Examples dividing by tenths and hundredths

WEBVTT
Kind: captions
Language: en

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- [Instructor] Welcome.

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So let's see if we can figure out

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what eight divided by 0.4 is.

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Pause this video and see if
you can work through that.

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All right.

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So we're trying to figure
out what eight ones

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divided by four tenths is.

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And so one way to think about it

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is to think about everything
in terms of tenths.

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So eight ones.

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One one is equal to 10 tenths,

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so eight ones is equal to to 80 tenths.

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So this is equal to 80 tenths.

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And I'm dividing it by four tenths.

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So divided by four tenths.

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So if I have 80 tenths,

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and if I wanna divide it into
equal groups of four tenths,

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how many equal groups am I going to have?

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Well I am going to have 20
equal groups of four tenths.

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That would make up 80 tenths.

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80 divided by four is 20.

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So eight divided by four
tenths is equal to 20.

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Another way that we
could've approached that

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is we could've thought about
it in terms of fractions.

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We could've said, hey, eight
divided by four tenths,

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that's the same thing as eight

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divided by four tenths,

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which is the same thing when
you divide by a fraction.

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It's the same thing as
multiplying by the reciprocal.

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So this is the same thing as eight times

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10 over four,

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or you could view it as the
same thing as eight over one.

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Eight ones times 10 over four.

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Times 10 over four.

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In your numerator you get eight times 10,

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so that is going to be equal to,

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I'll do it in this orange color,

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eight times 10 is equal to 80.

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And then in our denominator

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we're going to have one times four.

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I didn't circle the one.

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You're going to have
the one times the four.

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So that's just going to be equal to four.

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So 80 fourths, or 80 divided by four,

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once again that is
going to be equal to 20.

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Let's do another example,
but in this case,

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let's deal with hundredths.

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So draw a line here.

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And so let's say that someone
walks up to you on the street

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and says, "Quick,

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"what is 48

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"divided by

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"0.24?"

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Pause this video and see
if you can figure that out,

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and I'll give you a little bit of a hint.

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The techniques we just did might apply.

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Although now we're
dealing with hundredths.

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All right now let's work on this together.

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Well one way to work on it
is you could view this as 48.

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You could just convert
everything to fractions.

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As 48, and actually I can
just write it as 48 over one,

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divided by

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divided by

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24 hundredths

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24 hundredths.

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This division sign looks
a little bit weird.

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Let me make it look a
little bit more normal.

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There you go.

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And so when you divide by a fraction

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that's the same thing as
multiplying by the reciprocal.

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We've seen that before.

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So this is equal to 48 over one

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times 100 24ths.

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And so our numerator here

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is going to be equal to 48 times 100.

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Actually I'm just gonna write
that out as 48 times 100

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in the numerator.

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And then our denominator is
going to be 24 times one.

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Well that's just would equal to 24.

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And so what is this going to be equal to?

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48 times 100 divided by 24.

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Well we could divide the numerator

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and the denominator by 24.

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If we did that, 48 divided by 24 is two.

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And 24 divided by 24 is one,

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so we're left with two
times 100 divided by one.

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Well two times 100 is equal to 200.

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And then you divide that
by one, you still get 200.

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So there you have it.

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This is equal to 200.

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Now another way that you
could have approached this

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is you could've thought
about everything again,

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in terms of hundredths.

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And so 48 ones is the same
thing as 4800 hundredths.

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So it's equal to 48.

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4800 hundredths.

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So let me write that down.

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Hundredths.

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Divided by 24 hundredths.

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Divided by 24 hundredths.

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So if I have 4800 or 4,800 hundredths,

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and if I wanna divide it into
equal groups of 24 hundredths,

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how many of those equal
groups would I have?

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Well you would have 4,800
divided by 24 equal groups.

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So this is going to be equal to

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4,800.

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4,800.

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Divided by 24.

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Now 4,800 is exactly what we had up here.

