I started going over more word problems with my son. We jumped into the DRT governing equation: Distance = Rate * Time. I wanted to show him that you can treat units like factors that are multiplied and divided.
For example, if you have 30 mph, this is really 30 * miles/hour. The 30 and the miles are in the numerator and the hour is in the denominator. You should treat them just like any other factor in the equation. You can move them around using:
a*b = b*a
and cancel them with
a/a = 1
For simple equations, like D=RT, it's probably easiest to just make sure that your units are consistent and then you can ignore units. In other words, you don't want to multiply 30 mph times 60 minutes. However, if you carefully track your units, then it will jump out at you.
D = 30 mph * 60 minutes
This is
D = 30 * miles/hour * 60 * minutes
Everything is multiplied together, so I can separately combine the numbers and units to get
D = 1800 * miles * minutes / hour
This is a correct answer.
As an interesting side note, mathematicians use a dash instead of a multiply sign when two units are multiplied side by side. They also don't show the multiply sign between the number and the units. A proper form for the answer would be
D = 1800 miles-minutes/hour
or
D = 1800 minutes-miles/hour
It may be correct technically, but the nice thing about keeping the units in the equation is that you won't forget the conversion factors.
In this case, 60 minutes = 1 hour, or 1 hour/60 minutes = 1.
[a/b = 1 if a = b]
I can multiply D by 1 without change, so
D = 1800 * miles * minutes / hour * 1 * hour/(60 * minutes)
since a*b = b*a, I can move any of the factors around in the numerator and denominator (and ignore the 1).
D = 1800/60 * miles * (minutes*hour)/(minutes*hour)
(minutes*hour)/(minutes*hour) = 1, so
D = 30 miles
My son doesn't like it when I do this. He can do the problem in his head. I tell him that I want to teach him about units while the problems are easy and not wait until he has to deal with many other things.
I will do one more problem.
100 miles = R * 2 hours
or
100 * miles = R * 2 * hours
To solve, divide both sides by 2 and divide both sides by hours.
R = 100/2 * miles/hour
R = 50 miles/hour
This (anal?) approach seems weird for simple problems, but it can really help you see if you are doing a problem correctly.
My analytical chemistry teacher in college called this "dimensional analysis." I never saw this technique until I was in my last year (my second time around) of college, and that ol' lightbulb really went on! I was awestruck. Dimensional analysis was no less than an epiphany for me (I know, I need a life) and I went on to get a graduate degree in science. It sure became a lot easier! This was the teacher who also clued me in to how buffers really work (another epiphany), and believed God & science weren't mutually exclusive (I guess that's a real Epiphany!).
ReplyDeleteSo anyway don't feel bad if your soon-to-be 6th grader has trouble with it or isn't impressed. He just doesn't appreciate it yet!
That is how I was taught in soviet school - in math, physics, chemistry. Starting with simple problems, it helps a lot not to get lost in units in more complex ones. And that's how you check the answer.
ReplyDeleteI taught this method to my seven-graders in physics this year - they all seemed to think that units were not important, and didn't treat the units as numbers/fractions. Mph, rpm etc - didn't mean anything to them.
P.S. I just got back from Germany and Ukraine and braught some textbooks - math manuals for teachers starting from grade 1, physics for grades 6,7,8, geography for grades 5,6, 7,8,9. I will look at them closely later.
Isn't 30 * 60 = 1800?
ReplyDeleteDid I miss a step there?
"Isn't 30 * 60 = 1800?"
ReplyDelete"Did I miss a step there?"
That's a very kind way telling me.
Oops.
Thanks so much for writing this post. I've been teaching C. dimensional analysis for years, and he's got a decent ability to do dimensional analysis prolems accurately but not fluently.
ReplyDeleteBut I haven't gotten the concept across at all, and I've been fretting about it.
This summer we're working on fluency in the pre-algebra "math facts" AND we're working on word problems, word problems, word problems.
He has NO idea how to do a simple percent word problem. This is a kid one-third of the way through algebra 1.
He has NO idea how to do a fraction problem.
He has LESS THAN NO idea how to do a "reverse percent" problem, i.e. "John bought a shirt for $25, that had been reduced from the original price by 50%. What was the original price?"
It's horrifying.
He's had two years of pure memorization with essentially no practice and certainly no applications.
So he barely remembers what he's learned and when he does remember it he doesn't have a clue what to do with it.
