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## Worksheets to Support Teaching Dimensional Analysis

As of October 13, 2005, I have completed my set of worksheets. I still need to add solution files for the practice problems (DimWksheet100 and DimWksheet110). The idea is to work through the sheets in increasing numerical order of their titles. Start with DimWksheet010 and work through DimWksheet110. There are several accompanying answer sheets, too (DimAnswerXXX). In retrospect, I should have included the answers as follow-on slides in the same files as the problems. Well, I'm sure that is among many possible improvements.

Anyway, I think I have included a lesson to illustrate each of the techniques of dimensional analysis that I can think of. In the first several files, I have also tried to explain the underlying fraction manipulation techniques that we apply to the dimensions. The stuff at the beginning seems obvious, perhaps to the point that the exercises are laughably trivial. By the end, we get to problems that seem awfully contrived. In between, though, I'm hoping that there is enough meat to get the point across. I wouldn't be surprised if a user came away wondering what the big deal is. It might seem so simple as to produce a "Well, duh!" reaction. That wouldn't disappoint me. This stuff really is easy. In fact, that's why I want to help people to learn the techniques. Once you get it, the rate and conversion problems are easily solved by letting the units be your guide.

I don't know if twiki lets other people re-edit my attached files. I have left gaps in my numbering of the files, though, so that others can insert new stuff wherever it fits.

Enjoy.

Here at KTM, I have seen two basic types of comments in our discussions of dimensional analysis. One type comes from people who have never been introduced to it. They show varying levels of interest, but even the most enthusiastic can't just pick up the method in a snap. This is consistent with the experience of everyone who is familiar with it. We all had to work with it for a while before becoming fluent. Another factor, though, is that we commenters who are familiar with dimensional analysis have been singing its praises, highlighting our favorite aspects of it, and demonstrating it; but we haven't really been TEACHING it.

The second type of comment comes from those who are familiar with the concept. We pretty universally talk about how it makes problem solving easier. We also say that we learned it fairly late in our schooling (as undergrads or grad students). Several have suggested that it should be taught much earlier--in fifth grade or some time in middle school, perhaps. We are not aware, though, of any curriculum that includes it.

So, I want to put together materials to teach dimensional analysis. I slapped together one presentation that explains a way to "play" through dimensional analysis with tiles similar to dominoes. That is explained below. If that tool were refined, it might provide practice and insight to the user, but it still wouldn't be the thing to use to TEACH dimensional analysis. So, I've now begun to create some worksheets that could be used while teaching the concept to a middle school student. They are attachments to this page (way down at the bottom). The number worked into the file name shows what sequence I would present them. The first several are fairly short; they don't have lots of exercises. If you were really working this into a curriculum, you'd probably want to work these types of problems into the recurring cycles as in Saxon Math.

I'll be adding more layers over the next week or so. Please look them over and provide comments.

## Dimensional Dominoes Manipulative

I think that "Dimensional Analysis" may sound like too intimidating a title for many. In fact, it is a concept that greatly simplifies problems that involve units. In an attempt to illustrate this, I have designed a "game" similar to dominoes. In the game, units (e.g. hours, feet) are mapped to colors. Then, you can solve problems by simply getting the color patterns right. You don't worry about numerical values at all, yet the numbers come out right in the end.

I'm sure I'm now going too far the other way, and making it sound too easy. Then, if you're still struggling with it, you will think, "I can't even do this easy stuff." Well, it is easy, but it's probably the kind of thing that takes a couple of weeks to sink in. Work the game a little, then leave it alone for a few days. Then come back and work it some more. After while it will sink in. I will try to post a couple of new problems to practice over time.

Once you get it, you will see that the units (colors) are able to guide you right to the solution.

The link to the powerpoint file is at the bottom of this page--way down below the comment box. I'm not sure, but I think you can view it even if you don't have MS Office/Powerpoint software. I think the Internet Explorer browser can display it directly (but I'm not sure of anything!).

Dan K.

-- DanK - 25 Jul 2005

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DanK,

this is totally cool! I will definitely look forward to going through these!!!!

Thanks so much for making these available!

Carolyn

-- CarolynJohnston - 26 Jul 2005

Dan, I went through the first ppt and I am psyched to go over the others. These are very very clever!

The tally piles are a bit confusing, and I wonder whether there's a way to do without them -- perhaps by having detachable numerators and denominators.

-- CarolynJohnston - 26 Jul 2005

I AM LEARNING DIMENSIONAL ANALYSIS TOMORROW!!!!!

-- CatherineJohnson - 28 Jul 2005

I just looked at all of Dan's powerpoints, and I think the best place for adults to start getting the idea of dimensional analysis is worksheet 3.

Because dimensional analysis is sort of a 'lost art' (I don't know if it was ever found, except in science and engineering classes), I think it's critical to get the idea across to adults first.

