Hi Andy!

yup, that's what we got

you can switch the 9 and the 3 with each other, and the 5 and the 6.

which reminds me, I should probably have Christopher do that, to show that he knows there are a couple of other solutions.....

It's a ludicrous problem.

You do have to notice that there are 'shared' digits in each equation (that C appears in two equations & so on), and there's some reasoning as to the larger numbers (closer to 9) versus the smaller....

And that's it.

Given how desperately in need of practicing to mastery he is at this point, this is appalling.

We're going to be hounding the school district about Teaching To Mastery from now on.

In fact, I've already started. I copied the head of the school board on my Interim Report email today, and said that we have two critical issues facing us:

1. inadequate communications with parents
2. a policy of teaching to coverage instead of teaching to mastery

-- CatherineJohnson - 03 Jan 2006

"Also, couldn't the answers for A & B and H & I be switched and still be correct?"

Yeah, because the constraints aren't tight enough. It's a pretty stupid problem/

-- VerghisKoshi - 03 Jan 2006

I just had Ed write out every conceivable value for A, B, etc.

Looks like Letter F, or something, can have only one value.

IIRC, and I probably don't.

-- CatherineJohnson - 03 Jan 2006

We got answers of 1 and -1 as answers to #7, and we both feel we must be missing something.....

-- CatherineJohnson - 03 Jan 2006

The sum of the digits from 1 to 9 is 45. The sum of four thirteens is 52. This means that the numbers that are used twice, e.g. C, E, and G have to add up to 52 - 45 = 7. The only 3 different digits that add to 7 are 1, 2, and 4. You can't use either 1 or 2 for E because then you would have to use 10 for one of the other numbers which isn't allowed.

Once you discover this constraint, the problem is straightforward. It wasn't me, however, but another family member who finaly realized this.

-- KtmGuest - 03 Jan 2006

The sum of the digits from 1 to 9 is 45. The sum of four thirteens is 52. This means that the numbers that are used twice, e.g. C, E, and G have to add up to 52 - 45 = 7. The only 3 different digits that add to 7 are 1, 2, and 4. You can't use either 1 or 2 for E because then you would have to use 10 for one of the other numbers which isn't allowed.

We definitely didn't get this fancy.

The best reasoning we could come up with was.....I may be too tired to remember.

It was something about how you weren't going to be able to use 7, 8, or 9 as the pivotal numbers; something like that.

-- CatherineJohnson - 03 Jan 2006

wow

I'm going to have to study your answer.

Of course, this is a case where I'll get far more out of studying the manner in which someone else solves the problem than I would doing it myself.

-- CatherineJohnson - 03 Jan 2006

Maybe a stupid question - why are you spending so much time on these questions?

From what you say, the teacher isn't checking them, and with the exception of the first they don't illustrate any useful mathematics. They're fun to do if you like those sorts of things, but from the sound of things Chris isn't at the moment.

Why not just give Chris a fixed time limit, and anything he gets done in 1/2 an hour or whatever gets handed in?

-- TracyW - 03 Jan 2006

I couldn't get my anwswers to look right in plain HTML or MathML^?, so I just gave up and typed it in LaTeX^?. :)

Oh, and BTW, riding in a plane is, mathematically, not too different from riding in a boat. Specifically, it's like riding in a speed boat. Even more specifically, a hydroplane.

The main difference is that water is dense enough and provides enough buoyancy that ships don't have to go really fast and be special shapes to stay afloat.

-- PaulMiller - 03 Jan 2006

"We got answers of 1 and -1 as answers to #7, and we both feel we must be missing something....."

No, I don't think so. Given that a is an integer, and that 1/a is also an integer, 'a' must be either 1 or -1.

Infinity is a possibility, but is it an integer? And not in 6th-grade. Might throw it in to frag Mrs. Kahl, though. And what about 'i', the square root of -1? It ain't an integer in the normal sense of the word, but then it ain't a fraction either. Shouldn't we celebrate diversity by including it? :-)

-- VerghisKoshi - 03 Jan 2006

Being literal here (choose the point on the autism number line), but there is no possible way that "the sum of A+B+C, C+D+E, E+F+G, and G+H+I is equal to 13."

