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Do a made up example.
Let's convert 5.2 pounds to ounces.
And so we're beginning with 5.2 pounds.
And the question is, how do I decide how to convert that?
So from your little unit sheet, it says that one pound is 16 ounces.
And what I'm saying is a very good way to do this and a way that my
sister does it and the way that the book does it, so they never
screw up, is to take that thing, which are the same and make what's
called a unit fraction.
A unit fraction is just taking one of the relations in the numerator
and one in the denominator.
And you'll use one or the other, depending on your situation.
So the one on the left here would represent division.
The one on the right would represent multiplication.
And so for this example, I would want to use 16
ounces over one pound.
So I could cross-cancel the units in pounds.
And it's a nice--
That's how you determine it?
That's how I determine it.
One way I've shown this before--
we used to do longer conversion, like my sister does.
I'd bring in sticky notes.
And we would put them down until the one that we wanted cancelled.
I think for these one-steppers that I just want to get everybody
thinking about cancel the one that you want to go away and leave the
one that you want.
That's why I was stalling there.
So this one is 5.2 times 60, which is 312.
And the advantage here is it tells me 312 whats.
It's 312 ounces that are left inside fit the
conversion that I wanted.
[INAUDIBLE].
So the top one I divided because it was in the denominator.
So what I typed in on the top one was 40 divided by 16, which would
represent something in the denominator.
And what I typed in my calculator on this one was 5.2 times 16
because it was in the numerator.
I just typed it in wrong?
[INAUDIBLE].
Oh, right.
Would it be easier to think of it as if you have ounces, since it's
a smaller unit of measurement, if you pour it into a bigger one,
then you divide?
There's a whole another way to do this, just like that, where you
make decisions about multiplication and
division about size.
I don't recommend it.
I recommend doing unit fractions.
The book recommends it.
Your science teachers will recommend it because you get a
nice visual with practice on who cancels and where you're going.
And it makes it really easy for my sister to do when a doctor gives
her to do a drip and it's in a weird unit.
That doesn't mean you can't do it, but you would get through this
chapter just fine.
I'm still [INAUDIBLE].
Just try example five.
What my question was, I was like, how do you know which one to
multiply, when to do the division one?
That's just an understanding of fractions.
A denominator is divide.
A numerator is times.
That's it.
So when it's in the denominator, you divide it out.
When it's in the numerator, you multiply.
It can help to do this.
I want to talk again.
I typed it in wrong--
5.2 times 16.
Let's try another one to get everybody on the page.
Hopefully this one will help.
So example five, it wants to just to convert 120 yards, and they
want us to convert it to miles.
So you start with your beginning value.
We have something that your math book doesn't have.
Hey, everybody, even if you--
everybody attention, hello.
On your handout that you should have-- not everybody got one at
the moment-- but it says one mile is 1,760 yards.
Everybody see that?
One mile is 7,600--
Why don't we do the yard?
We'll do that again.
So we'll do it like the book did as well.
Is that what you're asking?
We don't have that many yards to make a mile.
[INAUDIBLE]
use the yards to repeat?
I said convert yards to miles.
So this would be your from yards to miles conversion.
Yeah, or no?
These are harder than they sound.
When you divide it, can we put that 17 [INAUDIBLE]?
Here's your two choices.
These are the possible unit fractions that you could use.
You're saying which one
would cancel, The mile.
But which of these two?
[INAUDIBLE].
Either way.
Every conversion makes two unit fractions.
[INAUDIBLE] miles.
Fraction, unit fraction.
11.2?
What did you do?
Divided it?
It's significantly easier than it sounds.
You take this conversion, and the denominator should cause
cancellation.
So I want the yards to cancel.
So I want the yards to go in the denominator.
So whatever the unit of measurement is, do you want to
have that as denominator?
So it cancels, like a fraction.
So on a fraction--
some of us are struggling because we don't see canceling as
cross-canceling with fractions.
Then you've got to go back and do some multiplication.
But you put it so that it cancels, like cross-canceling in a fraction
is multiplication.
And then you divide that?
So that's where people are trying to decide.
Is that divide or multiply?
I see it right away.
I don't think everybody in this class is seeing it.
But if it's in the denominator, it's division.
That's what the denominator is.
It's just saying 1 divided by 1760. .
So that's definitely a division.
So there's two skill sets here this is relying on--
cross-canceling and then seeing that a denominator is division.
So that is 120 divided by 1760, which is fairly small.
So if this one popped up on an exam, I'd have to give you a
rounding digit, which, by the way, some people need
some review as well.
