You’ve made some excellent points. I’d like to expand upon a few of them.

1. Everything can be defined as a fraction. Even a number like 25 can be written as 25/1. It's perhaps obvious, but I found it to be quite useful when multiplying and dividing numbers and fractions.

I’ve paid some attention to the Math Wars, but I’m not sure I have all the players’ positions well understood. I mean, I generally side against the constructivists, but I’m not sure what every plank in our party’s platform is. One issue in the Math Wars, I believe, is how much to pay attention to fractions. I think that one side claims that fractions are increasingly unimportant. For one thing, it is hard to do fractions on a calculator. Well, I think that the use of dimensional analysis highlights how valuable it is to be comfortable with the manipulation of fractions. Once you understand how to work the numbers in fractions, you can start to apply the same methods to the units. Then, you can solve a ton of real world problems.

2. If you have 25 feet per second, then this can be written as (25 feet) / (1 second). The units can be split between the numerator and denominator. These units can be multiplied, divided (and cancelled) just like the numbers. Another example is that the acceleration constant of gravity is 32.174 ft/sec^2. This can be written as (32.174 feet) / (1 sec^2)

Yes. This is where my inability to format Twiki bit me in one of my initial posts. I tried to write things like

1 mile
----------
5280 feet

to emphasize that the numerator includes the number AND the unit. The fact that I screwed up the formatting so much is one thing that led me to Powerpoint. I think my twiki formatting is improving, though.

5. One can have funny units like "ft-lbs" (for moment, and the dash does not mean subtraction!), which means that a length unit in feet is multiplied by a weight (force) unit in lbs. For example, a boy weighing 50 lbs who sits 4 feet out on a teeter totter exerts a moment of 50 lbs X 4 feet = 200 foot-lbs (or lbs-foot).

Keeping track of such funny units is a great reason to track the units throughout the problem, rather than trying to divine after the fact.

6. Units conversions (like the dominoes) work because you start with something like 60 seconds = 1 minute. Putting this in fraction form (60 seconds/1 minute) is the same as (1 = 1/1), which you can use to multiply any term of an equation. Some people think that units conversion is magic - something other than legal algebra.

Yes. If we look at it algebraically, we know that if x = y, then x/y =1, and y/x = 1, for all nonzero x,y. So, if x = 3 feet, and y = 1 yard, then we still have x = y, so x/y = 1, and y/x = 1.

7. If someone knows the governing equation, like distance = rate X time, perhaps it is easier to convert the numbers to the desired units before plugging them into the equation. If you want to use distance = miles; rate = miles/hour; and time = hours, then perhaps it is easier to first convert the numbers in the problem to these units before plugging them into the equation. Another example of units cancelling is the distance an object falls in gravity. The position (or distance) D = 1/2 X 32.174(ft/sec^2) X t^2, where 32.174 is the acceleration constant of gravity, time (t) is in seconds and distance D is in feet. When you multiply and divide the units on the right hand side of the equation, you have to come up with feet. D = (1/2) X (32.174 feet)/(1 sec^2) X (t^2 sec^2)/1.

This is an excellent point. In a lot of the examples given in our discussions of this topic here, we set the problem up right, then use the unit conversions to get the right answer. Here you point out that you don’t have to set the whole problem up before doing the conversions. Let’s say you want to know how far an object traveling at 3 meters per second goes in one hour. You might just begin by saying, I have the speed in m/s, but I need it in m/hr; how do I convert the rate to m/hr. You follow the dimentions to make the rate conversion you desire. Thereafter, you can put the solution together.

-- DanK - 25 Jul 2005

DanK^?,

I really like your idea of dominoes as fractions and your use of color. It's easy for people to see and match up the colors to do the units conversions.

My only comments (respectfully submitted) are that I have found many people have trouble with other basics that might interfere with learning your method. Here are a few of them.

1. Everything can be defined as a fraction. Even a number like 25 can be written as 25/1. It's perhaps obvious, but I found it to be quite useful when multiplying and dividing numbers and fractions.