Because "the sum of A+B+C, C+D+E, E+F+G, and G+H+I" to me means (A+B+C)+(C+D+E)+(E+F+G)+(G+H+I), which is equal to A+B+C+D+E+F+G+H+I+C+E+G. But we are told that "each letter represents a different digit from 1-9", so that means that sum(A thru I) = sum(1 thru 9) = 45. There's no way that 45+C+E+G can equal 13, if C, E, and G are all positive integers.

That's the point at which I stopped working on it and scrolled down to see what answers other people got.

People who write math problems in English ought to be able to write the freakin' language properly.

-- GoogleMaster - 03 Jan 2006

The fourths problem, on the other hand, is the kind I like.

-- GoogleMaster - 03 Jan 2006

Being literal here (choose the point on the autism number line), but there is no possible way that "the sum of A+B+C, C+D+E, E+F+G, and G+H+I is equal to 13."

no kidding

Ed and I both had to get over that hurdle

-- CatherineJohnson - 03 Jan 2006

From what you say, the teacher isn't checking them, and with the exception of the first they don't illustrate any useful mathematics.

You raise an interesting point.

She checks these problems.

Each one is worth 10 points.

It's HUGE.

The regular homework, the Mere Practice, is not checked, and is worth 2 points.

-- CatherineJohnson - 03 Jan 2006

These 3 problems may be worth 10 points altogether, I don't know.

I just know that the figure of TEN POINTS is attached.

-- CatherineJohnson - 03 Jan 2006

Paul

Oh, and BTW, riding in a plane is, mathematically, not too different from riding in a boat. Specifically, it's like riding in a speed boat. Even more specifically, a hydroplane.

Just as I suspected!

I take it this doesn't an autistic 11-year old can knock a plane out of the air.

-- CatherineJohnson - 03 Jan 2006

Just a thought . . .

Perhaps you could present Question No. 8 to the superintendent (or principal) and see what answer he gets. Be sure to ask him to show his work. A friend of mine used this technique with her daughter's principal and it was very effective.

-- KtmGuest - 03 Jan 2006

A friend of mine used this technique with her daughter's principal and it was very effective.

OOOH! I really like that idea.

-- CarolynJohnston - 03 Jan 2006

Regarding math:

I thought 7 and 9 were perhaps useful to check understanding of concepts. ("Integer" and "reciprocal" in 7, and division by fractions and conversion of mixed numbers to fractions in 9.)

I disliked 8 until I read Guest's solution; afterwards it looks like a clever puzzle. This says nothing about its value as a skill diagnostic, which I think to be really low.

Regarding flying:

I think word you want is "microburst" rather than "microshift". A microburst is a phenomenon associated with thunderstorms (usually) where you get sudden severe downdrafts.

Also, when you're above 30000', a 200' variance in altitude is pretty insignificant, except insofar as it causes you to leave your lunch behind. Frankly, when you're more than 1000 feet above the terrain, 200' bumps are just annoying.

(Before you think I'm minimizing the problem, I get motion sick, and have flown through "moderate to severe turbulence" in a Cessna 172. I was, ummm, annoyed. So were the other people in the plane; even those who didn't try to get a refund on breakfast.)

-- DougSundseth - 03 Jan 2006

She checks these problems.

Each one is worth 10 points.

I'd inquire into this. Is it ten points for having a go at the homework, or is it ten points for getting the homework right? (I've had marking schedules for both sorts in my time at school and uni.)

If it's the second, you have more ammo - students should not have their grades dependent on getting homework like this right unless they've had opportunities to ask their teacher for help.

-- TracyW - 03 Jan 2006

Yes, this does mean an autistic 11-year-old can't knock the plane out of the air by jumping around... unless he is flying it.

-- PaulMiller - 04 Jan 2006

Also, when you're above 30000', a 200' variance in altitude is pretty insignificant, except insofar as it causes you to leave your lunch behind. Frankly, when you're more than 1000 feet above the terrain, 200' bumps are just annoying.

One of my more thoughtful friends today told me that he'd heard of unseat-belted people getting plastered against the roof of the plane and badly hurt or killed. I don't see why the plane would fall any faster than the person unless it were driven into a dive.

