Math 20 - Lesson 10

Uploaded by PCCvideos on 09.09.2009

A Portland Community College mathematics telecourse.
A Course in Arithmetic Review
Produced at Portland Community College.
We have finished our first review
on whole numbers and their relationships.
We begin a second unit on primes and multiples.
Because some of the ideas in this next unit of study
are quite new,
one might be tempted to think of them as, well, strange
and resist some very important ideas,
ideas which will later help you to
be able to do, quite easily, problems like this.
Finding the lowest common denominator [ 5/128 + 7/244 + 11/156 ]
of these three messy fractions.
Most people, by the way can't do this.
If you follow through on this next unit,
three or four lessons very carefully,
you'll find out this is really rather routine.
This next unit will also consist of ideas
which will help those of you
who will go on to more advanced math.
So be very attentive to these new ideas.
The first concept we want to develop is the idea,
simple idea, of a multiple.
We will be asking you such questions as
Is 282 a multiple of 6?
Now, let's ask this question:
Is 282 divisible by 6?
Well, let's just see if we can divide 6 into it. [ 282 ÷ 6 ]
[ 282 ÷ 6 = 47 ] The answer is yes.
Now the answer itself is not relevant to this question.
The question was simply would this divide in evenly
or would I have a remainder?
If 6 divides into a number evenly with no remainder,
then we say that 'that number is a multiple of 6.'
How about this question:
Is 38 a multiple of 6?
Well, is 38 divisible evenly by 6? [ 38 ÷ 6 ]
And the answer is no. Just that simple.
That leads us into a general definition of 'multiple.'
Any whole number which is evenly divisible
by a given number here called 'N' is said to be 'a multiple of N'
Except zero. We'll exclude that.
Say you were given this assignment:
Find all the multiples of 8.
Well that's the same thing as saying
What are all the numbers that 8 will divide into evenly?
Well 8 will go into 8 once. [ 8 ÷ 8 = 1 ]
8 will go into 16, so 16 is a multiple. [ 16 ÷ 8 = 2 ]
8 will go into 24. [ 24 ÷ 8 = 3 ]
But there's an easier way of looking at this.
Let's just list the counting numbers:
1, 2, 3, 4, 5, 6... and so on forever.
Now notice, this first multiple of eight
is simply 1 times 8. [ 1 x 8 = 8 ]
2 times 8 gives me the next. [ 2 x 8 = 16 ]
3 times 8 gave me 24. [ 3 · 8 = 24 ]
And so on.
4 times 8 is 32. [ 4 · 8 = 32 ]
5 times 8 is 40. [ 5 · 8 = 40 ]
6 times 8 is 48. [ 6 · 8 = 48 ]
So if we're trying to find the multiples of 8,
all we have to do is take all the counting numbers
and one by one multiply it by 8.
And what we get is a string of all the numbers
that 8 will divide back into.
Well since the counting numbers go on forever,
the list of all the multiples of 8 is an impossible list.
It goes on forever.
So a point of fact,
the list of all the multiples of any given number is endless.
So you see the point of this lesson is very simple:
to develop a very strong clean feel
for the use of the word 'multiple.'
So if we wished, in this case,
to have the first five multiples of 12,
then we simply list the first five counting numbers
and multiply each one of them times 12.
1 times 12 is 12, [ 1 · 12 = 12 ]
[ 2 · 12 = 24 ] 24,
[ 3 · 12 = 36 ] 36,
[ 4 · 12 = 48 ] 48,
[ 5 · 12 = 60 ] 60.
And we have the first five multiples of 12.
And of course this list goes on forever.
But these are the first five. [ 12, 24, 36, 48, 60 ]
A simple idea?
Very simple idea but very, very important.
Now do you realize that this is a true statement?
All even numbers are multiples of two.
That is 2 will divide into every one of them.
That's what we mean by an even number.
So you see this lesson is primarily about language
and the use of one word in this math language. 'Multiple.'
Let's look at this problem.
9 divides into 108, 12 times. [ 108 ÷ 9 = 12 ]
9 goes into 108 evenly; therefore, we can say about 9 and 108
that "108 is a multiple of 9."
That is "it's divisible by 9,"
but that's just one way of saying a fact
that can be said several other ways.
If we had said "9 is a factor of 108" that means the same thing,
but now we're emphasizing this first.
This divides in evenly. It makes it a factor.
But we could have also said "9 divides 108 evenly."
