Merged: No more algebra?

[psionl0]

Originally Posted by Dinwar

Engineers don't like it when you ask "How do you correct for the fact that your walls are going to be at different angles, since they're on slightly different positions on the Earth's surface and you're using a plumb bob?"

Assuming a 100m wide building and an earth radius of curvature of 6350km, I calculate that the angle between the opposing walls (if plumbed) would be about 20 seconds of arc.

I doubt that this could be measured with any standard building tool.

[Careyp74]

Originally Posted by I Ratant

By measuring with a micrometer/plumb bob, marking the uprights with chalk, and cutting with an ax.
I was observing a new building going up near here.. only two stories, but the bare structure looked askew.. leaning to the north.
Had to be an optical illusion, because the windows fit!

You need new shoes. One wore out faster than the other.

[Dinwar]

Originally Posted by psionl0

Assuming a 100m wide building and an earth radius of curvature of 6350km, I calculate that the angle between the opposing walls (if plumbed) would be about 20 seconds of arc.

I doubt that this could be measured with any standard building tool.

Yeah, that's why no one worries about it--at the scales we build to, there's no need to worry about it (a mile-high building may be a different story, but we don't build those). Honestly, I just do it to see engineers who've never given the matter any thought start shooting smoke out of their ears.

Originally Posted by I Ratant

I was observing a new building going up near here.. only two stories, but the bare structure looked askew.. leaning to the north.
Had to be an optical illusion, because the windows fit!

I grew up in a house like that. Dad and I tried to put a door in once, and couldn't do it. We had to take a plain to it (well, we didn't HAVE to--Dad had a plain, and I had been a smart-mouth that week, otherwise we'd have used the power tools) to make it skewed just a bit. Fits perfectly now, and has the added advantage of holding itself in just the right position to be out of everyone's way, thanks to the slight incline.

The local hardware store once refused to sell us supplies until we brought in pictures of how screwy that house was. The owner thought we were lying to him...

[cwalner]

Originally Posted by Dinwar

Yeah, that's why no one worries about it--at the scales we build to, there's no need to worry about it (a mile-high building may be a different story, but we don't build those). Honestly, I just do it to see engineers who've never given the matter any thought start shooting smoke out of their ears.

Yup. That's why Euclidean geometry still dominates, because despite the technical inaccuracy for being on a (roughly) spherical surface, the calculations in Euclidean geometry are simpler, and the error is below the threshold of significance so it works just fine.

However, when you go large scale (say finding the shortest route to fly a plane from Chicago to Tokyo or calculating how to steer a spacecraft to Saturn) we need non-Euclidean geometries to get the best answers.

[I Ratant]

Originally Posted by Careyp74

You need new shoes. One wore out faster than the other.

.
I would have taken photos, as I do of everything, but this is the new police station, and photoing things like that are politically stupid now.

[psionl0]

Originally Posted by Dinwar

(a mile-high building may be a different story, but we don't build those).

If the aforementioned building was a mile high then the base would be 100m wide and the top would be 100.01m wide - a difference of 1cm. This is measurable but would make no difference to the number of bricks you lay at the top compared to those at the bottom.

Originally Posted by Dinwar

The local hardware store once refused to sell us supplies until we brought in pictures of how screwy that house was. The owner thought we were lying to him...

That sounds like a hardware store employee who doesn't get around much. This is hardly unusual at all.

At a house I once lived in the front door wouldn't even shut anymore. The reason was that the foundation pegs were sinking unevenly. The building manager had to use a truck jack to fix the problem.

[Ziggurat]

Originally Posted by AvalonXQ

I thought cab meters ran according to the amount of time it took, not the mileage driven. So if you're stuck in a cab in traffic or ask the cabbie to wait on you, she's not getting paid? That seems unfortunate.

Typically fare is assessed as a combination of time and mileage. Sitting in a stationary cab for 10 minutes isn't free, but it is cheaper than sitting in a driving cab for 10 minutes.

Estimates based on mileage use some assumptions about the time it takes. If they didn't have to make an assumption about the time involved, then they wouldn't have to give you an estimate, they could give you a quote.