Let's suppose I said round to thousandths place.
So my calculator is giving 0.0681.
So there's the thousandths place.
It's the a.
Stays the same.
Today's conversation when we get to percents running across the
football field from one yard end to the other yard end--
that's where the 120 yards comes from--
is about 6.8% of running a full mile.
We'll talk about that today.
This is where we're heading today.
Is that starting to look better?
You take your conversions and put them in such a way that you get
cancellation for the English measurements.
And in the book, they weren't as lucky as we were on that one.
They only knew that there's three feet in one yard.
And there's 5,280 feet in one mile.
And it's just because they didn't have this nice little conversion
that we do.
You see that?
This is not in the book.
This 1,760 yards is not in the book anywhere.
This will give you a feel for what my sister does.
When she's doing these major conversions, multi-step
conversions, she can take these, make them into fractions in such a
way that they'll cancel.
Take this and make it into a fraction in such a
way that it'll cancel.
How do you get that?
So we did two steps.
My yards cancelled.
My feet cancelled.
I'm left with miles.
My conversion is correct if I put in my calculator.
What's on the top of the 5,280?
What does that say?
One mile.
And that's this conversion that the book had.
They had that one mile.
The book only has these few pieces of information.
So the book couldn't do this nice single-step
conversion that we did.
They had to do a multi-step conversion.
The yards canceled.
The feet canceled.
And they're going to take the 120.
Are they going to multiply or divide by three?
Multiply.
Are they going to multiply or divide by 280?
Divide.
So if they type in 120 times 3 divided by 5,280, lo and behold,
they get the same decimal that we did.
They left it as a fraction because they're assuming that you're not
using your calculator like we are.
So you'll notice that a little bit now that they've the calculator,
but the book still shows more steps.
They don't really show the calculator steps.
And that'll be particularly true in the percent chapter.
They'll show you how to do stuff.
I just want you plucking away at the calculator for the next two
weeks to speed it up and to seriously make it easier.
So let's do one more example.
I had written down to do example nine.
Which page?
Page 451.
They want us to convert 105 minutes to hours.
Maybe this would have been a better starting place, because I
bet you've done a lot of conversions between minutes and
hours in your past.
And so if I got 105 minutes, and I want it to be in hours, you
probably know if you divide or multiply by 60.
I think more people know how to do that one.
Hey, I divide.
But for me, I want to show you that there's a way to know that
you're dividing by 60, so you don't make a mistake when you're
doing ones that you don't have that same level of understanding,
because not all of us know the units of yards and miles
and towns and tons.
And quarts and gallons is one that I don't even know very well.
If you can put your knowledge from the sheet into a fraction in such
a way that things canceled, like cross-cancellation, then I agree
with you all for sure now that I ought to divide by 60, because 60
is in the denominator.
So I take 105 divided by 60, and I get 1.75 hours.
The book gives the answer to that same problem as 1 and 3/4 hours.
An hour and 45 minutes.
Or an hour and 45 minutes would be a way to answer that same problem.
An hour and 45 minutes is more like dividing by hand and thinking
of the remainder.
I could use my calculator to get the 1 and 3/4 if I hit the
fraction key.
I took 105, Fraction, 60.
And then I left it as a fraction.
And then when I typed in 105 divided by 60,
it gave me the decimal.
So the fraction key would give it as a fraction.
The decimal key is the decimal.
Questions on that one?
Are you starting to see it?
Or you feel better about it with that one?
Because you're more used to that one.
So that one might have helped.
It may be good for the future to start with that one.
One last example--
maybe I'll start with this example in the future.
How many days in a year?
365.
And there's always, as mentioned, you can make it a decimal.
You can go 365, 364, somewhere.
365 days in a year.
It's 364.25.
And so sometimes they use the 24, the 25, or the 0.25 at
[INAUDIBLE].
That's right.
It's not 365.25.
I would cancel the years.
It doesn't matter if you use 365 or 364 or the fraction.
I haven't made a goal for this one yet.
How many hours in a day?
24.
How many minutes in an hour?
60.
How many seconds in a minute?
60.
This would be a way to tell how many seconds in a year.
So you consistently can see that you've got it correctly.
You're left with that unit.
So we're doing one steps.
So we're not doing a multi-step like that.
I was hoping to give you a feel for that.
I don't know if it helped.
But in any case, try that all by Wednesday.
Come in the morning sooner if you're struggling with it.
Let's look at 5.4.