Please discuss and comment.

I am well aware that once you're high over the terrain there is a ton of room, and bumps are no big deal. Still, I dislike 0-G or negative G a very great deal. As I've said before, FOF is beyond the capabilities of every a very rational mind to control; it's something that goes on in the reptile brain.

-- CarolynJohnston - 04 Jan 2006

"I don't see why the plane would fall any faster than the person unless it were driven into a dive."

Good point, but if a puff of air forced you down (a "microburst", no less), the aeroplane would be going down faster than you. "Driven into a drive", sure, but not by the pilot.

What you're talking about is free-fall, don't necessarily apply here.

-- VerghisKoshi - 04 Jan 2006

I want to know what the physics of microbursts are. Is a puff of air in fact forcing you into an accelerated dive?

Usually the problem with a drop is that a tailwind suddenly comes up and your relative airspeed drops precipitously, in which case you go into a freefall.

-- CarolynJohnston - 04 Jan 2006

Right. What we call "G-force" is really just a measure of acceleration. Gravity causes an accelleration of about 32 ft/s^2 toward the center of the earth, which keeps your feet on the ground, and, in a plane, your butt in the seat.

To experience weightlessness, the plane has to be diving at 1g, i.e. 32 ft/s^2. Any more than that, and you start experiencing negative g's. A 200' dive at 1g would take roughly 3 seconds (actually a teeny bit more), assuming the plane starts from level flight, which is pretty gentle.

Calculating what kind of acceleration you'd need to actually hit the roof of the cabin with your head in a standing position is a little beyond me, since I was always bad at physics. :)

-- PaulMiller - 04 Jan 2006

"Usually the problem with a drop is that a tailwind suddenly comes up and your relative airspeed drops precipitously, in which case you go into a freefall."

You mean a stall? Possible, but since jets fly at about 600 mph, stall at 150 mph, it's have to be quite a tailwind.

Paul's right about how long it would take to fall 200 feet (actually 3.53 seconds), but I don't think that's gentle. Think of falling off the roof of a 200-foot building.

To hit the roof, assuming a generous 4 feet from your head to the roof of the plane, and a time of 3.53 seconds, the plane should be going down at 32.65 ft/sec/sec.

But that's academic. What's more interesting is the speed at which you face meets your breakfast :-)

-- VerghisKoshi - 04 Jan 2006

Ouch, right, not gentle at all, considering you're going approximately 75mph in the downard direction at the end of the 3 sec. I made some kind of mistake calculating that before... somehow I got into my head that it was 45 mph, which would be pretty gentle.

OTOH, 32.65 ft/sec^2 compares well with what I was getting, assuming the cabin ceiling was 4 feet from your head (probably pretty realistic for a sitting position).

Let's see... yeah, you could get caught in a 450 mph tail wind... on Jupiter. ;)

-- PaulMiller - 04 Jan 2006

Vertical wind shear and turbulence:

Turbulence in aircraft occurs when the air the airplane is flying through is changing its vertical direction. What you feel in the airplane will depend on the frequency of the wind-speed change and the speed of the airplane. In a jet at cruising speed, high-frequency (small area) turbulence is often described as "chop". The airplane flies through the rising and falling air rapidly enough that it feels like bouncing. Longer-period turbulence, or a slower-flying airplane, causes the aircraft to rise or fall for longer times, and feels like swooping. For me, at least, it's the swooping turbulence that's the problem.

Turbulence is caused by several things:

In the summer (mostly), you can get columns of rising air that are caused by hot spots on the ground (newly ploughed fields, for instance). The air that rises has to come down somewhere, so it will fall at relatively cooler spots on the surface. At the cruising altitude of a jet, this isn't likely to be significant, but it can be very (very) noticeable in a light airplane.

If you have a lot of air rising in a somewhat coordinated fashion in this way, you can set up a sustained rotating system. As the air rises, it cools, then falls. When it cools, if it cools below the dew-point, you can get clouds, and with enough energy, you can get thunderstorms. Thunderstorms create a very strong vertical rotation, with falling air in the core of the storm and rising air in the surrounding area. All aircraft, other than some of the maniacs at NOAA and some military storm aircraft, stay away from thunderstorms, but you can still get pretty intense turbulence in thunderstorm weather.

