Mixed numbers and improper fractions

WEBVTT
Kind: captions
Language: en

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We're now going to learn how
to go from mixed numbers to

00:00:03.610 --> 00:00:05.680
improper fractions
and vice versa.

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So first a little
bit of terminology.

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What is a mixed number?

00:00:08.490 --> 00:00:10.350
Well, you've probably seen
someone write, let's

00:00:10.350 --> 00:00:13.850
say, 2 and 1/2.

00:00:13.850 --> 00:00:15.470
This is a mixed number.

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You're saying why is
it a mixed number?

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Well, because we're including a
whole number and a fraction.

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So that's why it's
a mixed number.

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It's a whole number
mixed with a fraction.

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So 2 and 1/2.

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And I think you have a sense
of what 2 and 1/2 is.

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It's some place halfway
between 2 and 3.

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And what's an
improper fractional?

00:00:32.086 --> 00:00:36.260
Well an Improper fraction is a
fraction where the numerator is

00:00:36.260 --> 00:00:37.410
larger than the denominator.

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So let's give an example
of an improper fraction.

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I'm just going to pick
some random numbers.

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Let's say I had 23 over 5.

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This is an improper fraction.

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Why?

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Because 23 is larger than 5.

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It's that simple.

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It turns out that you can
convert an improper fraction

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into a mixed number or a
mixed number into an

00:01:00.040 --> 00:01:01.360
improper fraction.

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So let's start with the latter.

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Let's learn how to do a
mixed number into an

00:01:05.380 --> 00:01:06.600
improper fraction.

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So first I'll just show you
kind of just the basic

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systematic way of doing it.

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It'll always give you the right
answer, and then hopefully I'll

00:01:13.190 --> 00:01:15.110
give you a little intuition
for why it works.

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So if I wanted to convert 2 and
1/2 into an improper fraction

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or I want to unmix it you could
say, all I do is I take the

00:01:22.900 --> 00:01:27.190
denominator in the fraction
part, multiply it by the whole

00:01:27.190 --> 00:01:30.150
number, and add the numerator.

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So let's do that.

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I think if we do enough
examples you'll

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get the pattern.

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So 2 times 2 is 4 plus 1 is 5.

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So let's write that.

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It's 2 times 2 plus 1,
and that's going to

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be the new numerator.

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And it's going to be all of
that over the old denominator.

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So that equals 5/2.

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So 2 and 1/2 is equal to 5/2.

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Let's do another one.

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Let's say I had 4 and 2/3.

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This is equal to -- so this
is going to be all over 3.

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We keep the
denominator the same.

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And then new numerator is
going to be 3 times 4 plus 2.

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So it's going to be 3 times 4,
and then you're going to add 2.

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Well that equals 3 times 4 --
order of operations, you always

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do multiplication first, and
that's actually the way

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I taught it how to
convert this anyway.

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3 times 4 is 12 plus 2 is 14.

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So that equals 14 over 3.

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Let's do another one.

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Let's say I had 6 and 17/18.

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I gave myself a hard problem.

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Well, we just keep the
denominator the same.

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And then new numerator is
going to be 18 times 6

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or 6 times 18 plus 17.

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Well 6 times 18.

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Let's see, that's 60 plus
48 it's 108, so that

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equals 108 plus 17.

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All that over 18.

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108 plus 17 is equal
to 125 over 18.

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So, 6 and 17/18 is
equal to 125 over 18.

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Let's do a couple more.

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And in a couple minutes I'm
going to teach you how to go

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the other way, how to go from
an improper fraction

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to a mixed number.

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And this one I'm going to try
to give you a little bit of

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intuition for, why what I'm
teaching you actually works.

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So let's say 2 and 1/4.

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If we use the -- I guess you'd
call it a system that I just

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showed you -- that equals
4 times 2 plus 1 over 4.

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Well that equals, 4 times 2
is 8 plus 1 is 9, 9 over 4.

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I want to give you an intuition
for why this actually works.

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So 2 and 1/4, let's actually
draw that, see what

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it looks like.

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So let's put this back into
kind of the pie analogy.

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So that's equal to one pie.

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Two pies.

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And then let's say
a 1/4 of a pie.

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A 1/4 is like this.

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2 and 1/4, and ignore
this, this is nothing.

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It's not a decimal point --
actually, let me erase it so it

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doesn't confuse you even more.

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So go back to the
pieces of the pie.

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So there's 2 and
1/4 pieces of pie.

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And we want to re-write
this as just how many 1/4s

00:05:03.000 --> 00:05:04.800
of pie are there total.

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Well if we take each of these
pieces -- I need to change the

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color -- if we take each of
these pieces and we divide it

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into 1/4s, we can now say how
many total 1/4s of

00:05:17.800 --> 00:05:19.060
pie do we have?

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Well we have 1, 2, 3, 4,
5, 6, 7, 8, 9 fourths.

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Makes sense, right?

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2 and 1/4 is the
same thing as 9/4.

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And this will work
with any fraction.

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So let's go the other way.

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Let's figure out how to go
from an improper fraction

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to a mixed number.

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Let's say I had 23 over 5.

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So here we go in the
opposite direction.

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We actually take the
denominator, we say how

00:05:53.290 --> 00:05:55.080
many times does it go
into the numerator.

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And then we figure
out the remainder.

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So let's say 5 goes into
23 -- well, 5 goes

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into 23 four times.

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4 times 5 is 20.

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And the remainder is 3.

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So 23 over 5, we can say
that's equal to 4 and in

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the remainder 3 over 5.

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So it's 4 and 3/5.

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Let's review what we just did.

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We just took the denominator
and divided it into

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the numerator.

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So 5 goes into 23 four times.

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And what's left over is 3.

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So, 5 goes into 23,
4 and 3/5 times.

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Or another way of saying that
is 23 over 5 is 4 and 3/5.

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Let's do another
example like that.

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Let's say 17 over 8.

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What does that equal
as a mixed number?

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You can actually do this
in your head, but I'll

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write it out just so
you don't get confused.

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8 goes into 17 two times.

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2 times 8 is 16.

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17 minus 16 is 1.

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Remainder 1.

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So, 17 over 8 is equal to 2
-- that's this 2 -- and 1/8.

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Because we have
one 8 left over.

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Let me show you kind of a
visual way of representing this

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too, so it actually makes sense
how this conversion is working.

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Let's say I had 5/2, right?

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So that literally means I have
5 halves, or if we go back to

00:07:37.780 --> 00:07:42.310
the pizza or the pie analogy,
let's draw my five

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halves of pizza.

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So let's say I have one half of
pizza here, and let's say I

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have another half
of pizza here.

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I just flipped it over.

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So that's 2.

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So it's 1 half, 2 halves.

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So that's three halves.

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And then I have a
fourth half here.

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These are halves of pizza,
and then I have a fifth

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half here, right?

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So that's 5/2.

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Well, if we look at this, if we
combine these two halves, this

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is equal to 1 piece, I have
another piece, and then I

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have half of a piece, right?

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So that is equal to 2
and 1/2 pieces of pie.

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Hopefully that doesn't
confuse you too much.

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And if we wanted to do this the
systematic way, we could have

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said 2 goes into 5 -- well, 2
goes into 5 two times, and

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that 2 is right here.

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And then 2 times 2 is 4.

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5 minus 4 is 1, so the
remainder is 1, and

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that's what we use here.

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And of course, we keep the
denominator the same.

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So 5/2 equals 2 and 1/2.

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Hopefully that gives you a
sense of how to go from one

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mixed number to an improper
fraction, and vice versa,

00:09:04.570 --> 00:09:07.520
from an improper fraction
to a mixed number.

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If you're still confused
let me know and I might

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make some more modules.

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Have fun with the exercises.


Converting fractions to decimals

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:03.770
I'll now show you how
to convert a fraction

00:00:03.770 --> 00:00:04.920
into a decimal.

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And if we have time, maybe
we'll learn how to do a

00:00:06.990 --> 00:00:08.730
decimal into a fraction.

00:00:08.730 --> 00:00:11.420
So let's start with, what
I would say, is a fairly

00:00:11.420 --> 00:00:12.480
straightforward example.

00:00:12.480 --> 00:00:15.210
Let's start with
the fraction 1/2.

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And I want to convert
that into a decimal.

00:00:17.390 --> 00:00:20.170
So the method I'm going to
show you will always work.

00:00:20.170 --> 00:00:22.850
What you do is you take the
denominator and you divide

00:00:22.850 --> 00:00:24.530
it into the numerator.

00:00:24.530 --> 00:00:25.510
Let's see how that works.

00:00:25.510 --> 00:00:29.110
So we take the denominator-- is
2-- and we're going to divide

00:00:29.110 --> 00:00:32.280
that into the numerator, 1.

00:00:32.280 --> 00:00:34.110
And you're probably saying,
well, how do I divide 2 into 1?

00:00:34.110 --> 00:00:37.010
Well, if you remember from the
dividing decimals module, we

00:00:37.010 --> 00:00:40.220
can just add a decimal point
here and add some trailing 0's.

00:00:40.220 --> 00:00:42.880
We haven't actually changed the
value of the number, but we're

00:00:42.880 --> 00:00:45.260
just getting some
precision here.

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We put the decimal point here.

00:00:50.260 --> 00:00:50.650
Does 2 go into 1?

00:00:50.650 --> 00:00:51.280
No.

00:00:51.280 --> 00:00:56.180
2 goes into 10, so we go 2
goes into 10 five times.

00:00:56.180 --> 00:00:59.060
5 times 2 is 10.

00:00:59.060 --> 00:01:00.050
Remainder of 0.

00:01:00.050 --> 00:01:01.150
We're done.

00:01:01.150 --> 00:01:06.675
So 1/2 is equal to 0.5.

00:01:10.570 --> 00:01:12.050
Let's do a slightly harder one.

00:01:12.050 --> 00:01:15.000
Let's figure out 1/3.

00:01:15.000 --> 00:01:19.190
Well, once again, we take the
denominator, 3, and we divide

00:01:19.190 --> 00:01:20.740
it into the numerator.

00:01:20.740 --> 00:01:25.470
And I'm just going to add a
bunch of trailing 0's here.

00:01:25.470 --> 00:01:27.800
3 goes into-- well, 3
doesn't go into 1.

00:01:27.800 --> 00:01:30.150
3 goes into 10 three times.

00:01:30.150 --> 00:01:32.452
3 times 3 is 9.

00:01:32.452 --> 00:01:35.720
Let's subtract, get a
1, bring down the 0.

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3 goes into 10 three times.

00:01:37.700 --> 00:01:39.700
Actually, this decimal
point is right here.

00:01:39.700 --> 00:01:42.710
3 times 3 is 9.

00:01:42.710 --> 00:01:43.930
Do you see a pattern here?

00:01:43.930 --> 00:01:45.070
We keep getting the same thing.

00:01:45.070 --> 00:01:47.350
As you see it's
actually 0.3333.

00:01:47.350 --> 00:01:48.830
It goes on forever.

00:01:48.830 --> 00:01:52.160
And a way to actually represent
this, obviously you can't write

00:01:52.160 --> 00:01:54.020
an infinite number of 3's.

00:01:54.020 --> 00:02:00.430
Is you could just write 0.--
well, you could write 0.33

00:02:00.430 --> 00:02:03.060
repeating, which means that
the 0.33 will go on forever.

00:02:03.060 --> 00:02:06.960
Or you can actually even
say 0.3 repeating.

00:02:06.960 --> 00:02:08.630
Although I tend to
see this more often.

00:02:08.630 --> 00:02:09.840
Maybe I'm just mistaken.

00:02:09.840 --> 00:02:12.410
But in general, this line on
top of the decimal means

00:02:12.410 --> 00:02:17.320
that this number pattern
repeats indefinitely.

00:02:17.320 --> 00:02:25.210
So 1/3 is equal to 0.33333
and it goes on forever.

00:02:25.210 --> 00:02:29.770
Another way of writing
that is 0.33 repeating.

00:02:29.770 --> 00:02:33.400
Let's do a couple of, maybe a
little bit harder, but they

00:02:33.400 --> 00:02:35.060
all follow the same pattern.

00:02:35.060 --> 00:02:36.890
Let me pick some weird numbers.

00:02:40.470 --> 00:02:41.890
Let me actually do an
improper fraction.

00:02:41.890 --> 00:02:49.050
Let me say 17/9.

00:02:49.050 --> 00:02:50.160
So here, it's interesting.

00:02:50.160 --> 00:02:52.260
The numerator is bigger
than the denominator.

00:02:52.260 --> 00:02:54.200
So actually we're going to
get a number larger than 1.

00:02:54.200 --> 00:02:55.270
But let's work it out.

00:02:55.270 --> 00:03:00.586
So we take 9 and we
divide it into 17.

00:03:00.586 --> 00:03:06.000
And let's add some trailing 0's
for the decimal point here.

00:03:06.000 --> 00:03:08.730
So 9 goes into 17 one time.

00:03:08.730 --> 00:03:11.260
1 times 9 is 9.

00:03:11.260 --> 00:03:14.040
17 minus 9 is 8.

00:03:14.040 --> 00:03:16.240
Bring down a 0.

00:03:16.240 --> 00:03:20.080
9 goes into 80-- well, we know
that 9 times 9 is 81, so it has

00:03:20.080 --> 00:03:21.830
to go into it only eight times
because it can't go

00:03:21.830 --> 00:03:23.230
into it nine times.