But there is the reason why we want you to use unit fractions.
It's to get that nice consistent so you can do longer problems in
the science class without mixing them up.
If you did all divide and multiply and just thought through them,
you'd probably be fine.
But I'd like you to think about unit fractions.
In 5.4, when we're doing metrics--
did I miss one?
I missed a decimeter.
When we're doing metrics, they have made your life easy.
If we lived in the metric world, I would say you
better memorize that.
Since we don't, I threw it on the sheet here.
Be careful not to do what I did, which is mark it up so bad that I
can't read it anymore.
But you'll get a nice, new sheet for the final with this on it,
because we're not as used to the metric world.
And the great thing about the metric world-- what do
the m's stand for?
Measure.
And that measures what kind of thing?
Distance If we want to measure volume in the metric world, we
change to what?
The base word is liters, which is then milliliters, because that's
one we use the most.
So in the volume world, it's liters.
And in the weight world, it's--
And you're used to hearing kilograms the most.
And kilograms would be right there.
You're used to hearing milliliters the most.
Milliliters would be right there.
Here's the great thing about the decimal world.
Once you know how to do something in the weight world, all you do is
change the root word, and you know how to do something in the
distance world.
So they're all consistent.
And you guys just know some of the--
you're more likely to have heard about kilograms.
And you're more likely to hear about milliliters.
But the base word is liters, grams, and meters.
Those are the base words.
Is that a good start?
Can you see that?
Don't need to memorize it, but notice I only gave you the meters
on your little cheat sheet.
They'll be asking you convert grams and liters, and you just
change the base letter for those conversions.
That make sense?
Everybody will have one of these in the final.
Everybody should, theoretically, have one right now.
I've just run out of them.
So a little conversation--
on page 456 at the start of the chapter, since we're not used to
them, they give you the distance ones and a way
to think about them.
So a kilometer is about 60 train, a car-long train.
That's probably a fair typical sized trained, is a kilometer.
Does anybody know if a kilometer is longer or shorter than a mile?
Longer.
Shorter, actually.
Yeah, but your numbers end up being bigger.
A meter is about the door handle's height.
A centimeter, for a lot of us, it's our pinky finger.
You might actually put your pinky finger on the
metric thing over here.
That's not a pinky finger sized shape.
Pretty close, actually.
Mine's exactly, if I go from my nails, mine's about as precise as
you're going to get.
Some of yours are smaller or bigger.
And then finally this is a good one.
A millimeter is a thickness of a dime.
That
I was actually trying this out to show somebody the other day.
I tried to use pennies.
Pennies are way bigger than dimes, it turns out.
It's a dime.
And that's important.
The dime is the narrow one.
Millimeters are pretty small.
So it's just good to know that.
You will not be asked anything about that.
The old book used to do these things.
It would say, if I was measuring the distance from here to Eugene,
should I use kilometers, meters, centimeters, or millimeters?
Kilometers.
Kilometer to go to Eugene.
If I was measuring the distance from here to my office?
Kilometers.
Probably meters.
Kilometers works, though.
Meters would be the most likely.
I'd say it's about 100 meters and maybe a little bit more.
So it's about a football field to my office.
If I was measuring the length of a pen, I'd go with centimeters.
And finally, if I was measuring the width of a pamphlet,
millimeters.
The old books used to ask those.
I didn't see much of that in your homework.
So I probably won't be asking that.
I'll be asking conversions.
And thank god, they're a quite a bit easier than the conversions we
were just doing.
So if I do example number six, which was on page 462--
convert 7.86 kilograms to grams.
Especially for those of us that don't use the metric system a
whole lot, I would recommend if I end up in a science class doing
this, that I find this information.
And it's really important to do this.
You'll notice that all of them in the book how the information from
the largest on the left to the smallest on the right.
I'm not going to write them down each time.
But all I have to do my little handout that I've mixed
up is put the grams.
So for this one, we're starting at 7.6 kilograms.
We're going how many steps?
So one, two, three-- do you see that-- to get to the grams.
Put kilo, [INAUDIBLE], one, two, three, steps to the right.
What do we do with the decimal?
Put it three spaces to the left onto--
Do the same.
Do the same with the decimal.
If you've written it from largest to smallest, and consistent--
not everybody learned it this way.
I didn't either.
So I always have to flip, because they're usually written backwards.
So the book's suggestion--
I like it--
they say write from the largest to the smallest, your units, so that
when you do your move from kilograms to grams, you go the
same way that your decimal is going to roll.
[INAUDIBLE].
