Sunday, December 8, 2013

Math Monday Blog Hop: Multiplication and Division

Cindy, over at love2learn2day, runs a weekly blog hop on elementary level math. This week's topic is multiplication and division. I'm writing this post to gather a few resources I like.

Most of the posts that usually gather in the blog hop are activities. Before we get to activities, we might want to ask ourselves some questions, like "What is multiplication?" Multiplication can be modeled in many ways, and it's important to give young kids a chance to think about it through lots of different frameworks. Your own conceptions of multiplication may grow as you think about it.

One resource that will help you broaden your perspective on multiplication (along with a number of other math topics) is the lovely book Moebius Noodles. (It's $15 for the paperback version, and whatever you decide to pay for the pdf version.) It is full of activities to share with young kids.

I also love the art connection made by Waldorf schools, with their multiplication stars. Beauty helps us love what we're doing, which helps us learn.

As we teach, the more we can see the child's understanding, and help that grow, instead of telling them what is so, the better off we are. I can think of two ways we might just tell children our adult understanding, without giving them enough opportunities for exploration. One is in the commutative nature of multiplication, noticing that 3x5 = 5x3. Kids don't see that at first. If we tell them, we take away their chance to discover it. I never knew that until I read ‘Third Graders Explore Multiplication’, a chapter by Virginia Brown in the book What's Happening In Math Class, Volume 1, edited by Deborah Schifter. [You might be interested to know that mathematicians study versions of multiplication that are not commutative. Matrices are a tool for solving systems of equations, if A and B are matrices, AB may not equal BA.]

The other example I'd like to share of this is our knowledge that to multiply by ten we just "add a zero". (Why does that work?) Here's an excerpt from the chapter 'Trust, Montessori Style', by Pilar Bewley in the soon-to-be-published book Playing With Math (edited by me):

Answer Versus Process 
Five-and-a-half-year-old Roland came to ask if he could multiply 8,696 times 10 using the stamp game. I was thrilled to see his interest in math taking off again after an unfortunate temper tantrum with the addition blank chart. I suggested he borrow a stamp game box from another classroom to supplement the one we have in ours, and he got started.

It took him a while to figure out where he would do the work, and then he painstakingly began to make 8,696 with the stamps… Six units, nine tens, six hundreds, and eight thousands… Leave a space and repeat… Six units, nine tens, six hundreds, and eight thousands…

He made the amount five times before it was time to go home, and he left his work out so he could return to it the next day. (Can you imagine what that looks like, once you’ve made 8,696 with little colored tiles ten times? How cool!)

“I’m going to get right back to work as soon as I change my shoes,” he declared before leaving that afternoon. “I won’t even talk to anybody!”

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten).

Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.”

My heart sank. Oh no, Dad! You robbed your son of such a cool experience! He was getting ready to see what ten times 8,696 looks like, and would have discovered the process that takes place during multiplication. He won’t be doing multiplication tables for at least two more years in public school - the answer doesn’t matter yet, but the process could have really taught him something valuable.

Although I firmly believe that math is about understanding, multiplication facts are one of the few things that need to be memorized. Many students come to college not knowing those basic facts. When I was teaching beginning algebra, I gave a quick quiz on all the basic multiplication facts, and required students to keep trying until they could get 85% right. Here are some of the suggestions I offered them:

Those suggestions were for adults who were likely to have emotional baggage about not yet knowing something basic. For kids, just have fun. Make up games (like John Golden does) using dice or cards, go on scavenger hunts, or tell stories. (Perhaps you could tell stories about the large family, who have to buy in bulk because they have so many children. Each child needs 3 pair of shoes, for dress-up, play, and beach. How many shoes are in that house?)

Hmm, all of that and I haven't even mentioned division. I guess that will have to wait for its own post.

Thursday, November 28, 2013

More Links: Good article, video, tool, problems, game, and activity

What I've stumbled upon in the past two days:

And some older ones (as I begin to slowly clear out my backlog...):

Tuesday, November 26, 2013

Links versus Real Writing

I used to share links more often. I used to write substantive blog posts more often, too. Since I've been writing less, I haven't been comfortable sharing lots of links. Didn't want this blog to descend into just a link-share. But it would be helpful to me to have them here. So maybe I'll start sharing my almost daily finds, even if it's not exciting for y'all.

These were the tabs I've kept open - some for weeks - hoping to figure out how to use some, how to find time to really process others.

Friday, November 22, 2013

Online Conversations: Math Communication, and Understanding Computer Graphing

I am enjoying two online conversations right now.

Michael Pershan asked:
Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math?

I am currently trying to grade my students arguments (as prosecutors) for the murder mystery. Some of them really got into it. Most still didn't explain the math well. My take on this is that students will write (maybe even well) if we give them a good enough context.

In the other conversation, Mr. Honner blogged about what happens when you zoom in super far on Desmos, looking for the hole in a rational function. It gets a bit crazy. The conversation got more interesting for me when Alan Eliasen started explaining "interval arithmetic", which I had never heard of.

Friday, November 15, 2013

Starting Circle Trig in Pre-Calc

I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And, once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.

That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.

The problem:
Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them? 

I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to them.)

To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.

It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.

When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.

Sunday, November 3, 2013

What are our intuitions about temperature?

I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.

Wednesday, October 30, 2013

What is Calculus? Part Two

In our first installment, we saw a bit of history, and began to think about the idea of rate of change (which can be visualized as the slope of a graph). Slopes of straight lines, central to algebra, are defined by rise over run, or the ratio of change in y over change in x. In calculus, we extend our definitions so that we can consider slopes of curvy lines. Since the slope is different for different parts of the graph, we can think of it as a new function. For each x value, there is a y value, and there is a slope. We call this new slope function the derivative of our original function. (This is the part of calculus called 'differential calculus' - differentiating means finding the derivative.)

