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One of my pet concepts is dimensional analysis. For many real world problems, the dimensions are your guide about when to use reciprocals. Take a problem like

A car is traveling at 40 miles per hour. How many feet does it travel in one second?

We know that there are 5280 feet in a mile and 3600 seconds in an hour. But do we multiply by 3600 or divide by it? The dimensions are our guide. Think of the conversion factors as fractions. Since 5280 feet is that same as 1 mile, the fraction (I hope my line breaks work!)

5280 feet

is equal to one. So, we can multiply by it (or its reciprocal) without changing the value.

We start with 40 miles per hour as a fraction and multiply by our conversion factor:

40 miles 5280 feet 40 * 5280 miles * feet

We cancel miles from both the numerator and denominator, and are left with 211200 feet per hour. Now, convert from feet/hour to feet/second.

211200 feet 1 hour 211200 feet * hours 58.667 feet

Notice how the hours cancelled out of the top and bottom of the fraction. So, if we observe for one second:

1 sec * 58.667 feet/sec = 58.667 feet.

Danica’s problem is done the same way.

1 car 1 car 7 cars

We’ve got one car to wash. So, we apply this rate to that one car in a way that our dimensions will turn out to be minutes.

1 car * 24 min 1 * 24 cars * minutes 3.43 minutes
--------- = -------------------------- =
7 cars 7 cars

The dimensions cars cancels out of the numerator and denominator, leaving us with minutes.

I could go on and on about the value of dimensional analysis. It guides you right through the problem. Unfortunately, Saxon Math starts out emphasizing dimensions. They make second graders write the units (2 cupcakes + 3 cupcakes = 5 cupcakes), but drop the ball when they get to multiplication (For five students with 6 pencils each, they want the answer written as 5 * 6 pencils = 30 pencils. It should be 5 students * 6 pencils/student = 30 pencils.)

-- DanK - 23 Jul 2005

Oh wow!

Dan K

Welcome to ktm! (I haven't seen you before, have I?)

I'm going to be reading ALL your comments!

-- CatherineJohnson - 23 Jul 2005

This reminds me.

Carolyn J found a page somewhere about making simple drawings on a web page....I have to look at it, because it might solve our problem here, at ktm, of illustrating problem solving.

Minor child crisis here; I'll be back later.

-- CatherineJohnson - 23 Jul 2005

"I don't understand why you would use the reciprocal to solve this problem."

I think it will become clear if you look again what the fractions tell you about which unit is involved (car vs. minute). In 7/24 the unit is minutes (with a fraction of a car). In 24/7 the unit is a car (with fractional minutes for completion).

The question asks how long it would take both to do ONE CAR. Car is the unit you want (24/7). You don't want a fraction of a car (where the unit is a minute). This is what 7/24 would give you. That's why you need the reciprocal.

I hope I made this a bit clearer. If not I need to hone hone my explanatory powers further.

Instructivist

-- KtmGuest - 23 Jul 2005

"dimensional analysis"

Wow!

I would call the approach described by DanK^? deep conceptual understanding. Real math involves computational fluency and conceptual understanding among other things.

The fuzzies arrogantly and ignorantly believe they have a monopoly on conceptual understanding.

Instructivist

-- KtmGuest - 23 Jul 2005

OK, people. I'm just a 5th grade teacher. I'm going to try to explain this like I would to a 5th grader because I think it's the easiest.

You can wash 7/24 of the car in 1 minute. How long will it take to wash the car?

Now let's take a different problem that you can reason through with no trouble.

What if you could wash 1/4 of the car in 1 minute. How long would it take to do the car? You already know the answer. (4 minutes)

You have just multiplied by the reciprocal. You probably didn't think, "OK, I'm going to choose a reciprocal." You just did it without much thought. You multiplied the 1/4 by 4 and knew that would be get you to one (1) car, and then you multiplied the 1 minute by 4 to get the answer, 4 minutes. (4 is the reciprocal of 1/4 and can be written 4/1) You deliberately chose the reciprocal because that would get you to one (1) car.

You can set the problem up as a ratio: (I'm going to see if I can get this to set up like a ratio.)

1/4 of a car
_____________
1 minute

Now remember we can multiply this ratio by any fraction that equals 1 without changing the value of the ratio. Let's multiply it by 4/4. Write it out -- it will help you see it. (1/4 over 1 x 4 over 4)

You are multiplying 1/4 by 4/1 and you are multiplying 1 by 4/1. You get the new ratio of 1 (4/4) car over 4 minutes. One car in 4 minutes.

Using a reciprocal is just a faster way to get the same answer. 1/4 x 4/1(reciprocal) = 1

Now back to the original problem:

7/24 x (what number) = 1

It will always be it's reciprocal, which is 24/7.