[xtifr]

What I find really amusing about this whole thread is that when it was pointed out to OP the many common cases where he(?) uses algebra day-to-day, he basically responded, oh, that's obvious--that's not algebra. Wrong! Basic algebra is simple and obvious. More simple and obvious, IMO, than basic arithmetic. I use algebra to convert my tip calculations from the tricky n x 0.15 to the much easier-to-do-in-my-head (n + 1/2n) / 10. I know the result is the same because...algebra!

I think the problem with algebra is not that it's difficult (it's not) but that it's taught poorly. You're expected to master basic arithmetic first, which is silly, and then it's presented as if it were the next, even-more-difficult step. It's not. You could start learning algebra as soon as you start learning to add. And, in fact, to some extent, you do, but it's so obvious and simple that nobody bothers pointing it out at that point.

[Dinwar]

Originally Posted by psionl0

That sounds like a hardware store employee who doesn't get around much. This is hardly unusual at all.

At a house I once lived in the front door wouldn't even shut anymore. The reason was that the foundation pegs were sinking unevenly. The building manager had to use a truck jack to fix the problem.

Sorry, was mixing two anecdotes. The time the owner refused to sell supplies to us was when we gutted the bathroom. We started talking about the different things wrong with it, and the guy simply refused to believe that a room could be that messed up and still standing (which was part of the reason we were gutting it--it was sloping pretty badly).

My grandfather's done the same thing to chicken coops. Does horrible things to the wire he uses when he does that trick to make it square; after the first time you don't use it for anything else, and after the second time it's junk.

[psionl0]

Originally Posted by Dinwar

My grandfather's done the same thing to chicken coops. Does horrible things to the wire he uses when he does that trick to make it square; after the first time you don't use it for anything else, and after the second time it's junk.

Darn this curved earth's surface!

[Mark6]

Originally Posted by xtifr

What I find really amusing about this whole thread is that when it was pointed out to OP the many common cases where he(?) uses algebra day-to-day, he basically responded, oh, that's obvious--that's not algebra. Wrong! Basic algebra is simple and obvious. More simple and obvious, IMO, than basic arithmetic. I use algebra to convert my tip calculations from the tricky n x 0.15 to the much easier-to-do-in-my-head (n + 1/2n) / 10. I know the result is the same because...algebra!

I think the problem with algebra is not that it's difficult (it's not) but that it's taught poorly. You're expected to master basic arithmetic first, which is silly, and then it's presented as if it were the next, even-more-difficult step. It's not. You could start learning algebra as soon as you start learning to add. And, in fact, to some extent, you do, but it's so obvious and simple that nobody bothers pointing it out at that point.

Agree completely.

In Russian schools the concepts of "equation" and "unknown quantity" are introduced in first grade. Seven year olds learn that:

x + 4 = 6
x = 6 - 4
x = 2

That's a year before they learn multiplication table.

They do not call it "algebra" -- the word is not introduced until 5th or 6th grade, -- but they do learn it.

[Blue Mountain]

Originally Posted by AvalonXQ

It can take you 90 minutes to travel 20 miles through Chicago, and then another 90 minutes to travel the next 100 miles south of Chicago. Miles aren't the relevant unit.

There's a highway in Illinois with a posted limit of 110 MPH (180 km/h)?

[I've rounded the numbers a bit]

[Hazel]

Originally Posted by Mark6

Agree completely.

In Russian schools the concepts of "equation" and "unknown quantity" are introduced in first grade. Seven year olds learn that:

x + 4 = 6
x = 6 - 4
x = 2

That's a year before they learn multiplication table.

They do not call it "algebra" -- the word is not introduced until 5th or 6th grade, -- but they do learn it.

I've been telling friends this for years but they just look at me crazy. They find it hard to believe anything that seems so complicated could be so simple. I am not a math genius. I am not a success at advanced math at all but I had no trouble with algebra. I keep wondering if I slept through most of it when I hear so many saying they flunked it. Now, geometry did slow me down. I got the basics and never went further. Thank goodness for grammar, literature and history.