I should add that I've taught dimensional analysis procedurally, apart from trying to drill in the idea that any value divided by itself is 1; hence you can cancel the units.
ReplyDeleteI've done that partly because I'm not adept enough yet to teach procedurally and conceptually, but partly because unit multipliers are such an ironclad procedure ("ironclad" meaning unit multipliers are bizarrely resistant to "careless error") that I wanted him to have them regardless of whether he understood them or not.
But we're past that point now.
He's got to start using formulas AND having a clue where the formulas came from & why.
As I've practiced with unit multipliers (which I learned because of ktm-1 -- remember, Dan K created a lesson & everyone talked about them - Saxon teaches them, too) I have written out rate unit multipliers as 60 m/1 hr
ReplyDeleteIt's been a help, and I had been planning to have C., do the same.
welcome back, exo
ReplyDeleteKEEP US POSTED.
Are any of these books available in English?
That's another project SOMEONE needs to take on - we need to get these Russian textbooks translated.
I fixed it. Thanks Lynn.
ReplyDeleteI never saw this technique until I was in my last year (my second time around) of college, and that ol' lightbulb really went on!
ReplyDeleteCarolyn said the same thing.
She may not have learned dimensional analysis until grad school - or have I got that wrong??
Dimensional analysis is a lot like comment lines in programming: it's a boring, repetitive annoyance right up until the moment where it becomes absolutely critical.
ReplyDeleteFurthermore, the advanced kids will likely assume they can get by without working at it - and usually succeed for a while - before learning the hard way.
["John bought a shirt for $25, that had been reduced from the original price by 50%. What was the original price?"]
ReplyDeleteIt's probably a good idea to avoid 50% at the beginning to contrast and highlight the relationship, e.g. 30% off leaves 70% of the original price.
BTW, what's the correct math language to express the relationship between 70% and 30%?
since a*b = b*a, I can move any of the factors around in the numerator and denominator (and ignore the 1)
ReplyDeleteI'm not sure whether I've stressed that the commutative property applies to numerators & denominators, too.
instructivist
ReplyDeleteThanks!
That's what I'm doing - but do you have a particular point at which you introduce the "inverse" word problem??
Right now I'm having him do the Saxon Math 8/7 "percent ovals."
Saxon tells students, in his text, that they CANNOT master percent without drawing it!
For most of us, I'm sure that's true.
So we are drawing percent ovals.
As a former chemistry teacher I would say you cannot overemphasize an understanding of dimensional analysis.
ReplyDeleteFor kids who don't see the point, ask them "backwards" questions such as how many feet are in an inch?
Also, have them do long chains such as determining how many centimeters in a mile. It's good to have figured these out in advance yourself so your student is instantly rewarded if they get the right answer.
Saxon does TONS of long chains.
ReplyDeleteWith Saxon you start learning unit multipliers by 8/7 (possibly 7/6 - I don't have the book).
8/7 is the 7th grade book for regular (non-accelerated) students.
You use them from then on, in ever-increasing complexity.
As for the per cent problems would it help to start by ensuring that the student understands how to do the same kinds of problems using fractions instead of percents?
ReplyDeletePer cent version:
3 is what per cent of 10?
Fraction version:
3 is what fraction of 10?
Per cent version:
What is 20% of 10?
Fraction version:
What is one-fifth of 10?
Per cent version:
The shirt was $40 after it had been marked down 20 per cent. What was the original price?
Fraction version. The shirt was $40 after it had been marked down by 1/5 of the original price. What was the original price?
As for the per cent problems would it help to start by ensuring that the student understands how to do the same kinds of problems using fractions instead of percents?
ReplyDeleteYes, definitely - and this is an excellent example of the sloppiness of "reactive teaching...."
Because my time is SO short I'm wagering I can "get away with" teaching the Algebra 1 percent word problems and the fraction bar models from Saxon 8/7 - while simultaneously attempting to point out that fractions, decimals, and percent are the same thing.
So we'll see.
btw, Susan - I think you and I emailed about Saxon's funky method of introducing variables:
ReplyDelete3 is what per cent of 10?
is set up as
3 = WP x 10?
"WP" stands for "what percent."
I know it's wildly incompatible with standard notation, but it really gets the job done.
He introduces x and y in Algebra 2, Lesson 94 (on functions and function notation).
It's wonderful.
He explains dependent and independent variables, and says x is customarily used to stand for the independent variable, y for the dependent variable.
A brilliantly smooth transition to the conventional notation (IMO, obviously).