You make a cylinder by curving the paper around and joining it; where you join it, (i.e., how much overlap in the join) however, tells you what the diameter of the circle is. So assuming that anyone could even get that far with the problem, kids have to assume that the two ends of the paper are joined with no overlap.

-- BarryGarelick - 14 Jun 2006

I wonder if she's pulling these questions out of the air just before giving the test. By asking for a ratio, you can do the calculation without specifying the value of pi (the term falls out), but I got an answer that's a repeating decimal:

Ratio = (222.75/pi)/(272.25/pi) = .818181... 

The interesting problem would have been to use a piece of paper with one side twice the length of the other. ( Why it's interesting is left as an exercise, blah blah...)

-- OldGrouch - 14 Jun 2006

Duh, the ratio is the same as the ratio of the page dimensions.

-- OldGrouch - 14 Jun 2006

OldGrouch^?, you must have used a calculator. ;-)

I got 9/11 by canceling out a pi(9)(11)/2 from numerator and denominator.

Of course, that leads to the next pitfall -- is she expecting an exact fractional answer like 9/11 or a repeating decimal answer like 0.8181... or a rounded decimal answer like 0.82?

Monte Hall says I'm more likely to win if I switch my answer. :-)

-- GoogleMaster - 14 Jun 2006

good lord

repeating decimals

and yes, the overlap business is a problem

-- CatherineJohnson - 14 Jun 2006

Well no, I solved it in steps (find R, then V) rather than setting up one grand formula. :-P

This would have been a fun thing to "discover" in a classroom demonstration (try some specific values, then "What can we say about the general case?") Maybe that's what she actually did yesterday, but if so it obviously went right past Christopher-- another :-P

-- OldGrouch - 14 Jun 2006

The result is exactly 11/9 (or 9/11; either ratio satisfies the problem as stated) when you reduce the fraction fully. It's a nice clean number in base 11 (or base 9, depending). (It's also the ratio of the height and width of the paper, of course.)

The problem with this, ummm, problem is that there are too many intermediate steps for students that are just learning the material. None of the steps is particularly difficult, but there are lots of places to make mistakes. And any mistake will keep you from seeing the rather elegant answer.

Bonus (Trick) Question: Assuming a perfectly butted rectangular piece of paper, what are the dimensions that will give the largest volume while still maintaining the same area (99 square inches)?

-- DougSundseth - 14 Jun 2006

Maybe that's what she actually did yesterday, but if so it obviously went right past Christopher--

I don't think so. He had a really good, clear set of notes.

She taught them that super-simple problem making a ratio between the cylinder with a radius of 1 and a second cylinder with a radius of 2.

-- CatherineJohnson - 14 Jun 2006

I'm now lost myself.

sigh

-- CatherineJohnson - 14 Jun 2006

I better go do some algebra

-- CatherineJohnson - 14 Jun 2006

and yes, definitely (I think definitely....) this could be a good inductive lesson in class

of course, that would be teaching

-- CatherineJohnson - 14 Jun 2006

I was talking to my neighbor about whether the universe of Things a 6th Grader Doesn't Know About Math is or is not a finite set.

Saxon says some people say a finite set is a set you could count if you lived long enough.

If you could count all the items in a set in half a million years, say, that's finite.

So the challenge for her last year, and for us at the end of this year, now that Ms. K has reverted to form, is to select the specific things Christopher does not know about area & volume of prisms & cylinders Ms. K will put on the test.

How big is that set?

-- CatherineJohnson - 14 Jun 2006

Christopher's year in algebra has such a familiar ring to it. I've talked to a couple of parents who have decided to hold their high math kids back from going on to Algebra 2.

One parent has tutored her daughter practically every night. She said she had enough when she realized that her daughter had no idea what volume was and it was the end of the year. She felt they were just cramming formulas in right and left and her daughter had lost many gains from earlier years. This girl has always been a top notch student and is now pretty devastated.

Another friend just had to make the same decision for her boy.