2. If you have 25 feet per second, then this can be written as (25 feet) / (1 second). The units can be split between the numerator and denominator. These units can be multiplied, divided (and cancelled) just like the numbers. Another example is that the acceleration constant of gravity is 32.174 ft/sec^2. This can be written as (32.174 feet) / (1 sec^2)

3. Numbers can only be added or subtracted if they have the same units. You can't add 1/2 pie with 1/4 apple.

4. Any units can be multiplied or divided, but it may not be meaningful.

5. One can have funny units like "ft-lbs" (for moment, and the dash does not mean subtraction!), which means that a length unit in feet is multiplied by a weight (force) unit in lbs. For example, a boy weighing 50 lbs who sits 4 feet out on a teeter totter exerts a moment of 50 lbs X 4 feet = 200 foot-lbs (or lbs-foot).

6. Units conversions (like the dominoes) work because you start with something like 60 seconds = 1 minute. Putting this in fraction form (60 seconds/1 minute) is the same as (1 = 1/1), which you can use to multiply any term of an equation. Some people think that units conversion is magic - something other than legal algebra.

7. If someone knows the governing equation, like distance = rate X time, perhaps it is easier to convert the numbers to the desired units before plugging them into the equation. If you want to use distance = miles; rate = miles/hour; and time = hours, then perhaps it is easier to first convert the numbers in the problem to these units before plugging them into the equation.

Another example of units cancelling is the distance an object falls in gravity. The position (or distance) D = 1/2 X 32.174(ft/sec^2) X t^2, where 32.174 is the acceleration constant of gravity, time (t) is in seconds and distance D is in feet. When you multiply and divide the units on the right hand side of the equation, you have to come up with feet.

D = (1/2) X (32.174 feet)/(1 sec^2) X (t^2 sec^2)/1

The sec^2 in the denominator cancels the sec^2 in the numerator because everything is multiplied and divided together. That leaves only feet on the right hand side of the equation. If you got something other than a length units, there would be something wrong.

After all of this, I want to repeat what I said in a post quite a while back. I was always good in math, but I felt that it took me until Algebra II or my trig course in my junior year before I got all of this algebra stuff straight. I take so much for granted now; things that took me a long time to figure out.

-- SteveH - 25 Jul 2005

SteveH^?,

Well stated.

I have created a new user page here at Kitchen Table Math called DimensionalDominoes. Please check it out. I'd be interested in your opinion of its usefulness.

-- DanK - 25 Jul 2005

I think that units (and consistency of units) should be taught from the earliest grades. I remember having to convert feet per seconds to furlongs per fortnight. We learned to multiply and divide units along with the numbers. The units had to be consistent. It's a great way to figure out if you are doing something wrong.

if someone is driving at a speed of 25 feet per (divided by) second, how far would they drive in two and a half days?

If distance = speed X time, you have to make sure that the units all agree. You can't just say

D = 25 X 2.5

25 should be written as (25 feet / 1 second) 2.5 should be written as 2.5 days

D = (25 feet/1 second) X 2.5 days X (24 hours/1 day) X (60 minutes/1 hour) X (60 seconds/1 minute)

If you multiply and divide the units along with the numbers and cancel out the units just like numbers, you get an answer in feet. If the units don't cancel out correctly, then something is wrong.

I remember learning about the formal process of Dimensional Analysis in college. It's a little bit more complicated.

"Dimensional Analysis is essentially a means of utilizing a partial knowledge of a problem when the details are too obscure to permit an exact analysis. It has the enormous advantage of requiring for its application a knowledge only of the variables which govern the result."

It goes on to discuss Dimensional Homogeneity. "Dimensional analysis rests on the basic principle that every equation which expresses a physical relationship must be dimensionally homogeneous. There are three basic quantities in mechanics - mass (M), length(L), and time(T). Other quantities, such as force, density, and pressure, have dimensions made up from these three basic ones."

For example, Area is L^2, Volume is L^3, and velocity is L/T.

My book goes on to describe an example where a general formula of the period of a pendulum is found.

I remember the first time I was (directly!) taught this process. It was like magic - coming up with an equation (perhaps with an unknown constant) just by selecting the proper variables.

-- SteveH - 25 Jul 2005

But it's time to take a break, and do something fun, and let your brain work on all of this

That is SO true.

Working on math has bred a tiny bit of patience in my character, because you really do have to let things percolate & come back to them.

I guess I'm always doing that in writing, too, but it's unconscious. (Obviously that's what I'm doing surfing the web instead of writing: I'M PERCOLATING!)