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And so we already saw

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that 4,800 divided by 24 is equal to 200.

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And if you just got to this point,

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and if you said, hey,
how could I work on this?

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You could just say, hey, let me divide

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the numerator and the denominator by 24.

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You could try to re express
4800 as 48 times 100.

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Or you could even say
that 48 is 24 times two,

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and then you're gonna
multiply that times 100

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to get to 4,800.

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And then the division is a
little bit more straightforward.

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But the big picture here is to realize

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that dividing by decimals
is not some mysterious thing

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when you think about it
in terms of place value

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or when you think about
it in terms of fractions.


Visually dividing decimal by whole number

WEBVTT
Kind: captions
Language: en

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- [Instructor] In this video,

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we're going to try to figure out

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what 4/10 divided by five is.

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So pause this video and see
if you can think about it

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before we work through it together.

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We're really going to think
about approaching this visually.

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All right, now let's work
through this together.

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Let's actually try to think
about what 4/10 looks like.

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So if you view this
entire square as a whole,

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you see that we've divided
it into 10 equal columns

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or 10 equal sections,
and four of those tenths

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are shaded in, so what you
see here in blue is 4/10.

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But how do we divide that into
five and make sense of it?

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Well, one way to think about it

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is to imagine 4/10 not just as 4/10,

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but to imagine it as 40/100,

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so this would be imagining it as 40/100.

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So we can re-write 4/10 divided by 5

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as 40/100 divided by 5,

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and now we can think about
taking these 40 hundredths,

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each of these little
squares is a hundredth,

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and divide it into five equal sections,

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and then we can say,
well, how many hundredths

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are in each of those five equal sections?

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So let's do that.

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So let's see, this is one, this is two,

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this is three, this is four

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and then we have five equal sections.

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So how many hundredths are in
each of those equal sections?

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Well we can see in each of them you have

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one, two, three, four,
five, six, seven, eight.

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So 40/100 divided by five

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is going to be 8/100, because
we have eight of these

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little squares in each of
those five equal sections.

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So 8/100 we would like this.

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And so 40/100 divided by five is 8/100,

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then 4/10 divided by five
is also equal to 8/100.


2 step estimation example

WEBVTT
Kind: captions
Language: en

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- [Instructor] We are told
a teacher bought 12 sheets

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of stickers to use on the
homework of her students.

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Each sheet had 48 stickers.

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At the end of the year,

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the teacher had 123 stickers remaining.

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Which is the best estimate

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for the number of
stickers the teacher used?

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So like always, pause this
video, and see if you can

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have a go at this before
we work on this together.

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All right, now let's
work on this together.

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So the first thing to appreciate is

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we just have to figure out an estimate.

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We don't have to figure
out the exact number

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of the stickers that were used.

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Let's see if we can do that.

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So let's see,

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we have 12 sheets of stickers,

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and each sheet had 48 stickers,

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48 stickers per sheet.

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So how many stickers did
the teacher start off with?

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Well, there were 12 sheets

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times the number of stickers
per sheet, so times 48,

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times 48.

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This is going to be the number
that they started off with,

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number to start.

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And then if we want to figure
out the number that are used,

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we just have to figure out, okay,

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from the number that was
started, how many are left over?

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And then that's how many were used.

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And so how many were left over?

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Well, 123 were remaining
at the end of the year,

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so that's this number right over here.

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So if we calculate that
first, the number to start,

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we subtract out the
number that are remaining,

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then that will be equal
to the number of stickers

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that the teacher used.

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Now once again, we don't
have to figure out exactly.

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We just have to estimate.

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And so I'm just going to try

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to figure out friendlier
numbers to work with.

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So instead of 12, let's imagine,

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let's imagine, actually,
I'll stick with 12.

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12 I can work with.

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But let's say that this is going

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to be approximately equal to,

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so in parentheses, instead of 48,

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I'll say it's roughly 50.