[Marras]

Originally Posted by Blue Mountain

[I've rounded the numbers a bit]

Perhaps a bit too much ...

[I Ratant]

In the late '50s, my aerodynamics professor Dr. Max Munk told us that the less capable in Germany were taught a binary arithmetic to do sums in the early 20th Century.
Mostly a number was divided by two until it was 1, 2, or 3. And the number of divisions were kept track of... if memory serves.

[xtifr]

Originally Posted by Blue Mountain

There's a highway in Illinois with a posted limit of 110 MPH (180 km/h)?

[I've rounded the numbers a bit]

More than rounded. If it takes you an hour and a half to travel 100 miles, you were not traveling 110 MPH!

I think your problem may have been forgetting that there's only 60 minutes in an hour.

[Hazel]

Originally Posted by Blue Mountain

There's a highway in Illinois with a posted limit of 110 MPH (180 km/h)?

[I've rounded the numbers a bit]

That's all right. The way people drive by here, I am quite certain it can be - and is being - done every day. Chicago is likely no different.

[AvalonXQ]

Originally Posted by Blue Mountain

There's a highway in Illinois with a posted limit of 110 MPH (180 km/h)?

[I've rounded the numbers a bit]

This is, perhaps, the best argument so far in this thread that math education is important.

Please show your work as to how you got 110 miles per hour from covering 100 miles in 90 minutes. (Clue: the correct answer is just over 65 mph).

[Hazel]

Originally Posted by AvalonXQ

This is, perhaps, the best argument so far in this thread that math education is important.

Please show your work as to how you got 110 miles per hour from covering 100 miles in 90 minutes. (Clue: the correct answer is just over 65 mph).

Or sticking with the term "miles" so people realize that miles are relevant.

[Rasmus]

Originally Posted by AvalonXQ

(Clue: the correct answer is just over 65 mph).

Just by looking at it, you should understand that 110 miles in 90 minutes is "just a little something over a mile a minute".

I am posting this because I am not sure just how much actual math I needed for that, and how much more for the realisation how much that "little something" would actually be.

[Hazel]

Originally Posted by Rasmus

Just by looking at it, you should understand that 110 miles in 90 minutes is "just a little something over a mile a minute".

I am posting this because I am not sure just how much actual math I needed for that, and how much more for the realisation how much that "little something" would actually be.

True. Doesn't the fact that you think distances in miles almost automatically alert you to the right or wrong of a statement? People who have no habitual concept of what miles are lose that advantage.

I think!

[AvalonXQ]

Originally Posted by Rasmus

Just by looking at it, you should understand that 110 miles in 90 minutes is "just a little something over a mile a minute".

I am posting this because I am not sure just how much actual math I needed for that, and how much more for the realisation how much that "little something" would actually be.

Agreed. Furthermore, you don't have to do the calculations to figure out that if you were travelling 110 mph, you'd cover 110 miles in the first hour -- so you're covering well over 100 miles in 90 minutes.

[AvalonXQ]

Originally Posted by Hazel

True. Doesn't the fact that you think distances in miles almost automatically alert you to the right or wrong of a statement? People who have no habitual concept of what miles are lose that advantage.

Not really. 100 furlongs in 90 minutes should still eyeball at "a little over a furlong a minute."

Where the intuition comes in is whether or not a furlong (or mile) per minute is a reasonable or an excessive speed.

[Dinwar]

Originally Posted by Hazel

People who have no habitual concept of what miles are lose that advantage.

They also lose the ability to easily plan around city driving. If all you knew was the distance through Chicago and you weren't familiar with the city you'd grossly underestimate the time it would take to get through the city (I've actually done it myself). If they tell you the time it takes to get through the city, however, you'll have a much more accurate view of the situation.

[Almo]

Originally Posted by AvalonXQ

I thought cab meters ran according to the amount of time it took, not the mileage driven. So if you're stuck in a cab in traffic or ask the cabbie to wait on you, she's not getting paid? That seems unfortunate.

If, in fact, the cabs are paid according to to the amount of time it takes, then estimating based on miles, especially in a heavy-traffic city, seems like a bad idea.

