Dimensional Analysis

When I went to undergrad, I had chem lab, and I was terrible at it. To be fair, my fine motor skills are not the best, and I did end up paying several hundred dollars for extra glassware, but the worst part was the math. We did a lot of dimensional analysis.

Dimensional analysis tends to be one of those things that math teachers think science teachers are going to cover, and science teachers think math teachers are going to cover, and often nobody covers it–which is what happened to me.

This is not unusual! Go fishing for dimensional analysis videos, and you’ll see videos made by long-suffering physics and chem and bio lab TAs running through basic dimensional analysis, step by step.

At the same time, I find that geometry is one of the most iffy things for my students. Whether it’s 3D-spatial visualization skill issues, or a lack of practice, or a lack of pre-requisite knowledge, I don’t know. But understanding geometry is really helpful for basic dimensional analysis!

So I spend a LOT of time reviewing basic concepts with an eye towards future problem solving. I start every geometry and/or dimensional analysis lesson like this:

What is this? Yes, it’s a point. How big is a point? It doesn’t actually exist! It’s a figment of our collective imagination, an agreement that we’re referring to a certain place. How many dimensions does a point have? 0! That’s right, good job.

• • • • • • •

What did I make when I put all these points together like beads on a string? A line, yes! How many dimensions does a line have? 1, good job! What do we call that dimension? “Length!” Excellent.

Now, suppose I put two lines like this. How many dimensions do I have now? 2, good! What do we call these dimensions? …… Ah, so this is trickier.

Do you see that we can measure how long each line is? When we do that both both sides, we create all these little squares, yes? Yes. This is how we measure area, in little itty bitty squares. All area is measured in how many squares it takes up. Whether it’s 93.5 square inches for a sheet of paper, or square feet of flooring, or even the size of the pond next door, we’re measuring it in squares. The squares are different sizes, depending on what we use–square centimeters or square meters or square yards–but area is always measured in squares. Because, well, that’s how we measure each side.

Who recognizes this from the coordinate system? Look at all of you! Good job remembering! Yes, this is the basic idea behind the coordinate plane. What’s a plane? No, I don’t mean the machine you fly around in. OK, a plane is like an infinite sheet of paper. It’s 2D, and it extends forever in all directions. That’s why we call it the Cartesian Planeit was René Descartes’ particular genius that led directly to you having to learn it now. Just like the idea of a 2D plane that extends forever, Descartes’ coordinate plane goes on forever.

Here, I’m moving on to 2D dimensional analysis. By the end of the year, my students will do 3D dimensional analysis, converting from the “customary” system to metric and back, including price analysis, but this is where we start.

Now, today, we’re going to learn about how to convert these. This actually pretty easy, to start. How many feet are in a yard? How many yards are in a mile? See, you got this! But, if we have, oh, I don’t know, say 396 square feet in my living room, and I need to go down to Lowes and pick up new carpeting, what am I going to do if they price it by the square yard? I need to figure out how many square yards that is, right? This is a solvable problem!

How many dimensions are in square feet? Two, because it’s? Area, right. So if I can convert one side of my living room to yards, and the other side to yards, I got this! Well, I have 18 feet, and remind me, how many feet are in a yard? 3, yes.

OK, so I have 18 feet, and 1 yard is 3 feet. Notice how I put the 3 on the bottom? Watch this:

Do you see what I see? What is anything divided by itself? 1, right! and 1 times anything? Is just the thing. Let’s do this:

Check that out! The feet have been canceled out, and we’re left with the fact that 18 feet equal 6 yards. So, what rule about setting up our conversion factor have we learned? Yes, the thing you want to cancel out goes on the bottom. What happens if you set up a fraction and you can’t cancel? Uh-huh, you did it wrong. Let’s not do it wrong!

Time to do the other side:

How are we going to set up that fraction? What is going to go on the bottom? The feet, yes. Why are we putting the feet on the bottom? So they cancel, right. What happens if you can’t cancel? You did it wrong! Good job.

Alright, notice how I canceled early, before we multiplied? You are allowed to do that! You don’t have to write it and then cancel. Unless you want to, and that is totally OK. You do what you need to do. It’s fine. So, here we are, with both sides in yards. Last but not least we’re going to?

Multiply, yes! And so we get 44 square yards. Remember, that means that there are 44 squares of a yard by a yard in my living room.

Now, I don’t know about you, but I would like to solve this slightly faster. Wouldn’t that be more helpful? And what if we don’t know the original sides of the room? What if only only know how many square feet it is? This, too, is a fixable problem.

Notice that it’s square feet. This gives us our first clue. How many dimensions is this? Two, yes! So how many times do we have to convert? Two, once for each dimension. Let’s see how that would look.

See how I separated out the feet squared? It’s just feet times feet. How many fractions did I use? Two! Where are the feet? On the bottom! Why? So they can cancel out. If we can’t cancel it….? We did it wrong, yes!

Did we successfully cancel? We sure did! Let’s do some multiplication.

Ta-dah! We got the same answer, without drawing! Can you draw? Sure, and it’s a good idea to always draw out your problems. Do you have to draw? Nah, it’s not required, but it sure is helpful.

So, what have we learned? We learned that a point is how many dimensions? Zero, right. A line is how many dimensions? One, yes. How many times do we have to convert a line? Just one, yes, because it’s one dimension. Area is how many dimensions? Two, excellent. How many times do we need to convert for area? Two, excellent!

Sneak peek for next lesson–how many dimensions is volume? Hmm, I’ll answer that next time!

Handling dimensional analysis like this does a couple of things. One, it reinforces the idea of dimensions, themselves. Points, lines, and planes underlie a significant portion of geometry, and getting a grip on how they’re composed and the ways in which we view them helps students immensely. Second, this gives students a bright-line rule for dimensional analysis, based on the number of dimensions at play. Even when they start using shortcuts, like 27 ft3 = 1 yd3, or

they’ll have a better sense of why that works.

One other tip is: most basic geometry formulas work easily with this. For example, C = πD has an exponent of 1, so it’s a little tip that it’s not the often-confused area of a circle. That is πr2 and the squared part reminds them that it’s 2D and therefore area. Then, when I ask students how many points are on a circle, we can have a conversation about infinity. “Points take up no space!”

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