It's actually a pretty straightforward idea -- once it clicks. It just says that units behave like numbers, specifically like fractions. Dan's third worksheet is a good intro that gets the point across.

Thanks again for putting these together, Dan!

-- CarolynJohnston - 28 Jul 2005

As I continue to put this sequence of worksheets together, my appreciation for the style of Saxon Math worksheets (as in grades 2 and 3) is reinforced.

Here, I am trying to teach one concept, starting with some fundamentals, then building upon them. The way the sequence flows, though, is as if to say:

Okay. Here's one building block. I'm going to explain it in three sentences. Then, I'm going to show you two examples. Then, I'm going to give you three problems to do. Did you get it? Did you? You should remember this lesson and understand it completely, because we're going to need it in a future lesson.

And in later lessons, it's like we're saying:

See this? Surely you remember this from a previous lesson, right? After all, you did three problems that showed this a few days ago. Let's build on it and move on. You see how this fits together, don't you? Don't you?

To go through a problem set of twenty problems of any one sub-topic would be repetitive, mindless busy work. But the student can't get a full appreciation of how the sub-topic applies in one sitting, and without working ahead. That's why gaining comfort with each sub-topic by doing two problems of that type every day, mixed in with other topics, as Saxon does, is such a great system.

-- DanK - 29 Jul 2005

Dan, I'm with you on the basic good horse sense of the Saxon math system. Even Prentice-Hall Math Course 1, which I basically like in spite of the awful busy layout, isn't as well-constructed in that respect.

I think Saxon is optimal in its presentation of material. In its pace, I'm not so sure -- I sometimes feel it is moving too slowly for my son, and Catherine, I know, sometimes thinks the same for hers. And I have heard that Saxon's high school courses are not as good for that age group as its lower level offerings are for theirs.

But for a kid who is struggling, who missed something way back when that he needed and has lost confidence as a result -- for that kid I would prescribe Saxon, every time.

-- CarolynJohnston - 30 Jul 2005

I'm finding it perfect for my LD son, even though we've done a tiny bit of skipping.

Saxon is also very helpful to the "unmathematic" parent. This is my complaint with Singapore. I think they need a lot more parent support directly pertaining to the teaching of the various concepts.

I think a lot of the reason you see intimidated parents trusting their kid's math education to the schools is because they didn't get a decent education themselves, or they flat-out forgot.

Even helping my "math kid" on some fairly advanced problem that he got wrong, I spotted a rare pencil/paper calculation on the side. (He tends to do everything in his head.)It was a simple multiplication problem, something like 45 X 25. He actually tried to solve it without moving his numbers over. He is going into 7th grade algebra next year so I found this to be of great concern.

Normally, I would have just said (as I was taught) that he goofed because he didn't move the numbers over, and that would have been that. Now, I remind him that the numbers move because he isn't just multiplying the top numbers to 2, but rather to 20, since the 2 is in the tens place. While the 0 is a "placeholder", it's also part of a legitimate number. Thus, the numbers move over.

I was never taught that when I learned multiplication.

The language has changed a bit since I was younger, and I think Saxon helps parents with that quite a bit.

-- SusanS - 30 Jul 2005

Now, I remind him that the numbers move because he isn't just multiplying the top numbers to 2, but rather to 20, since the 2 is in the tens place. While the 0 is a "placeholder", it's also part of a legitimate number. Thus, the numbers move over.

I was never taught that when I learned multiplication.

Me, neither!

That was one of the very first things I learned from Saxon 6/5, and it was GREAT!

That was a WOW, I BET THERE'S PLENTY MORE WHERE THAT CAME FROM moment.

I also figured out, I THINK on my own, that multiplication had to be repeated addition.

That probably sounds insanely simple, but I don't think it is when you're middle aged & you've overlearned the four operations as completely separate things.

Just about 5 seconds after starting to teach Saxon the relation of multiplication to addition popped out at me.

I've come to think the optimal approach is to use two textbooks, and will write about that at some point.

Diane Ravitch has recommended using 3 American history texts in high school: one liberal, one conservative, & one centrist.

I absolutely think she's right, and I think the same principal holds for math.

I'm learning SO much more than I was now that I'm seriously using two textbooks for me (Saxon 8/7 & Russian Math) while also doing Singapore bar models.

So.....at the moment I think choice of textbooks means that if you're in Susan's or my category, you probably need to go with Saxon as your main book, and supplement with one of the Challenging Word Problems books.

If you're in everyone else's category (i.e. you majored in math or are working in math-related fields) choose Singapore as the main text, but have your child doing SOMETHING from Saxon Math, as well (definitely the fast fact sheets, and I think also the mental math & the lesson practice.)

-- CatherineJohnson - 30 Jul 2005

By the way, the metaphor of the 'click' makes a huge amount of sense if you think that what's probably happening is that a student (that includes me) suddenly makes the 'hook-up' between what he or she is trying to understand and material he or she already does understand.