ETA: I see Zig got this part already, but I'll leave it anyway.

Cab meters read BOTH mileage and time. So if you're sitting still, it goes up a certain amount per minute. But if you're going fast, it goes up a certain amount per mile. One of these rates will be higher than the other depending on your speed, and the meter takes the higher of the two rates.

On topic, algebra and geometry are most certainly things everyone should learn, even if they can't pass a test on it 20 years later. These things help you learn how to think about abstract concepts in a structured manner, and that is a useful skill.

I don't understand this desire to teach as little as possible in such a competitive world.

[Hazel]

Originally Posted by Almo

ETA: I see Zig got this part already, but I'll leave it anyway.

Cab meters read BOTH mileage and time. So if you're sitting still, it goes up a certain amount per minute. But if you're going fast, it goes up a certain amount per mile. One of these rates will be higher than the other depending on your speed, and the meter takes the higher of the two rates.

On topic, algebra and geometry are most certainly things everyone should learn, even if they can't pass a test on it 20 years later. These things help you learn how to think about abstract concepts in a structured manner, and that is a useful skill.

I don't understand this desire to teach as little as possible in such a competitive world.

Thank you for meter explanation. I also don't understand this notion of teaching less. Well, I fear I do understand why they are doing it but I don't think it a very smart idea and I suspect I'm in good company there.

[Blue Mountain]

Originally Posted by xtifr

More than rounded. If it takes you an hour and a half to travel 100 miles, you were not traveling 110 MPH!

I think your problem may have been forgetting that there's only 60 minutes in an hour.

FACEPALM! Boy, am I embarrassed. Although, I kinda miss those old 90-minute hours ...

Sorry, there's no excuse such a silly slip-up rather than a bran fart of some sort. I wasn't drunk, stoned, or tired.

[Rasmus]

Originally Posted by Hazel

True. Doesn't the fact that you think distances in miles almost automatically alert you to the right or wrong of a statement?

Avalon already explained why the answer here is "probably not". Let me add that I don't think in miles, and 110 miles an hour seems like a perfectly reasonable driving speed.

Of course, *then* you'd have to remember that you're never going to average that over a journey and 65 miles is actually a very decent average, which makes me wonder just how someone would manage that between any two points in the USA...

Quote:

People who have no habitual concept of what miles are lose that advantage.

I think!

I know that common speed limits are around 60 or 70 miles or thereabouts.

[I Ratant]

On hundred mile or so freeway trips, I can average 80 mph... along with the rest of the traffic.
Phenomenal gas mileage at that rate, also!

[xtifr]

Originally Posted by Blue Mountain

FACEPALM! Boy, am I embarrassed. Although, I kinda miss those old 90-minute hours ...

Sorry, there's no excuse such a silly slip-up rather than a bran fart of some sort. I wasn't drunk, stoned, or tired.

Ok, now that the guy who made an elementary mathematical error has acknowledged it, maybe we can stop the derail about that error and get back to the topic of algebra. (And yes, I admit I participated in the derail--I'm not assigning blame here; merely observing that we are derailed.)

As I argued earlier, basic algebra is very simple. In my opinion, simpler than arithmetic. Much of it it is so simple that people use it all the time without even realizing it. At the same time, I am forced to admit that some people find the level of abstraction represented by algebra to be confusing or off-putting. My mother and one of her sisters felt as I did: math class got a whole lot easier when we reached algebra. On the other hand, her other sister had no problems with arithmetic, but felt like she hit a brick wall when she reached algebra. I don't fully understand it, but the three sisters never managed to resolve the issue, so clearly there really is something about trying to abstract mathematics to that level that some people have a hard time with.

Now I know my other aunt uses algebra a lot. She just doesn't seem to realize it. It's ok when you have specific instances in front of you that resolve to, and look like, pure arithmetic, but when you lay out the pure abstractions, she balks. If anyone has any suggestions for getting past that hump, I'd love to hear it. It's probably a little late for my aunt, but it might help others.

[Hazel]

Originally Posted by I Ratant

On hundred mile or so freeway trips, I can average 80 mph... along with the rest of the traffic.
Phenomenal gas mileage at that rate, also!