00:03:23.230 --> 00:03:27.010
8 times 9 is 72.

00:03:27.010 --> 00:03:29.560
80 minus 72 is 8.

00:03:29.560 --> 00:03:30.770
Bring down another 0.

00:03:30.770 --> 00:03:32.260
I think we see a
pattern forming again.

00:03:32.260 --> 00:03:35.990
9 goes into 80 eight times.

00:03:35.990 --> 00:03:40.820
8 times 9 is 72.

00:03:40.820 --> 00:03:44.350
And clearly, I could keep
doing this forever and

00:03:44.350 --> 00:03:46.790
we'd keep getting 8's.

00:03:46.790 --> 00:03:53.740
So we see 17 divided by 9 is
equal to 1.88 where the 0.88

00:03:53.740 --> 00:03:56.080
actually repeats forever.

00:03:56.080 --> 00:03:59.200
Or, if we actually wanted to
round this we could say that

00:03:59.200 --> 00:04:01.430
that is also equal to 1.--
depending where we wanted

00:04:01.430 --> 00:04:02.860
to round it, what place.

00:04:02.860 --> 00:04:05.990
We could say roughly 1.89.

00:04:05.990 --> 00:04:07.480
Or we could round in
a different place.

00:04:07.480 --> 00:04:09.310
I rounded in the 100's place.

00:04:09.310 --> 00:04:11.350
But this is actually
the exact answer.

00:04:11.350 --> 00:04:15.126
17/9 is equal to 1.88.

00:04:15.126 --> 00:04:17.380
I actually might do a separate
module, but how would we write

00:04:17.380 --> 00:04:20.730
this as a mixed number?

00:04:20.730 --> 00:04:23.030
Well actually, I'm going
to do that in a separate.

00:04:23.030 --> 00:04:24.390
I don't want to
confuse you for now.

00:04:24.390 --> 00:04:25.380
Let's do a couple
more problems.

00:04:28.560 --> 00:04:29.980
Let me do a real weird one.

00:04:29.980 --> 00:04:34.360
Let me do 17/93.

00:04:34.360 --> 00:04:36.710
What does that equal
as a decimal?

00:04:36.710 --> 00:04:39.130
Well, we do the same thing.

00:04:39.130 --> 00:04:45.630
93 goes into-- I make a really
long line up here because

00:04:45.630 --> 00:04:47.930
I don't know how many
decimal places we'll do.

00:04:50.570 --> 00:04:53.220
And remember, it's always the
denominator being divided

00:04:53.220 --> 00:04:54.930
into the numerator.

00:04:54.930 --> 00:04:56.950
This used to confuse me a lot
of times because you're often

00:04:56.950 --> 00:04:59.630
dividing a larger number
into a smaller number.

00:04:59.630 --> 00:05:02.580
So 93 goes into 17 zero times.

00:05:02.580 --> 00:05:04.080
There's a decimal.

00:05:04.080 --> 00:05:05.990
93 goes into 170?

00:05:05.990 --> 00:05:07.270
Goes into it one time.

00:05:07.270 --> 00:05:11.410
1 times 93 is 93.

00:05:11.410 --> 00:05:14.370
170 minus 93 is 77.

00:05:17.980 --> 00:05:20.360
Bring down the 0.

00:05:20.360 --> 00:05:23.700
93 goes into 770?

00:05:23.700 --> 00:05:24.660
Let's see.

00:05:24.660 --> 00:05:29.120
It will go into it, I think,
roughly eight times.

00:05:29.120 --> 00:05:33.330
8 times 3 is 24.

00:05:33.330 --> 00:05:35.970
8 times 9 is 72.

00:05:35.970 --> 00:05:39.730
Plus 2 is 74.

00:05:39.730 --> 00:05:42.186
And then we subtract.

00:05:42.186 --> 00:05:43.990
10 and 6.

00:05:43.990 --> 00:05:46.710
It's equal to 26.

00:05:46.710 --> 00:05:47.760
Then we bring down another 0.

00:05:47.760 --> 00:05:52.800
93 goes into 26--
about two times.

00:05:52.800 --> 00:05:57.020
2 times 3 is 6.

00:05:57.020 --> 00:05:58.704
18.

00:05:58.704 --> 00:05:59.920
This is 74.

00:06:03.120 --> 00:06:03.930
0.

00:06:03.930 --> 00:06:06.380
So we could keep going.

00:06:06.380 --> 00:06:08.030
We could keep figuring
out the decimal points.

00:06:08.030 --> 00:06:10.020
You could do this indefinitely.

00:06:10.020 --> 00:06:12.090
But if you wanted to at least
get an approximation, you would

00:06:12.090 --> 00:06:23.490
say 17 goes into 93 0.-- or
17/93 is equal to 0.182 and

00:06:23.490 --> 00:06:25.020
then the decimals
will keep going.

00:06:25.020 --> 00:06:27.170
And you can keep doing
it if you want.

00:06:27.170 --> 00:06:28.650
If you actually saw this on
exam they'd probably tell

00:06:28.650 --> 00:06:29.640
you to stop at some point.

00:06:29.640 --> 00:06:31.650
You know, round it to the
nearest hundredths or

00:06:31.650 --> 00:06:33.610
thousandths place.

00:06:33.610 --> 00:06:36.550
And just so you know, let's try
to convert it the other way,

00:06:36.550 --> 00:06:37.830
from decimals to fractions.

00:06:37.830 --> 00:06:40.090
Actually, this is, I
think, you'll find a

00:06:40.090 --> 00:06:42.300
much easier thing to do.

00:06:42.300 --> 00:06:49.810
If I were to ask you what
0.035 is as a fraction?

00:06:49.810 --> 00:06:56.845
Well, all you do is you say,
well, 0.035, we could write it

00:06:56.845 --> 00:07:05.130
this way-- we could write
that's the same thing as 03--

00:07:05.130 --> 00:07:06.300
well, I shouldn't write 035.

00:07:06.300 --> 00:07:10.700
That's the same
thing as 35/1,000.

00:07:10.700 --> 00:07:11.580
And you're probably
saying, Sal, how did

00:07:11.580 --> 00:07:14.120
you know it's 35/1000?

00:07:14.120 --> 00:07:18.590
Well because we went to 3--
this is the 10's place.

00:07:18.590 --> 00:07:20.230
Tenths not 10's.

00:07:20.230 --> 00:07:21.360
This is hundreths.

00:07:21.360 --> 00:07:23.230
This is the thousandths place.

00:07:23.230 --> 00:07:25.890
So we went to 3 decimals
of significance.

00:07:25.890 --> 00:07:29.260
So this is 35 thousandths.

00:07:29.260 --> 00:07:38.650
If the decimal was let's
say, if it was 0.030.

00:07:38.650 --> 00:07:40.140
There's a couple of ways
we could say this.

00:07:40.140 --> 00:07:42.490
Well, we could say, oh well
we got to 3-- we went to

00:07:42.490 --> 00:07:43.570
the thousandths Place.

00:07:43.570 --> 00:07:48.240
So this is the same
thing as 30/1,000.

00:07:48.240 --> 00:07:48.610
or.

00:07:48.610 --> 00:07:55.550
We could have also said, well,
0.030 is the same thing as

00:07:55.550 --> 00:08:02.710
0.03 because this 0 really
doesn't add any value.

00:08:02.710 --> 00:08:05.920
If we have 0.03 then we're only
going to the hundredths place.

00:08:05.920 --> 00:08:11.100
So this is the same
thing as 3/100.

00:08:11.100 --> 00:08:13.160
So let me ask you, are
these two the same?

00:08:16.330 --> 00:08:16.670
Well, yeah.

00:08:16.670 --> 00:08:17.680
Sure they are.

00:08:17.680 --> 00:08:20.065
If we divide both the numerator
and the denominator of both of

00:08:20.065 --> 00:08:24.890
these expressions by
10 we get 3/100.

00:08:24.890 --> 00:08:26.220
Let's go back to this case.

00:08:26.220 --> 00:08:27.550
Are we done with this?

00:08:27.550 --> 00:08:30.120
Is 35/1,000-- I
mean, it's right.

00:08:30.120 --> 00:08:31.660
That is a fraction.

00:08:31.660 --> 00:08:32.584
35/1,000.

00:08:32.584 --> 00:08:35.440
But if we wanted to simplify it
even more looks like we could

00:08:35.440 --> 00:08:38.530
divide both the numerator
and the denominator by 5.

00:08:38.530 --> 00:08:40.860
And then, just to get
it into simplest form,

00:08:40.860 --> 00:08:47.280
that equals 7/200.

00:08:47.280 --> 00:08:51.020
And if we wanted to convert
7/200 into a decimal using the

00:08:51.020 --> 00:08:54.150
technique we just did, so we
would do 200 goes into

00:08:54.150 --> 00:08:56.120
7 and figure it out.

00:08:56.120 --> 00:09:00.170
We should get 0.035.

00:09:00.170 --> 00:09:02.650
I'll leave that up to
you as an exercise.

00:09:02.650 --> 00:09:05.370
Hopefully now you get at least
an initial understanding of how

00:09:05.370 --> 00:09:09.320
to convert a fraction into a
decimal and maybe vice versa.

00:09:09.320 --> 00:09:11.840
And if you don't, just do
some of the practices.

00:09:11.840 --> 00:09:16.990
And I will also try to record
another module on this

00:09:16.990 --> 00:09:18.880
or another presentation.

00:09:18.880 --> 00:09:20.090
Have fun with the exercises.


Functions Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.040
Welcome to the second
presentation on functions.

00:00:03.040 --> 00:00:05.370
So let's take off where
we left off before.

00:00:05.370 --> 00:00:08.510
I still apologize -- in
retrospect that that

00:00:08.510 --> 00:00:09.770
whole foul food example.

00:00:09.770 --> 00:00:12.460
Well maybe it was helpful, so
I'm going to leave it there.

00:00:12.460 --> 00:00:15.050
Let's do some more problems.

00:00:15.050 --> 00:00:17.050
I think the best thing is to
keep doing problems with you

00:00:17.050 --> 00:00:19.050
and I think you'll see the
example, and hopefully

00:00:19.050 --> 00:00:21.960
you'll actually see that
functions are kind of fun.

00:00:21.960 --> 00:00:24.230
Let's do some more problems.

00:00:24.230 --> 00:00:25.840
Let's start off with an
example, not too different

00:00:25.840 --> 00:00:26.630
than what we saw before.

00:00:26.630 --> 00:00:47.210
Let's say that g of x is
equal to 1 if x is even, and

00:00:47.210 --> 00:00:56.030
it equals 0 if x is odd.

00:00:56.030 --> 00:01:15.595
And let's say f of x is equal
to x plus 3 times g of x.

00:01:18.240 --> 00:01:20.860
And let's say -- I'm going to
make it really complicated --

00:01:20.860 --> 00:01:22.130
well, actually I'm not going
to make it any more

00:01:22.130 --> 00:01:22.990
complicated now.

00:01:22.990 --> 00:01:24.680
So let's try some problems.

00:01:24.680 --> 00:01:26.800
So let's give an example.

00:01:26.800 --> 00:01:32.080
What is f of 5.

00:01:32.080 --> 00:01:34.180
Well, it's really pretty
straightforward.

00:01:34.180 --> 00:01:40.560
We take this 5 and we replace
it for x in the function f.

00:01:40.560 --> 00:01:49.890
So f of 5 is equal to 5 plus
3 times g of 5, right?

00:01:49.890 --> 00:01:52.190
We just literally took this
5 and replace it everywhere

00:01:52.190 --> 00:01:53.360
where we see an x.

00:01:53.360 --> 00:01:57.070
If instead of a 5, I had like a
dog here, it would be f of dog

00:01:57.070 --> 00:01:59.590
would equal dog plus
3 times g of dog.

00:01:59.590 --> 00:02:01.700
Not that that would necessarily
make any sense, but

00:02:01.700 --> 00:02:02.970
you get the idea.

00:02:02.970 --> 00:02:06.530
So f of 5 equals 5
plus 3 times g of 5.

00:02:06.530 --> 00:02:07.290
But what does that equal?

00:02:07.290 --> 00:02:12.000
So the 5 stays the same, plus 3
times -- well what's g of 5?

00:02:12.000 --> 00:02:16.750
Well, if we put 5 here,
if 5 is even we do 1,

00:02:16.750 --> 00:02:18.730
if five is odd we do 0.

00:02:18.730 --> 00:02:20.480
Well 5 is odd so it's a 0.

00:02:20.480 --> 00:02:22.270
So g of 5 is equal to 0.

00:02:22.270 --> 00:02:24.700
So this is 3 times 0.

00:02:24.700 --> 00:02:27.550
So this equals just 5,
right, because 3 times

00:02:27.550 --> 00:02:29.400
0 is equal to 5.

00:02:29.400 --> 00:02:35.050
Well what would be f of 6?

00:02:35.050 --> 00:02:42.090
Well, f of 6 would equal
6 plus 3 times g of 6.

00:02:44.650 --> 00:02:48.540
And once again, that equals 6
plus -- well, this time g of

00:02:48.540 --> 00:02:51.560
6 is, well, 6 is even, so 1.

00:02:51.560 --> 00:02:53.370
So g of 6 is equal to 1.

00:02:53.370 --> 00:02:56.300
So this equals 6
plus 3 times 1.

00:02:56.300 --> 00:03:01.080
So this equals 6 plus
3 which equals 9.