It's like a Lego snapping in place.

I also love the aural imagery of a perfectly engineered book lid clicking in place.

-- CatherineJohnson - 30 Jul 2005

"We also say that we learned [dimensional analysis] fairly late in our schooling (as undergrads or grad students). Several have suggested that it should be taught much earlier--in fifth grade or some time in middle school, perhaps. We are not aware, though, of any curriculum that includes it."

I suspect that it is true that DA is rarely taught in the middle grades. A pity!

I teach eighth grade math in a Chicago public school (an unusually good one) and have just finished teaching dimensional analysis. We use Prentice-Hall Middle Grades Math Course 3. The book has a chapter on DA (Ch 6-2).

You are right that it takes a while to sink in. Even some of my top students had trouble becoming fluent with it for a while. The notion of "one" in the conversion factor didn't click immediately. Then there was the question of which units of the conversion factor should be in the numerator and which in the denominator. Pointing to cross-canceling solved that. My genius kid balked at the need for a method when he could do conversions in his head, and so on.

-- CharlesH - 30 Sep 2005

Hey Charles!

Where do you live?!

I visit Chicago pretty often. (Well, not often. But more than I visit any other place...)

-- CatherineJohnson - 30 Sep 2005

My brother in law, who teaches chemistry at Evanston High (and has a Ph.D. in chemistry--began life as a researcher) said he teaches dimensional analysis to all his students.

He also said they come in knowing how to do ratios & proportions, but not having a clue how they work or why.

-- CatherineJohnson - 30 Sep 2005

Was Middle Grades Math the book Carolyn thought she was going to be using?

-- CatherineJohnson - 30 Sep 2005

hmm

I see that, back on July 28, 2005, I had a plan to learn dimensional analysis on.....July 29, 2005

-- CatherineJohnson - 30 Sep 2005

Was Middle Grades Math the book Carolyn thought she was going to be using?

Yes -- I was teaching Ben out of the first (6th grade) book this summer, in order to prepare him for his math class (which was to have used it) this fall. It's a reasonable text -- pretty good content and the material is laid out coherently -- but very busy visually (as I have complained about frequently). I am not sorry to have left it behind.

-- CarolynJohnston - 01 Oct 2005

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Attachment Action Size Date Who Comment
DimAnalysis01.ppt manage 77.5 K 25 Jul 2005 - 14:54 DanK This powerpoint file explains Dimensional Dominoes
DimWksheet010.ppt manage 26.0 K 26 Jul 2005 - 05:54 DanK Write Everything As Fractions
DimAnswer010.ppt manage 36.5 K 26 Jul 2005 - 05:57 DanK Write Everything As Fractions
DimWksheet020.ppt manage 25.5 K 26 Jul 2005 - 05:58 DanK Fractions That Are Equal to One
DimAnswer020.ppt manage 36.0 K 26 Jul 2005 - 05:59 DanK Fractions That Are Equal to One
DimWksheet030.ppt manage 21.5 K 26 Jul 2005 - 06:00 DanK Fractions as Conversion Factors
DimAnswer030.ppt manage 22.5 K 26 Jul 2005 - 06:00 DanK Fractions as Conversion Factors
DimWksheet040.ppt manage 25.5 K 29 Jul 2005 - 06:33 DanK Reducing Fractions by Cancelling
DimAnswer040.ppt manage 28.0 K 29 Jul 2005 - 06:34 DanK Reducing Fractions by Cancelling
DimWksheet050.ppt manage 35.5 K 04 Aug 2005 - 05:37 DanK Apply Cancellation to Units
DimAnswer050.ppt manage 31.5 K 04 Aug 2005 - 05:38 DanK Apply Cancellation to Units
DimWksheet060.ppt manage 29.0 K 29 Jul 2005 - 06:38 DanK Applying Unit Conversion Fractions
DimWksheet070.ppt manage 18.0 K 29 Jul 2005 - 06:38 DanK Dimensions in Simple Geometry
DimWksheet080.ppt manage 37.5 K 04 Aug 2005 - 05:39 DanK Addends' Units Must Agree
DimAnswer080.ppt manage 45.0 K 04 Aug 2005 - 05:40 DanK Addends' Units Must Agree
DimWksheet090.ppt manage 23.0 K 05 Aug 2005 - 05:38 DanK Problems Can be Attacked in Parts
DimWksheet110.ppt manage 20.0 K 05 Aug 2005 - 05:40 DanK Gratuitously Convoluted Practice Problems
DimWksheet095.ppt manage 28.5 K 13 Oct 2005 - 05:09 DanK Insert a Variable to Simplify Problem Setup
DimWksheet100.ppt manage 27.5 K 13 Oct 2005 - 05:10 DanK Reasonable Practice Problems