Math 20 - Lesson 3


Uploaded by PCCvideos on 08.09.2009

Transcript:
A Portland Community College mathematics telecourse.
A course in arithmetic review
produced at Portland Community College.
The Addition of Whole Numbers.
This lesson will assume that you can add whole numbers.
We simply wish to take a few moments to examine that process.
Let's recall the basic rule
that when adding
we add together only digits of the same place.
Many of us learn the rule that we line up the ends.
That's not the rule,
and that can be dangerous when we get to decimals.
It's not the ends that we line up
so much it is that we line up the ones digits,
the tens digits, [ 10s ]
the hundreds digits, [ 100s ]
and the thousands digits. [ 1000s ]
You might recall that in elementary school,
many teachers had you mark those places
to keep this idea very clear in your mind.
Then, because we have memorized the addition table,
we simply mentally add up the one's digit
so this is 12, 18, 22, 26.
26 now, but rather than writing the 20 here.
We wrote it up here and the 6 down here.
You see 26.
And the 20, of course,
belongs with the tens over in the next column.
We recall that every time we get 10 here, it becomes 1 over here.
10 of these is 1 of these.
10 of these is 1 of these, and so forth.
Now mentally adding these up we would think 10,
12,
17,
and 19.
There is my 'teen' or 10
and 9.
19.
And adding the next column 6,
10,
19.
Then finally 4.
And these tens digits from our adding we learn to call 'carries.'
The simple process and that's the one process
I'm sure that most people remember quite frequently
having done it so much in their lifetimes.
Frequently in business or science documents
you'll find your addition problem written horizontally.
Some people will actually learn to add horizontally.
It's not a bad trick to learn.
Most of us, however, prefer them written vertically.
So let's write this vertically
keeping the units places and the tens and the hundreds places
lined up
like this.
And if it will help you to draw lines
to keep your place values straight, then do so.
After all, most accountants do this
they're the professional ones at using numbers.
Then add quickly in each place column.
If you have difficulty in adding,
there are many, many tricks to memorizing additions
and learning to add in different arrangements than this.
It would take fully an hour or two-hour tape,
which we don't have time for.
So if you need some assistance
in memorizing your additions table,
then see the specialist at your school.
There are many, many devices that can be used for this.
So adding vertically very quickly we have 13,
22,
28,
34.
and the 30 comes to the next column as a carry.
Then 5,
10,
18.
So that teen came over here as a carry.
Then 5, and you're done.
You should be able to go just that fast,
and if you're going to be really professional
like an accountant, faster.
And of course nowadays
much of your arithmetic is done on calculators,
so it might be handy
if have you a calculator with you in this course
and learn to use it simultaneously.
Now of course in a calculator, these operation signs,
plus [+] become a button to be pushed,
and when you're all done pushing the buttons
you wish to know what the answer is,
so on most calculators, the equal sign, too [=] is a button.
So it is simply a matter of pushing these particular buttons.
The button 25, so 25. [2][5]
The button plus. [+]
8 plus, 3,5,9 plus. [8][+][3][5][9][+]
Now let's get this up to where you can see it better.
Eighty six, plus 106 and then equals. [8][6][+][1][0][6][=]
106 equals, and we have the result: 584. [1][0][6][=]
Now, that sounds so easy.
Seems so easy.
You might ask why are we doing it by hand?
Of course if you have your calculator, use it.
But there's an interesting fact that studies have found.
Studies show that a person who makes frequent mistakes
in hand arithmetic makes even more on calculators.
So therefore,
just because we have worked this problem with a calculator
does not guarantee that it's correct.
However, the calculator's main function
is to increase your speed.
So having done this problem so fast on the calculator,
it takes but a few more seconds to clear the calculator,
back off, and redo it once again to confirm our answer.
Therefore, always do all calculator computations
at least twice,
which I just did, and it checks.
Always twice. At least.
Now of course on a job or in personal daily life,
most problems will be given to you in words,
what some people call story problems,
word problems, or application problems,
so in fact you not only
have to be good at the basic math processes
but your language must be fairly good.
An example: The Alvarez family
has the following payments to make each month:
the rent, $430.
TV, $28.
A new stove, $78.
Car payments, $105.
What is the total of the payments that they make each month?
Now in all these words there are one or two key words.
In this case, 'total.'
