the tennis ball problem

A tennis ball can with radius r holds a certain number of tennis balls also with the same radius. The amount of space in the tennis ball can that is not occupied by the tennis balls equals at most the volume of one tennis ball. How many tennis balls does the can hold?

Barry sent me this problem months ago & I've been avoiding it because geometry scares me.

I finally shamed myself into attempting it just now & got an answer of 2. Unfortunately, I typed up my solution, loaded it to flickr, but flickr is on the blink so I can't post.

I solved it (assuming I did solve it) algebraically, then resorted to "logic and reasoning" to check.

Unfortunately, I'm confused by logic and reasoning at the moment.

I was thinking that because a sphere is 2/3 of a cylinder of same radius, with each ball you put inside a same-radius cylinder you end up with 1/3 of a ball's worth of empty space....which now implies to me that the answer should be 3 balls, not 2.

sigh


update (1):

I'm mixing things up

The 1/3 that's left over isn't 1/3 of a tennis ball. It's 1/3 of a cylinder with the same radius as the tennis ball.

I better forget the logic and reasoning & stick to algebra.

Assuming I didn't screw up the algebra, that is.


update (2):

OK, so in between dealing with screaming autistic youths, loading the dishwasher, & microwaving a taco for Jimmy, I realized that I don't need to know "how much of a tennis ball-sized volume is left over."

I just need to know how much empty volume is left over, period, then figure out how many multiples of that empty space add up to the volume of 1 tennis ball.

volume of tennis ball with radius r: 4/3πr^3
height of cylinder that fits just one tennis ball: 2r
volume of cylinder w/height of 2r: πr^2h = 2πr^3

vol. of cylinder - vol. of 1 tennis ball = vol. of empty space

2πr^3 - 4/3πr^3 = 2/3πr^3 empty space left over when 1 tennis ball is in cylinder

2 tennis balls leaves 2 empty spaces, each 2/3πr^3 in volume:

2/3πr^3 + 2/3πr^3 = 4/3πr^3, which is the volume of 1 tennis ball

so: 2 tennis balls

update (3):

Barry says the problem comes from Dolciani's Algebra 2! (I include the exclamation point because I'm happy to discover I am able to solve a problem from that book. cool.)

original wording:

A tennis ball can in the shape of a cylinder with a flat top and bottom of the same radius as the tennis balls is designed so that the space inside the can that is NOT occupied by the balls has volume at most equal to the volume of one ball. What is the largest nubmer of balls the can will contain?

wild goose chases

Found the quote -- it's from what is apparently a classic book on Mathematical Problem Solving by Alan Schoenfeld.

The Wild Goose Chase in Problem Solving

When solving standard mathematics problems, students normally recall and apply learned procedures in a straightforward way. However, if the problem is unfamiliar, some students simply pick a method and keep persistently on the same track for a long time without getting anywhere. Schoenfeld (1987) described this behaviour as chasing the wild mathematical goose.

That's not the meaning I remember.

I probably got it wrong.

knock on wood
true confession
wild goose chase

true confession

I found the wild-goose-chase problem, the one that took me many steps not to do while C. solved it at once using "logic and reasoning."

Find the value of r so the line that passes through each pair of points has the given slope.

44. (-2, 7), (r, 3), m = 4/3 ^*

C.'s solution:
7 - 3 = 4/3 (-2 - r)
4 = (-8 - 4r)/3
12 = -8 - 4r
20 = -4r
r = -5

My non-solution:
y = 4/3x + b
7 = 4/3(-2) + b
7 = -8/3 + b
21/3 + 8/3 = b
29/3 = b

I seem simply to have stopped at this point. I don't know why. Looking at my solution page now is a bit like examining a single-vehicle accident scene trying to determine how the driver flipped his car twice in broad daylight & nice weather.

Alternatively, I didn't stop with 29/3 = b (I remember having done more steps) but the rest of my perambulations are recorded on some other piece of paper, not this one.

We'll never know.

knock on wood
true confession
wild goose chase


^* source: Glencoe Algebra, p. 261

Elmore on the politics of education reform

"Hence the knowledge that is enacted in curriculum and pedagogy becomes a
byproduct of the political incentives that operate on teachers-discrete bits of information, emphasis on coverage rather than depth, diffuse and hard-to-understand expectations for student learning, little convergence between the hard day-to-day decisions about what to teach and the largely content-free tests used to assess student performance, and a view of pedagogy as a function of the personal tastes and aptitudes of teachers rather than as a function of external professional norms. Students who do well in such a system recognize that they are being judged largely on their command of the rules of the game, which reward aptitude rather than sustained effort in the pursuit of clear expectations. All systems have a code; the job of the student is to break it. Some do, some don't."

Richard Elmore
The Politics of Education Reform

knock on wood

C. is having a good math year.

I've hesitated to say so; I think all of us, the teacher included, are wondering whether this is real. But I think we're past the point at which it makes sense to wonder whether this is a fluke.