We need an example for this to make sense. Let's go back to that ball. From part one:
Suppose we throw a small ball straight up, and are able to measure its position perfectly every tenth of a second. We can make a graph for that - [its] horizontal axis will measure time ... in fractions of a second... The vertical axis will simply measure the ball's height. It starts out in my hand, about 4 feet off the ground. I think I can throw it about 20 feet high. We'll imagine together that I do. Working out what gravity does to that ball is one of the things we can do with calculus. Here's the graph (new version):
We see from the graph that the slope is positive until we reach the highest point, and after that it's negative. In fact, finding the highest point turns into finding out when the derivative (slope function) equals zero. This is a powerful tool for finding the most or least of anything that's defined as a function.

So how do we find these slopes? If I asked you to draw a tangent to this parabola at t=0.4 seconds, you would probably know what to do, even though the only definition of tangent line you may have seen before this is the tangent to a circle. Can you draw that tangent line? (I've drawn a picture below, but I'll discuss our next step now, so the picture won't show up right away. Please turn away from your computer, and draw a sketch of this graph with a tangent line added in.)

If we want a more precise description of what's happening to the ball, we need an equation. In this case, height in feet is determined by time in seconds, by h(t) = -16t2+32t+4. Notice that height at time 0 would be 4, as given in the story above. (The -16 is determined by the force of gravity on earth, and would be different under different gravitational conditions. This part can be understood more deeply by the end of a calculus course. The 32 is determined by how hard I throw the ball upward.)

The tangent line touches the graph at just one point. Using our equation for the curve, we can easily find the y-coordinate (height) at that point, but how would we find the slope? Slope requires finding the change in y and change in x, which takes two points. But all we have is one point. We're stuck.

One strategy mathematicians use might be called wishful thinking. Since we need two points to find the slope, let's pretend we have them - let's cheat! If we use a second point on our graph very near the first, and make a line that goes through both points, our line, called a secant, is pretty close to the tangent line we're seeking. What if we try to get closer and closer this way?

You may want to experiment here. Our one point is (0.4,14.24). If we use t=0.5 for our second point, we get (0.5,16). Now we can figure the slope of our tangent secant line. Slope is the ratio of change in y over change in x. We find those changes by subtracting, so we get:
That would be 17.6 feet per second. But that secant line is just a little bit less steep than the tangent line. We could get a better estimate by trying a second point at t=0.41. But we'll never get a perfect answer. Hmm ...

Ok, this is where it starts getting a little weird. We want to imagine the second point getting infinitely close to the first. The use of infinity is what separates calculus from algebra. Newton called his infinitely small quantities 'fluxions', and another mathematician of the time (George Berkeley) complained bitterly:
“What are these Fluxions? The Velocities of evanescent Increments. And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”
It was more than a hundred years later before mathematicians were able to develop a solid logical basis for calculus!  But if we're willing to trust our intuition, like Newton did, this method of taking points "infinitely close" to one another works amazingly well. If we call the distance between the time values h, and imagine h getting smaller and smaller, we would say we're "taking the limit as h goes to 0".

The algebra gets very ugly here, and it turns out it's easier to see for a generic time t than for a specific time (like t=0.4). We're imagining that t represents one time, and t+h represents another time very close to it. The height at time t+h would be
Still ugly. And it's going to get a bit worse before it gets better. Now we find the slope:
Here's where the logic gets tricky. We can't have zero on the bottom of the fraction (dividing by zero just doesn't work), but if h doesn't equal zero, no matter how close to zero it is, we can cancel it. What we're left with includes -16h, but h is so close to zero that this term doesn't really matter.  So we get that the slope is simply -32t+32 or 32(1-t). So the slope of the tangent line at time 0.4 is 21.6. We could also say that the ball is moving upward at that moment at 21.6 feet per second.

Finding an average speed (I traveled 122 miles in 2 hours, for an average of 61 miles an hour) is straightforward. It's a ratio of distance traveled over the time it took. But finding an instantaneous speed - how fast I was going at a particular moment - that takes calculus.

I think this is incredibly beautiful, and every semester I relish helping students understand this concept. But this is the first time I've written it all out without feedback from the 'students'. It's time to ask: "Any questions?"

Monday, October 28, 2013

What is Calculus? Part One

The New York Times has an online column called Numberplay, written each Monday by Gary Antonick. It usually features mathematical puzzles, but this week he was asked to explain 'differential calculus', and in his article he wrote:
I’ll lay out a few of my own thoughts but I’m especially looking forward to learning from others on this. Let’s make differential calculus our puzzle for the week.

He shares a delightful story of working in some very eccentric bakeries. Delightful though it was, I didn't feel like it would help anyone who didn't already get something about calculus. I want to offer up just a tiny bit of history, and then see if I can explain this for people who have at least a rudimentary understanding of algebra.

Isaac Newton and Gottfried Leibniz both invented calculus at about the same time. That sort of thing happens often in math. I think if neither of them had been around, someone else would have invented the methods of calculus - the problems being investigated by scientists of the time were just crying out it for it. Planetary orbits, gravity, the paths of projectiles, the shapes of lenses for microscopes and telescopes - understanding any of these requires calculus. (See the first six pages of this book.) The time was ripe.

Calculus mainly addresses two sorts of questions: How fast something is changing, and finding areas of irregular shapes. (The methods of calculus can also find volumes and the lengths of curves, among other things.)  Addressing these questions requires dealing carefully with the notion of infinity, mostly what it might mean to be infinitely close and what an infinitely thin slice might mean.  The Greeks (and probably others I don't know enough about) did some good work with these ideas. Even though they lacked the super-versatile tool of algebra, they made a good start on the area problem. Perhaps through lack of interest, they don't seem to have addressed the rate of change problem. Other cultures - and other individuals before Newton and Leibniz - made progress on both questions. What Newton and Leibniz did that got them famous was to find a simple link between the two, which is now called The Fundamental Theorem of Calculus.

The rate of change questions are addressed by what's called 'differential calculus' and the area questions are addressed by what's called 'integral calculus'. Since the question asked in Numberplay was about differential calculus, we'll start there. Even though the ancients did more with areas, rates of change are where we start in calculus courses these days.

If I made 3 loaves of bread a day (Are Gary and I making you hungry?) and saved them all, I could show how many loaves would be in my pantry with a simple graph, something like this ...

