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There's also an example here of dimensional analysis on SeekingAGoodPhysicsText.

-- GoogleMaster - 20 Feb 2006

Saxon is the only standard math book I've ever known to explicitly teach dimensional analysis.

-- CarolynJohnston - 21 Feb 2006

Whenever Ben does unit conversion problems, he wants to guess the conversion factor without going through all the steps.

Whenever he guesses, he gets it backward. No exception. 1.5 millimeters is 15 centimeters; that sort of thing.

Whenever he applies the unit multiplier, he gets it correct. No exception.

For months I've been pointing this out, but he always resists.

-- CarolynJohnston - 21 Feb 2006

When you think Chris is ready for it, you might want to try some goofier dimensional analysis problems. Perhaps, "What is the speed of light expressed in furlongs per fortnight?", to suggest a classic.

Or, "If Peter Piper picked a peck of pickled peppers, and peppers are $1.39/kg, how much are Peter's peppers worth? (The density of Peter's peppers is 0.92 grams/cubic centimeter.*)"

* Density chosen arbitrarily, and not necessarily related to the actual density of freshly picked or pickled peppers. Actual density left as an exercise for the student.

-- DougSundseth - 21 Feb 2006

Carolyn, does Ben know what a millimeter and a centimeter are? That is, does he have any frame of reference that would let him know that 1.5 tiny things cannot be the same as 15 bigger things?

Maybe the answer doesn't look ridiculous to him because millimeter and centimeter are just meaningless words to him.

Would he do the same if the problem were in feet and miles?

-- GoogleMaster - 21 Feb 2006

Whenever he guesses, he gets it backward. No exception. 1.5 millimeters is 15 centimeters; that sort of thing.

Whenever he applies the unit multiplier, he gets it correct. No exception.

This is what absolutely stumps me.

I see the same thing....CHRISTOPHER DIVIDING 75 BY 3!

In 'real life' if you asked him how far we'd get in 3 hours going 60 miles an hour, he'd get the answer.

On a practice test, kabloowey.

The unit multipliers cause kids to get the right answer — BUT THEY ALSO CONSTANTLY ILLUSTRATE AND TEACH THE CONCEPT.

Unit multipliers are like VOCAs for autistic kid; they're an 'assistive device' that also teaches.

-- CatherineJohnson - 21 Feb 2006

The metric system has MAJOR, BIG-TIME WICKELGRENIAN FLAWS.

Everything looks the same.

That's supposed to be the Big Advantage.

-- CatherineJohnson - 21 Feb 2006

Doug

COOL!

That's a great idea!

-- CatherineJohnson - 21 Feb 2006

One sticking point can be "What do you do if you have to invert something like the 3 hours?" We know that

         3                           1
     3 = -  and   the inverse of 3 = -
         1                           3

               3 hours                       
 but 3 hours = -------   and the inverse = ???        
               1 what?  

    75 miles/1 hour(s)    75 miles         1 
    ------------------ =  --------  * ----------- 
        3 hours             1 hour     3  hours

                       =  25 mile per 3 (hour*hour  ) 

                        1 hour
  inverse of 3 hours =  ------     ?
                        3 hours 

                         3 hours
   inverse of 3 hours =  -------   ? 
                         3 hours

My old professor suggested using something "made up that makes sense" to fill the blank and avoid this confusion. In this problem, "trip" works. That is,

    it's   3 hours      and we want to know  ?? miles
           -------                           --------
           1 trip                            1 trip

  ?? miles   75 miles/1 hour(s)   75 miles   1 trip
  -------- = ------------------ = -------- * ------
   1 trip      3 hours/1 trip     1 hour(s)  3 hours

While the correct setup looks like this:

     ?? miles     3 hour(s)   75 miles
     --------  =  --------- * -------- 
      1 trip       1 trip     1 hour(s)

-- OldGrouch - 21 Feb 2006

Would he do the same if the problem were in feet and miles?

Good question. My guess is that he would. I believe it happens because he is trying to get by without applying any thought at all, rather than that he has no frame of reference for mm vs. cm, but it's worth checking into, that's for sure! I'll report back.

-- CarolynJohnston - 21 Feb 2006

a-and when f= "number of feet"
and y = "number of yards" (say),
there's this near universal tendency to want
to believe that y = 3f
(evidently because "one yard = three feet")
rather than f = 3y
("there are three times as many feet as yards"),
which is actually true.

some writers call this as "the student-professor problem"
(because of some famous problem along the lines
"there's a single professor for every 30 students
on a certain campus. write an equation relating
s (# students) and p (# profs).").
but then, they're all student-professor problems ....

why is "dimensional analysis" rarely treated in texts?
—an important question in my opinion.
there comes a point where everybody
just seems to want to assume that it's obvious ...
but nobody seems to want to take time
from their own overcrowed schedule
to explain it ... maybe (he remarks cynically)
because it resists "spiralling"?

-- VlorbikDotCom - 21 Feb 2006

report of 1st experiment

I asked Ben to show me with his fingers about how big a centimeter is. He did so, correctly.

Then I asked him to show me with his fingers about how big a millimeter is. He did so, correctly.

Then I asked him to show me with his fingers about how big a mile is. He looked at me, dumbfounded, and said "I can't do that!"

"Good," I said. "You pass the test."

I tend to think what's going on is exactly what V said:

a-and when f= "number of feet"
and y = "number of yards" (say),
there's this near universal tendency to want
to believe that y = 3f

-- CarolynJohnston - 21 Feb 2006

The metric system has MAJOR, BIG-TIME WICKELGRENIAN FLAWS.

Everything looks the same.