Total simply means I want the results of having added,
and I want the results of having added payments.
So we simply identify all the payments
and that's what we're going to add.
So, here is our problem written in more traditional form.
Note that we've kept the ones digits in a column,
tens digits and a ones digit, the hundreds digit.
Now we can add these either on a calculator or by hand.
There are many tricks to add quickly mentally.
Perhaps your teacher will show you a few.
And if you're watching one of the extended lessons,
we'll show you a couple, next.
But adding this we would get 16, 21, carry the 2.
5, 7, 14, carry the 1.
6.
So the total Alvarez bill is $641 per month.
Now we leave it to you to do enough problems
to become comfortable and fast.
As in baseball or pinochle,
your comfort and your accuracy
is purely a matter of tons of experience.
Even a dozen problems won't do it
but ultimately hundreds of problems.
Where does the process or idea of 'carry' come from
in long addition?
Later courses in number theory or algebra
will examine this more formally, but by using the abacus,
we introduced in the last lesson,
we can actually see very clearly why we carry the way we do.
If we recall that what we do
with our numerals on a piece of paper
is simply a symbolic representation
of what we used to do with beads.
So let's examine what we now do symbolically.
Adding our units column
by having first memorized the addition tables,
we would get 10 plus 5 is 15. [ 10 + 5 = 15 ]
We would write the units digit here and carry the 1
or one 10 to the next column, and that's what we want to examine.
But adding here we get
10,
18,
26
writing the units digit here
carrying the 2, or 20, to the next column.
We would get 6 and 7 is 13 [ 6 + 7 = 13 ]
Again the units digit here [3] carry the 1 to this column,
and of course in that case, it falls straight through.
Let's look sort of historically at where these carries come from
and why they go to the next column.
Recall that the abacus
was used successfully for many, many centuries
before our ancestors actually realized
that these symbols stood for single large numbers.
This 95, to them, simply meant put 5 beads in this first column.
Which we have there.
Then put 9 more beads in this column,
which we learn to call the tens place or 10s column.
So there we have it.
We'd have to scoot it down to see, but 9 and 5
then only much, much later, actually centuries of time
did we think of this as one number, 95
and begin to write it in this manner.
Then this expression merely meant to this column
produce 6 more beads right on top of these.
We were literally adding them on top.
So 6 more beads here.
2,
4,
6.
Then in our second or tens placed, add 8 more beads.
So adding 8 more here
and then put 4 more beads here.
Of course there are no beads at all, and we put 4 more.
Recall that our ancestors one feat of abstraction, at this point,
was to realize that every time I get 10 beads here
or a hand of beads, they move back and became one over here.
And 10 here went back and became 1 over here.
So here we obviously have more than 10.
So if we were to count off 10,
2, 4,
6,
8,
10,
our rule of clearing says
those 10 go back and become 1 over here
and the same thing over here.
If we have 10 here,
2,
4,
6,
8,
10,
that goes back and becomes 1 over here.
So each time we get more than 10 here.
For each 10 that we have, it becomes one bead over here.
So if I have two groups of 10 here,
that would become two beads over here.
So continuing with our problem,
this says place 4 more beads here,
8 more beads here.
We produce 8 more,
and 7 more beads here.
But again, for every 10 we have in any column,
that becomes 1 in the next.
10 in this column,
1 in the next.
So literally when our ancestors were adding,
they were adding beads, not numbers.
With the one rule that every time they get 10 here, clear it,
and it becomes 1 over here, which we now call 'carry.'
If you have more than 10 here,
clear it and carry one over to here.
Then when they were through adding beads
and clearing groups of 10 and carrying,
they would simply write down their finished results.
So I have one bead in this column,
3 in this,
6 in this,
and 5 in this.
And this is exactly what it meant to our ancestors:
1 bead here,
3 beads here,
6 here
and 5 here.
Only after many, many centuries
did we finally learn to treat this as one number,
1365. One thousand, three hundred sixty five.
But can you begin to see physically
where this idea of carrying came from
and why it's always in groups of 10 that we carry?
And of course that's because we counted
originally on our hands,
which have 10 fingers.
We could have cleared by a different rule
but our numerals would then stand for a different number
and we will leave that for special courses
that you might take in the future.
Keeping this abacus in mind and how this works,
that is for every 10 in one column it becomes 1 in the next,
let's show you a shortcut that one could apply
to adding long columns of numbers.
