Math 20 - Lesson 16

Uploaded by PCCvideos on 09.09.2009

A Portland Community College mathematics telecourse.
A Course in Arithmetic Review.
Produced at Portland Community College.
In arithmetic, fractions seems to be the one thing
that bothers people more than anything else.
So let's return to the very lowest basics
and rebuild our understanding
through some very simple, progressive ideas.
Let's let this rectangle stand for the whole of something.
It can be a whole group of people, a whole barrel of apples,
a whole stock of inventory in a warehouse,
but it's a whole of something. All of it.
Now let's say whatever this is we're going to subdivide it.
If this is a class of people,
let's group them into groups that are more or less equal.
Let's say we chose to group them into groups of 12
for whatever the reason,
and that's not important to this discussion.
Now let's say out of this 12 groupings of the total,
the wholeness, we wish to take out for some reason, five of them.
And that leads us to the building of a fraction.
A fraction really is a code.
It's simply a device we invented.
This line says I'm going to divide this into two groupings.
First, the total number of pieces in my one group,
my whole group, is 12 and we'll put that on the bottom.
Then the number of pieces I'm taking out or working with
or whatever I wish to do will be expressed on the top.
So this is a code, just as this is a code [&] which means "and"
and this is a code [#] which means "number,"
a fraction is a code which simply means
I have taken 5 out of 12 pieces of my original total.
So here's our coded message.
Top number, how many are taken?
The fraction bar means 'out of.'
The bottom number,
the total subdivisions of a whole, in this case 12.
A supremely simple idea
but as we pointed out in the previous lessons,
we're going to take these very simple ideas and build them,
in bits and pieces, into a fairly complex structure.
Now notice here this fraction bar means 'out of.'
In a previous exercise it also meant 'divided by.'
And before we're through with this course
we'll find about half a dozen more things
that this very innocent fraction bar can mean.
So this code has two whole number parts.
A top number, which is more formally called a 'numerator.'
And a bottom number, which is more formally called a denominator.
Two numbers subtract separated by a bar,
and eventually we learn to treat this as a single number.
And it's that that perhaps confused you as a child
if you were confused.
And little wonder.
The ancients, even the brilliant ancients,
were confused by this concept.
And it took a long time for us to come to grips with this.
So let's accept it for these very simple notions
and begin to build it.
Now a mathematician by the way would define this differently,
but it's too abstract for us to consider here.
Some people get confused
by which one is the numerator and which is the denominator.
A favorite trick used by some instructors is
'down' starts with 'd,' so does 'denominator,'
so 'denominator' is the 'down' number.
Just a gimmick but it works.
Now sometimes we might start with a whole and subdivide it.
Let's start with 12 and ask somebody
How many of these 12 parts do you want?
And the person replies: All 12 of them.
So I have 12 out of 12. [ 12/12 ]
But this not only means 'out of,' it means 'divided by.'
And we know that 12 divided by 12 is 1. [ 12/12 = 1 ]
And a 1 is another way of saying 'whole' in some arguments.
So 12 out of 12 is the same thing as saying
take all the subdivisions that you have in your totality,
in other words, take 1 of the whole thing
that you had to begin with.
Sometimes you'll start with a whole, subdivided into 12 parts,
and you'll ask somebody: How many of those parts do you want?
A person might say 17. [ 17/12 ]
Well, how do I get 17 out of the 12?
Well, you might say I'll take all 12 of those,
but I need 5 more to get 17,
so I'll go to a second whole of the same thing, and take 5 more.
So I have 17 out of a wholeness subdivided into 12.
Now someone else might say, yes,
but you had to take one whole and in addition to that,
5 out of 12 of another of the same kind of wholenesses.
And of course traditionally, we write this [ 1 5/12 ] this way.
Are you beginning to see some sense into fractions?
It's a very simple thing really.
This leads us into three
traditional classifications of fractions.
If the top is less than the bottom,
we call that a, a 'proper fraction.'
Now if the top is larger than the bottom or equal to
catch this or 'equal to' the bottom,
then we traditionally call that an 'improper fraction.'