00:03:01.080 --> 00:03:02.820
I think you might be
getting the idea now.

00:03:02.820 --> 00:03:05.140
At first when you see a problem
with a lot of these functions,

00:03:05.140 --> 00:03:05.770
it seems very confusing.

00:03:05.770 --> 00:03:08.910
But if you just keep taking
what's inside of the

00:03:08.910 --> 00:03:11.960
parentheses and replacing that
for x and just keep moving

00:03:11.960 --> 00:03:15.690
along that way, you make a lot
of progress on these problems.

00:03:15.690 --> 00:03:18.260
Let's try a harder one.

00:03:18.260 --> 00:03:27.370
Let's say I said that f of x
is equal to x squared plus 1.

00:03:27.370 --> 00:03:45.440
Let's say that g of x is equal
to 2x plus f of x minus 3.

00:03:45.440 --> 00:03:53.325
And h of x is equal to 5x.

00:03:56.200 --> 00:03:58.440
Now I'm going to give
you a tough problem.

00:03:58.440 --> 00:04:04.620
What is h of g of x?

00:04:04.620 --> 00:04:05.500
No.

00:04:05.500 --> 00:04:10.660
What is h of g of -- let's
pick a number -- let's say 3?

00:04:10.660 --> 00:04:12.130
h of g of 3.

00:04:12.130 --> 00:04:14.310
Actually, we'll do examples in
the future where we actually

00:04:14.310 --> 00:04:16.520
could leave the x there
and we'll solve for it.

00:04:16.520 --> 00:04:21.810
But let's say this particular
example, what is h of g of 3?

00:04:21.810 --> 00:04:24.050
At first you might say wow,
this is crazy, Sal, I don't

00:04:24.050 --> 00:04:25.650
know how to even start here.

00:04:25.650 --> 00:04:27.200
But you just take
it step-by-step.

00:04:27.200 --> 00:04:28.170
What can we figure out?

00:04:28.170 --> 00:04:30.350
Can we figure out
what g of 3 is?

00:04:30.350 --> 00:04:31.540
Well sure.

00:04:31.540 --> 00:04:35.220
We could take the 3 and put
it into the function g and

00:04:35.220 --> 00:04:37.000
see what it spits out.

00:04:37.000 --> 00:04:39.460
So let's work on g of 3 first.

00:04:39.460 --> 00:04:46.400
So, g of 3 equals -- well it's
2 times 3, right, we're just

00:04:46.400 --> 00:04:48.450
replacing wherever we
see an x with a 3.

00:04:48.450 --> 00:04:56.970
So it's 2 times 3, so that's
6, plus f of -- what, we'll

00:04:56.970 --> 00:04:58.640
just replace the x again.

00:04:58.640 --> 00:05:02.140
3 minus 3, right?

00:05:02.140 --> 00:05:08.560
So this g of 3 is equal
to 6 plus f of what?

00:05:08.560 --> 00:05:12.500
3 minus 3 is 0.

00:05:12.500 --> 00:05:15.200
Now we have to figure
out f of 0 is.

00:05:15.200 --> 00:05:18.030
We have a definition here for
f, so we just figure it out.

00:05:18.030 --> 00:05:24.040
f of 0 is equal to -- well,
you replace the 0 here.

00:05:24.040 --> 00:05:26.810
So you get 0 squared,
which is 0 plus 1.

00:05:26.810 --> 00:05:29.130
So it's f of 0 is 1.

00:05:29.130 --> 00:05:32.910
So you take that and you
replace it for f of 0.

00:05:32.910 --> 00:05:39.360
So you get g of 3 is
equal to 6 plus 1.

00:05:39.360 --> 00:05:44.820
So g of 3 is equal to 7, right?

00:05:44.820 --> 00:05:46.490
Now we know what g
of 3 is equal to.

00:05:46.490 --> 00:05:49.470
We can substitute
that back here.

00:05:49.470 --> 00:05:52.430
So that's the same thing -- we
know g of 3 is equal to 7,

00:05:52.430 --> 00:05:56.920
so that's the same
thing as h of 7.

00:05:56.920 --> 00:06:03.360
And h of 7 is just equal
to 5 times 7 equals 35.

00:06:03.360 --> 00:06:06.060
So I think you're probably a
little confused here, and I

00:06:06.060 --> 00:06:08.270
would have been if I
was in your shoes.

00:06:08.270 --> 00:06:10.610
But the important thing is when
you first see this problem

00:06:10.610 --> 00:06:13.730
you're like what can
I tackle first?

00:06:13.730 --> 00:06:16.640
h of g of 3, it seems
very confusing.

00:06:16.640 --> 00:06:18.390
Well, g of 3, can
I tackle that?

00:06:18.390 --> 00:06:18.610
Sure.

00:06:18.610 --> 00:06:21.295
I have a definition of what
the function g does when

00:06:21.295 --> 00:06:24.280
it's given an x, or in
this case, was given a 3.

00:06:24.280 --> 00:06:25.190
And that's what we did.

00:06:25.190 --> 00:06:27.990
We figured out what
g of 3 was first.

00:06:27.990 --> 00:06:30.410
And g of 3, we just have to do
the 3, and we said well that's

00:06:30.410 --> 00:06:33.850
6 plus f of 3 minus 3, right?

00:06:33.850 --> 00:06:36.240
Because we just replaced
that x with that 3.

00:06:36.240 --> 00:06:37.630
And we just kept solving.

00:06:37.630 --> 00:06:40.070
We figured out what
f of 0 is up here.

00:06:40.070 --> 00:06:42.050
And we got g of 3 equals 7.

00:06:42.050 --> 00:06:44.290
Then we substituted that
back in right here.

00:06:44.290 --> 00:06:49.200
We got h of 7 is equal to 35
because it was 5 times 7.

00:06:49.200 --> 00:06:50.700
Let's do some more problems.

00:06:50.700 --> 00:06:54.670
Actually, let's do another
example with the same

00:06:54.670 --> 00:07:00.880
set of functions.

00:07:00.880 --> 00:07:02.980
I don't want to keep confusing
you with new functions.

00:07:02.980 --> 00:07:08.260
Let me it erase this
as fast as I can.

00:07:08.260 --> 00:07:10.925
I think I'm getting faster
at this erasing business.

00:07:15.160 --> 00:07:17.210
You can sit and think a little
bit about what we just

00:07:17.210 --> 00:07:18.110
did while I erase.

00:07:42.390 --> 00:07:44.530
So let's do another problem.

00:07:44.530 --> 00:07:56.530
What is f of h of 10?

00:07:59.450 --> 00:08:05.080
Well, first we want to figure
out what h of 10 is, right?

00:08:05.080 --> 00:08:06.510
Well, we could do it
in a different way

00:08:06.510 --> 00:08:07.170
as we'll see later.

00:08:07.170 --> 00:08:10.100
But we can figure out what
h of 10 is pretty easily.

00:08:10.100 --> 00:08:12.460
We take the 10,
substitute it in for x.

00:08:12.460 --> 00:08:14.830
h of 10 is equal to 5 times x.

00:08:14.830 --> 00:08:17.910
In this case x is 10
so it equals 50.

00:08:17.910 --> 00:08:22.910
So we know h of 10 equals 50.

00:08:22.910 --> 00:08:25.010
So we know h of 10 equals
50, so we substitute

00:08:25.010 --> 00:08:25.820
that back in here.

00:08:25.820 --> 00:08:29.080
So we say f of h of 10 is
the same thing as f of 50.

00:08:32.380 --> 00:08:34.890
And then f of 50 is, I think
pretty straightforward

00:08:34.890 --> 00:08:35.660
at this point.

00:08:35.660 --> 00:08:38.110
You just take that 50 and
replace it back here.

00:08:38.110 --> 00:08:40.150
Well, it's 50 squared plus 1.

00:08:40.150 --> 00:08:42.690
Well, 50 squared
is 2,500 plus 1.

00:08:45.750 --> 00:08:49.410
That equals 2,501.

00:08:49.410 --> 00:09:01.930
What is g of h of 1?

00:09:01.930 --> 00:09:08.920
Well, we take h of 1, h of 1 is
5, so this is equal to g of 5.

00:09:08.920 --> 00:09:12.840
And g of 5, we just replace the
5 here, so g of 5 is equal to 2

00:09:12.840 --> 00:09:16.810
times 5 plus f of 5 minus 3.

00:09:16.810 --> 00:09:19.230
We just take wherever we saw
an x and replace it with a 5.

00:09:19.230 --> 00:09:25.520
Well, that's equal to 2 times
5 is 10, plus f of 5 minus 3.

00:09:25.520 --> 00:09:27.730
Well 5 minus 3 is 2.

00:09:27.730 --> 00:09:29.950
Plus f of 2.

00:09:29.950 --> 00:09:31.470
What's f of 2?

00:09:31.470 --> 00:09:34.902
Well, 2 squared plus 1 is
5, right? f of 2 is 5

00:09:34.902 --> 00:09:36.630
-- 2 squared plus 1.

00:09:36.630 --> 00:09:41.400
So that equals 10 plus
5 which equals 15.

00:09:41.400 --> 00:09:43.340
If you're still
confused, don't worry.

00:09:43.340 --> 00:09:45.990
I'm about to record some more
problems that will give you

00:09:45.990 --> 00:09:49.780
even more examples of
function problems.

00:09:49.780 --> 00:09:51.680
See you in the next
presentation.

00:09:51.680 --> 00:09:52.980
Bye.


Functions (Part III)

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:01.220
Let's

00:00:01.220 --> 00:00:04.880
get going with more examples
of function problems, and

00:00:04.880 --> 00:00:07.200
hopefully as we keep doing
this, you're going to get

00:00:07.200 --> 00:00:08.665
the idea of how all
this stuff works.

00:00:08.665 --> 00:00:12.740
So let's do another problem.

00:00:12.740 --> 00:00:14.720
I'll use green this time.

00:00:14.720 --> 00:00:17.580
Let me clear everything.

00:00:17.580 --> 00:00:19.650
So I'll show you-- I showed you
that 1, you could define a

00:00:19.650 --> 00:00:22.920
function as just kind of a
standard algebraic expression,

00:00:22.920 --> 00:00:25.710
you could also do it a kind of
if number is odd, this is what

00:00:25.710 --> 00:00:27.980
you do, if a number is
this, is what you do.

00:00:27.980 --> 00:00:30.030
You could also define
a function visually.

00:00:30.030 --> 00:00:34.810
Let's say-- let me draw a
graph, and I'll use the line

00:00:34.810 --> 00:00:40.060
tool so it's a reasonably
neat graph-- that's

00:00:40.060 --> 00:00:43.480
the x-axis there.

00:00:43.480 --> 00:00:44.380
That's pretty good.

00:00:44.380 --> 00:00:48.830
And let's draw the f of x-axis,
or you might be used to calling

00:00:48.830 --> 00:00:54.650
that the y-axis, but-- OK.

00:00:54.650 --> 00:00:57.492
I almost had it vertical,
but let's see.

00:00:57.492 --> 00:00:59.490
Let's draw a few slashes here.

00:01:03.270 --> 00:01:05.740
And a couple here, like this.

00:01:09.970 --> 00:01:11.760
Sorry if you're getting bored
while I draw this graph.

00:01:11.760 --> 00:01:13.750
I should really have some
type of tool so that the

00:01:13.750 --> 00:01:15.410
graphs just show up.

00:01:15.410 --> 00:01:21.830
Let me draw a-- let's
say that-- let me

00:01:21.830 --> 00:01:22.500
draw this function.

00:01:22.500 --> 00:01:23.070
So this is what?

00:01:23.070 --> 00:01:28.950
This is 1, 2, 3, 4, 5, this is
negative 5, this is 5, this

00:01:28.950 --> 00:01:31.770
is 5, this is negative 5.

00:01:31.770 --> 00:01:38.410
And this is x-axis, and
this is-- we'll call

00:01:38.410 --> 00:01:39.770
this the f of x-axis.

00:01:39.770 --> 00:01:43.460
Now that might not seem obvious
to you at first, but all this

00:01:43.460 --> 00:01:47.210
is saying is let's say when x
is equal to negative 5, this

00:01:47.210 --> 00:01:50.450
function-- I'm creating a
function definition-- let's say

00:01:50.450 --> 00:01:59.436
it equals 2, that's negative 1,
that stays the same, that stays

00:01:59.436 --> 00:02:10.140
the same, then it goes to here,
and then it goes to here, to

00:02:10.140 --> 00:02:13.830
here, and then-- let's see.

00:02:13.830 --> 00:02:15.936
I hope I'm not boring you.

00:02:15.936 --> 00:02:18.450
And it just keeps moving up.

00:02:18.450 --> 00:02:19.910
Let me see, what would
this look like-- this

00:02:19.910 --> 00:02:21.820
would look like this.

00:02:21.820 --> 00:02:25.240
So if I-- you might think I'm
doing something very strange

00:02:25.240 --> 00:02:29.930
right now, but just bear
with me while I draw this.

00:02:29.930 --> 00:02:32.983
I hope I don't
mess up too much.

00:02:32.983 --> 00:02:35.640
And, see, one like that.

00:02:35.640 --> 00:02:37.220
See one like that.

00:02:37.220 --> 00:02:39.630
So we're like, Sal, this is a
very strange looking graph.

00:02:39.630 --> 00:02:40.570
And it is.

00:02:40.570 --> 00:02:42.820
But what this is, is this
is a function definition.