In algebra, we talk about the slopes of straight lines like this. For every one unit you go over, you need to go up three units to stay on the line, so we say the slope is 3. (You might remember "rise over run" - that's part of the definition of the concept of slope.) Well, slope and rate of change are pretty similar ideas. Rate of change just gets a bit more precise about the story that goes with the graph. In this case, we're adding 3 loaves of bread per day to the pantry.

The problem is, this notion of slope or rate of change only works with straight lines and constant rates of change, but most of the interesting things scientists want to study have changing rates of change. Like when we toss something into the air, and gravity pulls it back down. The methods of calculus helped Newton to understand how gravity affects objects, and it's a good place for us to start too.

Suppose we throw a small ball straight up, and are able to measure its position perfectly every tenth of a second. We can make a graph for that, too. Like our bread graph, the horizontal axis will measure time, but now it will be in fractions of a second instead of in days. The vertical axis will simply measure the ball's height. It starts out in my hand, about 4 feet off the ground. I think I can throw it about 20 feet high. We'll imagine together that I do. Working out what gravity does to that ball is one of the things we can do with calculus. Here's the graph. (I don't like that I showed the curve for negative time values. I wonder if I can fix that.)  I'll explain this one in part two.

Friday, October 18, 2013

A math circle...

I mentioned Pythagorean triples to my pre-calc students last month, and told them if enough of them were interested, I would run a math circle on this topic. Ten of them signed up, six came the first week, and three came to the second and third sessions. It's small, but it's going very well, and I may do something bigger next semester.

This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)

I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.

Week One
What examples can we come up with?  (3-4-5, 5-12-13, ...)
6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)
Maybe writing a list of all the perfect squares up to 400 will help us find more.
What patterns do we see?
  • Odd + Even = Odd
  • Middle number is a multiple of 4
  • c = b+1 (after which I added 8-15-17 to our list)

Week Two
One person was new, so we reviewed our first week's work for him.

We explored the "family" of triples with c = b+1. a2 + b2 = (b+1)2 becomes a2 = (b+1)2 - b2
= b2 + 2b + 1 - b2 = 2b-1. If  a2 = 2b-1, then b = (a2+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...

X noticed that in the triples
the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:

Week Three
Another new person came, so we summarized for her. Then we explored triples where c = b+2.

I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)2, and thought he might have to distribute.

We verified that we get all of the primitive Pythagorean triples with c=b+2 using:

Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.

Tuesday, October 15, 2013

Playing With Geometry

The Math Monday Blog Hop, at the love2learn2day blog, has a theme each week. This week, it's Geometry. At community college, we don't teach geometry as a course - not surprisingly, I feel it's my weakest area in math. So I love playing with games, puzzles, and problems that stretch me geometrically.

A Game
My favorite geometry puzzle/game is Katamino, which makes a two-player game out of pentominoes (I blogged about it here). I love the beautiful wood pieces. (I can't tell if the newer versions are wood or plastic. I recommend trying to find the wood.) Playing with this will work your visualization skills.

A Puzzle
Online, I recently stumbled on a site which offers 40 challenge problems in compass and straightedge geometry, implemented in an online puzzle.

Some Problems
I've recently come to love the Five Triangles blog, where mostly geometric problems are posted a few times a week. Their problems often challenge me to think in slightly new directions.

What's your favorite way to play with geometry?

Saturday, October 5, 2013

Playing With Math - The Book

I've been working on Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers for almost five years now. For years I've been saying "It's almost done." I have sometimes felt bad about how long it has dragged out. I'm not sure if the authors (over thirty of them!) trust that the book will really get into print. It will.

Last week I was perusing The World of Mathematics, by James Newman (a lovely four-volume mathematical encyclopedia), so I could write a description of it for my Book Picks section in Playing With Math. He mentioned in the preface that it took him fifteen years to put it together! Ok, now I don't feel so bad.

I've been sending out little updates about my progress on my facebook page for Playing With Math, and mentioned that there today. If you'd like to follow my progress, please 'like' that page.

Now I have something slightly bigger to say, which seems to fit better here on my blog. I'm working on answers to the puzzles Paul Salomon created - his Imbalance Problems. I am stuck on the last one we included. It may be that my brain is fried, and you'll all see an answer easily. But I just don't.

This puzzle was created by one of Paul's fifth grade students, Felix. Can you solve it?

Now I wish I had used these at the beginning of our inequalities unit in pre-calc. They really make you think about what > and < mean! I think I'll hand them out anyway - as a challenge.

Thursday, September 26, 2013

Groups Rule!

Maybe I'm slow, and you've all seen this sort of thing in action already, but my afternoon precalc class was a blast to watch in action today. I have 9 groups of 4, and gave each group a different word problem from the book. These were problems that needed the use of Law of Sines or Law of Cosines.

They were to solve the problem and be ready to be the experts on it when others asked them for help. I helped the groups who needed help, and as groups were getting close to done, I told each group their next problem, and who their personal experts were. I loved seeing one person walk over to another group to get help for their group. I loved how hard they all worked. It was great.

My morning class doesn't seem to get as engaged as this class, though. I'm trying to dream up a way to pull them in.

Sunday, September 22, 2013

Teaching Question: Students Overusing Proportional Reasoning

I have heard, from a colleague who works with prospective elementary teachers, that many of them are not good at proportional thinking. My students (in pre-calc) seem to be fine at it, but ... they're using it even when it doesn't apply. My question for you is how to help them see why proportional reasoning is not always a sensible choice.

Problem #54. Determining a Distance: A woman standing on a hill sees a flagpole she knows is 60 feet tall [yeah, right]. The angle of depression to the bottom of the pole is 14 degrees, and the angle of elevation to the top of the pole is 18 degrees. Find her distance x from the pole.

One student wanted to average the two angles at 16 degrees each. Another said the observer could stand on a stool to be a little higher, so the angles would be 16 degrees each. Their answers were very close. There were other good (but wrong) methods that all came down to assuming this relationship was linear in a way that it's not. Since their answers were very close, it was hard to help them see what was wrong with their reasoning.