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So this is going to be
approximately 12 times 50.

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Instead of 123, I'll say that's roughly,

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a friendlier number might
be 120 or might be 100.

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Let's just do 120, so minus 120.

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We could've done 100.

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And so we could figure out
what this is in our heads

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or with a little bit of paper.

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12 times five is 60,
so 12 times 50 is 600.

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And then 600, if we had 100 here,

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600 minus 100 would've been 500,

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or 600 minus 120 is 480.

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So what we want to do
is look at the choice

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and see which of these choices is closest

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to roughly 500 or roughly 480?

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And so let's see, out of all of these,

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actually they have exactly 480, which is,

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so they estimated exactly the
way we happened to estimate.

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Now not every person is going to do that.

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We could've chosen,

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instead of 123 becoming
120 in our estimate,

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we could've put 100 there, and
then we would've gotten 500.

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But even if 500 was our estimate,

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480 still would've been the
closest to that estimate.


Division with partial quotients example

WEBVTT
Kind: captions
Language: en

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- [Instructor] Let's say we
wanna figure out what 473

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divided by five is.

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And like always, why
don't you pause this video

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and try to work through it?

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And if you're familiar with the idea

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of division with partial quotients,

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I encourage you to try it out that way.

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All right, now let's
work on this together.

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And so let me just
rewrite this in a format

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that makes it a little bit easier for me

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to do my division with partial quotients.

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And so I'm just going to write 473

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divided by five.

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And what's neat about division
with partial quotients is

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that there's a lot of
flexibility in how you do it.

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There's no exactly one right way.

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And so you could say, all right,

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how many times can five go into 473

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without going over 473?

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You might be tempted.

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Someone might say, "Oh, maybe
you can go a hundred times."

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But you could quickly see that
if you said a hundred times,

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and they say, "How much do I left over?"

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you do a hundred times five.

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You get 500.

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And you'd say, "Oh, wait,
wait, I overshot 473."

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So a hundred is too high.

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So then you would back up a little bit.

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And then you could say,

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"All right, I'm gonna go for low numbers

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"and just slowly chip away at it."

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So you might say that five goes into 473.

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You know it goes more than 20 times.

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Let's just start with 20.

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Well, what is 20 times five?

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20 times five is 100.

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And let me be very clear what that is.

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That is 20 times our
five right over there.

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And so clearly we did not overshoot it,

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but how many more times can
five go into 473 than 20?

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To figure that out,

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we figure out how much more
we have left over to go,

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so we subtract the hundred from 473.

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And we have 373.

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So how many times can five go into 373?

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Well, if 20 times five was a hundred,

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well, gee, this is 300 here.

00:02:01.450 --> 00:02:03.650
So maybe we can go another 60 times.

00:02:03.650 --> 00:02:06.570
So let me put a 60 right over there.

00:02:06.570 --> 00:02:09.360
And so what is 60 times five?

00:02:09.360 --> 00:02:14.330
Well, six 10s times five
would be 30 10s, which is 300.

00:02:14.330 --> 00:02:17.780
And so how much further do we have to go?

00:02:17.780 --> 00:02:21.510
So we still have 73 left.

00:02:21.510 --> 00:02:25.900
373 minus 300 is 73.

00:02:25.900 --> 00:02:30.060
So how many times does five go into 73?

00:02:30.060 --> 00:02:31.480
Well, let's see.

00:02:31.480 --> 00:02:35.980
I know five times 12 is
60, so let me try a 12.

00:02:35.980 --> 00:02:38.950
So I'll put a 12 over here.

00:02:38.950 --> 00:02:42.110
And so 12 times five is 60,

00:02:42.110 --> 00:02:45.284
so how much further do we have to go?

00:02:45.284 --> 00:02:48.150
Let's see, 73 minus 60 is 13.

00:02:48.150 --> 00:02:49.910
And five can go into 13,

00:02:49.910 --> 00:02:52.430
so we can get a few more fives in there.