00:02:42.820 --> 00:02:46.200
This tells you whenever I input
an x, at least for the x's that

00:02:46.200 --> 00:02:49.660
we can see on the graph, this
graph tell me what

00:02:49.660 --> 00:02:51.030
f of x equals.

00:02:51.030 --> 00:02:56.830
So if x is equal to negative
5, f of x would equal plus 2.

00:02:56.830 --> 00:02:58.230
And we could draw a
couple of examples.

00:02:58.230 --> 00:03:04.550
f of 0, well we go to 0 on
the x-axis, and we say

00:03:04.550 --> 00:03:08.840
f of 0 is equal to 0.

00:03:08.840 --> 00:03:13.980
f of 1 is equal to-- well, we
go to x equal to 1, and we

00:03:13.980 --> 00:03:17.800
just see where the chart is,
well, it equals negative 1.

00:03:17.800 --> 00:03:18.620
I think you get the idea.

00:03:18.620 --> 00:03:21.990
This isn't too difficult, but
this is a function definition.

00:03:21.990 --> 00:03:27.960
So we've defined this graph
right here as f of x.

00:03:27.960 --> 00:03:31.330
So if that graph-- that's the
graph of f of x, and let's say

00:03:31.330 --> 00:03:41.230
that we define g of x is equal
to f of x-- let's say

00:03:41.230 --> 00:03:48.620
it's equal to f of x
squared minus f of x.

00:03:48.620 --> 00:04:03.090
And let's say that h of x
is equal to 3 minus x.

00:04:03.090 --> 00:04:18.640
So what if I were to ask you,
what is h of g of negative 1?

00:04:18.640 --> 00:04:21.120
So just like we did in the
previous problems, first we'll

00:04:21.120 --> 00:04:23.590
say, well, let's try to figure
out what g of negative 1 is,

00:04:23.590 --> 00:04:27.020
and then we can substitute
that into h of x.

00:04:27.020 --> 00:04:35.130
So g of negative 1 is equal
to-- and this is how I do it.

00:04:35.130 --> 00:04:36.390
There's no trick to it.

00:04:36.390 --> 00:04:38.360
Wherever you see the x, you
just substitute it with the

00:04:38.360 --> 00:04:40.940
number that you're saying
is now the value for x.

00:04:40.940 --> 00:04:47.680
So you say, well, that's equal
to f of negative 1 squared

00:04:47.680 --> 00:04:51.240
minus f of negative 1.

00:04:51.240 --> 00:04:53.203
All I did is at g of negative
1, I just substituted

00:04:53.203 --> 00:04:56.050
it wherever I saw an x.

00:04:56.050 --> 00:04:58.190
Well what's f of negative 1?

00:04:58.190 --> 00:05:01.740
Well, when x is equal
to negative 1, f of

00:05:01.740 --> 00:05:03.620
x is equal to 1.

00:05:03.620 --> 00:05:07.300
So f of negative 1-- let's
write that, f of negative

00:05:07.300 --> 00:05:09.660
1 is equal to 1.

00:05:09.660 --> 00:05:12.040
So g of negative 1 is equal
to-- well, that's just

00:05:12.040 --> 00:05:16.940
1 squared minus 1,
well that equals 0.

00:05:16.940 --> 00:05:20.050
Because f of negative 1 is
1, so it's 1 squared minus

00:05:20.050 --> 00:05:22.020
1 that equals 1 minus 1.

00:05:22.020 --> 00:05:23.500
0.

00:05:23.500 --> 00:05:25.940
So g of negative 1 is
0, so this is the

00:05:25.940 --> 00:05:29.410
same thing as h of 0.

00:05:29.410 --> 00:05:32.070
Because g of negative 1,
we just figured out is 0.

00:05:32.070 --> 00:05:35.960
h of 0, we just take that 0 and
substitute it here, so it's 3

00:05:35.960 --> 00:05:38.600
minus 0, so that just equals 3.

00:05:38.600 --> 00:05:40.660
And we solved the problem.

00:05:40.660 --> 00:05:42.710
Let's do another example, and I
don't want to erase my graph

00:05:42.710 --> 00:05:47.470
since I took four minutes to
actually draw it, let me

00:05:47.470 --> 00:05:51.230
erase what we just did here.

00:05:51.230 --> 00:05:53.500
And what you might want to do
after you watch it the first

00:05:53.500 --> 00:05:55.710
time-- and this isn't true just
of this video, actually of all

00:05:55.710 --> 00:05:58.080
the videos-- but especially the
functions, after watching it

00:05:58.080 --> 00:06:01.890
once, you might want to rewatch
it and pause it right after I

00:06:01.890 --> 00:06:04.380
give you the problem and try to
do it yourself, and then see--

00:06:04.380 --> 00:06:06.600
and if you get stuck, you can
play it, or if you get an

00:06:06.600 --> 00:06:09.150
answer, just you can play the
video and make sure that

00:06:09.150 --> 00:06:11.630
we did the same way.

00:06:11.630 --> 00:06:12.205
Let's see.

00:06:18.370 --> 00:06:19.750
I'm going to create
another definition

00:06:19.750 --> 00:06:22.640
for g of x this time.

00:06:22.640 --> 00:06:26.565
Let's say that g of x-- oh
whoops, I was trying to write

00:06:26.565 --> 00:06:38.440
in black-- let's say that
g of x is equal to f of x

00:06:38.440 --> 00:06:51.000
squared plus f of x plus 2.

00:06:53.530 --> 00:07:01.380
So now, in this case, what is
g of-- let's pick a random

00:07:01.380 --> 00:07:06.320
number-- what is g of minus--
no, let's pick a, let's

00:07:06.320 --> 00:07:08.760
say-- what is g of minus 2?

00:07:08.760 --> 00:07:10.290
After we try and pick a
number that we could find

00:07:10.290 --> 00:07:11.820
an actual solution for.

00:07:11.820 --> 00:07:16.010
Well g of minus 2, wherever
we see the x, x is not

00:07:16.010 --> 00:07:17.610
going to be minus 2.

00:07:17.610 --> 00:07:25.230
That is equal to f of
minus 2 squared plus

00:07:25.230 --> 00:07:29.080
f of minus 2 plus 2.

00:07:29.080 --> 00:07:31.380
All we did is wherever we
saw an x, we substituted

00:07:31.380 --> 00:07:33.090
it, minus 2 there.

00:07:33.090 --> 00:07:34.120
And let's simplify that.

00:07:34.120 --> 00:07:38.110
Well, f of minus 2 squared, we
know what minus 2 squared is,

00:07:38.110 --> 00:07:45.270
that's the same thing as f of
4, plus f of minus 2 plus 2.

00:07:45.270 --> 00:07:46.670
That's 0.

00:07:46.670 --> 00:07:48.720
Plus f of 0.

00:07:48.720 --> 00:07:51.680
And now we just figure out
what f of 4 and f of 0 is.

00:07:51.680 --> 00:07:55.120
Well, f of 4, we go where x
equals r, it's right here,

00:07:55.120 --> 00:07:59.220
and when x equals 4,
f of 4 is equal to 2.

00:07:59.220 --> 00:08:03.210
So this is equal
to 2 plus f of 0.

00:08:03.210 --> 00:08:05.710
And just as a reminder, this
is the definition of f.

00:08:05.710 --> 00:08:10.560
We didn't define it in terms of
an algebraic expression, we

00:08:10.560 --> 00:08:13.320
defined in terms of an
actual visual graph.

00:08:13.320 --> 00:08:16.290
So what's f of 0? f of 0 is 0.

00:08:16.290 --> 00:08:20.580
When x is equal to 0-- f of 0
is 0 so that's 2 plus 0-- so g

00:08:20.580 --> 00:08:23.880
of negative 2 is equal to 2.

00:08:23.880 --> 00:08:26.110
An interesting thing, you might
want to make problems like this

00:08:26.110 --> 00:08:28.330
for yourself and keep
experimenting with different

00:08:28.330 --> 00:08:30.420
types of functions, and a very
interesting thing would

00:08:30.420 --> 00:08:33.870
actually be to graph g of x,
and actually that's a

00:08:33.870 --> 00:08:34.670
good idea, I think.

00:08:34.670 --> 00:08:37.510
I think maybe we'll do that in
the future modules to kind of

00:08:37.510 --> 00:08:39.330
play with functions and
actually to try graph

00:08:39.330 --> 00:08:42.090
the functions and see
how they turn out.

00:08:42.090 --> 00:08:45.580
I will-- I don't know if I have
enough time-- actually, I'm

00:08:45.580 --> 00:08:48.320
going to wait until the next
lecture to do a couple

00:08:48.320 --> 00:08:49.250
more examples.

00:08:49.250 --> 00:08:51.940
I want to do as many examples
on the functions as I can with

00:08:51.940 --> 00:08:54.410
you, because I think as you
keep watching and watching the

00:08:54.410 --> 00:08:58.220
function problems and seeing
more and more variations on

00:08:58.220 --> 00:09:01.170
functions, you'll see both how
general of a concept this is,

00:09:01.170 --> 00:09:03.900
and hopefully you'll get an
idea of how the functions

00:09:03.900 --> 00:09:05.170
actually work.

00:09:05.170 --> 00:09:07.290
Well, I'll see you in
the next lecture.

00:09:07.290 --> 00:09:08.780
Have fun.


Introduction to functions

WEBVTT
Kind: captions
Language: en

00:00:01.200 --> 00:00:03.740
Welcome to the presentation
on functions.

00:00:03.740 --> 00:00:06.330
Functions are something that,
when I first learned it, it

00:00:06.330 --> 00:00:08.800
was kind of like I had a
combination of I was 1,

00:00:08.800 --> 00:00:11.280
confused, and at the same time,
I was like, well what's even

00:00:11.280 --> 00:00:12.710
the point of learning this?

00:00:12.710 --> 00:00:16.310
So hopefully, at least in this
introduction lecture, we can

00:00:16.310 --> 00:00:19.070
get at least a very general
sense of what a function is

00:00:19.070 --> 00:00:21.470
and why it might be useful.

00:00:21.470 --> 00:00:23.560
So let's just start off
with just the overall

00:00:23.560 --> 00:00:24.570
concept of a function.

00:00:24.570 --> 00:00:27.450
A function is something that
you can give it an input-- and

00:00:27.450 --> 00:00:29.350
we'll start with just one
input, but actually you can

00:00:29.350 --> 00:00:32.160
give it multiple inputs-- you
give a function an input,

00:00:32.160 --> 00:00:35.030
let's call that input x.

00:00:35.030 --> 00:00:39.420
And you can view a function
as-- I guess a bunch of

00:00:39.420 --> 00:00:41.080
different ways you can view it.

00:00:41.080 --> 00:00:41.820
I don't know if you're
familiar with the

00:00:41.820 --> 00:00:42.860
concept of a black box.

00:00:42.860 --> 00:00:45.060
A black box is kind of a box,
you don't know what's inside of

00:00:45.060 --> 00:00:49.340
it, but if you put something
into it like this x, and let's

00:00:49.340 --> 00:00:52.440
call that box-- let's say the
function is called f, then

00:00:52.440 --> 00:00:56.760
it'll output what
we call f of x.

00:00:56.760 --> 00:00:59.180
I know this terminology might
seem a little confusing at

00:00:59.180 --> 00:01:03.820
first, but let's make some-- I
guess, let's define what's

00:01:03.820 --> 00:01:05.670
inside the box in
different ways.

00:01:05.670 --> 00:01:14.330
Let's say that the function
was-- let's say that f of x is

00:01:14.330 --> 00:01:19.240
equal to x squared plus 1.

00:01:19.240 --> 00:01:22.280
Then, if I were to say
what is f of-- let's

00:01:22.280 --> 00:01:25.870
say, what's f of 2?

00:01:25.870 --> 00:01:28.050
Well that means we're taking
2 and we're going to

00:01:28.050 --> 00:01:30.360
put it into the box.

00:01:30.360 --> 00:01:32.970
And I want to know what
comes out of the box

00:01:32.970 --> 00:01:34.450
when I put 2 into it.

00:01:34.450 --> 00:01:38.430
Well inside the box, we know
we do this to the input.

00:01:38.430 --> 00:01:42.060
We take the x, we square it,
and we add 1, so f of 2 is 2

00:01:42.060 --> 00:01:44.500
squared, which is 4, plus 1.

00:01:44.500 --> 00:01:47.040
Which is equal to 5.

00:01:47.040 --> 00:01:47.900
I know what you're thinking.

00:01:47.900 --> 00:01:50.510
Probably like, well, Sal, this
just seems like a very

00:01:50.510 --> 00:01:55.470
convoluted way of substituting
x into an equation and just

00:01:55.470 --> 00:01:56.580
finding out the result.

00:01:56.580 --> 00:01:59.400
And I agree with you right now.

00:01:59.400 --> 00:02:02.770
But as you'll see, a function
can become kind of a more

00:02:02.770 --> 00:02:05.420
general thing than
just an equation.

00:02:05.420 --> 00:02:09.010
For example, let me say-- let
me actually-- actually not,

00:02:09.010 --> 00:02:10.530
let me not erase this.

00:02:10.530 --> 00:02:13.150
Let me define a
function as this.

00:02:13.150 --> 00:02:24.560
f of x is equal to x squared
plus 1, if x is even,

00:02:24.560 --> 00:02:29.820
and it equals x squared
minus 1 if x is odd.

