kitchen table math, the sequel: Dimensional Analysis

Sunday, May 24, 2009

Dimensional Analysis

We've talked about this subject in the past, but I thought I would bring it up again.

While working with my son, I've been trying to get him to look at units carefully, but unfortunately, I don't get a lot of help from his Glencoe Algebra I textbook ("units" is not even in the index). It's not as if they think that units are unimportant, but they don't go into details. For most problems, they take care of themselves if you think about it a little bit, so my son doesn't like it when I try to get more formal.

For example, if they talk about 60 miles per hour, I try to get him to think of it as

60 * 1 mile / 1 hour

where the units can stand alone and are multiplied and divided just like regular numbers. I don't want him thinking that there is a units called "miles per hour". I want him to treat units like numbers, and manipulate them like numbers. The base units are "miles" and "hours"; distance and time.

This came up when we were doing some more complicated D=RT problems. When problems are simple, units are simple. When problems get complicated, then units can cause your brain to explode. I told my son that units are your friends. They can tell you if you are trying to add apples with oranges.

Therefore, I've been pushing the idea that units are just like factors that are multiplied and divided and squared. If you calculate the area of a rectangle, you might multiply 5 ft times 7 ft. I want him to look at it like this:

5*(1 ft) * 7*(1 ft)

Where the units are separate and can be manipulated like numbers. I told him that after a while, he can think of "1 ft" as simply "ft". Since everything is multiplied together, you could think of it as:

5*7 * ft*ft


35 *(1 ft)^2

By now, he is annoyed and wondering why this formality is necessary. I explain that if he keeps the units with the numbers at all times and multiplies and divides them just like the numbers, the units will tell him if he is making a mistake. This is hard to justify with easy problems where the units are obvious, but you don't want to wait until the problems get difficult to add in these concepts of units.

I also explained to him that if you talk about something like moment: weight times distance, you can get "ft-lbs" for the units. I told him that the dash is commonly used, but it really means multiply. If you want the moment generated by a 60 lb. child 5 ft out at the end of a seesaw, then you would calculate moment by multiplying the two together:

60*(1 lb) * 5*(1 ft.)

which would give you

60*5 * lb *ft


300 ft-lb

But it really should be thought of as


Also, if you look at units as things you multiply and divide, then it's easy to see how to convert units using identities. If you want to convert 10 feet per second to miles per hour, you start with:

10 * ft/sec

I can multiply this by 1 and not change it (1*a = a). I also know that

3600 * sec = 1 * hr

I can rewrite this as:

(3600 *sec) / (1*hr) = 1

Or, I could invert it to get:

(1*hr) /(3600 *sec) = 1

I can then use this as my "1" to multiply the 10 ft/sec. So:

10 ft./sec * (3600 sec)/(1 hr)

To get:

10*3600 * (ft * sec) / (sec*hr)

Since the units are all multiplied and divided, I can "cancel out" the sec/sec because anything divided by itself is just 1, and then get:

36000 ft/hr

This kind of formality might seem unnecessarily difficult because we are only half way there, but once you practice it a bit, it becomes easy. It is also very necessary when the problems and units get much more complex.

For example, how about the mass unit called a slug?

1 slug = 1 lbf*sec^2/ft

Where lbf is pounds(force) or a weight. If you don't know how to formally deal with units, you will be lost.


Kai said...

I have the same conversations with my son. He gets vastly annoyed when I start doing math on the units and even more annoyed when I make *him* do it. I don't understand why; I think it's kind of fun. I keep telling him that if the units come out right there's a better chance that he has done the problem correctly. Then he rolls his eyes and informs me that he had gotten the right answer a *long* time ago.

SteveH said...

It's easy to understand when the problems are simple. When you eventually have to deal with physics problems that use gravity, lbs-mass, and lbs-force, it won't be complicated by units.

Jo Anne C said...

The math instruction I received never covered the concept of treating units as another variable in the equation.