And of course if the top is equal to the bottom,
it's also called a fraction
that's simply equal to 1, a whole number.
Notice that this whole number 1 fraction
is also an 'improper fraction.'
Many beginning students confuse these
saying it's either this or it's this.
No, a 1 fraction is also an 'improper fraction.'
This is rather unfortunate that we call this an improper fraction
because it's very useful and very correct.
And just as an aside, a child was learning
to call it an improper fraction
at the very time he was trying to be taught socially
to not do anything improper.
So psychologically, I rather suspect that this built up
a little bit of a hesitancy towards this type of a fraction.
But an improper fraction is an absolutely legitimate number.
It's simply a number which has the form
of a top being greater than or equal to the bottom.
A very simple device.
So this lesson is to simply take us
right back to very rock bottom
to begin to build basics and its accompanying vocabulary.
So you will see problems like this
where they'll give you a series of fractions
and simply ask questions, not to see if you know how to do it,
but to see if you know the vocabulary.
Because if you know the vocabulary, it will tell you what to do.
Okay. Which of these are proper fractions?
That's really asking: Do you know what a proper fraction is?
And hopefully you'll say yes,
any fraction where the top is less than the bottom.
So that one is [ 5/8 ] that one is not [ 13/12 ]
That one is not. [ 6/6 ]
That one is not. [ 9/1 ]
That one is. [ 1/9 ]
The tops are less than the bottom.
Now this is not asking do you know how to find improper fractions
it's asking you really,
Do you know what an improper fraction is?
It's a fraction
where the top is greater than the bottom or equal to.
So that's an improper fraction, [ 13/12 ]
that's an improper fraction, [ 6/6 ]
that's an improper fraction, [ 9/1 ]
and that's an improper fraction. [ 1/1 ]
In these two cases, [ 6/6, 1/1 ]
the top is equal to the bottom, that makes it improper.
And in these two, [ 13/12, 9/1 ]
the top is greater than the bottom,
and that makes it improper.
Now which fractions are equal to 1?
Well, those are those in which the tops are equal to the bottoms.
That one [ 6/6 ] just another way of saying 1,
and that one [ 1/1 ] another way of saying 1.
So those that are equal to 1, are also improper fractions.
And again we reminded you
that there's nothing improper about these,
these are perfectly good, useful expressions,
and we will need them many, many times.
And we trust that by now you realize
that a basic vocabulary of concepts
is an important necessity to mathematics.
To all sciences and arts, really.
But in math it can cause harm the lack of it can cause harm
faster than in perhaps any other field of study.
So learn these basics now that you have the chance.
In order to stress the point that vocabulary is important,
this and many textbooks will give you problems
using more words than numbers or so it seems.
Such problems are really to try to illustrate
and to make the point that vocabulary is important.
Because see, I am teaching you now on this videotape with words.
The book is communicating to you with words.
So we need the words to communicate to each other
about how to view the problems.
So let's give ourselves a few problems
for the remainder of this lesson, just to make the point
that we want you to be comfortable with
words and directions within a math textbook.
So here, how would you draw a picture
to illustrate the fraction 3/7?
Well, a person might say this.
I have a room. In this room I have 7 people.
So I have a group of 7.
So in fractions, the bottom tells us
how large the whole of whatever we're talking about is.
We have a whole group of 7.
Now the top number tells us how many of those
we want to take for something or other or to do something with.
So this says out of those 7, I want 3.
Let's say we want 3 of them to have blue heads.
So we can say 3 out of the 7 have blue heads, or 3/7.
So the bottom again, illustrates
the total pieces within our group,
and the top tells us how many of those pieces
we want to talk about or to use in some manner.
How about questions like this?
Again they're not asking
if you know how to find the denominator and numerator,
but do you know what they mean?
And you hopefully will recall that denominator starts with 'd,'
'd' is down, so the denominator is the down number.
So 51 we call the denominator of the fraction.
And numerator is the other one, which is 38.
So again, be comfortable with the word denominator and numerator.
