Without just imagining being a teacher, but as a former high school math teacher speaking from experience, I have successfully explained units to high school students as follows:
(Note that this is a very interactive format; in general you can help someone learn better if you coax them to ask and answer questions—in other words, to look—than if you just talk at them.)
"What's something that you can measure?"
"I don't know. Um, a table."
"Okay. What about it would you measure?"
"What do you mean?"
"Well, let's say you measure this table. What measurement do you get?"
"Maybe five feet."
"Okay. So what about the table are you measuring?"
"How many feet it is?"
"Okay, great. So what quality of the table are you measuring?" (I usually avoid the word "attribute" unless dealing with someone fairly bright.)
"Good! Now, length is a distance, right? If you measure its height or its width, you would still measure those using 'feet,' right?"
"What else can you use to measure a distance?" (The time it usually takes for a student to answer this question is surprising. Keep at it, be patient. At first they tend to only think of metric units vs. imperial units. You will not get "light years" or "nanometers" as your first answers.)
(After much coaxing) "Feet, yards...oh, meters! Um, inches. Centimeters. Millimeters."
"Okay, how about you think BIG?"
"Oh, miles! And kilometers." (Meanwhile I'm writing down all of these on a sheet of scratch paper under the heading "DISTANCE.")
"Okay, that's fine. All of these things are UNITS. Each of these is a specific AMOUNT of distance. Right?"
"All right. So these are units of distance. Can you think of any other type of unit? Something else you can measure, besides distance?"
"Um...not really. Length? Wait, no, that's distance. Height...um...oh! You can measure weight!"
"Good!" (Write another heading, "WEIGHT," next to "DISTANCE.") "What are some units you would use to measure weight?"
"Pounds, ounces, kilograms. Um, grams also." (DON'T get into an argument about weight vs. mass. More confusion than you need at this stage.)
"Okay, good." (Writing them all down under "WEIGHT.") "Now, you can convert between different units of the same type of thing, right? Like for instance, how many feet in one yard?"
"Good, and how many inches in a foot?"
"Good. How many grams in a kilogram?"
"One hundred. Oops, I mean a thousand!"
"Right. So, how many inches in one pound?"
"Uh...what? That doesn't make sense!"
"Right! You can't do that. It just doesn't make sense. You can only convert between units of the same type of thing, whether it's distance, or weight, or...what other kinds of units are there? What other properties of things can you measure?"
"Sure! And the units?"
"I can only think of degrees."
"Yep. But there are two kinds of degrees, right? Celsius and Fahrenheit. Actually there's another kind, also, but we don't need to get into that right now." (If they question about how many Celsius degrees in a Fahrenheit degree, then I explain that the thing you are really measuring here is heat, and so the zeros don't line up because you aren't really counting something. And then move on.) "There's something else you can measure. But first, can you check how much time we have left?"
"Um, thirty minutes to lunch time. Hey! Time is something you can measure!"
"Good!" (Writing down the header "TIME" and the unit "minutes.") "And what other units can you use to measure time?"
"Hours, and also seconds. Oh! And days, weeks, months, years. Centuries."
"Good! And decades, millennia. Anything smaller than a second?"
"Okay." (Writing it down.) "Now what are some other things you can measure? There's a lot of things. How about the surface of this table? How much surface it has?"
"Yeah, it's about five feet, like I said."
"Okay, but remember your geometry? It's five feet long, but how wide is it?
"About three feet."
"Good, so five feet by three feet is...?"
"Fifteen square feet. You can measure its square feet!"
"Okay, good! But square feet is just another unit. What kind of unit is it? What are you measuring? It's not really just distance; it's...?"
"Right! Square feet is a unit of AREA." (Writes it down under its header.) "How about another unit for area? Do you know how property is measured, like how big a field is?"
"Sure, that's a good unit for area. But if you're going to buy a house—maybe you didn't know—you can find out how much area the property has, and it's usually measured in acres."
"Oh, right, acres."