00:02:52.430 --> 00:02:53.420
Just to be clear what we're doing,

00:02:53.420 --> 00:02:55.210
we've already said,
hey, we can go 20 times

00:02:55.210 --> 00:02:58.510
and then another 60 and
then another 12 times.

00:02:58.510 --> 00:03:01.150
And if we go into 473 all of those times,

00:03:01.150 --> 00:03:02.960
we still have 13 left over.

00:03:02.960 --> 00:03:04.140
So we could go into,

00:03:04.140 --> 00:03:06.500
we can even fit a few more fives in there.

00:03:06.500 --> 00:03:08.480
Now, how many times does five go into 13?

00:03:08.480 --> 00:03:09.770
Well, five times two is 10.

00:03:09.770 --> 00:03:11.470
Five times three would be too much.

00:03:11.470 --> 00:03:13.410
So we could say two times.

00:03:13.410 --> 00:03:16.540
So that gets us two times five is 10.

00:03:16.540 --> 00:03:20.750
And so there what we have
left over is a three.

00:03:20.750 --> 00:03:22.830
And then we can't fit any more fives in.

00:03:22.830 --> 00:03:25.010
We can't divide five into three,

00:03:25.010 --> 00:03:26.930
so that's going to give us our remainder.

00:03:26.930 --> 00:03:30.700
So how many times does five go into 473?

00:03:30.700 --> 00:03:33.250
Well, we were able to
get it in there 20 times

00:03:33.250 --> 00:03:36.560
and then another 60 times
and then another 12 times

00:03:36.560 --> 00:03:38.270
and then another two times.

00:03:38.270 --> 00:03:41.330
So what we can do is we can
add all of these together,

00:03:41.330 --> 00:03:42.163
and what do we get?

00:03:42.163 --> 00:03:45.520
Two times plus 12 is 14,

00:03:45.520 --> 00:03:50.153
plus 60 is 74, plus 20 is 94.

00:03:51.450 --> 00:03:53.550
And then we have a remainder of three.

00:03:53.550 --> 00:03:56.340
So this is equal to 94,

00:03:56.340 --> 00:03:58.840
and then I'll write this in
red, with a remainder of three.

00:03:58.840 --> 00:03:59.800
Did I do that right?

00:03:59.800 --> 00:04:01.630
Two plus 14, let's see.

00:04:01.630 --> 00:04:03.530
Two plus 12 is 14.

00:04:03.530 --> 00:04:05.310
74, yeah, 94.

00:04:05.310 --> 00:04:06.810
Now let me just show you that there's more

00:04:06.810 --> 00:04:08.300
than one way to do this.

00:04:08.300 --> 00:04:09.420
I'll draw a little line here

00:04:09.420 --> 00:04:11.510
so we don't get it too confused.

00:04:11.510 --> 00:04:15.040
We could have done something like this.

00:04:15.040 --> 00:04:18.490
We could've said what is 473

00:04:18.490 --> 00:04:21.810
divided by five?

00:04:21.810 --> 00:04:25.260
And we might've immediately
recognized that,

00:04:25.260 --> 00:04:28.500
all right, five goes into 40 eight times.

00:04:28.500 --> 00:04:32.930
So five would go into
400 80 times at least,

00:04:32.930 --> 00:04:36.320
probably more times than that,
but let's just start with 80.

00:04:36.320 --> 00:04:40.060
And so how close does that get us to 473?

00:04:40.060 --> 00:04:40.920
Well, to figure that out,

00:04:40.920 --> 00:04:44.970
we multiply 80 times five to get 400.

00:04:44.970 --> 00:04:48.740
And so can we fit any more
fives into the difference?

00:04:48.740 --> 00:04:50.120
So let's figure out the difference.

00:04:50.120 --> 00:04:51.770
So we will subtract.

00:04:51.770 --> 00:04:55.490
And just to be very clear,
this was 80 times five.

