And that’s where you have the intersection of two separate educational disciplines, English and mathematics. Because the answer isn’t a math answer, it’s an English answer. And that answer is because “worms” is the direct object of the sentence. The question being posed is “how many”, and the object of that question is “worms”.
That’s where sentence problems in mathematics can get tricky. The problem is telling a story, and you have to translate the regular language of the problem into math, solve the math, then take that solution and translate it back into regular language to give an answer. In this case, the answer is, “Bob ate forty worms over a period of five days.”
The reason why that’s important is that when you need to apply mathematics into real world solutions, that’s how they end up. You aren’t given a piece of paper with a math formula and a blank space after an equal sign. You get a question like, “How many units are we going to need to order next month, and how is that going to affect the quarterly budget?” This is how the real world works. And while presenting it as a real world problem can make it easier to understand, that added information to provide context can also obfuscate things.
It’s like taking a simple diagram and writing notes all over it. Those notes provide guidance to help you, but also make it more visually complicated.
OK, then let’s go back to my example featuring time, speed, and distance:
You say you immediately grasp how to solve that problem using multiplication. Why? Can you explain in words, without mentioning the units involved, why 60 miles per hour multiplied by two hours equals 120 miles?
What about calculating speed, based on having covered 120 miles in two hours’ time? Can you explain (again, with no reference to units) why the correct answer is 120/2, and not, say, 120*2, or 2/120?
The units inform how to arrange the factors in a multiplication or division operation. Without units, the inputs (and output) are just dimensionless abstractions of quantity with no relation to the real world.
What you are asking me to do, to explain how I understand math without using units, is completely the opposite of how I think mathematically. Attaching the units to the numbers (or, more precisely, attaching stories to the ) IS how I understand it, and in my 18 years teaching math to folks, it is overwhelmingly the way that other people learn the concepts, too.
Without sarcasm, it is genuinely cool that’s not how your brain works. But if I am trying to explain a concept to someone who doesn’t understand it, I am going to focus on the concrete.
It seems we’re on the same page here: units are da bomb. But upthread, you wrote:
So based on that it sounded to me like you were claiming an intuitive understanding of the solution that didn’t involve units at all. And I was wondering what that looked like. Sorry for the misunderstanding.
I get your point. As you said, units are “da bomb”! Your “unit cancellation” is what I’ve typically called a “unit sanity check” to make sure a calculation makes sense. In this case, the units are miles/time x time. “Time” cancels out leaving “miles”, the distance traveled.
As another example one can sanity-check Einstein’s famous equation E=mc2 thusly:
The units are energy = mass x velocity2, which expands to:
energy = mass x distance2/time2
The SI unit of energy is the joule. A joule is the force of one newton acting through one meter. A newton is a force accelerating 1 kg to 1 m/s2.
Thus a joule is the force accelerating 1 kg to 1 m/s2 over a distance of one meter. So energy can be defined as:
mass x distance/time2 x distance,
or simply
mass x distance2/time2
Thus E=mc2 passes the units test. What on earth the speed of light has to do with mass-energy equivalence is a whole 'nuther question!
Oh, I see. It’s not units that present the difficulty for my understanding, it’s the specific concept of unit cancellation. That idea is really abstract to me. When Stoid was asking for help understanding how to represent multiplying by zero, I was saying that I didn’t think unit cancelation would help with the understanding. Units themselves are great. Sorry for the confusion!
You are correct, sir. The very concept of a unit is an abstraction. Afaik, a unit is something we’re numerically representing as 1. We just have to remember that it’s not actually a number 1, it’s something else: some other number, a worm, a mile, a frisbee, what have you. Hence you use something other than a numeral lest you forget.
I think it would help to be more explicit with this expression . First let’s stick with the singular just to make it extra clear that we’re treating those as if they are the number one but let’s remember that they actually aren’t. Let’s also remember that “divided by day” means “times the multiplicative inverse of day,” so 1/day.
5 days × 8 worms/day = 5 × day × 8 × worm × 1/day
It’s easy to see now that your going to be multiplying day by 1/day and that equals one. An actual, factual one. In effect, unit cancellation is finding the “real” ones, if there are any.
“Unit cancellation” or what is sometimes referred to as “dimensional analysis” is a very useful tool in working out real-world math questions. I use it all the time. But units are not numbers. How can we treat them like numbers and be certain everything is legitimate? The Buckingham π Theorem is a good place to start. Buckingham π theorem - Wikipedia.
I have heard non-academic instructors teach that if you use “unit cancellation” or “dimensional analysis” when solving a problem and the units in your final answer are correct then you have solved the problem correctly. While it is a step in the right direction, I don’t think it is a 100% guarantee that you have solved the problem correctly. No counterexamples come to mind right away and I really should get back to work. Can anyone think of a good counterexample?
Maybe we should go back to basics in the OP.
Stoid, what does multiplying a frisbee mean to you? Like what real-world action is representative of multiplying by zero?
Hmm… we have to make a choice one way or the other, but isn’t it typical to put
3 x 9 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3
Not this way, though! What I mean is, to define multiplication inductively you want to start with a × 0 = 0 and a × (1 + b) = a + a × b, not the other way around.