That's supposed to be the Big Advantage.

How's it a Wickelgrenian flaw?

(I grew up on the metric system and can't tell you off the top of my head how many feet are in a mile).

As for the millimeter/centimeter thing - perhaps have Ben work out a lot of problems with a ruler?

E.g. get him to draw a line that is 1.5 cm, and then draw a line that is 15 mm?

I do metric conversions by a ruler in my head. Which is interesting when I'm working in gigawatts and megawatts, but that's the advantage of rulers in your head. They can get very big.

-- TracyW - 21 Feb 2006

Tracy

I have no idea whether the metric system has 'Wickelgrenian flaws' for a person who grew up with it.

For a person coming to it, after years with the customary system (is that what it's called), it's a WICKELGRENIAN NIGHTMARE!!!

-- CatherineJohnson - 21 Feb 2006

V

I never thought of it that way.

(y = 3f)

-- CatherineJohnson - 21 Feb 2006

I'm very curious about this question.....

I'm convinced that unit multipliers are a procedure that teaches.

(This goes back to Andrew's Dynamo. A Dynamo isn't like a wheelchair, although it's classified as assistive technology. A Dynamo assists a handicapped person, but it also teaches. A wheelchair doesn't teach a person to walk.)

I think unit multipliers teach.

When you see it all laid out like this.....you learn.

I'm very curious about the origins of the decision not to include unit multipliers — and I'm curious about why John Saxon did include them.

-- CatherineJohnson - 21 Feb 2006

For a person coming to it, after years with the customary system (is that what it's called),

It's the imperial system. Because it was British and used by the British empire, as compared to the metric system introduced by Napolean.

-- TracyW - 21 Feb 2006

I'm very curious about the origins of the decision not to include unit multipliers — and I'm curious about why John Saxon did include them.

My guess--and, of course, that's all it is--is the link to science. Saxon's scope went beyond math to physics, didn't it? Any time spent with units of force, acceleration, velocity, etc. make clear the value of dimensional analysis. The conversions from grams to moles and back in chemistry are where I was first taught dimensional analysis.

As to why it isn't taught more...my guess goes back to the general math and science phobia of educationists. People drawn to elementary education are self-selected away from comfort with math and science. To really be sold on the value of dimensional analysis, someone has to encounter problems that are a little trickier than D = R * T.

-- DanK - 22 Feb 2006

To really be sold on the value of dimensional analysis, someone has to encounter problems that are a little trickier than D = R * T.

It would help enormously with problems like D=RT. Divide or multiply? Kids (and many adults) find this very confusing and hard to remember.

-- CarolynJohnston - 22 Feb 2006

Linking to earlier discussion on this topic:

http://www.kitchentablemath.net/twiki/bin/view/Kitchen/RussianMathProblem624

-- SusanJ - 22 Feb 2006

I finally looked it up...it's called 'Customary' system here.

I think we should start calling it the Imperial system!

-- CatherineJohnson - 22 Feb 2006

Dan

My guess--and, of course, that's all it is--is the link to science. Saxon's scope went beyond math to physics, didn't it?

You know - that makes sense.

Yes, he wrote a physics text (which I'm contemplating buying, since Carolyn's contact at Saxon says it's out of print).

Of course, that makes it even weirder that a curriculum like TRAILBLAZERS doesn't teach it, since the entire point of TRAILBLAZERS is to teach math integrated with science. (It was originally called TIMS, standing for something like 'integrated math & science.')

ALL of TRAILBLAZERS is 'data collection.'

But no unit multipliers.

-- CatherineJohnson - 22 Feb 2006

To really be sold on the value of dimensional analysis, someone has to encounter problems that are a little trickier than D = R * T.

Boy, I don't know.

I don't know how much farther I got than D = R * T and I love these things.

Though I must say, I was aghast the first time you brought them up.

The very phrase - 'dimensional analysis' - sounded complex, arcane, and BEYOND THE PALE.....

-- CatherineJohnson - 22 Feb 2006

"I finally looked it up...it's called 'Customary' system here. I think we should start calling it the Imperial system!"

While I usually use the term "Imperial", there are actually differences between US "Customary" and British, et al. "Imperial" systems. See, for instance, "Quart":

US liquid quart: 57.75 in.³

US dry quart: 67.201 in.³

British imperial quart: 69.355 in.³

Let's just say that there are plenty of reasons to use metric. (Though there are two standards there, too: CGS and MKS. At least they're easier to convert back and forth.)

-- DougSundseth - 22 Feb 2006

If you're a beer drinker, you want to use the Imperial system rather than the US system. In the US, you'll only get 16 ounces in your pint, but an Imperial pint will give you 20 ounces.

-- GoogleMaster - 22 Feb 2006

Convert inches per second to furlongs per fortnight.

I remember this question from one of my classes.

I have several little booklets (8 1/2" by 3 1/2") on hundreds of conversion factors. Things like:

To convert "horsepower-hours" to "joules", you have to multiply by 2.684 x 10^6.

-- SteveH - 23 Feb 2006

an Imperial pint will give you 20 ounces

That works!

-- CatherineJohnson - 23 Feb 2006

To convert "horsepower-hours" to "joules", you have to multiply by 2.684 x 10^6

I love it!

Christopher ripped through his unit multiplier problems last night (he did some 2 ft squared to square inches ones.....I have NEVER been able to do those!)

He's also doing reasonably OK on proportion.

Thank God for state tests.

He's got his Glencoe book of test prep, and these are the first word problems he's done all year. (He's done one or two, but never an extended batch. Ever.)

-- CatherineJohnson - 23 Feb 2006