Can anyone help me here?

Monday, September 16, 2013

Math and Children's Literature: My Favorite Mathy Picture Books

I love kids' books and I love math. So I've gathered together quite a collection. My son, who has gone to free schools where he isn't required to do math lessons, has probably gotten more math out of reading these books than he has from any formal math lessons.

His favorites are probably a few from the I Love Math series, published back in the early 90's. Although they are out of print, inexpensive copies of most of them are available online. My son and I especially enjoy the stories (in every volume, I believe) about Professor Guesser, a cat detective who solves mysteries using mathematical reasoning. She's featured in the title story of The Case of the Missing Zebra Stripes: Zoo Math. Some of the zebras are missing their stripes, and Professor Guesser figures out what's really going on. These twelve books feel like math magazines, even though they're hardcover, because they have so many different sorts of content - they're full of stories, games, mazes, riddles, and lots of math.  (I think this series is good for ages 4 to 12. On all of my age ranges, I have just used my own judgment.)

Here are the other picture books you'll find on my Math Books page (tab above):

The Opposites, by Monique Felix (ages 2 to 6)
One of the earliest math skills, more basic perhaps than counting, is noticing attributes. This book has no words, and yet it tells dozens of stories, each about opposites. Noticing the one attribute that shows opposites in the detail-filled pictures is a math game your child will want to play again and again.

Quack and Count, by Keith Baker (ages 2 to 7)
This is a board book, so it's good for the youngest child who will sit and listen to a story. And it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, 4 plus 3, 3 plus 4, and so on.) It would be great to have a book like this for each number, showing all the number pairs that make it. If I ever get to teach math for elementary teachers again, I'd love to get my students to make books like this one.

Anno's Counting House, by Mitsumasa Anno (ages 2 to 7)
Everything I've seen by Mitsumasa Anno is delightful. There is so much to see in his books, many of which have no words. In this book, ten people are moving from one house to another. In each two-page spread you can see one more person who's moved from the left house to the right, along with lots of furniture and other small items. 
Anno's Mysterious Multiplying Jar will appeal to older readers. There is one island with two counties, which have three mountains each ..., until we get to ten jars within each box - a lovely, very visual representation of factorials. Anno's Magic Seeds does have words, and tells a fascinating story, of a plant whose seed, when baked, will keep you from being hungry for a full year. The plant grows two seeds in a year, and one needs to be used to grow a new plant... You may also enjoy Anno's Math Games. Anno has written over 40 books, most available in English.

Two of Everything, by Lily Toy Hong (ages 3 to 7)
A poor old farming couple in China find a mysterious pot. When a hairpin drops in, they scoop two out.The math isn't discussed in the story, but it's pretty easy to add your own thoughts to this delightful tale of doubling.

How Hungry Are You? by Donna Jo Napoli and Richard Tchen (ages 3 to 12)
There are lots of great of great books on sharing equally. My favorite used to be The Doorbell Rang, by Pat Hutchins, but this one is even more delightful. The picnic starts with just two friends, rabbit is bringing 12 sandwiches and frog is bringing the bug juice. Monkey wants to come, "My mom just made cookies. I could take a dozen." They figure out how much of each goody each friend will get. In the end, there are 13 of them, and the sharing becomes more complicated. One of the delights of this book is the little icons showing who’s talking. Those would help kids to create a delightful impromptu play.

One Grain of Rice, by Demi (ages 5 to 12)
The greedy raja is gently outsmarted by a wise village girl named Rani. This is a very sweet take on the story of grains of rice put on a chessboard. (One grain on the first square, two on the next, then 4, 8, 16, …, until the board is filled. How much rice is that, anyway?)

 The Cat in Numberland, by Ivar Ekeland (ages 5 to adult)
The story starts when Zero knocks on the door of the Hotel Infinity. He’d like a room, but they’re all full (with the number One in Room One, and so on). Turns out that’s no problem. The cat who lives in the lobby gets confused - if the hotel is full, how can the numbers make room for zero just by all moving up one room? Things get worse when the fractions come to visit. This story is charming enough to entertain young children, and deep enough to intrigue anyone. Are you ready to learn about infinity with your 5 year-old?

You Can Count on Monsters, by Richard Evan Schwartz (any age)

Each number from 1 to 100 is a monster, and each one gets its picture on its own page. All of the numbers (except poor 1) are made up from their prime parts. The pictures are colorful, full of intriguing detail, and amusing. The pages in the front and back that explain prime factorization are unassuming, waiting for the reader to decide it’s time to find out more. This and Powers of Ten would both make great coffee table books, to peruse over and over.

Go to my Math Books page for reviews of chapter books for older kids (and books suitable for adults). There are lots of other good mathy kids' books, but these are my favorites.

I also love the idea of creating math lessons from good children's literature even when it wasn't intended for the purpose, but I've never done that myself. (I did find a good math lesson for my adult calculus students in the book Holes, by Louis Sachar.) Julie Brennan does wonders with this genre. Here's an excerpt from one of her chapters in my soon-to-be-published book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers:
I recall my daughter, Hannah*, running across the old question, “How do you split two things evenly among three people?” She has two very memorable experiences to draw on. One is a PBS Cyberchase episode she watched when she was five or six, where the kids had to split two apples exactly evenly among the heads of the three-headed dog or they were in trouble. She never forgot that each apple was split into thirds, and each dog got two thirds. Second, when she was around seven or eight, we were reading a Laura Ingalls Wilder book aloud, and we ran into the story of Laura and Mary getting two cookies from someone. On the way home, they agonized over wanting to eat the cookies themselves, but knowing they needed to share with their sister Carrie, and not knowing how to evenly divide the cookies. In the end, they erred on the side of caution, split one cookie between themselves, and gave the other whole cookie to Carrie. My daughters found it so funny that they didn’t think to divide the cookies up into thirds! Between these anchors, this idea of dividing and sharing proportionately is very real, and it gives a real sense of what 2/3 can be - two wholes divided three ways.

However you do it, I hope you'll enjoy finding math in great children's books!
[This post was written to share in the Math Monday Blog Hop. Thanks, Cindy.]

