Trigonometric Enrichment

Check your answers, Ben, I said.

He rolled his eyes and opened his math book with great labor. There was silence for a while and then some groaning and then exasperated complaints of "What!?" as he discovered there was more work to do.

He proceeded to redo his homework. I began preparing his enrichment -- a derivation of the sum-of-angles / difference-of-angles identities for sine and cosine.

It turns out that they just teach the trigonometric identities without showing where they come from. The kids just have to memorize them as self-evident truths. That's not too shocking, I suppose. I don't remember anything different from when I was that age. But those identities are so much less mysterious (and trigonometry so much less distasteful) when you learn that they really follow directly from the Pythagorean Theorem, the Distance Formula, you know: the square of the hypotenuse is ....

So I was drawing triangles and writing equations as he was reworking his problems. And I had about five pages of diagrams and equations and was just about ready when he looked over.

What are you doing? he asked.

Preparing a lesson, I said.

He groaned.

Don't worry, I said, it won't take long.

Somewhere on page four, he stopped me.

That's not the right identity, he said. That should be a minus sign.

It should? I asked.

Yes. The sine of a minus b is ... and you should have a minus sign there instead of a plus sign.

Are you sure? I asked.

Yes, he said. And he opened his book and showed me the identities printed on the inside cover.

So let's find the mistake, I said.

So we worked backwards line by line. Page 4. Page 3. Page 2. I stepped him back thru each line in the derivation. For the first time, I think he really began to understand why showing each step really is important. And then he found it.

That should be... and he pointed at my paper.

I looked down for a second and looked up at him and said, You're right. Go ahead and work the correction back down to page 4, and see if that fixes the mistake.

I walked off to answer the ringing telephone, and when I came back he was just about finished — three lines from the bottom.

Does it work? I asked.

Two lines from the bottom. Change a plus sign to a minus. One line from the bottom. Do it again. Final line, pull the minus out of the last term.

Yes! he said.

Are you sure? I asked.

He didn't know how to interpret that.

Um... he looked at it again. Yes, it fixes it.

You're right.

And I swear, as he was packing up his backpack a few minutes later, I heard him whistling the kind of whistle that he whistles only when he's proud of some piece of homework he's finally mastered -- and relieved to have finished.

10:34:42 PM   permalink: []   feedback:   comments: []  

Peter's Anthem

It was months ago, but being without a television and without much interest in baseball, I'm only hearing about it today.

Tens of thousands had come to Fenway Park to see the Red Sox play. (I understand it was a good year to live in Boston.) The game was about to begin.

There were players on the field, hats over their hearts. The fans were one their feet. Peter stood near home plate with a microphone before him and began to sing. It wasn't particularly good rendition of the national anthem, but it was Disabilities Awareness Day, and Peter has autism, and the fact that he was singing under the lights in front of thousands was very, very impressive.

Halfway thru... maybe he saw a face he knew, or maybe he started to forget the words -- he giggled. Loudly and directly into the microphone, he giggled. His laughter was so surprising and so genuine that the crowd erupted in laughter with him. And it really was with him: they clapped to encourage him and then cheered as he continued. "O'er the ramparts we watched..." He giggled again.

And then an amazing thing happened.

Whereas the crowd had been mostly silent up to then, with "so gallantly streaming," they began to sing. Faintly at first they backed him up. And then the crowd sang louder. At "the rockets red glare," the voices of the crowd filled the stadium so that although Peter was still singing (and giggling), he was but one of many. And as they reached "the land of the free," the fans began to shout. At the "home of the brave," the song was ended, the giggling didn't matter anymore, and everyone in Fenway Park was on their feet smiling and clapping and cheering — for Peter.

I was in tears.

When I sat down this evening, I had something different to say. Something with, shall we say, a different tone. But I think Peter's national anthem and the crowd at Fenway Park that night in late June is a better way to end the day.

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Watch the video.
A description.
More info (scroll waaay down to "A Very Special National Anthem at Fenway").

11:01:59 PM   permalink: []   feedback:   comments: []  

Losing Track of Time

It's good to see you. Where've you been.

I'm not quite sure. I kind of lost track of time. I remember I was in the woods in North Carolina and the leaves were turning. And I remember I was back here sitting outside on a balmy day with golden Ash leaves falling about me as a wind blew thru. (I was going to tell you about that.) And now fall is gone and it is cold outside.

Yes, it's been a while. Why don't you pick up where you left off in North Carolina.

Ok. Give me a chance to regather my thoughts.

Take your time ... or maybe I shouldn't have said that.

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Graveyard Fields II

The dark path led to some stairs that went down to the river and a wooden foot bridge across. And then we came to a set of steeper stairs to Second Falls. We could hear the rushing water from the top. At the bottom, we sat on the rocks for a while, watching the falls.

From the maps we had seen, it wasn't far to the Upper Falls. After climbing back up the stairs, we figured we didn't have far to go. The trail ran thru grassy moors and heather. It wound around bogs and pools and trickling streams where the Rhododendrons cast dark shadows along the banks. We passed blazing Maple trees and Huckleberry bushes with berries long gone.

And we passed five ladies hiking back from where we were going. They had determined looks in their eyes and were breathing hard. The last one said, Five miles!

Is that how much farther it is? I asked, for we were beginning to wonder.

No, but it seems like it, she said as she passed us.

We walked and we walked, and the path began to climb out of the lowlands. At each turn, we thought we heard the Upper Falls only to keep going. The ground got steeper, and we began picking our footsteps between rocks. On the ridge of a hill, we found ourselves in a Birch grove. The trees were growing out from between the massive boulders that were strewn across the hillside on either hand. Just beyond this grove, we finally found the Upper Falls.

We stopped, sat down on the rocks, drank our water, and ate some snacks. The sun began to show thru the clouds, and as we began the long hike back, we found that our shadows were stretching out before us.

The day had just begun, it seemed, but it was already getting late.

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Graveyard Fields
Blue Ridge Parkway, North Carolina

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Graveyard Fields I

Bob and his wife suggested that this was not a good day for the parkway. It was overcast and had been drizzling, and the forecast called for rain. Not a good day for a scenic drive in the mountains, they said. They must have been thinking of blue sky and clear vistas, but it seemed to us that fog on the mountains and driving above the clouds might be fine. So we decided to go.

Graveyard Fields at mile marker 418 snuck up on us. At first, it seems to be just (just!) another overlook, but the parking area was larger than most, and when you pulled in it was clear this place was the place we were looking for.

Below us was the source of the Yellowstone Prong of the Pigeon River. And across the valley, the hillsides were lit in yellow and orange and peach and brilliant red that blended as watercolors with the lifting mist.

The forest here was completely logged in the early 1900s, leaving bald mountain tops littered with stumps that evidently resembled gravestones and gave the place its name. Subsequent fires sterilized the soil, and only now, a century later, are trees coming back where once a dark forest of Spruce and Fir stood. The color on the far hillside was from Maples large and small.

Not far from where we stood surveying this scene was a trail that descended into a tall, dark thicket of Rhododendron and led to the river's edge. Although the sky was still overcast, the drizzle and rain were gone. The trail is why we had come.

We started down.

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Graveyard Fields
Blue Ridge Parkway, North Carolina

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