"Now what about a really small area? Like a sheet of paper? It's smaller than even one square foot."
"You can use square centimeters, right?"
"Yep! Now, centimeters are a unit of what?"
"Good. So DISTANCE times DISTANCE equals AREA." (Write it down under "AREA." Let them look it over.) "So you can use any unit for distance, times a unit for distance, and get a new unit for area."
"Square miles, square meters, square kilometers?"
"Sure. It doesn't even have to be the same unit twice. What if I have an area that's one foot wide and one yard long?"
"Three square feet."
"Right—or, one foot-yard." (Write down "1 foot x 1 yard = 1 foot-yard.") "Why not? It's distance times distance, right?"
"Or what about if you're making a spaghetti farm, so you want to buy a property with an area of one mile-inch? Yeah, that's a joke. It's out of Garfield. But it's a valid unit of area."
"So you could convert that to square feet?"
From there I would cover volume as the next logical step. (And don't forget to include gallons and liters amongst your volume units.)
Next after that I would cover speed.
Then I would discuss how you can get volume from distance times distance times distance, or you can get it from distance times area.
Then I would discuss changing speeds on the freeway, or when going onto the freeway or off the freeway, or when coming to a sudden stop. The student would bring up a time he was in a car that was coming to a squealing halt and everything fell on the floor. Then I would ask him how fast was the car going (roughly), then how long did it take to stop.
Then I would go into the fact that a change in speed from (say) 40 mph to 0 mph in 5 seconds can be shown using units of SPEED over (divided by) TIME. And write out "40 mph / 5 seconds."
Then I would bring up the idea of change of an amount as distinct from an amount itself. I would stand up and ask:
"How far away am I from you?"
"Okay, now how far?"
"About ten feet."
"Good. How long did it take me to get here?"
"About a second."
"Okay, so that's ten feet per second—or is it?"
"Yeah. Wait, no...you didn't go ten feet."
"But I'm ten feet away from you now, right?"
"Now it's been another second; how far away am I?"
"So that's fifteen feet per second, right?"
"No! You started from ten, so it's only five feet per second!"
"Good!" (Sit back down.) "The point is that change of distance is different from distance, even though you measure them with the same unit. Change happens across time. So the position now is 10 feet, then it changes to 15 feet in a one second time period, that's only five feet per one second because it's a change that I'm counting here. Got that?"
Next I would discuss how acceleration is a change in speed that happens across time. And look at the formula ACCELERATION = SPEED / TIME, and then point out that properly speaking, the formula is ACCELERATION = CHANGE IN SPEED / TIME, or ACCELERATION = (CURRENT SPEED - ORIGINAL SPEED) / TIME IT TOOK TO CHANGE SPEED. But that ACCELERATION = SPEED / TIME is an acceptable way to write it, and get the student's agreement that this is acceptable and makes sense.
Next comes the jump into "square time" which confuses so many students. I would point again to "distance times area" for the "volume" formula, and that area is distance times distance. Then I would show that since speed is distance (change in distance) over time, acceleration is:
(distance/time) / time
And then write it as (d/t)/t and make the student simplify it algebraically.
They would get d/t^2, and then I would write:
distance / (time x time)
Then I would emphasize that it's really change over time of the rate that distance itself is changing. Not just the rate of change of distance—but the rate of change of speed.
From there it's a short leap (albeit an important one) to get that: "If your position (distance) is changing at 5 miles per hour, and you wait 2 hours, how much will your distance have been changed?"
"Ten what, ten gallons? Ten chickens? What's the unit?"
Then write out the algebra for it.
(5 miles / hour) x 2 hours = 10 miles
Definition of a unit: Anything you can count.
Non mathematician teachers sometimes argue about this definition, but it's true. Since you can count poops, "poops" is absolutely a valid mathematical unit.
I dare you to do the above with a high school student and NOT wind up with them understanding units. It'll be hard work.
And after they've been through the above, always, always, always insist that your students include the correct units in their answers to their math problems.