The year started badly - dreadfully, in fact - with a D- on the first test.

Things picked up a bit when I pitched in with multiple-houred sessions of parent reteaching, but I figured I was looking at another school year of emergency preteaching, reteaching, and all the rest of the fol de rol.

I wasn't.

Those days are behind us (knock on wood!) I don't preteach, I do very little reteaching, and -- this is the interesting part -- when I do need to reteach, C. picks up the concept rapidly.

This vacation I'm doing a fair amount of reteaching because C. missed 4 days of the last 9 full days of school. He could go in for Extra Help and ask his teacher to reteach the material, but since I want him to return to school after break having completed all 4 of the missed homework assignments, I also need to teach the lessons. In other words, this is something I want to do, not something I absolutely have to do.

Going into the break, I was a bit worried. I had found the homework assignments challenging myself. Somehow the point-slope formula was mixing me up, and I'd never done a word problem involving linear functions.

So I had a time of it figuring the concepts out myself. How was C going to handle it?

He handled it fine.

Yesterday he correctly and efficiently did a point-slope problem I had hosed entirely, then explained his success by saying, "I used logic and reasoning."

Yeah, well, I used logic and reason, too, but I went on a wild goose chase instead of seeing the simple and should-have-been obvious solution. ^* C. looked at the problem and saw the solution immediately.

C. also missed the classroom lesson on finding the equation of a line parallel to another line. That seemed like a fairly big concept to me.

No problem. He instantly saw that a parallel line would have to have the same slope in order to be parallel -- and that it would have to have a different y-intercept if it wasn't going to be the same line. Looking at a line on the coordinate plane while thinking all this over, he had one of his "Oh, yeah" moments. After that he knew how to find the equation without further instruction.

We owe all of this to C's teachers. He has two math teachers this year because 8th graders have "Math Lab" 3 days a week. Both are terrific as far as I can tell. They're also experienced. The main teacher has been at the school for close to 10 years (I think); I'm not sure how long the other teacher has been here but I do know that he taught in NYC schools prior to coming to Irvington. He may also be in the 10-year category or close to.

They've done a fantastic job.


Math A archived exams
Math A Regents prep

knock on wood
true confession
wild goose chase


^* A few years back I read a study that distinguished between great math students and OK math students that I probably can't find again. The difference between the two groups was that the great students produced efficient and elegant solutions while the OK math students went on wild goose chases. They'd solve the problem, but it wasn't pretty.

Mmm... Tasty Brains...

Just what we've been searching for - a scientific explanation for all of the recipe posts at KTM: cooking may have been a significant driver of human evolution. On the other hand, this theory seems only to deepen the mystery of how some tribes managed to flourish without ever having learned to cook.

I tend to think of the advent of cooking as having a huge impact on the quality of the diet. In fact, I can't think of any increase in the quality of diet in the history of life that is bigger. And repeatedly we have evidence in biology of increases in dietary quality affecting bodies. The food was softer, easier to eat, with a higher density of calories—so this led to smaller guts, and, since the food was providing more energy, we see more evidence of energy use by the body. There's only one time it could have happened on that basis; that is, with the evolution of Homo erectus somewhere between 1.6 [million] and 1.8 million years ago. [/SNIP]

...Homo erectus is the species that has the biggest drop in tooth size in human evolution, from the previous species, which in that case was Homo habilis. There wasn't any drop in tooth size as large as that at any later point in human evolution. We don't know exactly about the gut, but the normal argument is that if you reconstruct the ribs, you have reduced flaring of the ribs. Up until this point you have ribs that went out to apparently hold a big belly, which is what chimps and gorillas are like, and then at this point [when Homo erectus arose] the ribs go flat, meaning you've got now a flatter belly and, therefore, smaller guts. And then you have more energy being used; people interpret the locomotor skeleton as meaning that the distances traveled every day are much farther. And the brain has one of its larger rises in size.

*I originally intended to title this post with a pithy reference to Brillat-Savarin, but George Romero is really much more my style.

Jeanne d'Arc

We went to a party on the Jeanne d'Arc helicopter carrier last night.

Men in uniform.

Woo hoo!

today's factoid 2

...somewhere between 97 and 98 percent of American voters have never looked at a blog in their lives...
source:
Foggy Bloggum by David Frum

Does that make us outliers?

in Pakistan

John Moore, photographer, describes the day.

I'm going to show this to C.

Siouxsie and the Banshees - Il Est Né le Divin Enfant

I'd never heard of Siouxsie, but I like this version....

Il est né le divin enfant

Benazir Bhutto, rip

Here's Hitchens:

The sternest critic of Benazir Bhutto would not have been able to deny that she possessed an extraordinary degree of physical courage.

long division in the time of computers

"with apologies to Gabriel Garcia Marquez" (pdf file)

"When we assert that “this is the factorization of a number into primes,” the Fundamental Theorem of Arithmetic is lurking in the background."