And we could have even said "108 is divisible by 9."
These are four ways of saying the same thing.
It's just a matter of language.
Here we're concentrating
on one particular aspect of this language. [ multiple ]
But again going back to what many students say
at the beginning of many math courses,
they will say, "I cannot understand the teacher"
or "I cannot understand the book."
Invariably what they're really saying is this:
"My language is not up to that person." "My language."
After all, if I cannot speak or understand German,
I cannot understand somebody else
no matter how simple they are saying it in German.
In reality, most statements in mathematics are quite simple
if, and that's a big important 'if,'
if the listener is equipped with the right language.
That's what this course seeks to develop along with the review.
The two, the review and the language, simply belong together.
You might ask why can't scientists and mathematicians
speak in an everyday street language?
The fact of the matter is the kind of things they need to say
later in your math career
cannot easily be reduced to street language.
It needs a technical language.
Therefore, this lesson simply seeks to familiarize you
with the technical meaning of one word: 'multiple.'
We will use that concept along with three or four others
to explain some fairly complicated processes.
Unless you know those words, then they become fairly simple.
And we do this by asking some rather simple questions
or asking you to do certain things.
For instance, here "List the first three multiples of 75."
If one has to ask the question: How do I do this?
Then this should tell you
that you don't yet know what we mean by the word 'multiples.'
So it's not a matter of how to, in this case.
It's a matter of what do we mean by this word?
And 'multiples' means that we want to take the counting numbers,
in this case the first three, and multiply each one times 75.
So 1 times 75, of course is 75. [ 1 · 75 = 75 ]
2 times 75 is 150. [ 2 · 75 = 150 ]
three times 75 is 225. [ 3 · 75 = 225 ]
So these we call the first three 'multiples of 75.'
Now notice the number we want to find the multiple of,
75 in this case, [ 75 · 1 = 75 ] is always the first multiple.
Now you can take one times it, two times it, [ 75 · 2 = 150 ]
three times it, [ 75 · 3 = 225 ]
or you can take 75 plus 75 to get the next one, [ 75 + 75 = 150 ]
plus another 75 to get the next one, [ 150 + 75 = 225 ]
plus another 75 to get the next one, [ 225 + 75 = 300 ]
which is the fourth multiple.
Now notice in this case if I took the fourth multiple of 75,
which is 300,
and divided the fourth multiple, 300,
by the number I'm finding the multiples of, [ 300 ÷ 4 ]
it comes out to be that. [ 300 ÷ 4 = 75 ]
And naturally; if 4 times 75 is this, [ 4 · 75 = 300 ]
then 75 into this must be 4. [ 300 ÷ 75 = 4 ]
But realizing that very simple relationship
will be expressed in your understanding
of such a question as this.
The question: Is 138 a multiple of 23?
That question can be restated in the form
"Is 138 evenly divisible by 23?"
Or we could have said "Will 23 divide into 138?"
Or, we could have said "Is 23 a factor of 128?"
Now this point is easy to see
a student can become somewhat perturbed and think
just words, words, words, words,
but you must realize yes, that's exactly it,
and that's what we're trying to stress here
is to begin to develop a vocabulary
so you can state an idea from several different points of views
using different words without getting lost in the meaning.
So in this case, to ask, Is 138 a multiple of 23?
which is where the focus of our vocabulary building is now,
is to ask
if I took 138, could I divide it by 23 and get zero remainder?
See, that's yet a fifth way of saying this.
And it looks like it will go in 6 times.
It will.
So the fact that I get a zero remainder
is what tells me the answer to this question is: Yes.
In fact, 138 is the sixth multiple of 23.
That is, if I were to start with 23, which is my first multiple,
then 2 times 23 would give me 46, so 46 is the second multiple.
Then 3 times 23 would be 69 [ 3 · 23 = 69 ]
or the third multiple of 23.
Then 4 times 23 would be 92, [ 4 · 23 = 92 ]
so 92 is a fourth multiple.
5 times 23, 15 carry the 1, [ 5 · 23 ]
11, so 115 is the fifth multiple [ 5 · 23 = 115 ]
and we're claiming that the sixth multiple of 23 is 138.
So we can take 6 times 23 [ 6 · 23 ]
or simply add 23 more. [ 115 + 23 ]
And of course if we add this,
[ 115 + 23 = 138 ] we do indeed get 138.
So see, over and over
we're not so much telling you how to do something.