* pseudonym

Sunday, September 8, 2013

Online Course: WOW! Multiplication

Maria Droujkova is very fun to work with. I have only recently met Yelena McManaman - I suspect she will wow you too.  This is very different from most online courses. First, it's very short - two weeks, two hours a week. Second, it's very participatory - you learn about some interesting activities the first week, then try them out with your kids (or anyone, really) the second week. Third, you actually contribute to research in math education when you do this.

The course begins tomorrow, so if you're interested, today is your last chance to sign up.

Saturday, September 7, 2013

Estimation and Orders of Magnitude

I just read a blog post by Jonathan Claydon titled Year of Estimation. Like me, he's loving the site I hadn't paid attention to how the items on the site build up from earlier to later, going from tissues in a travel pack to a box to a bigger package. I may use that feature if I can get myself using the site more often. (There's so much I'd like to do, and not enough time for it all.)

I make my own estimation activities too. This past week in pre-calc I brought in a jar of little origami stars and asked my students to estimate how many there were. This worked into Kate's absolute value lesson. After I announced the correct number of stars, I asked them each to write down how far off they were, their error. Of course some of them had to do Actual Number - Guess, or A-G, and others had to do G-A (leading us to absolute value). I put all the guesses on a spreadsheet, showed my students how I wrote a formula to have excel figure the error, and then graphed guess versus error, giving an absolute value graph. We decided that the good guesses were the ones with an error less than 20, giving us a reason to solve | 225 - x | < 20 (225 was the actual number of stars and x was the value of a guess).

Jonathan pointed to an online quiz that I found intriguing, on the relative sizes of things. I also found it frustrating, because with one wrong answer you had to start over. The quiz has some hard comparisons. Here are some I got wrong:  Which is bigger, ...
  • the Eiffel Tower or the Great Pyramid of Gaza?
  • the width of Uluru Rock (in Australia) or the height of Angel Falls in Venezuela?
  • the Milky Way or the Crab Nebula?
  • the moon or Pluto?
  • Russia's east to west length or the moon's diameter?

I wanted an easier version - with only items my students would know about - that they could use to think about orders of magnitude. So I created one myself. I alphabetized so it would be easier to search the list, and I lettered them for easier reference. Are any on my list still hard? Could you add in any and keep it relatively easy? Where am I jumping the most between orders of magnitude?

Quiz Yourself on Estimation
Put these in order from smallest to largest.
a. blue whale
b. California
c. carbon atom
d. dog
e. Earth
f. egg
g. eiffel tower
h. electron
i. giraffe
j. human
k. Jupiter
l. Milky Way galaxy
m. moon
n. Mount Rushmore
o. Niagara Falls
p. Oregon
q. Pacific Ocean
r. proton
s. red blood cell
t. soccer ball
u. sun
v. sunflower seed
w. tennis ball
x. United States
y. water molecule
z. white house

If you are working on estimation or orders of magnitude, you may also like some of these resources:

Wednesday, September 4, 2013

Week 3 of a Great Semester

I am still trying to squeeze out time to work on the book (Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers), so I seldom have the time to blog these days. But I just now had so much fun in my Pre-Calculus class, I have to write about it.

I've been noticing that I'm enjoying all four of my classes this semester. I usually have a favorite, and I am often struggling with a few disengaged students in at least one class. Somehow that hasn't materialized this semester. (One high school student, R, was being goofy the first day, and I called him on it in a puzzled sort of way. Turns out he is a great math student. Yay!)

In pre-calc, the first unit I do comes from the parts of the review chapter that I thought were worth focusing on: Lines, Circles, and Inequalities. (I have them look in the first six sections of the text for problems that would get them stuck, and we work a bit on those, but I don't lecture on all those details.) Although my three topics seem unrelated, I find small ways in which they connect.

We are starting on inequalities, and I was explaining interval notation: "For x ≥ 4, we write [4, ∞). The bracket means we include the 4, and the parenthesis means we do not include infinity, which we never do, because infinity is not a number." R felt that infinity is a number, and explained why, using the phrase 'infinity principle'. I'm not sure what he meant by that, but it actually helped another student think about infinity as "a principle of numbers, not a number itself". I lent R The Cat in Numberland after class.

Yesterday I had used part of Kate's lesson to help them see absolute value as distance. Today I described | x - 5 | < 3 as meaning the distance between our number and 5 is less than 3. A student asked if that was the same as |x| - 5 < 3. I said "Great question. What do you think?" The class was divided. I asked if anyone could give a reason for why it might be the same or different. Someone said that absolute value is a grouping symbol. I agreed and asked how that made the two inequalities different. No one had an answer to that. S then said that the first one is always positive (on the left side), and the second one can be negative (if x=2, for example). I told her after class that that's called a counterexample, and is used often when we're trying to prove something isn't true.

At that point I said I was falling in love with this class, and called them mathematicians. I'm sure they think I'm a bit nuts, but hopefully "in a good way".

Getting back to the task at hand, we picked numbers on the number line for which our first inequality was true, and I got them to tell me I could make it solid (coloring in all the points between 2 and 8). After we did this very concrete process, I showed them the algebraic way to "solve it". I told them we read the solution, 2 < x < 8 as 2 is less than x, which is less than 8.

We then looked at | x - 5 | ≥ 3, and I got them to tell me points that worked first, and then walked them through the steps to get the solution of x ≤ 2 or x ≥ 8. Someone asked if it was ok to write 2 ≥ x ≥ 8. I replied that this says 2 ≥ 8, so it doesn't work.

I won't know until the next quiz how much of this is really making sense to them. It seems great right now, but I am often terribly disappointed once test time comes. The downfall of a good lecture is that it looks and sounds better than it really is.