"Average"? Which means you are sometimes going far faster than 80? Heaven help us!

[I Ratant]

The 80 average is just life-preserving!
I hate being run over from behind!
There's people in a hurry out there on the freeways!
Middle lane, watch the mirrors...

[superfreddy]

Originally Posted by Blue Mountain

FACEPALM! Boy, am I embarrassed. Although, I kinda miss those old 90-minute hours ...

Sorry, there's no excuse such a silly slip-up rather than a bran fart of some sort. I wasn't drunk, stoned, or tired.

Yeah, I heard bran can give you gas...

[Careyp74]

I just had a flashback to uses of algebra in every day life. Someone mentioned sets earlier. I believe that if it weren't for the teaching of the concept, I wouldn't have the skeptical mind I have today. I think this applies to my wife too. She took the same courses I did in getting her Masters Degree, which included Logic, Statistics, and Algebra.

Now, this isn't really algebra, but the concept transfers. There was a discussion about the use of phrases like "up to 40% off or more." One of those commercials came on the other day and my wife criticized it. This got me thinking.

Many people believe that this means you will either save 40%, or you will save a percentage that is more. However, looking at the set of all percentages included in that statement, we find that 39%, all the way down to 0% also fit into the statement.

With our complete 'set' understood, we can evaluate the statement more exactly. This process isn't obvious, talk to other people that aren't as critical of a thinker as yourself and you will see.

Sure you can teach this without going into in depth algebra, but they don't do that. There are probably other tools picked up this way also. Lastly, going further into a subject helps solidify the simpler concepts in your mind, which is why I never had a problem with having to learn everything that was taught.

[cwalner]

Originally Posted by Careyp74

I just had a flashback to uses of algebra in every day life. Someone mentioned sets earlier. I believe that if it weren't for the teaching of the concept, I wouldn't have the skeptical mind I have today. I think this applies to my wife too. She took the same courses I did in getting her Masters Degree, which included Logic, Statistics, and Algebra.

Now, this isn't really algebra, but the concept transfers. There was a discussion about the use of phrases like "up to 40% off or more." One of those commercials came on the other day and my wife criticized it. This got me thinking.

Many people believe that this means you will either save 40%, or you will save a percentage that is more. However, looking at the set of all percentages included in that statement, we find that 39%, all the way down to 0% also fit into the statement.

With our complete 'set' understood, we can evaluate the statement more exactly. This process isn't obvious, talk to other people that aren't as critical of a thinker as yourself and you will see.

Sure you can teach this without going into in depth algebra, but they don't do that. There are probably other tools picked up this way also. Lastly, going further into a subject helps solidify the simpler concepts in your mind, which is why I never had a problem with having to learn everything that was taught.

That's more statistics than algebra, but statistics are often used in very misleading ways by people with agendas (political, commercial or ideological).

Some of my favorites common abuses:

'Our drug is clinically shown to be twice as effective as their drug'
(it works 2% of the time instead of 1%)

Misuse of 'average'. Is it Mean, Median or Mode that you are referring to? And remember, the 'average' human has 1 ovary and 1 testicle.

[Careyp74]

Originally Posted by cwalner

That's more statistics than algebra, but statistics are often used in very misleading ways by people with agendas (political, commercial or ideological).

well, there isn't really any statistics involved in my example, it is more about what the expectation of the consumer is, and the disappointment when he finds that he doesn't save anything at all, because he doesn't realize that the set of values in the statement "up to 40% or more" also includes 0.

[Modified]

Originally Posted by Careyp74

well, there isn't really any statistics involved in my example, it is more about what the expectation of the consumer is, and the disappointment when he finds that he doesn't save anything at all, because he doesn't realize that the set of values in the statement "up to 40% or more" also includes 0.

It also doesn't make much sense. If it's up to 50%, you should say "up to 50%". The "or more" is just an attempt to mislead.

[cwalner]

Originally Posted by Careyp74

well, there isn't really any statistics involved in my example, it is more about what the expectation of the consumer is, and the disappointment when he finds that he doesn't save anything at all, because he doesn't realize that the set of values in the statement "up to 40% or more" also includes 0.