00:02:32.800 --> 00:02:35.390
I know this would have been--
this is something that we've

00:02:35.390 --> 00:02:36.430
never really seen before.

00:02:36.430 --> 00:02:40.320
This isn't just what I would
call an analytic expression,

00:02:40.320 --> 00:02:43.540
this isn't just x plus
something squared.

00:02:43.540 --> 00:02:45.470
We're actually saying,
depending on what type of x you

00:02:45.470 --> 00:02:47.850
put in, we're going to do a
different thing to that x.

00:02:47.850 --> 00:02:49.480
So let me ask you a question.

00:02:49.480 --> 00:02:53.470
What's f of 2 in this example?

00:02:53.470 --> 00:02:57.930
Well if we put 2 here, it says
if x is even you do this one,

00:02:57.930 --> 00:02:59.470
if x is odd you do this one.

00:02:59.470 --> 00:03:01.480
Well, 2 is even, so
we do this top one.

00:03:01.480 --> 00:03:05.830
So we'd say 2 squared plus
1, well that equals 5.

00:03:05.830 --> 00:03:09.230
But then, what's f of 3?

00:03:09.230 --> 00:03:12.180
Well if we put the 3 in
here, we'd use this

00:03:12.180 --> 00:03:13.930
case, because 3 is odd.

00:03:13.930 --> 00:03:18.410
So we do 3 squared minus
1. f of 3 is equal to 8.

00:03:18.410 --> 00:03:22.590
So notice, this was a little
bit more I guess you could

00:03:22.590 --> 00:03:25.720
even say abstract or
unusual in this case.

00:03:25.720 --> 00:03:27.527
I'm going to keep doing
examples of functions and

00:03:27.527 --> 00:03:32.590
I'm going to show you how
general this idea can be.

00:03:32.590 --> 00:03:35.380
And if you get confused, I'm
going to show you that the

00:03:35.380 --> 00:03:37.250
actual function problems
you're going to encounter are

00:03:37.250 --> 00:03:38.560
actually not that hard to do.

00:03:38.560 --> 00:03:41.000
I just want to make sure that
you least get exposed to kind

00:03:41.000 --> 00:03:44.840
of the general idea of
what a function is.

00:03:44.840 --> 00:03:49.920
You can view almost anything
in the world as a function.

00:03:49.920 --> 00:03:53.740
Let's say that there is a
function called Sal, because,

00:03:53.740 --> 00:03:54.835
you know, that's my name.

00:03:58.620 --> 00:04:00.950
And I'm a function.

00:04:00.950 --> 00:04:08.490
Let's say that if you
were to-- let me think.

00:04:08.490 --> 00:04:14.190
If you were to give me
food, what do I produce?

00:04:14.190 --> 00:04:17.740
So what is Sal of food?

00:04:17.740 --> 00:04:23.050
So if you input food into
Sal, what will Sal produce?

00:04:23.050 --> 00:04:24.770
Well I won't go into some of
the things that I would

00:04:24.770 --> 00:04:30.710
produce, but I would
produce videos.

00:04:30.710 --> 00:04:33.690
I would produce math videos
if you gave me food.

00:04:33.690 --> 00:04:36.370
Math videos.

00:04:36.370 --> 00:04:37.670
I'm just a function.

00:04:37.670 --> 00:04:40.900
You give me food and-- and
maybe, actually, maybe

00:04:40.900 --> 00:04:41.920
I have multiple inputs.

00:04:41.920 --> 00:04:48.140
Maybe if you give me a food
and a computer, and I would

00:04:48.140 --> 00:04:51.830
produce math videos for you.

00:04:51.830 --> 00:04:54.215
And maybe you are a function.

00:04:57.010 --> 00:04:57.880
I don't know your name.

00:04:57.880 --> 00:05:00.540
I would like to, but I
don't know your name.

00:05:00.540 --> 00:05:09.440
And let's say if I were to
input math videos into you,

00:05:09.440 --> 00:05:12.840
then you will produce-- let's
see, what would you produce?

00:05:12.840 --> 00:05:17.080
If I gave you math videos, you
would produce A's on tests.

00:05:23.580 --> 00:05:24.910
A's on your math test.

00:05:24.910 --> 00:05:28.150
Hopefully you're not taking
someone else's math test.

00:05:28.150 --> 00:05:29.000
So it's interesting.

00:05:29.000 --> 00:05:31.690
If you give-- well, let's
take the computer away.

00:05:31.690 --> 00:05:33.820
Let's say that all
Sal needs is food.

00:05:33.820 --> 00:05:35.450
Which is kind of true.

00:05:35.450 --> 00:05:38.640
So if you put food into
Sal, Sal of food, he

00:05:38.640 --> 00:05:42.050
produces math videos.

00:05:42.050 --> 00:05:45.930
And if I were to put math
videos into you, then you

00:05:45.930 --> 00:05:49.120
produce A's on your math test.

00:05:49.120 --> 00:05:51.710
So let's think of an
interesting problem.

00:05:51.710 --> 00:05:59.080
What is you of Sal of food?

00:06:02.720 --> 00:06:05.620
I know this seems very
ridiculous, but I actually

00:06:05.620 --> 00:06:08.640
think we might be going
someplace, so we might be

00:06:08.640 --> 00:06:10.510
getting somewhere with
this kind of idea.

00:06:10.510 --> 00:06:14.000
Well, first we would try to
figure out what is Sal of food.

00:06:14.000 --> 00:06:17.340
Well, we already figured out if
you put food into Sal, Sal of

00:06:17.340 --> 00:06:19.080
food is equal to math videos.

00:06:19.080 --> 00:06:25.630
So this is the same thing as
you of-- I'm trying to confuse

00:06:25.630 --> 00:06:32.120
you-- you of math videos.

00:06:32.120 --> 00:06:34.230
And I already determined, we
already said, well, if you put

00:06:34.230 --> 00:06:37.830
math videos into the function
called you, whatever your name

00:06:37.830 --> 00:06:42.130
might be, then it produces
A's on your math test.

00:06:42.130 --> 00:06:47.480
So that you of math videos
equals A's on your math test.

00:06:52.100 --> 00:06:57.180
So you of Sal of food will
produce A's on your math test.

00:06:57.180 --> 00:06:59.000
And notice, we just said
what happens when you

00:06:59.000 --> 00:07:02.150
put food into Sal.

00:07:02.150 --> 00:07:04.660
This could-- would be a very
different outcome if you put,

00:07:04.660 --> 00:07:08.630
like, if you replaced food
with let's say poison.

00:07:12.570 --> 00:07:19.810
Because if you put poison into
Sal, Sal of poison-- not that I

00:07:19.810 --> 00:07:23.940
would recommend that you did
this-- Sal of poison

00:07:23.940 --> 00:07:28.510
would equal death.

00:07:28.510 --> 00:07:31.610
No, no, I shouldn't say
something, so no no no no.

00:07:31.610 --> 00:07:33.190
Well you get the idea.

00:07:33.190 --> 00:07:35.720
There wouldn't be math videos.

00:07:35.720 --> 00:07:36.600
Anyway.

00:07:36.600 --> 00:07:38.400
Let me move on.

00:07:38.400 --> 00:07:41.510
So with that kind of-- I'm not
so clear whether that would be

00:07:41.510 --> 00:07:46.810
a useful example with the
food and the math videos.

00:07:46.810 --> 00:07:50.700
Let's do some actual
problems using functions.

00:07:50.700 --> 00:07:54.990
So if I were to tell you that I
had one function, called f of x

00:07:54.990 --> 00:08:00.610
is equal to x plus 2, and I had
another function that said g

00:08:00.610 --> 00:08:11.850
of x is equal to x
squared minus 1.

00:08:11.850 --> 00:08:19.790
If I were to ask you
what g of f of 3 is.

00:08:22.780 --> 00:08:27.440
Well the first thing we want to
do is evaluate what f of 3 is.

00:08:27.440 --> 00:08:35.070
So if you-- the 3 would replace
the x, so f of 3 is equal to

00:08:35.070 --> 00:08:39.210
3 plus 2, which equals 5.

00:08:39.210 --> 00:08:45.440
So g of f of 3 is the same
thing as g of 5, because f

00:08:45.440 --> 00:08:46.390
of three is equal to 5.

00:08:46.390 --> 00:08:49.080
Sorry for the little
bit of messiness.

00:08:49.080 --> 00:08:50.340
So then, what's g of 5?

00:08:50.340 --> 00:08:53.890
Well, then we take this 5, and
we put it in in place of this

00:08:53.890 --> 00:09:02.760
x, so g of 5 is 5 squared, 25,
minus 1, which equals 24.

00:09:02.760 --> 00:09:07.110
So g of f of 3 is equal to 24.

00:09:07.110 --> 00:09:08.950
Hopefully that gives you a
taste of what a function is all

00:09:08.950 --> 00:09:12.000
about, and I really apologize
if I have either confused or

00:09:12.000 --> 00:09:16.730
scared you with the Sal
food/poison math video example.

00:09:16.730 --> 00:09:19.780
But in the next set of
presentations, I'm going to do

00:09:19.780 --> 00:09:22.420
a lot more of these examples,
and I think you'll get the idea

00:09:22.420 --> 00:09:25.110
of at least how to do these
problems that you might see on

00:09:25.110 --> 00:09:27.350
your math tests, and maybe get
a sense of what functions

00:09:27.350 --> 00:09:29.120
are all about.

00:09:29.120 --> 00:09:30.720
See you in the next video.

00:09:30.720 --> 00:09:32.020
Bye.


Integer sums

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:03.900
Welcome to the presentation
on finding sums of integers.

00:00:03.900 --> 00:00:06.670
You're probably wondering why
are we doing this within

00:00:06.670 --> 00:00:08.150
the context of averages.

00:00:08.150 --> 00:00:10.240
Well, if you think about it,
all an average is is you take

00:00:10.240 --> 00:00:13.030
a sum of a bunch of numbers
and then you divide by the

00:00:13.030 --> 00:00:14.930
number of numbers you have.

00:00:14.930 --> 00:00:16.770
What we're going to do here is
do a couple of algebra problems

00:00:16.770 --> 00:00:19.230
that involve just the sum parts
first, and actually they can

00:00:19.230 --> 00:00:23.550
carry over into average
problems as well.

00:00:23.550 --> 00:00:25.360
Let's get started
with a problem.

00:00:25.360 --> 00:00:32.800
Let's say I told you that I had
the sum of five consecutive

00:00:32.800 --> 00:00:48.580
integers is equal to 200.

00:00:48.580 --> 00:00:56.005
What is the smallest -- I
apologize for my handwriting

00:00:56.005 --> 00:01:03.300
-- what is the smallest
of the five integers?

00:01:03.300 --> 00:01:06.750
Well there's a couple of ways
to do this, but I guess the

00:01:06.750 --> 00:01:09.760
most straightforward way is
just to do it algebraically,

00:01:09.760 --> 00:01:11.080
I would say.

00:01:11.080 --> 00:01:15.315
So let's say that x is the
smallest of the integers,

00:01:15.315 --> 00:01:16.650
right, so x is actually
what we're going to

00:01:16.650 --> 00:01:17.610
want to figure out.

00:01:21.050 --> 00:01:24.810
Well if x is the smallest, what
are the other four going to be?

00:01:24.810 --> 00:01:26.090
We have a total of five.

00:01:26.090 --> 00:01:27.580
Well, they're consecutive.

00:01:27.580 --> 00:01:29.620
Consecutive just means that
they follow each other,

00:01:29.620 --> 00:01:32.100
like 5, 6, 7, 8, 9, 10.

00:01:32.100 --> 00:01:34.040
All of those are consecutive
integers, right?

00:01:34.040 --> 00:01:36.140
And if you remember, integers
are just whole numbers, so it

00:01:36.140 --> 00:01:38.110
can't be a fraction
or a decimal.

00:01:38.110 --> 00:01:41.420
So if x is the smallest, so
then the next integer is

00:01:41.420 --> 00:01:44.260
going to be x plus 1.

00:01:44.260 --> 00:01:47.360
And the one after that's
going to be x plus 2.

00:01:47.360 --> 00:01:50.400
And the one after that's
going to be x plus 3.

00:01:50.400 --> 00:01:54.050
And the one after that's
going to be x plus 4, right?

00:01:54.050 --> 00:01:56.150
It might seem confusing I'm
writing all of these x's.

00:01:56.150 --> 00:01:59.920
But if you think about it, if x
was 5, then this would be 6,

00:01:59.920 --> 00:02:03.030
this would be 7, this would
be 8, and this would be 9.

00:02:03.030 --> 00:02:05.150
And that's all I'm
writing here, right?

00:02:05.150 --> 00:02:07.960
So these would be, assuming
that x is the smallest of the

00:02:07.960 --> 00:02:12.550
integers, the five integers
would be x, x plus 1, x plus

00:02:12.550 --> 00:02:15.320
2, x plus 3, and x plus 4.

00:02:15.320 --> 00:02:18.030
And we know that the sum
of these five consecutive

00:02:18.030 --> 00:02:19.900
integers is 200.

00:02:19.900 --> 00:02:24.620
What is the sum of these
five, I guess we could say

00:02:24.620 --> 00:02:27.110
numbers or expressions?

00:02:27.110 --> 00:02:30.670
Well let's see, we have
five x's -- 1, 2, 3, 4, 5.

00:02:30.670 --> 00:02:36.662
So x plus x plus x plus x
plus x is equal to just 5x.