-- SusanS - 15 Jun 2006

I've had many test questions over the years that were designed to see if students could extrapolate their knowledge to new problem variations. The problem is that this is very difficult for a teacher to do well. What seems like a great idea (simple extrapolation) for the teacher may have very mixed results for the kids. The results may not correlate with how the kids did in the rest of the class.

The worst problems are those where everything depends on seeing the trick or key angle or idea. Figure out the trick and you get the problem correct; miss the trick, and you don't get any partial credit. Since these are usually larger problems, there are fewer of them on a test, and therefore, each one has a greater impact on the test grade. Larger extrapolations are really not fair within the limits of a stressful timed test.

Usually, problem variations are kept rather small, rather than jump off into problem domains that the students have never seen before. Teachers have to be very careful about the tests they prepare. Unfortunately, many are not. Teachers should keep in mind that test scores are also a reflection of their teaching ability.

-- SteveH - 15 Jun 2006

Teachers should keep in mind that test scores are also a reflection of their teaching ability.

-- SteveH^? - 15 Jun 2006

ITA! It would be very helpful if a teacher send home a sheet with every graded test that listed the scores for the entire class.

It would say something like: June 9 Test Results ..

Percentage of the class scoring: 95-100% correct - 5% 85-94% correct - 25%

etc...

I suppose this type of analysis would help the teacher identify a problem in his or her teaching or a problem with the student.

-- NicksMama - 15 Jun 2006

Teachers should keep in mind that test scores are also a reflection of their teaching ability.

-- SteveH^? - 15 Jun 2006

ITA! It would be very helpful if a teacher send home a sheet with every graded test that listed the scores for the entire class.

It would say something like: June 9 Test Results ..

Percentage of the class scoring: 95-100% correct - 5% 85-94% correct - 25%

etc...

I suppose this type of analysis would help the teacher identify a problem in his or her teaching or a problem with the student.

-- NicksMama - 15 Jun 2006

Steve's comment really nailed it as far as the two cylinder problem.

Ignoring the possibility that Ms K is just plain nuts, my guess is that she doesn't understand math herself.

fantastic comments!

Ms. K did send home class averages usually - Christopher was usually below the class average, but not always - but she never, ever gave the spread (is spread the word I want?)

Definitely, in all 3 of her classes, there are a couple of kids who score up towards 100....so we can't tell what Christopher's grades mean.

I have to say, though, that towards the end here even the class brains were dropping into the Bs.

-- CatherineJohnson - 17 Jun 2006

Christopher has said several times that he's in a cluster of boys considered not-that-smart.

We always ask who's smart, wanting to know how many kids are in that group. It seems as if there are maybe 3 or 4 kids in a class of 17 who are seen by all as smart.

Also, Christopher seems clearly not to be at the bottom of the class.

As far as we can tell, it's the standard 3-part class, which 2 or 3 teachers have now told me all classes have, no matter how they're put together. Top, middle, bottom.

Christopher seems to be in the middle.

Hope so, anyway.

-- CatherineJohnson - 17 Jun 2006

Doug I worked through the problem you suggested, and was having trouble figuring out why it the formula I got was linear when I realized that it doesn't care what the area of the end "caps" are.

So volume = 99w/4pi, where w is the width of the paper. In other words, max volume is that of the infinitesmally thick cylinder with infinite (well, arbitrarily close to infinite) radius.

The question that is more interesting, but probably just as unsuitable, is "what is the largest possible volume enclosed by a cyliner with a surface area of 99 square inches?"

-- AndyLange - 17 Jun 2006

Anybody else not seeing this post on the WebHome page?

-- OldGrouch - 17 Jun 2006

Anybody else not seeing this post on the WebHome page?

-- OldGrouch - 17 Jun 2006

uh-oh - is it still gone?

I'm going to check....

-- CatherineJohnson - 21 Jun 2006

heck

I don't see it

-- CatherineJohnson - 21 Jun 2006