So one, two, three--
you get 7,860 grams.
So what is measured in a big thing?
Kilograms.
Now it's measured in a smaller thing-- grams.
And we have more of them.
Does that make sense?
[INAUDIBLE].
Which unit is?
It's a decigram.
A decigram?
The question up here is, what do all the names mean?
In the book, if you look on each of the pages, we're
used to using kilograms.
I've never used hecto or decagrams.
I just know that they're spaces on the move.
I've used grams.
I've never used decigrams or centigrams, but I have used
milligrams.
Why don't you try one?
Let's have you--
That's the starting place?
So I'm going to use two methods to do this conversion.
I'll call the first one the shortcut.
And then I'll do unit analysis with the unit fraction.
So there really is a shortcut to use this thing.
One of the questions that came up is, this is in m's, meters.
How do I get the same thing to do with liters?
And I didn't explain that very well.
I think you just drop these down.
They won't let her change this.
The m, which stood for meters, changes to l, which stands for?
Liters.
Liters.
And that's what I was trying to say, but not super clearly, I
guess here was that the base words are all that change.
And you can use that same base formula.
So the meters drops down, becomes liters, or grams that are totally
interchangeable in metric.
That's why the metric is easier and better
than our current system.
So the real question said, convert six liters to milliliters.
Do you agree with that now?
Come through it?
So on here, I got six l's--
that's an l now.
And I'm going to millimeters, or milliliters.
How many spaces should I move?
Three.
That's three spaces.
You see those three spaces?
So you go from l to dl to cl to ml, which is three spaces.
What do I do with my decimal?
Where is my decimal to begin with?
To get your implied decimal, to get 6,000 milliliters.
The other way to do this same problems is
with the unit fraction.
If we lived in the decimal world, you wouldn't even probably ever
learn about fractions.
You'd just learn about moving your decimals back and forth and forget
unit fractions.
But I want you to notice that on your conversion sheet, it says
that one millimeter is 0.001 liters.
And there's also another conversation up higher that you
could use as well.
But say, hey, what about that one?
So they had six liters.
And that one that we just read off said that one
milliliter is 0.001 liters.
I wouldn't have known whether to multiply or divide on that one at
all, because I don't have the same intuition on metric.
But if I had made a unit fraction with it, it's saying take 6
divided by 0.001, and I get 6,000 just the same
using the unit fraction.
Which would you rather do?
Move the decimal to the same number of places as a little
formula, or make a unit fraction with metrics?
It's a lot easier, I think, to move the decimal.
I think I better give you another one.
If you buy 12 bottles of wine at 750 milliliters each, how many
liters would that be?
I'm giving back the second exam, and you're a little stressed.
I should write out more stuff on these.
We'll put a few more notes here.
I want you to do some practice.
You won't find this very hard once you get some practice in.
If I have 12 bottles at 750 milliliters, where do I start with
those two things?
[INAUDIBLE].
Which is 9,000 what?
Meters.
Milliliters.
We haven't done our conversion yet.
There's 9,000 and I mean 750--
I think I got it out of the book.
It's typical.
This one's 591.
So there's 591 in here.
750 is just a bigger.
It's about a wine bottle size.
I think that's right, according to the book.
So now we got 9,000 milliliters.
I'll write down more directions.
But I was hoping that you'd see right away there's milliliters.
I want to go to liters.
How many places do I need to move?
Three.
So that's one, two, three spaces to the left.
And what do I do with the decimal?
[INAUDIBLE].
One, two, three spaces to the left, making nine liters.
And then if you did the fraction one, which is that the milliliters
have [INAUDIBLE].
If you took the 9,000 milliliters and wanted to use the fraction
that I had circled here, I would want the milliliters
to cancel on that.
So would you multiply or divide by 0.001?
Multiply.
Because its in the numerator, we'll multiply it.
9,000 times 0.001 is also going to give me 9.
So you can use the unit fractions or the other.
Why don't you go ahead and head on [INAUDIBLE].
So first of all, English measurements, not so nice.
With a lot of those measurements, some of you know based on size, as
Betty was saying, when you should multiply or divide.
I highly recommend and so does the book and so will your science
teachers that you start working on this thing called a unit fraction.
It might not be necessary here.
It really, really helps if you start practicing on
the one-step ones.
I was trying to show you that-- but it didn't come through very
well today--
that when you were converting years into seconds, or my sister's
doing cubic meters per second into drops per minute for IVs, she
wants to be nice and consistent.
There's a reason to learning unit fractions.