Linear Algebra
Today I was 'covering' linear independence. The book gives a definition, and I wanted the students to see a need for the definition before I put it up. We had previously seen an example of a vector that was a linear combination of two other vectors. I used a similar example - two of the vectors would make a plane, while the third vector (a linear combination of the first two) would contribute nothing new. So we call this a linearly dependent set of vectors. (And, by our textbook's definition, this happens when there is at least one non-zero ci in the equation c1a1+c2a2+...+cnan = 0.) Naturally, linear independence is defined to be the opposite situation. If c1a1+c2a2+...+cnan = 0 has only the trivial solution (all the c's = 0), then the set {a1, a2, ... an } is linearly independent.

That may not sound exciting, but I love how the various concepts in linear algebra all weave together. I couldn't stop myself from mentioning dimension today, even though the book doesn't get to that until the next chapter.

Last night, we figured out a few derivatives (which I like to also call the slope function, to help the students keep their eyes on the meaning) using the definition. I keep asking and they keep telling me - it's just change in y over change in x, but I'll only know whether or not they really see that after the first test.

During the first two weeks, they were very confused. It's beginning to come together for them, I think.

I feel very lucky to be teaching students who are willing to play around with math. I also am seeing how my work with math circles, my work on the book, and my blogging have all contributed to my enthusiasm and my steadily increasing skills, even after 25 years of teaching.

Sunday, August 18, 2013

Once Again, Day One

We start classes tomorrow. No, I'm not ready yet. I've been learning how to use our online site, which comes from "Desire 2 Learn." (I hate that name. I feel like I'm spouting propaganda every time I say it. I'll be calling it d2l, and mentioning the problem of the name in each class. Gag.) I want students to be able to see material early, get copies of material when they miss class or lose their copies, etc. I'm having trouble uploading one particular file, and can't figure it out. Sigh.

I'll be teaching two sections of pre-calculus, one of calculus, and one of linear algebra. In each of my classes, I'll move the desks into groups of four, and hand out the syllabus and a sheet summarizing unit one and listing the homework. I'll also pass around a phone list for them to put their name and info on, which I'll copy and hand out the next day. And they will put their name on a 3x5 card, which I'll use to call on people randomly. Here are my planned day one activities for each class:


We'll be using (Thank you, Andrew Stadel.)
Our first problem: Breaths in a day (my guess: 8000, google, 17,000 to 28,000) I was 9000 off, 9000/17000 = 53% low

Visual Patterns
We'll be using (Thank you, Fawn Nguyen.)
Our first problems: #2 (easier) and #1, pretty hard

Equations for Graphs
We'll be using Daily Desmos. (Thank you, Team Desmos and all the contributors.)
Our first problem: 110a2 (use two eqns, or challenge yourself to find just one)

Graphing Stories
We'll be using and what I downloaded from Dan Meyer's blog. (Thank you, Dan Meyer.)

I have had endless trouble with the projection systems in my classrooms, so I plan to test it all out in both of them today. I plan to do one of these activities (or a quiz) with my students each day at the start of class. I think this will fill my hour up quite nicely.

Linear Algebra

Calculus (starts Tuesday evening)

On Screen:
Screen 1: What is the meaning of acceleration? (Write what you think it is on your paper.)
Screen 2: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
Think about it on your own, without discussing it yet. We’ll vote, then discuss, then vote again.
Screen 3: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
A. up B. down C. none D. not enough information

I don't want to bother with clickers, so I need to stop by the copy shop to get my vote cards made up. (I got this idea from Kate. Mine will be a bit different. I'll post more if it works out, and maybe even if it doesn't. Kate has a good calculus question at that post.)

Tangent Task
For the past two semesters, on day one, I had students carefully graph y=x2, and then show a line tangent to the graph at x=2. After that they were supposed to estimate the slope of the tangent line. It worked great in the fall. But in the spring, a bunch of them knew the derivative 'rule' and that destroyed the activity. I'm going to use a circle and a few graphs I've drawn this time, so they can't use 'rules'.

This class meets for 2 1/2 hours, so I'll have lots more planned, but I get to work on that tomorrow and Tuesday.

On the first day or two I also:
  • Explain how Donut Points work (Every time someone in class catches me in a math mistake, that's a donut point. When the class has gotten 30 donut points, I bring in donuts.);
  • Talk about the difference between what people think math is and what it really is;
  • Talk about mindsets, stereotype threat, and how neurons grow when we learn new things  (I like talking about myelin growth);
  • Explain how to get cheap textbooks (online, used, I allow older editions);
  • Explain that I will stamp homework each day;
  • On day one I ask them to find interesting things on the syllabus, on day two I ask them to share, giving me the opportunity to explain test retakes.

You may find other helpful ideas at my previous Day One posts, here, here, and here.

Saturday, August 10, 2013

Book Review: String, Striaghtedge, and Shadow: The Story of Geometry, by Julia Diggins

This book was recommended on Living Math Forum. I can see why. The storyline is very engaging overall, and gets you thinking about the history as if it were happening while you watch. But...

My biggest issue is that it is so gendered:
"A string can usually be found in a boy's pocket..."
"Ancient men discovered the ideas and constructions of elementary geometry ..."
"Through the ages, men have searched to find the secrets of the universe."
"A long, long time ago primitive men observed the lines and curves and other forms of nature."
"It was from this inner sense - man's sensitivity to the order and harmony of the universe - that geometry really began."

I don't think writers do that so much these days. (This book was written in 1965.) Writers sometimes still say 'he' when they mean all of us, and still sometimes say 'man' to mean people, but not often. Reading this book made me think modern writers must be avoiding this construction, even if unconsciously.

Why does it matter? Research shows that we need to feel a part of  a community in order to do our best thinking. Women and girls are shut out by this sort of writing. As much as I might like the content of this book, it sets me up as an outsider (even though the author is a woman!), and that's part of how stereotype threat happens.

I haven't read the whole book, but I did discover one error, I believe. The discovery of the fact that the square root of two is irrational seems to be described incorrectly:
"Then was it a ratio of whole numbers between 1 and 2? ... They tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2."

There are two problems here. One, they couldn't have tried 'every possible ratio', because there are an infinite number of possibilities. More importantly, it wasn't about giving up. If I understand the history correctly, they actually proved that no such ratio can exist. This notion of proof is a very important foundation - it's part of what mathematics is. So her version of this story takes away some of its drama.

