roll off the table and it shatters. The student is broken up about it and worried, so John consoles him. If you do what you're planning on, you'll pay society back many times over.

At this student's 50th reunion, John gets to see him after his PhD in biology, and a career working in medical science all over the world. He got to collaborate with two Nobel prize winners. John shares the story from high school, and all are amazed and touched.

It turns out John only became a teacher because of getting a second chance when he was in high school. But the moral is the same, I think. We're investing in people. Teach your students, love your students, and you have no idea where your adventures will lead you or them.

Good night!

ps. if you ever get to meet John, ask him about the dynamite.

Not Subtracting

I love how Twitter works some days. OK, most days. #MTBoSfanboy

I retweet Scott's video talking about multiplication with his daughter. (He has lots of nice videos along these lines.)

 Martin responds with a connection.

 Brad shares the video.

 I respond, prompting a (as usually, intriguing) Dan tweet. Which will double as the moral of this blogpost.

/Explaining/ a trick is good math!

Which Marilyn responds to... (talk about fanboy). And we will be looking into this trick.

Cool stuff in Nicholas' trick shared by Ms. Burns.

Links: Scott's post with his video, Brad's video, Marilyn's blog with her trick.

Since this wound up here, I thought I'd share how I subtract.

When I was in 2nd grade, Mrs. Schultz told us you can check subtraction by adding the result back. 6 year old mathchico thinks that if this is always correct, why don't we do that in the first place?

132 - 49? 3 plus 9 is 12 so 1+4 (now I'd say ten + forty) is 5, and 8 plus 5 is 13. Done, 83.

Over the years I've gotten in a lot of trouble for not borrowing.

One more:

 6+5=11.

 8+(6+1)=15

 7+(4+1)=12

 2+1=3.

Solved and checked, Mrs. Schultz!

Peace be to any teacher who had me in class, up to and including my advisor, Nigel Higson.






And Dan writes my signoff:

Queen of Quilts

Actually, Quilts wouldn't make a bad 5th suit...

Elizabeth Clark is a friend and mentor to all of us in my family in many ways, but this post is about the specific inspiration of her quilts to me.

We were over today and she showed this amazing quilt: a Sierpinski gasket stitched over in lissajous and something like the Mystery Curve.

Wow.

For some reason this gave me a flash for how to do a similar thing in GeoGebra. Have a square where you set the pattern, and then use translations and dilations to propagate it. When you click a square it toggles a true/false value, which controls the color. Then in some of the patterns, I use the same Boolean value to determine of the translation happens.

I'm pretty happy with the results. I tried one generation further of the fractal, but it didn't show up well and increased computing time a lot. I know other people have made better and slicker versions of this with programming, but it's fun to make for yourself, too.

Of course, the interactive is on GeoGebraTube if you want to play for yourself.

Hopefully I'll get a chance to blog more about Elizabeth because she gave me one of my favorite problems, and she is endlessly creative.

Ode to a Circle

Ode to a Circle

One of Michal Krasnopolski's cool
minimalist movie posters.

First seen in nature
spreading waves from a stone
or seen in the sky as sun
and moon, perfect essence of round.

As soon as we knew distance
we knew it as a kind
of a unity
from the center.

We named aspects
of interaction,
chord and secant, arc, sector
radius, diameter.

Thales found great beauty within,
the first theorems with
the birth of our dear
right triangle.

And then bound by
number with surprise
as the fundamental
Pythagorean sum.

Trigonometry within,
called out by Hipparchus,
but glimpsed by Euclid,
Archimedes.

These lead to a
science of the rhythm,
periodicity,
Fourier's analysis,

Built upon the beauty bare,
of a series approach
as Euler alone
saw you in e.

-John Golden

I don't know where this idea came from exactly, but it was a Thales kind of week, with the start of my math history course, Dan Meyer trying an activity in Desmos and my son studying circles in geometry.

Looking for a movie poster, I found those cool minimalist images. Fellowship of the Ring would have been a good name for this poem!

What's the Worst that Could Happen?

Fun book, terrible movie.

So it happened. I missed a day in the #MTBoS30 challenge. Yeah, I had a Tumblr post, but that doesn't count, right? Because ... well, I guess I made up some rules and that's one of them. For an okay reason, maybe. I post to Tumblr regularly, but this blog has fewer entries every year. Why do the number of entries matter? On the one hand, they don't. On the other hand, I know that this blog has been important for connecting to people I love and with whom love to work, and it's been an important tool for reflection. I ask my students to write for reflection, so I should be doing it, too.

Is missing a day a reason to stop? Obviously not. But... it feels like it? What is this secret perfectionist hiding inside of me that tells me to stop #MTBoS30 because it's damaged somehow.

It's when I hear that kind of thinking that it really makes me stop and wonder where else am I doing that?

For teaching, did I give up on something I was trying to do (better or feedback, more individualized support, reworked activities, notes about what I tried) because I was inconsistent or had an off day? Or because I tried and it didn't work? Why would I expect something new I tried to work the first time?

For parenting, shoosh. Currently I'm loosing it at least once a day with one of the kids. But today, I'm going to try it again. I can be calm and discuss things, I can. Tomorrow I will start again, despite the zero consecutive successful days.

So I'm blogging again today, one short of a dozen.

In Donald Westlake's Dortmunder books, the not quite small time crook asks 'What's the Worst That Could Happen?' because of a deep and abiding belief that this is exactly what will happen. But it doesn't stop him trying the heist. Because you have to make a living, right? In a way, it makes him a better crook, because he plans contingencies, and keeps calm when things go wrong.

Wrong Hands

I hope you give it a go, whatever you want to try. What's the worst that could happen?

What is math?

My capstone class started today, Math 495: the Nature of Mathematics. We take a historical look at math, and the first assignment is to blog shortly on what is math and 5 milestones that they know
already.

Since, for some reason, I am trying to blog everyday this month, I thought I'd join them.

Mathematics to me is noun and verb.

Noun: it's hard to do better than Eugenia Cheng's description, "the logical study of logical things." Quantities, their properties, operations on them, their properties, treat the operations as the new object and repeat. Pretty soon you have cohomological sheaves and categories dancing with your sugar plum fairies. "Mathematics is beautiful."

Verb: reasoning about, playing with, representing or describing mathematical objects or posing or solving the problems and questions that you have about them. "Hey, watch out, I'm doing mathematics over here."

Students first responses:

Milestones:

1. Invention of number systems to record quantities, especially place value systems (Babylonians, Mayans, Indians)
2. First application problems connected with measurement. (Egyptian and Babylonian stand out to me.)
3. Mathematization: the study of the mathematics itself. Greeks, of course. Brahmagupta, Liu Hui, also.
4. The Persian synthesis of algebra and geometry. Al Khwarizmi in the House! (House of Wisdom that is.)
5. Descartes and the invention of analytic geometry. 

Very hard to pick five. Not having Gauss or Euler is painful. But I'm big on synthesis. My favorite modern mathematics is the Atiyah-Singer Index Theorem.

p.s. If you're interested in the course, here's the Google doc of what we did the last time around.
p.p.s. Found this movie poster on this page. Haven't ever heard of this movie, but I'll be looking for it!
p.p.p.s Erin found the movie as released, which lost the cool x+y title.