We will use these words to try to direct your attention
to where we wish to focus at any particular moment.
Or how about this,
can you see the fraction question within this statement?
If 108 people applied for a job and only 7 were hired,
what fractional part of the applicants were hired?
So this is our fraction bar.
Our whole of the applicants was 108.
So again, remember that the denominator, the down number,
is the one that's referring to how many pieces,
in this case people, constitute the whole.
Then we're asked, What part of this whole are we talking about?
Well, we're talking about those hired.
Only 7 of that whole were hired,
so we say 7/108 were hired.
Another one.
Let's say that our goal for a charity collection was $500.
$307 was collected.
What fraction of the goal was reached?
Now notice that there's sort of a pattern to this.
As soon as we want the fraction of a goal
notice that both problems use the word "of" and a goal.
This is sort of pointing to what our whole is.
So our whole is the goal. And our goal was $500.
So the whole amount of money that we want is 500.
But of that which we wanted, that which we got was 307.
So the fraction 307/500 of our goal was reached.
However, what happens if in another situation
where we wanted $500, we actually went out and collected $615?
Now what fraction of the goal was reached?
Well, the goal that we're forming the fraction from, is still $500.
That's the whole of what I want.
So it goes to the denominator again.
But this time we got more than we wanted,
but nevertheless it still is the top.
So we received 615/500 of our goal.
Please note then this gives us a feel for what's happening.
If the bottom in this case was what we want
and the top was what we got,
then if the top is less than the bottom,
obviously we got less than what we want.
If the top is bigger than the bottom,
then that means we actually worked hard
and got more than the whole that we wanted.
And if the top is equal to the bottom,
then that means we actually reached the wholeness that we wanted,
which we knew is equal to 1.
So 1 is frequently used to refer to as 'all of.'
1 of something.
That's what we mean by 1 of something, all of something.
Now for a few minutes let's anticipate the kind of situations
that will come within the next few lessons.
Let's let this be the whole,
and let's take one of these subdivisions.
So I want 1 part out of the whole consisting of 6 such parts.
So we are taking 1/6 of the whole,
which is this small shaded portion right here.
However, note this,
we could have divided this into even smaller subgroups.
We could have divided across here.
And taking the small, smaller squares,
we can say now instead of having 6, we've got 1,
But of the 12 smaller squares,
now I want 2 to capture all of that green piece.
So 2 out of 12 of the very small squares
is still the same amount
and we'll say it with an equal sign
as 1 of the larger square as far as being the part of the whole.
So these two numbers must in some sense be the same number
or at least the same amount.
we could have gone even further and divided it even smaller.
Now we have very tiny squares subdividing the one large whole.
So we could ask, How many very tiny squares do we have?
Well, we have 1,
4 by 1,
so 4 times 6 is 24. [ 4 · 6 = 24 ]
But now in our little piece of the whole
there are 4 of the very tiny squares.
So we say okay, we're taking 4 tiny parts of the 24 tiny parts.
But that's still the same portion that I had originally.
So we begin to get the feeling that fractions,
the same amount can be written many, many different ways.
That 1/6 of the whole is the same as 2/12 of the whole
is the same as 4/24 of the whole. [ 1/6 = 2/12 = 4/24 ]
but you also begin to notice
that had I multiplied top and bottom by 2,
I would have gotten this. [ 2/2 · 1/6 = 2/12 ]
Or had I multiplied top and bottom,
numerator and denominator, by 4 in this case, [ 4/4 · 1/6 ]
I would have ended up with this form [ 4/24 ] of the fraction.
or over here had I multiplied this one top and bottom by 2,
[ 2/2 · 2/12 = 4/24 ] I would have had 4/24 too.
So we ask ourselves: Is it possible
that if I have any fraction with a top number and a bottom number
whatever it is,
that if I multiply top and bottom by the same number,
which I'll call 'n' here, [ n/n ]
that whatever I get will look different because I did that,
but even though it looks so different we begin to ask ourselves,
will it be the same amount?
And the answer is going to be yes.