One of these days I will be a person who knows what that means.

[pause]

Well, I have now skimmed the entire pdf file and I have no idea what it's talking about, or what the author's views on the place of long division in the curriculum is or is not.

So that was enlightening.


the long version (pdf file)

from anonymous...

Thanks for reminding me why we homeschool.


chuckle

edu-jargon

EDUCATIONAL JARGON GENERATOR

I'm very amused by this.

Educational Jargon Generator.

Singapore Math v. Everyday Math -- You Be The Critic

I recently had a fantastic discussion with an administrator at my local public middle school. This person is new to the Town and school. My impression is that she is less pedagogically dogmatic than most I have met. She does not have an direct involvement with EM. In Middle School she see the pre-algebra and algebra math of 7th and 8th grade. She hears from many other teachers and administrators that EM is fantastic and wonderful and perfect for our school. However, she, and most other educators in town notice that this "wonderful" elementary math program isn't connecting well to middle school and high school. Kids aren't doing all that great on those high school courses.

The reigning wisdom has been that the problem can't be EM. It must be the middle school math program (with are traditional pre-algebra and algebra courses).

There is a failure to analyze their underlying assumptions. Nobody is willing to consider that EM might not be the best preparation for advancing in math.

But this administrator has shown an interest. I gave her my opinion on the matter at a forum a couple weeks ago and she was interested in what I had to say. I advised that she put EM and Singapore Math side by side and compare to cut through all the rhetoric. She could make up her own mind about it. I finished by saying if there was one thing I could convince this district to do, it would be to teach bar models as a means of problem solving.

She had just read an article about bar models. She wants to know more.

So here is your chance, everyone. If you had limited time available with an interested administrator, and you had 1st through 6th grade Singapore and Everyday Math at your disposal -- where would you start? What pages, links, sections would you highlight or focus on?

Chocolate Pecan Pie & Fractions

I have been baking chocolate pecan pie almost every Christmas for over 20 years. Tonight my 5th grade daughter helped me with this recipe:

CHOCOLATE PECAN PIE
1 pie shell, unbaked
Filling:
1/4 cup (1/2 stick) unsalted butter
2 ounces unsweetened chocolate
3 large eggs
1 cup sugar
3/4 cup dark corn syrup or sugar cane syrup
1/2 teaspoon pure vanilla extract
3 tablespoons bourbon or rum (optional)
1/4 teaspoon salt
1 1/2 cups pecan halves
Preheat the oven to 350 degrees F.
To make the filling: melt the butter and chocolate in a small saucepan over medium-low heat, remove from heat and let cool. Beat the eggs in a large mixing bowl until frothy and then blend in the sugar. Stir in the syrup, vanilla, bourbon, salt, and the melted butter mixture until well blended.
Arrange the pecans on the bottom of the pie crust and carefully pour the egg mixture over them. Bake until the filling is set and slightly puffed, about 45-50 minutes. Test for doneness by sticking a thin knife in the center of the pie, if it comes out pretty clean, you're good to go. Transfer the pie to rack and cool completely before cutting.

We made two pies because we’re having 15 guests for dinner on Christmas Day. My daughter instantly converted all the fractions to the quantities needed for two pies. She was faster than I was. It may not seem like such a great achievement to some, but to me it was wonderful.

Thank you, Kumon!

where does tofu come from?

Kitchen Chemistry at MIT

MIT physics lectures on the web

Walter H. G. Lewin, 71, a physics professor, has long had a cult following at M.I.T. And he has now emerged as an international Internet guru, thanks to the global classroom the institute created to spread knowledge through cyberspace.

Professor Lewin’s videotaped physics lectures, free online on the OpenCourseWare of the Massachusetts Institute of Technology, have won him devotees across the country and beyond who stuff his e-mail in-box with praise.

“Through your inspiring video lectures i have managed to see just how BEAUTIFUL Physics is, both astounding and simple,” a 17-year-old from India e-mailed recently.

Steve Boigon, 62, a florist from San Diego, wrote, “I walk with a new spring in my step and I look at life through physics-colored eyes.”

Professor Lewin delivers his lectures with the panache of Julia Child bringing French cooking to amateurs and the zany theatricality of YouTube’s greatest hits. He is part of a new generation of academic stars who hold forth in cyberspace on their college Web sites and even, without charge, on iTunes U, which went up in May on Apple’s iTunes Store.

At 71, Physics Professor is a Web Star

Or, if physics doesn't interest you, you can go audit the fancy-shmancy Yale course on death.


Chronicle of Higher Education on Yale online courses
Yale Offers Free Online Courses
Open Learning Initiative at Carnegie Mellon
MIT Open Courseware

the key to life

beautiful

show your kids

Intercepts reports that Will Smith and his wife homeschool their children.