00:02:36.662 --> 00:02:38.920
Or you could just
say 5 times x.

00:02:38.920 --> 00:02:42.540
And then that's plus 1
plus 2 is 3, 3 plus 3

00:02:42.540 --> 00:02:45.600
is 6, 6 plus 4 is 10.

00:02:45.600 --> 00:02:48.490
So the sum of these five
integers is going to be 5x plus

00:02:48.490 --> 00:02:51.520
10, and all I did is add up the
x's and added up the constants.

00:02:51.520 --> 00:02:56.450
And we know that that
is going to equal 200.

00:02:56.450 --> 00:02:58.330
Now this is just a level
two linear equation.

00:02:58.330 --> 00:03:00.440
We can just solve for x.

00:03:00.440 --> 00:03:06.250
So we get 5x is equal to 190
-- I just subtracted 10

00:03:06.250 --> 00:03:08.020
from both sides, right?

00:03:08.020 --> 00:03:15.090
And then x is equal to --
let me divide 5 into 190.

00:03:15.090 --> 00:03:19.870
5 goes into 19 three
times, 3 times 5 is 15.

00:03:19.870 --> 00:03:23.160
9 minus 5 is 4,
bring down the 0.

00:03:23.160 --> 00:03:25.500
5 goes into 40, eight times.

00:03:25.500 --> 00:03:26.940
So x is equal to 38.

00:03:29.700 --> 00:03:32.010
Pretty straightforward
problem, don't you think?

00:03:32.010 --> 00:03:36.790
Now what if I were to ask you
what is the average of the

00:03:36.790 --> 00:03:38.770
five consecutive numbers?

00:03:38.770 --> 00:03:41.020
Well now, there's two
ways of doing this.

00:03:41.020 --> 00:03:43.250
Now that we already know that x
is 38, we know that the other

00:03:43.250 --> 00:03:52.130
numbers are going to be -- well
this is 38, 39, 40, 41, 42.

00:03:52.130 --> 00:03:53.960
Well we could just average
these four numbers.

00:03:53.960 --> 00:03:57.900
You could just say 38 plus
39 plus 40 plus 41 plus 42.

00:03:57.900 --> 00:03:59.360
And well we already know
what those -- I don't

00:03:59.360 --> 00:04:00.060
even have to do the math.

00:04:00.060 --> 00:04:03.320
You already know that they
average up, they sum up to 200

00:04:03.320 --> 00:04:06.860
and then we divide the sum by
5, because there are 5 numbers.

00:04:06.860 --> 00:04:09.310
So the average is 40.

00:04:14.380 --> 00:04:16.490
There are a couple ways you
could think about that.

00:04:16.490 --> 00:04:19.420
One, you see 40's just a middle
number so that makes sense.

00:04:19.420 --> 00:04:22.230
And the only time we can really
say it's the middle number

00:04:22.230 --> 00:04:25.800
is when the numbers are
distributed evenly around 40.

00:04:25.800 --> 00:04:27.620
If we had a number that was
much smaller than 40 or

00:04:27.620 --> 00:04:29.270
something, you couldn't
just necessarily pick

00:04:29.270 --> 00:04:29.740
the middle number.

00:04:29.740 --> 00:04:32.090
But in this case these are
consecutive and makes sense.

00:04:32.090 --> 00:04:34.230
Another way we could have done
this problem, if you were, say,

00:04:34.230 --> 00:04:37.840
taking the SAT and they were to
ask you the sum of five

00:04:37.840 --> 00:04:41.500
numbers is 200, what's the
average of the numbers?

00:04:41.500 --> 00:04:43.800
Well you say, well, all I have
to do is divide that 200

00:04:43.800 --> 00:04:46.270
by 5 and I'll get 40.

00:04:46.270 --> 00:04:47.700
Let's do another problem
and I'll make it a

00:04:47.700 --> 00:04:50.070
little bit harder.

00:04:50.070 --> 00:05:08.540
Let's say the sum of seven odd
numbers, and let me make up a

00:05:08.540 --> 00:05:12.060
good -- I hope this one works,
I'm going to try to do it in

00:05:12.060 --> 00:05:27.535
my head -- is 217, what
is the largest number?

00:05:30.750 --> 00:05:35.640
I shouldn't say number
-- seven odd integers.

00:05:35.640 --> 00:05:38.210
Actually it becomes a much
harder problem if it was just

00:05:38.210 --> 00:05:40.840
seven odd -- well actually, the
only thing that could be odd

00:05:40.840 --> 00:05:42.900
are integers anyway, so you
could almost assume it.

00:05:42.900 --> 00:05:45.400
But the sum of seven
odd integers is 217.

00:05:45.400 --> 00:05:50.160
What is the largest
of the integers?

00:05:50.160 --> 00:05:53.200
As you can tell I'm
doing this on the fly.

00:05:53.200 --> 00:05:55.750
Actually my wife just diagnosed
me with, she thinks I

00:05:55.750 --> 00:05:57.030
have benign vertigo.

00:05:57.030 --> 00:05:59.490
I got very dizzy this morning
when I went to work, so you

00:05:59.490 --> 00:06:00.850
have to forgive me
for that as well.

00:06:00.850 --> 00:06:03.010
That's impairing me even more.

00:06:03.010 --> 00:06:05.080
So let's do this problem.

00:06:05.080 --> 00:06:08.385
Let's say that x
is the largest.

00:06:12.080 --> 00:06:14.400
Then what would the
number right below x be?

00:06:14.400 --> 00:06:16.550
Would it be x minus 1?

00:06:16.550 --> 00:06:19.990
Well, if x is an odd
number, x minus 1 would

00:06:19.990 --> 00:06:21.750
be an even number.

00:06:21.750 --> 00:06:26.850
So in order to get the number
right below it, we have to

00:06:26.850 --> 00:06:30.290
do x minus 2 to get
another odd number.

00:06:30.290 --> 00:06:32.630
My apologies -- it should
say the sum of seven

00:06:32.630 --> 00:06:34.170
consecutive odd.

00:06:34.170 --> 00:06:36.160
I don't know if
you assumed that.

00:06:36.160 --> 00:06:39.130
I'm trying my best
today to confuse you.

00:06:39.130 --> 00:06:43.840
The sum of seven consecutive
odd integers is 217.

00:06:43.840 --> 00:06:46.530
What is the largest
of the integers?

00:06:46.530 --> 00:06:49.420
So if x is the largest, then to
next smallest one would be x

00:06:49.420 --> 00:06:53.030
minus 2, right, because it's
consecutive odd numbers,

00:06:53.030 --> 00:06:54.250
not just consecutive.

00:06:54.250 --> 00:06:58.020
So consecutive odd numbers are
like 1, 3, 5, 7 -- you're

00:06:58.020 --> 00:06:59.050
skipping the evens, right?

00:06:59.050 --> 00:07:01.470
So that's why you're going up
or down by two, depending

00:07:01.470 --> 00:07:02.740
how you view it.

00:07:02.740 --> 00:07:05.345
So the next one down would be
x minus 2, then we'll have x

00:07:05.345 --> 00:07:13.450
minus 4, x minus 6, x minus
8, x minus 10, x minus 12.

00:07:13.450 --> 00:07:14.060
I think that's it.

00:07:14.060 --> 00:07:16.700
One, two, three, four,
five, six, seven, right.

00:07:16.700 --> 00:07:18.060
Those are seven numbers.

00:07:18.060 --> 00:07:19.300
They're separated by two.

00:07:19.300 --> 00:07:21.990
X is the largest
of them, right?

00:07:21.990 --> 00:07:24.210
We can assume that they're
odd because apparently

00:07:24.210 --> 00:07:26.310
the problem will work
out so that they're odd.

00:07:26.310 --> 00:07:28.320
So what is the sum of
these seven numbers?

00:07:28.320 --> 00:07:31.850
Well the seven x's
just add up to 7x.

00:07:31.850 --> 00:07:38.630
And then let's see, 2 and 4 is
6, 6 and 6 is 12, 12 and 8 is

00:07:38.630 --> 00:07:43.330
20, 20 and 10 is 30,
30 and 12 is 32.

00:07:43.330 --> 00:07:51.840
So 7x minus 32 is equal to 217.

00:07:51.840 --> 00:07:53.520
We just solved for x.

00:07:53.520 --> 00:07:58.310
7x is equal to -- let's see,
if we add 32 to both sides

00:07:58.310 --> 00:08:04.380
of this equation we get 249.

00:08:04.380 --> 00:08:10.070
Let's see, 7 goes into
249 -- is that right?

00:08:13.060 --> 00:08:13.750
Right.

00:08:13.750 --> 00:08:17.685
So 7 goes into 249 -- did I
do this addition properly?

00:08:17.685 --> 00:08:19.070
I want to make sure.

00:08:19.070 --> 00:08:28.770
2 plus 4 is 6, 6 plus 6 is 12,
12 plus 8 is 20, 20 plus 10

00:08:28.770 --> 00:08:32.820
is 30, 30 plus 12 is 42.

00:08:32.820 --> 00:08:34.210
Oh, here you go.

00:08:34.210 --> 00:08:37.320
See, my mathematical spider
sense could tell that something

00:08:37.320 --> 00:08:39.090
was fishy about this.

00:08:39.090 --> 00:08:41.470
So that's 7x minus 42.

00:08:41.470 --> 00:08:46.870
So if we add 42 to both sides
it's 7x is equal to 259.

00:08:46.870 --> 00:08:49.900
See how brave I am, I do
this thing in real time.

00:08:49.900 --> 00:08:50.680
259.

00:08:50.680 --> 00:08:55.530
So 7 goes into 259 -- let's
see, 7 goes into 25 three

00:08:55.530 --> 00:09:01.900
times, 3 times 7 is 21, 49 --
it goes into it 37 times.

00:09:01.900 --> 00:09:06.530
So we get x is equal
to 37 and we're done.

00:09:06.530 --> 00:09:09.630
So just to review because I
think had a lot of errors in

00:09:09.630 --> 00:09:11.460
this problem when
I presented it.

00:09:11.460 --> 00:09:13.830
The question was the sum
of seven consecutive

00:09:13.830 --> 00:09:16.900
odd integers is 217.

00:09:16.900 --> 00:09:18.830
What is the largest
of the integers?

00:09:18.830 --> 00:09:21.390
I said x is the largest, and
then if x is the largest, the

00:09:21.390 --> 00:09:23.650
next smaller one
will x minus 2.

00:09:23.650 --> 00:09:25.140
Because we're not saying
just consecutive integers,

00:09:25.140 --> 00:09:28.010
we're saying consecutive
odd integers, right?

00:09:28.010 --> 00:09:32.320
So if x is 37, which is what we
solved for, then x minus 2 is

00:09:32.320 --> 00:09:38.490
35, this is 33, this is 31,
this is 29, this is

00:09:38.490 --> 00:09:41.490
27, this is 25.

00:09:41.490 --> 00:09:46.050
And then we just added up all
the x's and I'll add up all

00:09:46.050 --> 00:09:48.910
the constants and said,
well they add up to 217.

00:09:48.910 --> 00:09:50.600
And then we just solved for x.

00:09:50.600 --> 00:09:53.460
I think you're now ready to
try some of these problems.

00:09:53.460 --> 00:09:54.940
Have fun.


Basic addition

WEBVTT
Kind: captions
Language: en

00:00:01.160 --> 00:00:04.730
Welcome to the presentation
on basic addition.

00:00:04.730 --> 00:00:06.810
I know what you're thinking,
Sal, addition doesn't

00:00:06.810 --> 00:00:08.260
seem so basic to me.

00:00:08.260 --> 00:00:10.940
Well, I apologize.

00:00:10.940 --> 00:00:13.170
Hopefully by the end of this
presentation or in a couple

00:00:13.170 --> 00:00:14.840
of weeks it will seem basic.

00:00:14.840 --> 00:00:16.990
So let's get started
with, I guess we could

00:00:16.990 --> 00:00:18.390
say, some problems.

00:00:18.390 --> 00:00:23.100
Well let's say I start
with an old classic.

00:00:23.100 --> 00:00:25.950
1 plus 1.

00:00:25.950 --> 00:00:29.440
And I think you already know
how to do this, but I'll kind

00:00:29.440 --> 00:00:31.960
of show you a way of doing this
in case you don't have this

00:00:31.960 --> 00:00:35.050
memorized or you haven't
already mastered this.

00:00:35.050 --> 00:00:40.930
You say, well, if I have 1,
let's call that an avocado.

00:00:40.930 --> 00:00:45.030
If I have 1 avocado and
then you were to give me

00:00:45.030 --> 00:00:48.870
another avocado, how many
avocados do I now have?

00:00:48.870 --> 00:00:49.340
Well, let's see.

00:00:49.340 --> 00:00:52.300
I have 1, 2 avocados.

00:00:52.300 --> 00:00:54.740
So 1 plus 1 is equal to 2.

00:00:54.740 --> 00:00:55.910
Now, I know what
you're thinking.

00:00:55.910 --> 00:00:58.390
That was too easy, so let
me give you something a

00:00:58.390 --> 00:01:00.410
little bit more difficult.

00:01:00.410 --> 00:01:01.840
I like the avocados.

00:01:01.840 --> 00:01:04.130
I might stick with that theme.

00:01:04.130 --> 00:01:09.420
What is 3 plus 4?

00:01:09.420 --> 00:01:11.980
This is, I think, a more
difficult problem.