These might not be good examples.
I'm not going to ask anything harder than these.
These are the samples that we're looking at.
And the idea is the units that 12 inches equals one foot, if I were
to write it as a fraction, because they're the same, that's the
number one.
That's why they call it unit fractions.
The numerator and the denominator are the same.
12 inches divided by one foot is just one.
One foot divided by 12 inches is just one.
I don't think about it at that level.
What think about on this, if I want to convert 65 inches, you're
trying to do something with 65, and so I write down
my beginning value.
And if I want to convert 3.4 feet, the same thing.
I'm going to write down my beginning value.
This fraction represents multiplying by 12.
And this fraction represents dividing by 12.
The reason that--
I think you this.
It's just stress today.
I've got 12 in the numerator, and I'm multiplying.
And I have 12 in the denominator below one, I'm dividing.
So this would be dividing by 12, and that would be
multiplying by 12.
And the thing with the unit fractions is, if I screw up
accidentally--
damn it, I screwed up--
I'll know it.
How do I know I screwed up on that?
You can't cross-cancel.
I can't cross-cancel.
So I get this visual and other things that are bad.
That one was an oops.
And I've got to try again.
And I try the reciprocal, of the flipped.
And I feel pretty good this time, because they cross-cancel.
On this example, I want to divide by 12.
So 65 divided by 12--
And what I try to do on exams, when they're calculator, is give
you a rounding digit on these examples.
I don't always know that they're going to need to be rounded.
So let's say the tenths place on that one.
Did I type it in right?
Yeah, you're fine.
[INAUDIBLE].
That's a different conversation entirely.
I want you to say 5.4.
[INAUDIBLE].
Or the hundreds.
It didn't say.
That's what I did.
And Daryl changed the [INAUDIBLE].
That's a conversation for a different day.
And 3.4 feet--
I want to make a decision about which unit fraction to put there.
What do I want to have cancel?
Feet.
Feet.
So I put 12 inches over 1 foot to get the feet to cancel.
So you can see a whole bunch more examples like this in the book,
but I'd like you to work on doing it with unit fractions.
So do these conversions with unit fractions, even if they seem like
they make the one-steppers a little harder.
[INAUDIBLE].
And so 3.4 times 12.
It might help somebody who may have this issue if you make the
whole numbers into fractions so you can see them multiple across.
And if that helps you, I made the original one into a fraction to
help me see it.
That's 40.8 inches.
And as I showed earlier with the metrics, you can totally use the
unit fractions.
And in that little handout that you'll get on the final, it even
provides a bunch of the unit fraction information.
But I would find it easier to use this stuff, which
will also be provided.
Now, Jackson?
[INAUDIBLE]
starting unit and then move to your goal unit.
Begin at your starting unit and move to your goal unit.
The decimal moves to the same number of places.
I was just trying to see what to translate.
So that's supposed to say being at your starting unit.
Move to your goal unit.
The decimal moves the same number of places.
And so I tried to explain, but didn't very well, that meters,
liters, and grams are symmetric across these.
They change which letter is there.
Maybe I should redo this thing and put all three versions there, but
I'm not going to, because I'd like you to know this,
that they're symmetric.
And so if I'm asked to convert 437.4 centimeters to meters, I
start at my starting unit.
And I move to my goal.
Where's my goal at?
[INAUDIBLE].
I move.
So start--
which way am I moving?
To the left.
I move to the left two places.
It's my handwriting more than your writing.
So the reason to write them from big to small, which isn't normally
how we would do something like this, is so that your move is
precisely the same way.
When I learned that we wrote them the order that you would expect, I
had to this little flip-flop, and it was always the opposite order,
so I did get them mixed up when I first learned this.
I think this is the way to go.
I like the book's consistency.
If you write this--
now, mind you, you can ignore the word "memorize," because I'm
giving it to you.
But in a science class, you may not be so lucky.
So in a science class, you'd work on memorizing that.
So that's all I'm going to talk, unfortunately, about 5.3 and 5.4.
But on Wednesday, bring your homework.
I'll answer questions.
Hopefully, I'll remember just [INAUDIBLE] better next time.
I haven't taught this in awhile.
I hardly ever, out of the book, got enough time to teach it.
But I think it's an important section.
I have taught, on the other hand, percents a whole bunch.
Does anybody know what percent means?
Times 100 [INAUDIBLE]
parts per hundred.
Or how does it go?
Any of these.
I'm hearing lots of good ones.
Parts per hundred, out of 100--
that kind of statement.