-- CatherineJohnson - 24 Jul 2005

Yes, your click was just right.

But it's time to take a break, and do something fun, and let your brain work on all of this.

-- CarolynJohnston - 24 Jul 2005

As with converting fractions to have common denominators, the trick is to multiply the answer by 'one'. In this case, the conversion factor will be (60 minutes)/(1 hour). (You see why this is really 'one'?)

Hey!

You're turning into Danica McKellar!

I think I'm going to have to knock off, and approach this in the morning, when I'm fresh.

I did have a 'click' just now, albeit a tenuous one: I suddenly made the connection to the 'multiply by one' when you drew the analogy to converting fractions by multiplying by '1' (renamed as 3/3 or 4/4 or whatever....)

Now that I've quickly (too quickly) read through 4 different explanations of dimensional analysis, which until today I had never even heard of, I'm thinking that a key attribute of pedagogical knowledge is knowing where to insert those 'pauses.'

I just read an explanation of dimensional analysis at Math Forum that I found utterly impenetrable--and it was geared to my level.

So I think we're doing pretty well here!

-- CatherineJohnson - 24 Jul 2005

dimensional analysis: why and how to use it

DanK brought up dimensional analysis in this thread, and it's such a useful idea that I thought we should have a thread to explain what it is, and talk about it and its possible uses in math education.

Here's a very simple example, where dimensional analysis can help you get the right answer.

Suppose a man drives 60 miles in 50 minutes. How fast is he driving?

There are two answers a kid is likely to come up with: the first (and correct) one is 60/50, but a kid might very well come up with 50/60 and not notice he's made a mistake.

Here's how dimensional analysis could help this student get the right answer: he knows he wants a rate for an answer; distance per unit of time. If he thinks of the 60 as '60 miles', and the 50 as '50 minutes', then his two choices are:

(60 miles)/(50 minutes) = 60/50 miles/minute

or

(50 minutes)/(60 miles) = 50/60 minutes/mile.

This gives him more context to help him choose the right answer. Miles per minute are units that make sense for this answer: minutes per mile don't.

In addition, dimensional analysis is the tool to use to make unit changes. If the question requires the answer to be given in miles per hour, then 60/50 is not the right answer, because the units are miles per minute. How to do the conversion to miles per hour?

As with converting fractions to have common denominators, the trick is to multiply the answer by 'one'. In this case, the conversion factor will be (60 minutes)/(1 hour). (You see why this is really 'one'?)

Thus the answer in miles per hour is:

(60 miles)/(50 minutes) x (60 minutes)/(1 hour).

Notice that (60 minutes/1 hour) is actually 1, expressed in different units in the numerator and denominator!

Now for the trick. Move the units around a little, just as though they were numbers in fractions being multiplied, and you get

(60 miles/1 hour) x (60 minutes/50 minutes).

Now the minutes cancel in that second term, and you are left with 60/50 (otherwise known as 6/5) as a dimensionless number. (A dimensionless number is a number without any units attached. For example, all ratios are dimensionless).

So the answer is: 60 miles/hour x 6/5, or 72 miles/hour.

There's even more that you can do with dimensional analysis. As Dan points out, it's a very handy concept, but hardly any math text uses it to the fullest extent they could.

At the undergrad level, it's something engineers and scientists learn explicitly. They have to know it in order to make unit conversions. I was a graduate student when I learned it in a geochemistry (i.e., thermodynamics) class; I had already had a complete undergraduate math education. I taught that whole class of geochemists how to do differential calculus; in return, they taught me dimensional analysis, and I think I got the better end of the deal.

So: when are kids ready to learn, and to start using, dimensional analysis?

Manipulating dimensions is a lot like manipulating fractions, and largely uses the same skills. You can't add dimensioned quantities, for example, unless the dimensions are the same: for example:

x miles/hour + y meters/minute = x+y miles/hour

doesn't make any sense unless you first convert the y term to miles/hour. Identical units can cancel (as the first example showed, when I canceled minutes in the numerator and denominator). So right about the age Ben and Christopher are now -- tennish or elevenish -- is about the earliest kids could really start using it, and it's also about the time that math texts stop emphasizing units (as DanK pointed out).

Plus, if the parents don't know it, how can they teach it?

Once again, it's the internet to the rescue.

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