Hope this helps.

-- CarolynMorgan - 23 Jul 2005

Two little additional points:

1. Instructivist: I’ve read many of your posts, and I respect your opinion very much. However, you overstate the depth of the dimensional analysis concept. Understanding the simple concept of dimensions unlocks a bunch of seemingly deeper concepts like “rate problems”, “metric conversion”, and even lends insight into differentiation. I didn’t learn about dimensional analysis until high school, when my chemistry teacher sang about “grams to moles and moles to grams (and little lambs eat ivy).” It is hugely important in physics, where acceleration is in units of meters per second per second. I’d imagine that high school science teachers’ lives would be much easier if fifth graders were taught dimensions as soon as they learn to multiply fractions. Even if a student thinks he can do a rate problem in his head, it’s always a good post-check to ask “Did the units come our right?”

2. I was never an avid viewer of “Boy Meets World,” but my guess is that the math teacher was claiming that two boys working together might never get the car washed, because they would get into water fights, etc. 

Dan K.

-- DanK - 23 Jul 2005

CarolynM^?, that helped me. Thanks. We math phobes need the even simpler explanation.

I've clearly regressed to a 5th grade level in math. I swear I got through college algebra.

-- SusanS - 23 Jul 2005

DanK,

thanks so much for bringing up dimensional analysis!

I've put up a post about it. I really feel it's an easy tool, and one that's totally underutilized. I'd like to figure out how to make it easy enough for parents to teach and kids to learn, that it can enjoy (perhaps) a little revival.

SusanS and CarolynMorgan,

It's interesting to me that Susan found it easy to follow Carolyn's explanation. I found it kind of confusing! It just goes to show you that different ways of explaining things are needed for different people.

One thing Carolyn did that is a GREAT teaching idea (definitely one for the tips and tricks thread), is to simplify the problem slightly so that the answer is obvious, and then try to help the kid reason toward the answer.

-- CarolynJohnston - 23 Jul 2005

I'm still confused, but I'm tired, I've had a beer, and....hmm.

Apparently I'm out of excuses.

I'm going to come back tomorrow morning and read through these explanations carefully.....

Also, I have to try to remember (and find) my various notes on reciprocals, and why it is they're mystifying to me.

I've also been wondering about what, exactly, conceptual understanding actually is.

I don't know if I can explain this, but often there are math concepts that I 'feel' I don't understand, but that I act as if I do understand.

I have procedural fluency with them.... but if I were just watching myself from the outside I'd say that I have more than simple procedural fluency...

I'll have to notice an example the next time one crops up.

So....conceptual understanding seems to involve not just understanding but the feeling of understanding!

(We've talked about the 'click,' which is another example of a 'feeling' of understanding.)

Another aspect of this is that there's one whole genre of explanations that don't work for me, and that is 'demonstrations' or perhaps even proofs.

Often someone will try to explain something to me by showing me that it works.

But that doesn't help.

I can see that it works; my problem is that I can't see why it should work. (I find people start getting impatient at that point, and telling me things like, 'You're a verbal thinker.')

One thing I've noticed, as I've gone along, is that, pace Daniel Willingham, simple repetition of a concept, and of the problems associated with that concept, will cause me to begin to accept, or see as natural, the fact that something I'm not any closer to being able to explain.

I can't think of good examples right now, but the next time one occurs to me, I'll write it down.

It's a process of naturalizing what has been a strange and magical concept.

This is why I'm probably going to have Christopher (and me, too) start practicing 7 fact families....I'd put money on it that if every day I did a set of exercises that demonstrate to me that the reciprocal 'works' or 'shows up' in this or that problem-solving situation, after a while it would 'make sense.'

I based the 7 fact families on Saxons 4 fact families, which I like tremendously. He drills his fact families into kids' heads (into mine, too). I didn't know why he was doing that; I thought it was just to develop 'number sense,' and to teach the operations as inverses of each other.

But heading into pre-algebra I'm having a voila experience, because Christopher is now solving all kinds of simple equations of the form:

a + 7 = 10

or

s X 3 = 12

It took him about 5 seconds to figure out that these equations can be solved using inverse operations.

(Is using inverse operations a more elegant approach than 'doing the same thing to both sides'? It seems so to me, but maybe I'm wrong.)

In any case, Saxon 6/5 radically naturalizes the concept of inverse operations; I now almost can't think of addition without also thinking of subtraction.

So I'm going to experiment with taking the same approach to reciprocals.

And I'm going to figure out dimensional analysis first thing tomorros.

-- CatherineJohnson - 24 Jul 2005

"I'm still confused, but I'm tired..."

I thought you were going to say: I'm still confused, but on a much higher level.