The Pythagoreans believed, as she says, "that the universe was ruled by whole numbers." So to prove that a length exists which cannot be described by a ratio of whole numbers was extremely unsettling to them.

How do we prove that the square root of two cannot be a ratio of whole numbers? There is more than one way to do it. You might like Kate and Justin's way more than the one I usually use. It's less dependent on being comfortable thinking with variables. Here's the way I think of it:

If the square root of 2 could be represented by a fraction, we could writes that fraction in simplest terms as a/b. Then we'd have (a/b)2=2, or a2 = 2*b2 . Since the right side of this equation is even, the left side must be, too. If a2 is even, a must itself be even. Let's call it 2c. Then our equation becomes (2c)2 = 2*b2 , or 4*c2 = 2*b2 , or 2*c2 = b2 .  Now the left side of this new equation is even, so the right side must be too. And that means we can write b as 2d. But if both a and b are even numbers, then the fraction can be simplified. We started out with what we thought was a fraction in simplest terms, and found out that it could be simplified. This is a contradiction. It happened because we tried to write the square root of 2 as a fraction - it can't be done, and this proves it.

Proof by contradiction is a bit weird. I think I might like Kate and Justin's proof better myself.

Well, you might like String, Straightedge, and Shadow, even with its flaws. I might, myself. But I have decided not to include it in my Book Picks section.

Thursday, August 8, 2013

Book Review: How to Count Like a Martian, by Glory St.John

Once again, my work on the last bits of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is getting me to write things that belong here.

Today I'm working on the Book Picks section, one of the resources you'll find at the back of the book. Much of it comes from what I've already written for the Math Books page of this blog. (You can see the tab for it above.) But there are some great books I hadn't written up yet.

Right now, I'm writing a description of How to Count Like a Martian, by Glory St. John. It's running too long for the Book Picks section, so I'm posting it here. I'll pare it down afterwards.

A really good way to understand place value is to work with other number bases. How to Count Like a Martian is a detective story in which the history of other number systems plays a starring role.

 “Out of the depths of the dark and starry night come the first of the faint and mysterious sounds … At your radio telescope, you are expertly tuning the dials.” You have just received a message from Mars. “You know that this is not a message in words. Martians and Earthlings would have too much trouble trying to find the same words to succeed that way. But there is another kind of language that both Martians and Earthlings understand.”

Numbers… And so you research the number systems that have been used on Earth, hoping that will help you decipher this message. The book proceeds to explain eight different counting systems, including the abacus, and computers. 

In the process, the concepts of place value (she just calls it place), base, and zero are explored. By the end of the book, you can see that the beeps and bee-beeps of the message you received are just the counting numbers, Martian style.

How to Count Like a Martian was written in 1975, when there were still dials and tape recorders. those two items may be the only evidence of its age. I wonder if any young kids will like it as much as I do. Please let me know if your kid loves this book.

Wednesday, August 7, 2013

KenKen: A Simple Puzzle That Goes Deep

In the conclusions I wrote for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, I mentioned KenKen*. Our copy editor asked what it was. As I searched for a good reference to put in a footnote, a memory began to surface of an article I read long ago. Memory is funny. It turns out I had read the article over four years ago, and I still remembered a particular word from it - midnight. That led me to the article I had read, but it's unfortunately behind a paywall. Luckily, I had downloaded it the first time I read it, and finally found it on my own computer.

I had saved it back then because I was using KenKen with the kids at Wildcat Community FreeSchool. The puzzles I was sharing with the kids seem pretty easy, but to solve them requires holding addition facts in your head while also thinking logically about the relationships. This is a good way to deepen your hold on those facts. I wanted the parents to understand how valuable this simple puzzle was, so I copied the article.

'Midnight' was in the first paragraph, in the dramatic opening of the story written by Leo Lewis for The Times of London:
At one minute to midnight every September 30, the decrepit, cluttered schoolroom of Tetsuya Miyamoto stands frozen in time. Breaking the sepulchral silence of the Yokohama side street, the clock ticks over into the first day of October and a fax machine in the corner shudders to life. 

Throughout the rest of the night, page after page spews out of the machine, each one representing a different seven-year-old child, each one an application form pregnant with parental hopes and fears. 

The class these parents so desperately wanted their kids in consisted of puzzle-solving sessions. Tetsuya Miyamoto provided the KenKen Puzzles he had invented, and the children would then work alone for 40 minutes on up to three puzzles. The first is a 4 by 4 grid, the next is a harder 5 by 5 grid, and the third one is harder yet. After they've worked on the puzzles alone, the group works together on a puzzle Tetsuya Miyamoto puts on the board. He calls on a student for a number, then says right or wrong. That's it (according to Leo).

Before I go any further, I'd better share a KenKen puzzle with you.  Here's today's puzzle from the New York Times. If you like it, go there for more puzzles. Since this puzzle is 4 by 4, each row and column will have the numbers 1 to 4 in it. The clue on the middle top, 6X, means that the two numbers for that outlined box must multiply to 6. We know we can't use 1x6, because 6 won't be used in this puzzle. Is that enough to get you started?

Tetsuya Miyamoto designed his KenKen puzzles to draw students in and get them sweating:
Every puzzle, says Mr Miyamoto, contains a “trick, a discovery – a story”. The puzzle works in his classroom, he says, only because the children want to root out the clues and persevere with the discovery process. “As the feeling of achievement increases, so too does the level of concentration,” he says.
By combining the four main mathematical functions of addition, subtraction, multiplication and division, the brain is forced to dart between competing theories. The puzzle, he says, is impossible to solve without the scientific process of trial and error. 
The puzzle, Mr Miyamoto says, draws out the primal, self-starting learning instinct of human beings – an instinct that is notoriously suppressed by the fact-cramming teaching methods of the Japanese education system, but which he says needs to be encouraged in people of all backgrounds.

I think U.S. schools are headed toward that same sort of fact-cramming. It was never a good idea, but it seems clear to me that we need particular facts less than ever with the internet at our side. What we need are understandings of how it all fits together.