00:04:55.490 --> 00:04:56.550
The reason why we did that,

00:04:56.550 --> 00:04:59.580
you see how close did we get to 473?

00:04:59.580 --> 00:05:01.500
And now we wanna figure out
how much do we have left over?

00:05:01.500 --> 00:05:03.440
How many more fives can we fit in?

00:05:03.440 --> 00:05:06.330
And so you subtract 400 from 473,

00:05:06.330 --> 00:05:10.240
and we are going to get 73.

00:05:10.240 --> 00:05:14.490
Well, we can for sure
fit more fives into 73.

00:05:14.490 --> 00:05:15.340
We might say, hey,

00:05:15.340 --> 00:05:18.020
maybe let's try to fit
another 10 fives in there.

00:05:18.020 --> 00:05:20.210
So let's stick another 10 fives in there.

00:05:20.210 --> 00:05:21.630
Now how close are we?

00:05:21.630 --> 00:05:24.740
Well, 10 times five is 50.

00:05:24.740 --> 00:05:26.600
And to be clear, what we just did here

00:05:26.600 --> 00:05:30.370
is we multiplied 10 times five.

00:05:30.370 --> 00:05:32.650
And now can we fit any more fives in?

00:05:32.650 --> 00:05:36.970
Well, to figure that out,
we would need to subtract

00:05:36.970 --> 00:05:39.860
and figure out what we have left.

00:05:39.860 --> 00:05:44.740
So 73 minus 50, that's going to be 23.

00:05:44.740 --> 00:05:47.950
And we can for sure
fit more fives into 23.

00:05:47.950 --> 00:05:51.210
We know that five times
four is equal to 20.

00:05:51.210 --> 00:05:53.810
So let's put four more fives in there.

00:05:53.810 --> 00:05:56.400
So four times five is going to be 20.

00:05:56.400 --> 00:05:57.960
Let's see how close we are now.

00:05:57.960 --> 00:06:02.960
And to be very clear, that
is four times our five.

00:06:03.090 --> 00:06:04.350
And so what do we get to?

00:06:04.350 --> 00:06:05.587
What do we have left?

00:06:05.587 --> 00:06:07.690
Can we fit any more fives in?

00:06:07.690 --> 00:06:10.370
Let's see, 23 minus 20 is just three.

00:06:10.370 --> 00:06:12.120
And that's less than five,

00:06:12.120 --> 00:06:15.150
so we can't fit any more
fives in, so there we have it.

00:06:15.150 --> 00:06:18.040
We were able to get 80 fives into 473.

00:06:18.040 --> 00:06:19.700
We saw that we still had some left over.

00:06:19.700 --> 00:06:22.330
We were able to fit another
10 into that leftover.

00:06:22.330 --> 00:06:25.040
And then we still saw that
we had some left over.

00:06:25.040 --> 00:06:28.410
We were able to fit another
four into that leftover.

00:06:28.410 --> 00:06:29.950
And then we had a leftover of three

00:06:29.950 --> 00:06:31.930
and we weren't able to
fit any more fives in.

00:06:31.930 --> 00:06:33.730
And so in total,

00:06:33.730 --> 00:06:37.740
we were able to fit 94 fives

00:06:37.740 --> 00:06:41.190
into 473, and then we had three left over,

00:06:41.190 --> 00:06:43.333
which is exactly what we had there.


Dividing a decimal by a whole number example

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.740
- [Instructor] Let's see if we can compute

00:00:01.740 --> 00:00:06.740
what 1.86 divided by two is.

00:00:07.000 --> 00:00:10.610
And like always pause this
video and have a go at it.

00:00:10.610 --> 00:00:12.700
And I'll give you a hint,
see if you can think

00:00:12.700 --> 00:00:17.380
about 1.86 as a certain
number of hundredths,

00:00:17.380 --> 00:00:18.903
and then divide that by two.