It will just look different.
so you begin to realize that fractions have different forms.
So we will have to pay attention to this and ask ourselves
if we have several fractions that look different,
are they really?
Or are they all really the same number
which simply have different forms for that same amount?
So this is one of the questions
we're going to have to ask within the next couple of lessons.
And fortunately it's a very easy question to answer.
Other types of questions we might ask
could be illustrated in this manner.
Let's say you've got a certain plot of land out in the country,
and you are going to sell this much of it.
So we could ask what fractional part of the land
were we going to sell?
Let's say you have some way of determining what it is.
So let's say if we were to somehow subdivide this land
into 24 pieces, just a little piece of it is only 5/24 of it.
Now we don't actually see the little subdivision
because the land is very, very crooked,
and it would be hard to illustrate that.
But let's say later we wish to sell another portion of our land,
and of the total land we were able to somehow determine
that we only sold 1 piece out of 7 such pieces.
Then the question we could have is now that we've done this
in such a strange manner,
what part of the whole land did we sell totally?
And total means I'm going to add the first part I sold [ 5/24 ]
to the second part I sold. [ 5/24 + 1/7 ]
And I now want to know:
What is this as a part of my total land now?
That is, as a single fraction.
Or in short, we want to review for you how do you add fractions,
and in particular how do you add them
if the denominators weren't the same.
Because had I said that this is 5 out of 24,
then I want to add to that let's say 3 more out of 24,
then common sense without any arithmetic knowledge
would tell me that, gee, if I take 5/24 out of 24
and then 3 more out of 24, totally I have taken 8 out of 24.
So if the denominators were the same and we wanted to add,
it seems common sense
that we would simply leave the denominator alone
because we still have the same number of original subdivisions,
and we simply add the top. [ 5/24 + 3/24 = 8/24 ]
But now what do we do
if the bottoms or denominators weren't the same?
You see, that's not quite an
as easy a question as this is, isn't it?
But we're still trying to do this same thing,
we're still trying to add two fractions.
So in a few lessons we will be doing this,
but we will having, be having to build you up
very, very slowly to do this because this will not be so easy.
Especially if my denominators are quite messy in a sense.
Very large numbers and very much unlike.
Then how will we add such fractions?
the reason I mention this here, some lessons
before we begin to learn this,
is what we're going to do is actually use the knowledge
we taught in the last chapter of this textbook.
So you might want to spend a few of your moments
while you're doing this very easy lesson
and the next few easy lessons
to review the lessons on prime factorizations
because we're going to have you
to prime factor each of the denominators,
and then using this prime factorization of each denominator
to find the least common multiple of the denominators.
And having done that, we will find that adding two fractions,
no matter how messy they look,
really are quite routine and not at all hard.
So if you were good at these two abilities
from the previous chapter,
when we get to this, this will be
somewhat longish, but not at all hard.
On the other hand if you had some difficulty
a few lessons ago on these,
use your time over the next lesson or so
to review and review until you are comfortable.
Then you're ready for this.
In the meantime, build your vocabulary.
A fraction is simply a code that takes a written bar,
and the number you write on the bottom
represents the parts in the whole
of whatever you're talking about.
And that number in that position we call the 'denominator.'
And the number you write in the top is
represents the parts out of that whole
that you wish to take for some kind of consideration.
And the number that we write on the top we call the 'numerator.'
And if the top number is less than the bottom number,
and if the top number is less than the bottom number
as it is here, we call the whole fraction a 'proper fraction.'
It's simply a name.
There is nothing 'proper' about it.
It's just a name for that kind of a fraction
where the top is less than the bottom.
On the other hand, if the top is larger than the bottom,
or if the top is equal to the bottom,
either of those two fractions we called 'improper fractions,'
recognizing that technically
there is nothing improper about these.
There'll be many, many times when this is the form that we need.
And of course, where the top is equal to the bottom,
that special improper fraction is equal to 1.
If the top is greater than the bottom,
that fraction will have a value greater than 1.
So stressing the importance of that simple basic vocabulary,
this is your host, Bob Finnell.