First, to keep the columns straight, frequently helps
if you take your ruled paper and turn it sideways
and now we have some free columns
for better representing our columns of numbers
or our place values.
Now there are quite a stack of numbers even for a calculator,
but let's look back at our addition facts
and think about them a little bit differently.
For a moment, when we see something like 5 and 8,
let's not think 13. Let's just think 3
and let that teen
simply be a sort of a side thought.
So 5 and 8, [ 5 + 8 ] we'll think 3, with a teen over here.
7 and 7 is 4. [ 7 + 7 = 4 ]
And of course 8 and 1 is still 9. [ 8 + 1 = 9 ]
But 9 and 4 is 3 [ 9 + 4 = 3 ]
with that teen coming over.
And remember from our abacus
that teen always goes to the next column.
Generally when a person has difficulty
with adding a long column of numbers,
it's not the number of numbers that's bothering them
because at any one moment you're only adding two numbers.
7 and 5. [ 7 + 5 ]
That's just two numbers, which is 12. [ 7 + 5 = 12 ]
12 and 7. [ 12 + 7 ]
That's only two numbers, which is 19. [ 12 + 7 = 19 ]
19 and 4 is 23. [ 19 + 4 = 23 ]
Just two numbers.
So at any one moment, on paper or mentally,
you're only adding two numbers.
So what really bothers one with a long column addition are,
is holding in your mind all those carries.
Let's think about that a moment.
All the carries ultimately
are going to come over to this next column anyway.
So let's think this way.
7 and 5 is 12. [ 7 + 5 = 12 ]
Well that teen will come over here
so I will just put a little tic mark here.
12. That leaves 2.
Let's keep going with just two.
2 and 7 is 9. [ 2 + 7 = 9 ]
9 and 4 is 3. [ 9 + 4 = 3 ]
Actually it's 13.
Teen comes over here.
3 keeps on going.
3 and 8 is 11. [ 3 + 8 = 11 ]
Teen comes here, 1 keeps going.
1 and 4 is 5. [ 1 + 4 = 5 ]
5 and 2 is 7, [ 5 + 2 = 7 ]
7 and 9 is 6 with that teen. [ 7 + 9 = 6 ]
Now forget these just a moment.
We've got them already over to the next column
where they're going to go.
so, with a little bit of practice you'll actually go this fast.
Now watch me here.
8 and 9 is 7. [ 8 + 9 = 7 ]
7 and 3 is 10. [ 7 + 3 = 10 ]
6 and 7 is 3. [ 6 + 7 = 3 ]
9, 9 and 3 is 2. [ 9 + 3 = 2 ]
2 and 8 is 0. [ 2 + 8 = 0 ]
Now can you see what I was doing?
All of these marks are the teens
and I just took the units digit and kept on going.
8 and 9 is 17. [ 8 + 9 = 17 ]
Teen, took the 7 and keep on going.
7 and 3 is 10. [ 7 + 3 = 10 ]
Teen, zero, keep on going.
6 and 7 is 13. [ 6 + 7 = 13 ]
Teen, take the 3, keep on going.
3 and 6 is 9. [ 3 + 6 = 9 ]
9 and 3 is 12. [ 9 + 3 = 12 ]
There's my teen.
2, 2 and 8 is 10. [ 2 + 8 = 10 ]
There's my teen and zero.
So with very little practice you're going like this.
1,
5,
3,
2,
8,
3,
6.
Now all of these tic marks, there are 1, 2, 3, 4, here.
Well, they were the carries
that were coming over to this next column anyway,
which is where I put them.
And here I have 1, 2, 3, 4, 5 that was coming over to this column
And here I have 1, 2, 3, 4, which was coming to this column.
So I'm adding all of my carries
when I'm done, rather than as I go,
and that saves me having to hold them in my head as I go along.
Rather than holding them in my head,
I was just putting them down here one at a time as I get to them,
so now I get 6,
4,
11.
Carry the 1 in its simple case, and I get my answer.
With very little practice,
one can make this procedure enormously fast.
Interestingly enough,
after I was done I added these columns and numbers
on a rather large sophisticated calculator,
and of course I got the original, the same numbers,
but originally, when I added on a calculator,
I got a different number here than I did on paper.
So naturally your first thought is I made a mistake here,
but actually the mistake was made on the calculator, not here.