00:00:19.940 --> 00:00:22.340
All right now let's work
through it together.

00:00:22.340 --> 00:00:24.320
So as promised, let's see if we can

00:00:24.320 --> 00:00:27.950
think of 1.86 as a certain
number of hundredths.

00:00:27.950 --> 00:00:29.700
So if we go to the hundredths place,

00:00:29.700 --> 00:00:31.880
we only see six there, but we could

00:00:31.880 --> 00:00:35.680
view 8/10 as 80 hundredths.

00:00:35.680 --> 00:00:40.350
So you could view the
.86 as 86 hundredths,

00:00:40.350 --> 00:00:44.170
and each one is 100
hundredths, so we really

00:00:44.170 --> 00:00:47.520
have 186 hundredths.

00:00:47.520 --> 00:00:49.630
So this number right over here I'm going

00:00:49.630 --> 00:00:54.630
to rewrite as 186 hundredths, hundredths.

00:00:56.700 --> 00:01:01.420
And that is what we need to divide by two.

00:01:01.420 --> 00:01:06.320
Now, if you're dividing
186 of something by two,

00:01:06.320 --> 00:01:08.670
well it's just gonna be 186 divided by two

00:01:08.670 --> 00:01:09.800
of that something.

00:01:09.800 --> 00:01:14.800
So we could rewrite this
as 186 divided by two,

00:01:16.450 --> 00:01:18.340
and then that's the number of hundredths

00:01:18.340 --> 00:01:21.600
this is going to be, so I'll
put that in parenthesis.

00:01:21.600 --> 00:01:25.453
So it's going to be that
many hundredths, hundredths.

00:01:26.830 --> 00:01:31.380
All right, so what is 186 divided by two?

00:01:31.380 --> 00:01:33.070
Some of you might be able
to do it in your head,

00:01:33.070 --> 00:01:36.660
some of you might wanna use some paper.

00:01:36.660 --> 00:01:40.130
The way I like to think
about it is if I were to take

00:01:40.130 --> 00:01:43.760
the six ones and divide it by two,

00:01:43.760 --> 00:01:45.800
well that's going to be three ones.

00:01:45.800 --> 00:01:50.180
And then if I were to take the 18 tens

00:01:50.180 --> 00:01:53.830
and divided by two, that's
going to be nine tens,

00:01:53.830 --> 00:01:56.120
so that gets us to 93.

00:01:56.120 --> 00:01:58.710
Now you could've also
thought about it like this,

00:01:58.710 --> 00:02:00.680
you could've said, how many times does two

00:02:00.680 --> 00:02:02.680
go into 186?

00:02:02.680 --> 00:02:04.240
Let's see, it doesn't go into one,

00:02:04.240 --> 00:02:05.870
how many times does it go into 18?

00:02:05.870 --> 00:02:08.750
Nine times, nine times two is 18,

00:02:08.750 --> 00:02:11.970
we could subtract, and then you have zero,

00:02:11.970 --> 00:02:13.430
you bring down that six.

00:02:13.430 --> 00:02:16.870
Two goes into six three
times, three times two is six.

00:02:16.870 --> 00:02:18.530
You subtract and you get no remainders.

00:02:18.530 --> 00:02:20.550
So either way you're going to get 93.

00:02:20.550 --> 00:02:22.280
And it's 93 what?

00:02:22.280 --> 00:02:25.860
Well it's 93 hundredths, is
what we're talking about.

00:02:25.860 --> 00:02:28.850
93 hundredths.

00:02:28.850 --> 00:02:32.030
And now we just wanna
express 93 hundredths

00:02:32.030 --> 00:02:33.340
as a decimal.

00:02:33.340 --> 00:02:34.740
Well how do we do that?

00:02:34.740 --> 00:02:36.532
Well that's gonna be 0 point.