00:01:11.980 --> 00:01:14.070
Well, let's stick
with the avocados.

00:01:14.070 --> 00:01:17.710
And in case you don't know what
an avocado is, it's actually

00:01:17.710 --> 00:01:18.890
a very delicious fruit.

00:01:18.890 --> 00:01:21.210
It's actually the fattiest
of all the fruits.

00:01:21.210 --> 00:01:22.340
You probably didn't even
think it was a fruit,

00:01:22.340 --> 00:01:25.470
even if you ate one.

00:01:25.470 --> 00:01:32.270
Let's say I have 3
avocados-- 1, 2, 3.

00:01:32.270 --> 00:01:35.120
And let's say you were to
give me 4 more avocados.

00:01:35.120 --> 00:01:38.620
So let me put this 4 in yellow
so you know that these are

00:01:38.620 --> 00:01:40.610
the ones you're giving me.

00:01:40.610 --> 00:01:45.780
1, 2, 3, 4.

00:01:45.780 --> 00:01:48.980
So how many total
avocados do I have now?

00:01:48.980 --> 00:01:55.790
That's 1, 2, 3, 4,
5, 6, 7 avocados.

00:01:55.790 --> 00:01:59.000
So 3 plus 4 is equal to 7.

00:01:59.000 --> 00:02:00.290
And now I'm going to introduce
you to another way of

00:02:00.290 --> 00:02:01.070
thinking about this.

00:02:01.070 --> 00:02:02.180
It's called the number line.

00:02:02.180 --> 00:02:05.490
And actually, I think this is
how I do it in my head when I

00:02:05.490 --> 00:02:08.920
forget-- if I don't
have it memorized.

00:02:08.920 --> 00:02:11.450
So number line, I just write
all the numbers in order.

00:02:11.450 --> 00:02:13.850
And I go high enough just
so all the numbers I'm

00:02:13.850 --> 00:02:15.470
using are kind of in it.

00:02:15.470 --> 00:02:18.000
So you know the first number
is 0, which is nothing.

00:02:18.000 --> 00:02:20.020
Maybe you don't know,
but now you know.

00:02:20.020 --> 00:02:36.250
And then you go to 1, 2,
3, 4, 5, 6, 7, 8, 9, 10.

00:02:36.250 --> 00:02:37.600
Keeps going-- 11.

00:02:37.600 --> 00:02:39.360
So we're saying 3 plus 4.

00:02:39.360 --> 00:02:41.740
So let's start at 3.

00:02:41.740 --> 00:02:45.500
So I have 3 here and we're
going to add 4 to that 3.

00:02:45.500 --> 00:02:48.430
So all we do is we go up the
number line, or we go to the

00:02:48.430 --> 00:02:50.850
right on the number
line, 4 more.

00:02:50.850 --> 00:02:59.130
So we go 1, 2, 3, 4.

00:02:59.130 --> 00:03:01.360
Notice all we did is we just
increased it by one, by

00:03:01.360 --> 00:03:02.610
two, by three, by four.

00:03:02.610 --> 00:03:04.090
And then we ended up at 7.

00:03:04.090 --> 00:03:06.030
And that was our answer.

00:03:06.030 --> 00:03:07.180
We can do a couple
of different ones.

00:03:09.690 --> 00:03:14.030
What if I asked you
what 8 plus 1 is?

00:03:14.030 --> 00:03:14.960
Well, you might
already know it.

00:03:14.960 --> 00:03:16.390
You know, plus 1 is
just the next number.

00:03:16.390 --> 00:03:18.225
But if we look at the
number line you start

00:03:18.225 --> 00:03:22.500
at 8 and you add 1.

00:03:22.500 --> 00:03:26.340
8 plus 1 is equal to 9.

00:03:26.340 --> 00:03:27.445
Let's do some harder problems.

00:03:30.370 --> 00:03:32.830
And just so you know, if you're
a little daunted by this

00:03:32.830 --> 00:03:34.550
initially, you can always
draw the circles.

00:03:34.550 --> 00:03:36.060
You can always do
the number line.

00:03:36.060 --> 00:03:39.670
And eventually, over time, the
more practice you do-- you'll

00:03:39.670 --> 00:03:41.330
hopefully memorize these and
you'll do these problems

00:03:41.330 --> 00:03:42.440
in like half a second.

00:03:42.440 --> 00:03:43.240
I promise you.

00:03:43.240 --> 00:03:46.250
You just got to
keep practicing.

00:03:46.250 --> 00:03:48.380
I want to draw the number line
again, actually, I have a line

00:03:48.380 --> 00:03:50.740
tool, so I should give you
all those ugly looking lines

00:03:50.740 --> 00:03:52.260
that I've been giving you.

00:03:52.260 --> 00:03:52.840
Look at that.

00:03:52.840 --> 00:03:54.660
That's amazing.

00:03:54.660 --> 00:03:56.810
Let me see.

00:03:56.810 --> 00:04:00.230
Look at that.

00:04:00.230 --> 00:04:01.490
That's a nice looking line.

00:04:01.490 --> 00:04:04.650
I'm going to feel bad
to erase it later on.

00:04:04.650 --> 00:04:06.920
So let me draw a number line.

00:04:06.920 --> 00:04:28.590
0, 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15.

00:04:28.590 --> 00:04:30.060
So let's do a hard problem.

00:04:33.180 --> 00:04:35.290
I'm going to do it in
different colors now.

00:04:35.290 --> 00:04:38.550
5 plus 6.

00:04:38.550 --> 00:04:40.690
So if you want, you could
pause the video and try this.

00:04:40.690 --> 00:04:42.720
You might already
know the answer.

00:04:42.720 --> 00:04:45.140
And the reason why I say this
is a hard problem is because

00:04:45.140 --> 00:04:48.660
the answer has more numbers
than figures, so you can't

00:04:48.660 --> 00:04:50.770
necessarily do it
on your fingers.

00:04:50.770 --> 00:04:53.980
So let's get started
with this problem.

00:04:53.980 --> 00:04:55.780
Actually, my phone is ringing,
but I'm going to ignore the

00:04:55.780 --> 00:04:58.900
phone because you're
more important.

00:04:58.900 --> 00:05:00.980
OK, let's start at the 5.

00:05:00.980 --> 00:05:05.400
So we start at the 5 and
we're going to add 6 to it.

00:05:05.400 --> 00:05:13.980
So we go 1, 2, 3, 4, 5, 6.

00:05:13.980 --> 00:05:16.640
And we're at 11.

00:05:16.640 --> 00:05:19.670
So 5 plus 6 is equal to 11.

00:05:19.670 --> 00:05:22.475
Now I'm going to ask you a
question, what is 6 plus 5?

00:05:26.690 --> 00:05:28.540
Well, we're now
going to see that.

00:05:28.540 --> 00:05:30.890
Can you switch the two numbers
and get the same answer?

00:05:30.890 --> 00:05:32.260
Well, let's try that.

00:05:32.260 --> 00:05:33.600
And I'm going to try it in
a different color so we

00:05:33.600 --> 00:05:34.520
don't get all confused.

00:05:34.520 --> 00:05:36.930
So let's start at 6.

00:05:36.930 --> 00:05:39.880
Ignore the yellow for
now and add 5 to it.

00:05:39.880 --> 00:05:45.040
1, 2, 3, 4, 5.

00:05:45.040 --> 00:05:47.040
We get to the same place.

00:05:47.040 --> 00:05:48.840
And I think you might want
to try this on a bunch of

00:05:48.840 --> 00:05:50.600
problems and you'll see
it always works out.

00:05:50.600 --> 00:05:53.900
That it doesn't matter what
order-- 5 plus 6 is the

00:05:53.900 --> 00:05:55.910
same thing as 6 plus 5.

00:05:55.910 --> 00:05:56.670
And that makes sense.

00:05:56.670 --> 00:05:58.960
If I have 5 avocados
and you give me 6,

00:05:58.960 --> 00:05:59.740
I'm going to have 11.

00:05:59.740 --> 00:06:01.845
If I have 6 avocados and
you gave me 5, I'm going

00:06:01.845 --> 00:06:05.350
to have 11 either way.

00:06:05.350 --> 00:06:07.190
Since this number line is so
nice, I want to do a few

00:06:07.190 --> 00:06:08.170
more problems using it.

00:06:08.170 --> 00:06:10.720
Although as I use it I'm sure
I'll just continue to confuse

00:06:10.720 --> 00:06:12.460
you because I'll write
so much on top of it.

00:06:12.460 --> 00:06:14.290
But let's see.

00:06:14.290 --> 00:06:17.250
I'll use white now.

00:06:17.250 --> 00:06:22.320
What is 8 plus 7?

00:06:22.320 --> 00:06:26.830
Well, if you can still read
this, 8 is right here.

00:06:26.830 --> 00:06:28.820
We're going to add 7 to it.

00:06:28.820 --> 00:06:36.910
1, 2, 3, 4, 5, 6, 7.

00:06:36.910 --> 00:06:38.110
We go to 15.

00:06:38.110 --> 00:06:39.390
8 plus 7 is 15.

00:06:42.450 --> 00:06:45.550
So hopefully that gives
you a sense of how to do

00:06:45.550 --> 00:06:48.200
these types of problems.

00:06:48.200 --> 00:06:50.570
I guess this and you're going
to learn multiplication in a

00:06:50.570 --> 00:06:53.880
little bit, but these types of
problems are-- when you're

00:06:53.880 --> 00:06:55.830
getting started off in
mathematics, these kind of

00:06:55.830 --> 00:06:58.390
require the most practice and
to some degree, you have to

00:06:58.390 --> 00:06:59.430
start memorizing them.

00:06:59.430 --> 00:07:03.150
But over time, when you look
back, I want you to remember

00:07:03.150 --> 00:07:05.650
how you feel while you're
watching this video right now.

00:07:05.650 --> 00:07:09.780
And then I want you to watch
this video in like 3 years and

00:07:09.780 --> 00:07:11.930
remember how you felt when
you're watching it now.

00:07:11.930 --> 00:07:13.340
And you're going
to be, oh my God.

00:07:13.340 --> 00:07:16.080
This was so easy because you're
going to learn it so fast.

00:07:16.080 --> 00:07:19.940
So anyway, I think
you have an idea.

00:07:19.940 --> 00:07:22.500
If you don't know the answer to
any of the additional problems

00:07:22.500 --> 00:07:25.735
that we give in the exercises
you can press the hints and

00:07:25.735 --> 00:07:28.560
it'll draw circles and you can
just count up the circles.

00:07:28.560 --> 00:07:30.020
Or if you want to do it
on your own so you get

00:07:30.020 --> 00:07:32.260
the problem right, you
could draw the circles.

00:07:32.260 --> 00:07:34.370
Or you could draw a number
line like we did in

00:07:34.370 --> 00:07:36.710
this presentation.

00:07:36.710 --> 00:07:40.330
I think you might be ready to
tackle the addition problems.

00:07:40.330 --> 00:07:41.870
Have fun.


Level 2 Addition

WEBVTT
Kind: captions
Language: en

00:00:01.040 --> 00:00:04.980
Welcome to the presentation
on level two addition.

00:00:04.980 --> 00:00:07.590
Well I think we should get
started with some problems, and

00:00:07.590 --> 00:00:10.100
hopefully as we work through
them, you'll have an

00:00:10.100 --> 00:00:13.491
understanding of how to do
these types of problems.

00:00:13.491 --> 00:00:18.650
Let me make sure the
pen tool is correct.

00:00:18.650 --> 00:00:27.480
Let's say I had 11 plus 4.

00:00:27.480 --> 00:00:32.570
So first you'd say hey, Sal, 11
plus 4, I don't know how to

00:00:32.570 --> 00:00:34.250
add two-digit numbers yet.

00:00:34.250 --> 00:00:35.960
Well there's a couple of ways
we can think about this.

00:00:35.960 --> 00:00:39.720
First I'll show you how all you
have to know is how to add

00:00:39.720 --> 00:00:42.210
one-digit numbers and you can
use something called carrying

00:00:42.210 --> 00:00:43.510
to solve the whole problem.

00:00:43.510 --> 00:00:45.610
Then we'll actually try to
visually represent it to show

00:00:45.610 --> 00:00:47.100
you how you could actually
do this type of problem

00:00:47.100 --> 00:00:48.640
in your head as well.

00:00:48.640 --> 00:00:51.860
So what you do with these types
of problems is you first look

00:00:51.860 --> 00:00:56.040
at the right-most
digit on the 11.

00:00:56.040 --> 00:00:57.510
We call this the
ones place, right?

00:00:57.510 --> 00:01:00.410
Because this 1 is 1, and we
call this the tens place.

00:01:00.410 --> 00:01:04.180
I know I'm going to confuse
you a lot, but that's when

00:01:04.180 --> 00:01:06.570
everything looks
easier later on.

00:01:06.570 --> 00:01:08.780
So you look at this ones place,
you say there's a 1 there.

00:01:08.780 --> 00:01:11.820
You take that 1 and you add it
to the number right below it.

00:01:11.820 --> 00:01:13.850
So 1 plus 4 is 5.

00:01:13.850 --> 00:01:16.670
You knew that, right?

00:01:16.670 --> 00:01:20.920
You know that 1 plus
4 is equal to 5.

00:01:20.920 --> 00:01:21.680
That's all I did here.

00:01:21.680 --> 00:01:25.730
I just said this 1 plus
this 4 is equal to 5.