Again, the point that we made at the beginning of this lesson,
for a beginner on a calculator
you will tend to make more mistakes on the calculator
than you will by hand.
The calculator is simply quite fast.
So after doing it once on the calculator
and writing your answer down on a piece of paper,
then clear it,
and do it once again to see if you can verify the same answer.
And you'll be surprised how accurate your hand arithmetic is
to your calculator arithmetic
until you've had much, much experience with the calculator.
Again let us remind you that as you're adding numbers
which are given to you in a row,
as they usually will be in written documents,
if you're going to rewrite them in column form,
do remind yourself to keep the places lined up.
See here we have the units all in the same column
the tens [ 10s ] all in the same column,
the hundreds [ 100s ]
and the thousands. [ 1000s ]
but again see how your eye begins to want to wander?
So again it's to your advantage if either turn your paper on end
or give yourself some clear columns in which to work.
Then if you're using this trick that we just used,
every time you get over 10,
if you put the 10 over into this column as a little tic mark
and keep your units digit and keep right on going.
So 5 and 6 is 1 [ 5 + 6 = 1 ]
with a teen over here. [ 5 + 6 = 1 + 1 teen. ]
1 and 4 is 5. [ 1 + 4 = 5 ]
And 9 is 4 [ 5 + 9 = 4 ]
[ 5 + 9 = 4 + 1 teen ] with my teen over here.
So 4 and 8 is 12.
2, keep going.
2 and 8 is 0.
Then I have 1, 1,
here I have 1, 2 coming over to here.
And here I have 1, 2, coming over here.
And of course some of you might prefer
to just pick these teens up as you go down the column.
That's fine, too.
Or another trick that many of you might have learned
in your early school years,
particularly with long columns of numbers
is to scan the column first
and find combinations that will give you 10.
So there's 10, [ 7 + 3 = 10 ]
which I bring over to this column.
Now do you see any other combinations that will give you 10?
There's another 10 [ 8 + 2 = 10 ]
and that will leave you now with 7 here. [ 6 + 1 = 7 ]
Then scan this looking for combinations that will give you 10.
Well, there's one. [ 6 + 4 = 10 ]
There's one. [ 9 + 1 = 10 ]
And now if you don't see any others
then just add what you have including these carries.
So there's 1, 6, 14, 21. [ 1 + 5 + 8 + 7 ]
Now over here look for combinations of 10 once again.
There's ten; 8 and 2 is 10. [ 8 + 2 = 10 ]
5 and 5 is 10. [ 5 + 5 = 10 ]
Okay, no more obvious ones.
So there's 2, 7, 11, 15. [ 2 + 5 + 4 + 4 ]
Carry the 1 again
and 3.
So see here we've shown you two tricks
looking for combinations of 10 or just straight out addition,
but every time you get a combination greater than 10,
you put the teen here,
take the units digit and keep on adding
until you get another teen,
then you add all the teens in after the fact.
Use these short cuts if they seem convenient to you.
If you're one of the fortunate students
that can simply add straight out and carry in your head,
then you will probably want to stay with that kind of an approach
if it's been giving you so much success for so many years.
For instance, a really experienced person
would simply think this way: 7,
15,
24,
25,
32.
11,
14,
16,
20,
29.
7,
11,
20,
27,
35.
And possibly even faster than that.
So if you are one of the fortunate few
that can add in that manner and that quickly,
we say bless you stay with it.
But all of these techniques
are predicated on the underlying assumption
that you have your addition table
and later your subtraction and multiplication tables memorized.
Do you?
If not, then do see your instructor
or your school's tutorial center.
Let them know that you don't have these tables memorized
so that they can give you the extra help, drill,
and outside practice to have these basic tables memorized.
It's the start of all of our thinking in arithmetic.
Without it, it just isn't possible.
But do you get a feel for how man has gone? From the abacus
to counting beads to carrying
to eventually memorizing a table
which is simply the memorized results
of what happened on this bead board.
Then from there
to doing in our minds what this did,
and then finally through all of that
to the calculator and computer.
So practice and use the practice
as a bit of a lubrication exercise for your thinking
so that as we move into more advanced arithmetic
and later algebra and business math,
you're quite smooth and comfortable in these basic operations.
So until your next lesson on subtraction,
this is your host, Bob Finnell, saying,
practice and become comfortable.
We'll see you then.