In some weather conditions, the air at different altitudes can be moving at different speeds. (You can sometimes see this in the summer, when different layers of clouds are moving at different speeds.) When this happens, you can get turbulence at the boundary layer between different-speed streams of air. At the altitudes where jets cruise, you can get near or into the "jet stream", and the boundary layer turbulence near that can be pretty intense.

Another cause of vertical wind-direction changes is mountain waves. The classic sign of a mountain wave is a stationary band of clouds on a windy day downwind of the mountains (called "lenticular clouds"). They can be really nice for gliders and sailplanes, but not so much for powered aircraft.

(There can be other causes, but these dominate the air conditions for aircraft.)

If you are descending through the rising air to maintain altitude, then get a sudden change in air direction, you can drop pretty quickly. And it is possible (though very uncommon) to pull negative g's when this happens.

Horizontal wind shear:

Carolyn mentioned horizontal wind shear, which is generally more dangerous. Many of the things that can cause vertical wind shear can also cause horizontal wind shear.

Now, as Paul mentioned, when you're flying at 600 mph, a change in wind speed is not normally significant. But landings and takeoffs are made in the 140-200 mph range, where horizontal wind-speed changes can be very significant. If you have a headwind or tailwind, your ground speed (speed relative to the ground) and air speed (speed relative to the air you are flying through at that moment) can be quite different.

The ability of the airplane to fly is based on its air speed; below the "stall speed" of an airplane, its wings stop producing much of their lift. (Note that in an airplane a stall is a wing effect, it has nothing to do with the plane's engines.) If the wind changes velocity (speed or direction) dramatically, your air speed can drop even though your ground speed is unchanged or increasing. If you are close to the stall speed, as you must be when you are landing and taking off, your air speed can drop below the airplane's stall speed. If you are flying close to the ground when this happens, you can be in trouble.

Fortunately, we understand this condition much better now than we did even a few years ago, and can predict it well.

The upshot:

Turbulence is only somewhat predictable and sometimes unavoidable, but it's just uncomfortable, not dangerous. You would do well to wear your seatbelt when flying, though. Horizontal windshear is dangerous when landing and taking off, but predictable and avoidable.

-- DougSundseth - 04 Jan 2006

My real fear when turbulence starts is that THIS is going to be the day when metal fatigue finally has its way with the wings of the 35-year-old 737 I am riding in. I am afraid that I am going to look out the window and see that the left wing has shaken itself right off of the airplane. I know a plane can make it home on one engine -- but one wing?

I told you it wasn't rational.

-- CarolynJohnston - 05 Jan 2006

Did you ever see the Twilight Zone episode where the guy sees the creature on the wing trying to rip off the engine cover? He was on his way home after being released from a mental institution and it was a dark and stormy night. He closes the window cover. Finally, his curiosity gets the best of him and he opens it to come face to face with the creature. The last scene shows him being carted away back to the mental institution.

I was 10 and had a morning paper route at the time. It was dark and I remember expecting the creature to peer out at me from every dark window.

-- SteveH - 05 Jan 2006

Did you ever see the Twilight Zone episode where the guy sees the creature on the wing trying to rip off the engine cover?

Maybe I did... and that's the source of my problem.

-- CarolynJohnston - 05 Jan 2006

"I told you it wasn't rational."

I (think I) understand. Aerodynamics, for most people anyway, is a black box, though. Even if the above didn't help you, I hope it helps someone else whose fear is more based on ignorance than the mysterious operations of the subconscious.

Plus I think it's interesting, so I inflicted it on all of you.

-- DougSundseth - 05 Jan 2006

I'm positive I'm going to feel MUCH BETTER about TURBULENCE as soon as I read this thread.

-- CatherineJohnson - 09 Jan 2006

oh my gosh!!!

I just found the attachment!!

-- CatherineJohnson - 21 Feb 2006

• ktm.pdf:

Attachment	Action	Size	Date	Who	Comment
ktm.pdf	manage	50.5 K	03 Jan 2006 - 04:54	PaulMiller