00:01:25.730 --> 00:01:27.300
Now I go to this 1.

00:01:27.300 --> 00:01:29.520
This 1 plus -- well there's
nothing here other than a plus

00:01:29.520 --> 00:01:30.870
sign and that's not a number.

00:01:30.870 --> 00:01:33.030
So this 1 plus nothing is 1.

00:01:33.030 --> 00:01:35.560
So we put a 1 here.

00:01:35.560 --> 00:01:40.770
And we get 11 plus
4 is equal to 15.

00:01:40.770 --> 00:01:44.590
Just so you know that this
system actually works, let's

00:01:44.590 --> 00:01:46.490
actually draw it out in a
couple of different ways

00:01:46.490 --> 00:01:48.620
just to give you the
intuition of 11 plus 4.

00:01:48.620 --> 00:01:55.260
So if I had 11 balls -- one,
two, three, four, five, six,

00:01:55.260 --> 00:01:58.910
seven, eight, nine,
ten, eleven.

00:01:58.910 --> 00:01:59.720
That's eleven, right?

00:02:15.760 --> 00:02:17.840
So that's 11 and we're
going to add 4 to it.

00:02:17.840 --> 00:02:22.740
So one, two, three, four.

00:02:22.740 --> 00:02:26.430
So now all we have to do is
count how many total circles

00:02:26.430 --> 00:02:28.060
or balls we have now.

00:02:28.060 --> 00:02:34.000
That's one, two, three, four,
five, six, seven, eight,

00:02:34.000 --> 00:02:40.570
nine, ten, eleven, twelve,
thirteen, fourteen, fifteen.

00:02:40.570 --> 00:02:42.130
I don't recommend that you
do this every time you do

00:02:42.130 --> 00:02:44.000
a problem because it'll
take you a long time.

00:02:44.000 --> 00:02:46.070
But hey, if you ever get
confused, it's better to take a

00:02:46.070 --> 00:02:48.350
long time than to get it wrong.

00:02:48.350 --> 00:02:50.650
Let's think about another way
of representing this, because

00:02:50.650 --> 00:02:53.300
I think different visual
approaches appeal in different

00:02:53.300 --> 00:02:54.480
ways to different people.

00:02:54.480 --> 00:02:56.210
Let's draw a number line.

00:02:56.210 --> 00:02:58.010
I don't know if you've seen
a number line before but

00:02:58.010 --> 00:03:00.930
you're going to see it now.

00:03:00.930 --> 00:03:03.690
And a number line, all
I do is I draw out all

00:03:03.690 --> 00:03:04.795
the numbers in order.

00:03:04.795 --> 00:03:14.310
So 0, 1, 2, 3, 4, 5, 6 -- I'm
doing them small so I know I

00:03:14.310 --> 00:03:35.910
can get to 15 -- 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18.

00:03:35.910 --> 00:03:36.450
And so on.

00:03:36.450 --> 00:03:39.610
And these arrows mean that
the numbers keep going

00:03:39.610 --> 00:03:41.180
in both directions.

00:03:41.180 --> 00:03:42.923
I know this is a little early
for you to learn this, but

00:03:42.923 --> 00:03:43.890
actually the numbers can
actually keep going to the

00:03:43.890 --> 00:03:45.260
left below zero as well.

00:03:45.260 --> 00:03:46.810
I'll leave you to
think about that.

00:03:46.810 --> 00:03:48.990
But anyway, so let's go
back to this problem.

00:03:48.990 --> 00:03:52.180
So we have 11, so let me circle
11 -- let me see where 11

00:03:52.180 --> 00:03:52.980
is on the number line.

00:03:52.980 --> 00:03:55.210
11 is here, right?

00:03:55.210 --> 00:03:56.820
This is 11.

00:03:56.820 --> 00:03:58.520
And we're adding 4.

00:03:58.520 --> 00:04:03.020
So when you add, that means
we're increase 11 by 4.

00:04:03.020 --> 00:04:05.120
So that when you increase we're
going to go up the number line,

00:04:05.120 --> 00:04:07.010
right, or we're going to go the
right on the number line

00:04:07.010 --> 00:04:08.410
because the numbers
are getting bigger.

00:04:08.410 --> 00:04:15.300
So we'd go 1, 2, 3, 4 -- bam.

00:04:15.300 --> 00:04:16.910
We're at 15.

00:04:16.910 --> 00:04:19.930
Once again, this takes a long
time, but if you ever get

00:04:19.930 --> 00:04:23.450
confused or you forget what 1
plus 4 is, although I don't

00:04:23.450 --> 00:04:26.620
think you should, then you
could just do it this way.

00:04:26.620 --> 00:04:29.500
Let's do some maybe
harder problems now.

00:04:32.740 --> 00:04:41.300
Let's do 28 plus 7.

00:04:43.840 --> 00:04:46.660
8 plus 7 -- I'll tell you,
frankly, even to this

00:04:46.660 --> 00:04:50.560
day, I sometimes get
confused with 8 plus 7.

00:04:50.560 --> 00:04:53.700
If you know the answer then you
already know how to do this

00:04:53.700 --> 00:04:55.450
problem, you can just write
whatever the answer

00:04:55.450 --> 00:04:55.770
is down here.

00:04:55.770 --> 00:04:58.570
But let draw it on the number
line, because I think a little

00:04:58.570 --> 00:05:01.620
bit of more basic addition
practice isn't unwarranted

00:05:01.620 --> 00:05:02.750
at this point.

00:05:02.750 --> 00:05:04.920
So we could do out with
the number line again.

00:05:04.920 --> 00:05:05.690
8 plus 7.

00:05:09.030 --> 00:05:11.570
I'm not going to start at 0,
I'll start at like 5, because

00:05:11.570 --> 00:05:13.440
if you keep going you'll
get to 0 eventually.

00:05:13.440 --> 00:05:29.600
So let's see 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15,

00:05:29.600 --> 00:05:33.276
16, 17, 18 and so on.

00:05:33.276 --> 00:05:35.500
It'll keep going all the
way to 100 and 1,000 and a

00:05:35.500 --> 00:05:37.480
million billion trillion.

00:05:37.480 --> 00:05:38.600
So what are we doing?

00:05:38.600 --> 00:05:40.450
We start at 8 because
this is 8 plus 7.

00:05:40.450 --> 00:05:44.405
We want to figure out
what 8 plus 7 is.

00:05:44.405 --> 00:05:46.780
So we start at 8.

00:05:46.780 --> 00:05:47.790
We're going to add 7 to it.

00:05:47.790 --> 00:05:49.540
Let me change colors.

00:05:49.540 --> 00:05:58.030
So you go 1, 2, 3, 4, 5, 6, 7.

00:05:58.030 --> 00:05:59.580
Oh, that's 15 shows up again.

00:05:59.580 --> 00:06:02.590
So 8 plus 7 is equal to 15.

00:06:02.590 --> 00:06:05.500
Over time as you do practice, I
think you'll memorize these

00:06:05.500 --> 00:06:09.240
that 8 plus 7 is 15 or
whatever, 6 plus 7 is

00:06:09.240 --> 00:06:10.710
13 or any of these.

00:06:10.710 --> 00:06:13.420
But in the interim it actually
doesn't hurt to do this number

00:06:13.420 --> 00:06:16.370
line because you actually are
visualizing what's happening.

00:06:16.370 --> 00:06:18.260
And you can do it with
the circles as well.

00:06:18.260 --> 00:06:19.590
So we know 8 plus 7 is 15.

00:06:19.590 --> 00:06:22.280
So this is a new thing you're
going to learn right now.

00:06:22.280 --> 00:06:24.260
You don't write the
whole 15 down here.

00:06:24.260 --> 00:06:29.010
You write the 5 -- you write
this five right here.

00:06:29.010 --> 00:06:33.300
And then that 1,
you carry the 1.

00:06:33.300 --> 00:06:35.170
You put it up there.

00:06:35.170 --> 00:06:38.270
I think in a future
presentation I'll explain why

00:06:38.270 --> 00:06:41.840
this works and maybe you might
even kind of have an intuition

00:06:41.840 --> 00:06:45.862
because the 1 is in the tens
place, and this is

00:06:45.862 --> 00:06:47.320
the tens place.

00:06:47.320 --> 00:06:48.810
I don't want to confuse you.

00:06:48.810 --> 00:06:50.870
So you have that 1 and
now you add it to the

00:06:50.870 --> 00:06:54.670
2, and you get 35.

00:06:54.670 --> 00:06:59.060
Because 1 plus 2 is
equal to 3, right?

00:06:59.060 --> 00:07:00.050
So you're done.

00:07:00.050 --> 00:07:02.090
35.

00:07:02.090 --> 00:07:03.760
You might ask, well, does
that make sense that

00:07:03.760 --> 00:07:06.720
28 plus 7 is 35?

00:07:06.720 --> 00:07:09.000
And there's a couple of ways
I'd like to think about this.

00:07:09.000 --> 00:07:12.800
Well, 8 plus 7 we know is 15.

00:07:12.800 --> 00:07:14.240
And I don't know how
comfortable we are

00:07:14.240 --> 00:07:15.000
with higher numbers.

00:07:15.000 --> 00:07:18.500
But 18 plus 7 -- so
look at this pattern.

00:07:18.500 --> 00:07:22.640
8 plus 7 is equal to 15.

00:07:22.640 --> 00:07:25.360
18 plus 7 -- you're probably
saying, Sal, where are you

00:07:25.360 --> 00:07:27.720
getting the 18 from, but
take my word for it.

00:07:27.720 --> 00:07:30.380
18 plus 7 is 25.

00:07:30.380 --> 00:07:36.930
28 plus 7 is equal to 35, which
is the one that we just did.

00:07:36.930 --> 00:07:38.710
This is a check mark.

00:07:38.710 --> 00:07:41.890
If you kept going, you
said 38 plus 7, that

00:07:41.890 --> 00:07:43.380
actually equals 45.

00:07:43.380 --> 00:07:45.860
So you might see a little
pattern here, and then you can

00:07:45.860 --> 00:07:47.910
just sit and think about this
for a little bit if you like.

00:07:47.910 --> 00:07:49.740
Maybe you pause the video.

00:07:49.740 --> 00:07:52.350
Another way you could think
about this if you still don't

00:07:52.350 --> 00:07:57.870
believe me is you say OK, if I
have 28, if I add 1 I get 29,

00:07:57.870 --> 00:08:01.970
if I add 2 I get 30,
if I add 3 I get 31.

00:08:01.970 --> 00:08:05.410
If I add 4 I get 32.

00:08:05.410 --> 00:08:08.425
If I add 5 I get 33.

00:08:08.425 --> 00:08:11.550
If I add 6 I get 34.

00:08:11.550 --> 00:08:14.110
And if I add 7 I get 35 again.

00:08:14.110 --> 00:08:16.150
Right, all I did is I kept
saying oh, if I had one more

00:08:16.150 --> 00:08:19.800
I'll get a larger little bit --
the number a little bit larger.

00:08:19.800 --> 00:08:21.230
Let's do some more problems,
and I think we'll

00:08:21.230 --> 00:08:21.940
do a couple more.

00:08:21.940 --> 00:08:24.680
Let's do it a little faster
because you might get what

00:08:24.680 --> 00:08:26.110
we're doing here now.

00:08:26.110 --> 00:08:27.580
Let's do a hard one.

00:08:27.580 --> 00:08:33.300
Let's do 99 plus 9.

00:08:33.300 --> 00:08:35.400
So what's 9 plus 9?

00:08:35.400 --> 00:08:37.670
So if you don't know what it
is, you can work it out

00:08:37.670 --> 00:08:40.720
either using the number line
or drawing the circles.

00:08:40.720 --> 00:08:41.990
That's a fair way to do
it, although you should

00:08:41.990 --> 00:08:44.070
eventually kind of know it.

00:08:44.070 --> 00:08:46.110
9 plus 9 it turns out is 18.

00:08:50.770 --> 00:08:55.320
You put the 8 down here
and you carry the 1.

00:08:55.320 --> 00:08:56.810
And now you just say 1 plus 9.

00:08:56.810 --> 00:08:58.150
Well you know what 1 plus 9 is.

00:08:58.150 --> 00:09:01.480
1 plus 9 is equal to 10.

00:09:01.480 --> 00:09:04.640
So there's nowhere to carry
this 1, so you write the

00:09:04.640 --> 00:09:07.460
whole thing down here.

00:09:07.460 --> 00:09:12.280
So 99 plus 9 is equal to 108.

00:09:12.280 --> 00:09:15.820
Let's do one more problem.

00:09:15.820 --> 00:09:22.450
Let's say 56 plus 7.

00:09:22.450 --> 00:09:23.780
Well what's 6 plus 7.

00:09:23.780 --> 00:09:30.070
Well 6 plus 7 is 13, right?

00:09:30.070 --> 00:09:32.650
If you get confused, draw
out everything again.

00:09:32.650 --> 00:09:33.980
And then you get 1 plus 5.

00:09:33.980 --> 00:09:35.570
1 plus 5 is 6.

00:09:35.570 --> 00:09:36.960
63.

00:09:36.960 --> 00:09:38.770
And you might want to give
yourself a bunch of problems

00:09:38.770 --> 00:09:41.580
and I think you're also now, if
you understand what we did,

00:09:41.580 --> 00:09:45.060
ready to try the level
two addition problems.

00:09:45.060 --> 00:09:46.580
Have fun.