It just reminded me of another misleading use of percents that I consider within statistics.

this would be an example. Bob has 25% more M&M's than Amy, and Carol has 20% fewer M&M's than Bob. Who has more M&M's: Amy or Carol?

ETA: This can actually be solved easily using Algebra to bring it back to the OT.

[Careyp74]

Originally Posted by Modified

It also doesn't make much sense. If it's up to 50%, you should say "up to 50%". The "or more" is just an attempt to mislead.

That is marketing for you. One good reason for critical thinking.

Originally Posted by cwalner

It just reminded me of another misleading use of percents that I consider within statistics.

this would be an example. Bob has 25% more M&M's than Amy, and Carol has 20% fewer M&M's than Bob. Who has more M&M's: Amy or Carol?

ETA: This can actually be solved easily using Algebra to bring it back to the OT.

I am glad you realized that.

[Minoosh]

Is algebra necessary?
http://www.nytimes.com/2012/07/29/op...anted=all&_r=0

I came across this article recently and found it vaguely disturbing, even though as a short-lived high school algebra teacher I sometimes wondered myself if the near-universal requirement for freshman algebra did more harm than good. Where I taught, the normal freshman algebra flunk rate was 50 percent. My state had recently upped the high school math requirement to four years, and in practice students got algebra twice, as freshmen and juniors; they did geometry in 10th grade and the required fourth year was "financial math" or "consumer math," although calculus and or precalculus were also available.

My first, 29-year-long career was in journalism, and I actually used math a lot. Most of that time was as an editor and I saw very intelligent "word people" mangle statistics and struggle with the concept of area (!) Example, a reporter writing about a billboard ordinance said the size of boards would be cut in half. She didn't realize that halving each dimensions would cut the permitted area by three-quarters. A fellow editor, extremely sharp, told me he had wanted to become a geologist, but that he "couldn't do fractions." Really?

This may be more appropriate for the education subforum, but my question is broader than what public education policy should be. It has to do with the (IMO) potentially crippling effects of innumeracy. My hunch is that math phobia has become so ingrained in the U.S. - over two or three generations - that adults unwittingly carry the message that math is a bogeyman, a semi-sadistic struggle we inflict on children "for their own good," and that the expectation students will fail becomes a self-fulfilling prophecy.

A clue: When I earned teacher certification in my late 40s, the bright young "kids" in my college cohort panicked somewhat at having to prove they could do middle-school math. This wasn't even algebra, for the most part. More like decimals, fractions and percentages. Area or volume, maybe. The Pythagorean theorem.

The author seems to think that if we focused on statistics kids would see the relevance and therefore become more engaged, and this may be true, but I don't see how you get past the fact that math learning is largely sequential, sometimes abstract and mechanical, and that skimping on the foundations can put students at a disadvantage in ways they may not appreciate at age 14.

I now work as a paraprofessional at a small charter school which introduces algebraic concepts in third grade. "Find the missing addend." Most kids seem to have no problem with that. The text doesn't call this "algebra," and it doesn't tell students they're "solving for n," but that's what they're doing.

At this school, 7th graders are using the text my old school used in 9th grade. And for the most part they succeed. Both schools are in low-income neighborhoods. I don't think my school "cherry picks" kids; it just warns parents or guardians that the accelerated curriculum might not be the best fit for some students. Tutors are liberally applied where needed, and there is a schoolwide culture of discipline that IMO reduces distractions quite a bit.

And, things get followed up on.

I hadn't meant to write all that, I just suspect the author of this article has things backwards and I can't quite put my finger on why. Maybe it's his assumption that algebra is confusing - but why should it be? Yes, kids have to learn that x does not mean "multiply," but freshman algebra is usually pretty straightforward - you make $6 an hour, the motorcycle costs $300, how many hours do you have to work to buy the motorcycle? How do you say that in words, how do you say it in numbers and symbols? Why should that be considered confusing, arcane, irrelevant? Is that too much for kids to handle? If so, does it say anything about future U.S. competitiveness?