These are all things hopefully you know already.
We're playing with words until you become comfortable
with sliding from one set of words to another.
Until soon using the vocabulary
we will be able to tell you how to do this very, very simply.
A matter which you probably have not been told
in all of your education, up to now,
not because it's complicated,
but because you have yet to build your vocabulary
so that we can use those technical words
to tell you in one, two, three, four fashion
exactly what things to do
to find that one number that all three of these can divide into,
which is a great large number.
But we will indeed end this chapter in just three or four lessons
by being able to tell you how to do this,
and it will be quite simple at that time,
if we keep working on this vocabulary.
So while doing these lessons of learning new vocabulary,
let's as each, a part of each of these lessons
constantly review a bit of our past vocabulary
so we can tie all of that together in the near future.
Some simple ones to get warmed up.
Find the sum of anything.
Remember the 'sum' is simply a vocabulary word
which indicates 'the results of having added.'
Added what? Well in this case, these five numbers.
So, this key word [ sum ] is telling us to do this.
And in doing that, can you see
we also get some practice in adding?
See if I can do this quickly and see if you can follow me.
Carry the 3.
Carry the 2,
Now I went through that rather rapidly,
so let's see if it checks on a calculator.
So, 53 added to 85, [5][3][+][8][5] added to 9, [+][9]
added to 106, [+][1][0][6] added to 47 [+][4][7]
equals 300. [=] 300
So do practice your long-hand arithmetic.
At the same time we encourage you
to begin to become comfortable with the calculator
to check your longhand practice,
but most importantly to realize, in this case,
the word 'sum' means 'the results of having added.'
Always as you're reading, be on the lookout for that key word.
'Difference' means 'the results of having subtracted.'
And always the first number that's given is written first,
and the second number is written secondly,
and if you're going to do it on a calculator,
this is as easy a way to write it as there is.
If you're going to do it by hand,
the vertical form is perhaps better.
So 5 from 10 is 5. [10 – 5 = 5 ]
9 from 9 is zero. [ 9 – 9 = 0 ]
8 from 9 is 1. [ 9 – 8 = 1 ]
4 from 7 is 3. [ 7 – 4 = 3 ]
Did you follow me? So again by hand.
Then check by calculator
8000 minus 4895 is? [8][0][0][0][-][4][8][9][5][=]
3105. It checks.
But the key word here is 'difference,'
which means 'the results of having subtracted,'
and the first number given is the first one, or the top one.
Second number given is the second one, or the bottom one.
Then subtract in the exact order that it's given to you
when you use the word 'difference.'
Other words might give us a different order to these listings.
Another one. What's the key word here?
And of course it's the word 'product.' Isn't it?
The 'product' meaning 'the results of having multipled,'
in this case, 13 and 19. [ 13 x 19 ]
So however you're going to do it,
the key to this problem is not getting an answer.
See we learn how to do this in elementary school
and reviewed it in the last chapter.
What we're trying to do now is tie down vocabulary.
And of course, if you know what the vocabulary is saying,
you know exactly what to do.
So, on these types of problems,
it's not the answer that's important.
In reading this statement did you know what to do?
If you did, then that and that alone is the lesson.
Of course it does give us some practice which we can always use.
Then along the way we begin to give you more complicated,
what you and I probably used to call 'story problems,'
not to see if you know how to do this,
but to see are you developing a good enough vocabulary
so that in reading this you will know what they're talking about
and hence you will know what to do.
So in this case let's see
if we can begin to get a feel for what's being said here
and what's being asked for.
I have a storage tank containing 2090 gallons of gasoline.
Now sometimes it will help you
if you draw a little bit of a sketch of what's happening.
Okay. So in this case I've got 2090 gallons
in a large storage tank.
If 12 gallons are pumped into each of 35 cars,
so we've got 35 cars here,
and each of them are going to have
every single day, 12 gallons in each.
Then how long, how many days
will this large storage tank last this supplier?
Well first, this right here stands for a day. Doesn't it?
So we have 12 gallons being filled in 35 times.
So, each day I will use 35 times 12 gallons of gas in these cars.
Now are you beginning to see what's happening here?
And see that I'm using a diagram in this case
to sort of feel my way through what they're saying
if I don't see it in the words alone.
Of course with practice you will see it in the words alone.
So, multiplying these two [ 35 x 12 ]
we get 420 gallons, but for what? [ 35 x 12 = 420 ]
Well, 420 gallons of gas per day in those 35 cars.