Mr Miyamoto’s theory is that the brain – of a child or adult – is failed by conventional teaching. By concentrating on a “third way” of problem-solving, he believes that the mind becomes a more potent tool for dealing with the rest of life...
I wish I knew what second way is implied here. I'm assuming fact-cramming is the first way.
For both children and adults, runs Mr Miyamoto’s theory, the brain feeds on what it has worked out for itself rather than what it has been told to focus on.
This important idea has been stated many ways by many excellent teachers. I am reminded of the quote the Kaplan's use to define their math circle philosophy:
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."    --G. C. Lichtenberg

There are other ways to make arithmetic challenging and appealing, some of which you'll find in Playing With Math. But KenKen is a particularly easy one to bring into your life. Enjoy!

*The name KenKen is trademarked. Because of this, you can also find these puzzles under other names. Calcudoku seems to be the most common.

Saturday, August 3, 2013

Self-Referential Puzzle, by Jack Webster

If you've been following our Facebook page, you know that Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is nearing the end of copy editing. We still need to add some artwork and get the pages laid out nicely. Then we'll be done.

Today I'm working on solutions to all of the puzzles in the book. I just spent two or three hours lost in the craziest puzzle. I had solved it a few years ago when I first saw it, but of course I don't have notes from back then.

If I hadn't solved it before, I'm not sure I would have believed I could do it. When you first look at it, it doesn't seem possible that the few clues Jack gives could be enough. They are. I have a page-long explanation of the logical steps I took to solve it. I wonder if it's possible to solve it in any other order.

If you like challenges, this is a good one. And if you like this, Jack has others on his site.

Sunday, July 21, 2013

Sixth Annual Math Circle Summer Teacher Training Institute

I've been to five out of the six, and this was the best one so far for me personally. One of the changes was to have the participants (who are learning how to run math circles) stick with a group of young people through the week. I facilitated the high school team. We had 8 adult participants, counting me, who ran math circles each afternoon for about 15 mostly high school age kids. Staying with one group of kids was great - we could really see the group of them coalescing over the course of the week.

One thing that was missing was the beauty and skill the Kaplans bring to this event each year. (There was a health problem, and rest needed to recuperate, so they couldn't make it.) I expect them to be back - as good as ever - next year, and hope to once again have the best time ever next July. 

We got to all participate in a big math circle together each morning.

On Monday, Sam Lichtenstein led us in an exploration of how much overhang you can get on a stack of dominoes. I had seen images of this sort of thing before, so I didn't start out hugely excited. But as we worked on it, my interest and excitement grew. I never felt like I had complete clarity on the physics involved, but we did come up with a mathematical solution that satisfied me.

On Tuesday, Nathan got us working on a puzzle that started out with a hydra on our lawn. You probably know that if you cut off a hydra's head, it will grow two heads in place of the one. Our lawn began at the bottom left corner of a chessboard with an infinite number of spots upward and to the right. The hydra was in that first spot, and when we chopped its head off, it grew two new ones - one to its right, and one just above it. We could chop off one of those and again get heads to the right and above that one. The other head was now blocked, though.

We used Othello boards to model the situation, and had lawns as shown. We got those two squares cleared and worked hard to clear the next diagonal. No one had yet done it when we gathered back together to discuss it. We went back and forth enthusiastically between trying to figure out how to do it, and trying to figure out a way to prove that we couldn't do it. Seems to me that's one sign of a good problem. A few of the high school age students attended morning sessions, and one of them wrote up a lovely solution to this problem - very clear and lively, with excellent illustrations.

On Wednesday, Amanda led us in exploring Rational Tangles. Many of us had played with this amazing puzzle/game before, but I think everyone loved having another chance to explore. You need four volunteers, two ropes, and a caller. The dancers start, each holding a rope end, with the ropes parallel. On "turn 'em round", the group rotates 90 degrees, so everyone is in the position where the person to their left had been standing. On "do si do", the person at the back left holds up their rope and comes forward, with the person at the front left going under and to the back. The ropes get all tangled up after a few random calls. And if you know the number secret, you can always get them untangled! The mathematical thought that goes with this is delightfully tangled.

On Thursday, Amanda led us through some interesting number puzzles. A number is onederful if it it composed of all ones in some (integer) base. 3 is onederful because it is represented as 11 in base two. Which numbers are onederful? Along with this exploration, we looked at divisibility rules: Why is a number divisible by 9 when its digits add up to a multiple of 9? What are the divisibility tests for 2, 3, 4, 5, and 6, and why do they work? 7 is the first number that has no really simple divisibility test. We came up with a great test together, by using modular arithmetic.

Suppose we have a number abc (where a, b, and c are the digits). Algebraically, our number = 100a+10b+c. Modular arithmetic is a way to pay attention only to the remainders. 100 is equivalent to 2 mod 7, since 100 = 7*14+2. (This is usually written using a three bar equal sign. I'll use a regular = with a 7 subscript for the rest of this post.) So 100a+10b+c =7 2a+3b+c. Let's look at just two digits. 10a+b =7 3a+b. If we multiply both sides by two, we change what we're looking at, but we don't change whether or not it's a multiple of 7. So now let's look at 6a+2b =7 -a+2b = -1*(a-2b). Once again, we can ignore the multiplier if we're just interested in whether or not we have a multiple of 7. One more step: Imagine a number with many digits, where the last digit is b. Use an A to represent the rest, so Ab means 10A+b. The same logic we used above applies, and if A-2b is a multiple of 7, so is the original number. (And if one is not a multiple of 7 the other also is not.)

So our test has us take off the last digit, multiply it by 2, and subtract it from the number formed by the remaining digits. Is 123456789 a multiple of 7? 12345678-18 = 12345660, 1234566 -2*0 = 1234566, 123456-12 = 123444, 12344-8 = 12336, 1233-12 = 1221, 122-2=120, which is not divisible by 7. That means 123456789 is not a multiple of 7. Wild! (Did I do that right? Please correct me if I messed up.)

I'm going to end this too long post here. Maybe later I can describe Friday's circle, and some of the evening entertainments.
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