Instructivist

-- KtmGuest - 24 Jul 2005

Oh, Catherine. The teacher in me can't rest until it "clicks" with you.

Do these mentally.

If you can wash 1/5 of a car in 1 minute, how long would it take you to wash the car?

You know it, don't you. You deliberately chose the reciprocal (even without thinking) of 1/5 because you knew it would be the entire car, it would get you to 1 complete car.

(And you multiply the 1 minute by the reciprocal (5) also and get 5 minutes.) This part you do consciously, deliberately, don't you.

If you washed 1/3 of a car in 1 minute, how long would it take you to wash the car?

You deliberately chose the reciprocal of 1/3 (not thinking about choosing a reciprocal) because you know it would make you wash the entire car. Then you multiply the 1 minute by the reciprocal (3) also and get 3 minutes.) This you do consciously, deliberately, to get the answer, 3 minutes.

You can repeat the same procedure with any simple fraction (1/2, 1/8, etc.) and the answer will be easy to compute mentally.

So now, a harder one.

If you washed 2/9 of the car in 1 minute, how long would it take you to wash the car?

Now this time you have to deliberately and consciously choose the reciprocal of 2/9 because that will get you to the one car, the 1 entire car. AND you also must multiply the 1 minute by the reciprocal (9/2) which gives you 9/2 minutes, which is 4 and 1/2 minutes.

Sometimes I think writing things out and talking about it as you go -- getting all of the senses involved -- helps.

I hope this makes sense and helps.

-- CarolynMorgan - 24 Jul 2005

My $0.02:

In one sense at least, the reciprocal does not need to be "used" to solve the problem--it "is" the solution.

I made no use of the CONCEPT of the reciprocal in solving the problem; I simply divided 24 by 7.

But one can think of Danica's way of approaching the problem as setting up a distance = rate x time formula.

If we consider distance to be the number of cars the boys can wash, then we can translate the situation into the obvious:

"The number of cars the boys can wash is equal to the rate at which they wash multiplied by the time they spend washing."

The boys was 1 car (distance = 1) at a rate of 7/24 cars per minute (rate = 7/24) in t minutes (time = t):

1 = 7/24 x t

HERE one can make a smooth transition into using the concept of the reciprocal, because if a x b = 1, then b must be the reciprocal of a (and vice versa).

-- JdFisher - 27 Jul 2005

I STILL have not sat down and worked my way through this thread, BUT I've had some major breakthroughs on reciprocals, which I ought to be recording.

One of them came because of Dan's observations, and I THINK it was what JD said about suddenly perceiving that it 'is' the solution: i.e. in some cases you would write 4 to 7 (4/7) and in others you would write 7 to 4 (7/4) depending on .... which unit you're looking for??? (I forget.)

Also, a web page explanation AND the Russian Math demonstration were major breakthrough moments. (I'll post the web page at some point. I've GOT to start turning math pages into gifs I can upload...)

JD, if you're still around: how do you put your equations up??

-- CatherineJohnson - 30 Jul 2005

If they require fractions or something else I can't represent in HTML, I actually go through the ridiculous process of typing them up in Word (using MathType^?), printing them out, scanning them in, editing the picture, and then uploading them to Flickr.

I'd be glad to know a simpler way. I'd also like to know how I can remove those ridiculous frames around my JPEGs. I feel like I've tried everything in HTML.

-- JdFisher - 30 Jul 2005

Perhaps I can help you save a couple of steps...

JD, if you have picture-editing software, you probably have a way to take a screenshot (i.e., a picture of what's in a window on your computer). What picture-editing software do you use? You should look up "screenshots" in the picture-editor software's help pages.

Once you know how to take a screenshot, you can skip the steps where you print out and then scan in. Just take a screenshot of the Word window with the math in it, then edit the screenshot, and save as a jpeg or whatever. Voila.

-- CarolynJohnston - 31 Jul 2005

If they require fractions or something else I can't represent in HTML, I actually go through the ridiculous process of typing them up in Word (using MathType^??), printing them out, scanning them in, editing the picture, and then uploading them to Flickr.

yup, that's what I've been doing.

Except I haven't been doing it. (I haven't gotten adept yet at downsizing the image, and that's caused me to procrastinate scanning more images in.)

-- CatherineJohnson - 31 Jul 2005

btw, Ed, who uses tons of slides in lectures & books (I know I just said he didn't.....what he doesn't use is PowerPoint slides or the equivalent. Only images, which he doesn't keep up permanently on the screen.)

Anyway, Ed just takes digital photographs of text and it works perfectly.

I'm going to try it. You skip many steps that way, because it goes straight into iPhoto (or whatever you're using). You can downsize it, and put it up.

-- CatherineJohnson - 31 Jul 2005