I would have likely ignored the dimensional analysis problems in the Saxon books (as being a trivial matter) had I not read your earlier posts on the subject and printed out all the links KTM provided.

Thanks Steve & all you other KTM contributors for the fantastic posts!

I just got my copy of "Warriner's English Composition and Grammar the Complete Course" (discussed on another thread).
ISBN# 0-15-311736-2 (1988 version)

What a fantastic reference book! Thanks again KTM!

VickyS said...

I didn't encounter "dimensional analysis" until a college analytical chem class. It was a revelation! And necessary for chemistry!

But I'm pretty sure I wouldn't have appreciated it if I had been introduced to it does seem to make a "simple" problem very clunky. Perhaps it's better to introduce this in more complicated problems when the kid can really see that it's an asset.

This reminds me of the "show your work" fight I always get in with my son. If he doesn't show his work for the easy problems, he inevitably gets lost on the hard ones b/c as they get progressively harder he keeps thinking he can still do it all in his head. My kids don't seem to want to do anything that adds more time to the work even if it adds more accuracy!

ChemProf said...

There is a good class of problem that is pretty easy if you keep track of units, but not if you don't. I use them to review units at the beginning of General Chemistry (where units are really critical, as VickyS noted!)

An example:
A roll of aluminum foil is 12 inches wide and 25 yards long, and contains 12 ounces of aluminum. The density of aluminum is 2.70 g/mL. What is the thickness of the aluminum foil in mm? Conversion factors: 2.54 cm = 1 inch; 28.3 g = 1 ounce; 1 cm = 10 mm; 1 mL = 1 cm^3

This is a bit of a mess units-wise, but that's sometimes how real data comes out. I've found that someone who can deal with this problem can definitely handle first semester chemistry. Those who can't are going to struggle. But if you don't keep track of your units, you are sure to get a mess at the end!

(The answer, by the by, is 0.018 mm)

SteveH said...

"I didn't encounter 'dimensional analysis' until a college analytical chem class."

I didn't have it until college either. "Dimensional analysis" is really a specific analytic process, and I was really talking about just plain "units". I called it dimensional analysis because that's what many people call it. I probably shouldn't do that. It's just "units".

When I learned the formal process in college, I thought it was magical. You could seemingly figure out a formula just by knowing the base units of the variables. The example in my college textbook is too complex, and I can't figure out an easy one to use here that isn't too trivial. The process involves determining the unknown exponents of the variables based on consistency of the units.

r. r. vlorbik said...

words well said on an important topic.

"dimensional analysis" is good enough for me
(and doesn't mean a *lot* of other things...
as "units" *does*... so it seems like it's one
of those "let the sloppy usage win" deals...).

it's a pleasure to quibble over a triviality
with steve h. amazing how much we agree on really.

sandra said...

It's a luxury to have the time to split hairs over the rudiments of units. Time and maturity usually take care of this understanding but if you have the time and patience to choose this as one of your "battles", I support you! There are so many things to understand at depth that you have to make choices with time. Do you spend time explaing WHY you cannot divide by 0, for instance, or do you just tell them that "eggs can't be on the bottom of the sack" or they will break. You'd be surprised how many students use that mnemonic with no understanding at all!

concernedCTparent said...

Despite the fact that our math spine is Singapore Math, my children also work through Saxon mostly because it does take the time to define the terminology, explain why you cannot divide by zero (for example), and it emphasizes such concepts as *unit multipliers* through distributed practice. It's been a good way to help move the many concepts into long-term memory.

However, at times (often actually), my kids get a little irritated and want to jump the various steps and just get to the answer. As often as I have the presence of mind to do, I explain that their patience and persistence will pay off later.

My oldest child is experiencing the payoff now that she's working through algebra I; some of the habits that she's developed (such as using unit multipliers) has helped her turn what initially seems like a daunting task into something manageable. It's kind of like saying "I told ya so," without really having to.