So we're, now we're asking
how many of these [ 420 gal/35 ] are in here [ 2090 gal ]?
Here's one 420, here's another 420 etc.,
so we're asking, "How many times will 420 go into this [ 2090 ]?"
Here's once, here's twice.
So basically we're asking at this point
In 2090, how many times will 420 go into it? [ 2090 ÷ 420 = ? ]
And it looks like about four. Doesn't it?
See 0, four 2s is 8, [ 4 · 2 = 8 ]
four 4s is 16. [ 4 · 4 = 16 ]
Subtracting. [ 2090 - 1680 ]
We have 0, [ 0 - 0 = 0 ]
[ 9 - 8 = 1 ] 1,
[ 20 - 16 = 4] 4.
So if this is how many are going in each day,
this will go in 4 times, leaving me when I'm all done, 410 gallons.
So the answer to my question:
How many full days will the tank last?
Four days.
And of course you can see very, very easily
that this is the type of problem
that a service station owner has to be aware of
or any grocery operator.
They must know ahead of time when things are going to run out,
so they can project ahead and order ahead of time.
You can't wait until you're out and then order it.
The business would go broke trying to do that.
So what a businessman must do is anticipate what's happening,
anticipate what he's had,
and be able to answer questions like this.
So this type problem is being done a multitude of times
every single day
in dozens of places in your very neighborhood.
So the point we're making is this:
In math, a good vocabulary is a must.
It's not nice. It is a must. It is impossible without it.
So we will constantly nudge you on vocabulary.
And in this particular lesson, our key vocabulary word
along with all of these we just reviewed is:
Do you know what is meant by the word
the 'multiple' of a given number?
If so, you could answer this question very nicely.
What is the sixth multiple of 9? Well it's simply
6 times 9 which is 54. [ 6 · 9 = 54 ]
Now what do we mean by multiple?
Well it means if I started with 9, and counted by 9s
so 9 and 9 is 18. [ 9 + 9 = 18 ]
18 and 9 is 27. [ 18 + 9 = 27 ]
27 and 9 is 36. [ 27 + 9 = 36 ]
36 and 9 is 54. [ 36 + 9 = 54 ]
Then if I counted by 9s, that the sixth number I would get to
by thus counting by 9s is 54. [ 9, 18, 27, 36, 45, 54 ]
So we're saying more than the fact
that 9 will divide into 54 to be a multiple of 9 [ 54 ÷ 9 ]
to say that the sixth multiple means that in fact
it is the sixth number in my list
that I could divide evenly by 9.
And it's that kind of sequencing
that this word 'multiple' allows us to focus in on
by our communications.
So if I were to ask to list
the first four multiples of any number
without even knowing what this number is,
you know immediately you're going to take one times it,
two times it, three times it, and four times it.
So if somebody were to say
I want the first four multiples of say 20,
you're going to take 1 times 20, which gives me 20. [ 1 · 20=20 ]
2 times 20, which is 40. [ 2 · 20 = 40 ]
3 times 20, which is 60. [ 3 · 20 = 60 ]
4 times 20, which is 80. [ 4 · 20 = 80 ]
So these are my first four multiples of 20 [ 20, 40, 60, 80 ]
and of course 20 itself is the first.
And furthermore, and this is important to our future lessons,
this list goes on forever.
It's an infinite set of numbers.
And of course
a good teacher can check your understanding of the vocabulary
by rephrasing questions using that word many, many, many ways
to see if you can hang with the meaning of this.
So what is the sum of the first three multiples of 14?
Well, the first three multiples
would be 1 times 14 is 14. [ 1 · 14 = 14 ]
2 times 14 is 28. [ 2 · 14 = 28 ]
3 times 14 is 42. [ 3 · 14 = 42 ]
And this says, What is the sum of those first three multiples?
And 'sum' means 'the results of having added.'
And of course that sum is 84. [ 14 + 28 + 42 = 84 ]
Now please note as you pursue this and future courses,
the answer to this arithmetic is somewhat irrelevant.
It's merely a vehicle to see if you, the student,
really caught the vocabulary.
If you need to ask somebody else How do I do this problem?
Realize, being honest,
you have missed the whole meat of the lesson
because if the lesson is do you know what this vocabulary is?
If you do, you will know what to do.
Problems will change all the time, but the vocabulary,
that's with us for the rest of our math career.
This is your host, Bob Finnell.