00:02:36.532 --> 00:02:41.532
The 90 hundredths is
the same thing as 9/10,

00:02:42.470 --> 00:02:46.000
and then the three hundredths,
is of course 300 hundredths.

00:02:46.000 --> 00:02:48.010
Three in the hundredths place.

00:02:48.010 --> 00:02:50.680
So it's 0.93, and so we
can write it up here.

00:02:50.680 --> 00:02:54.140
This is going to be equal to 0.93,

00:02:54.140 --> 00:02:55.163
and we're done!


Visually dividing a whole number by a decimal

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.570
- [Instructor] In this
video, we're going to see

00:00:01.570 --> 00:00:06.570
if we can compute what three
divided by 0.75 is equal to,

00:00:11.410 --> 00:00:12.850
and I'll give you a little bit of a hint

00:00:12.850 --> 00:00:14.830
before I tell you to pause the video.

00:00:14.830 --> 00:00:19.420
So let's imagine three
wholes right over here,

00:00:19.420 --> 00:00:22.430
and what if we were to
think of these three wholes,

00:00:22.430 --> 00:00:24.560
these three ones, and what if we were to

00:00:24.560 --> 00:00:26.990
think about them in terms of hundredths?

00:00:26.990 --> 00:00:30.000
Well then we could, once we
express these as hundredths,

00:00:30.000 --> 00:00:34.590
we could say, well how many
groups of 75 hundredths

00:00:34.590 --> 00:00:35.990
can we divide this into?

00:00:35.990 --> 00:00:39.570
How many equal groups of 75 hundredths?

00:00:39.570 --> 00:00:41.020
To help you think about that,

00:00:41.020 --> 00:00:43.580
here's a little bit of a visualization

00:00:43.580 --> 00:00:48.580
where three is divided into
equal groups of 75 hundredths.

00:00:49.740 --> 00:00:51.960
So pause this video and use what I have,

00:00:51.960 --> 00:00:54.760
this little visual here, as a big hint

00:00:54.760 --> 00:00:59.253
to figure out what three
divided by 75/100 is.

00:01:00.630 --> 00:01:02.660
All right, now let's do it together.

00:01:02.660 --> 00:01:05.830
So as I said, we started with three ones,

00:01:05.830 --> 00:01:08.170
or three wholes, however
you want to think about it,

00:01:08.170 --> 00:01:11.290
and you could divide division by 75/100

00:01:11.290 --> 00:01:15.530
as saying, well how many
equal groups of 75 hundredths

00:01:15.530 --> 00:01:17.760
can you divide these three wholes into?

00:01:17.760 --> 00:01:19.930
And so we thought about these three wholes

00:01:19.930 --> 00:01:23.490
in terms of hundredths, and
the way it's been color coded,

00:01:23.490 --> 00:01:25.550
it actually has already been divided

00:01:26.617 --> 00:01:30.420
into four equal groups of 75 hundredths.

00:01:30.420 --> 00:01:31.830
You can count these, so this is

00:01:31.830 --> 00:01:35.500
10, 20, 30, 40,

00:01:35.500 --> 00:01:38.710
50, 60, 70,

00:01:38.710 --> 00:01:40.900
and then five hundredths.

00:01:40.900 --> 00:01:45.900
So this right over here, this
is one group of 75 hundredths.

00:01:47.930 --> 00:01:52.360
Then in green you have another
group of 75 hundredths.

00:01:52.360 --> 00:01:56.050
So you have 10, 20, 30,

00:01:56.050 --> 00:02:00.137
40, 50, 60, 70,

00:02:01.470 --> 00:02:04.710
and then 75, so that right over,

00:02:04.710 --> 00:02:06.380
let me use a color I can see.

00:02:06.380 --> 00:02:08.560
This right over here is our second group.

00:02:08.560 --> 00:02:10.620
This is our first group right over there.

00:02:10.620 --> 00:02:12.610
Then that's our third group,

00:02:12.610 --> 00:02:15.490
these are all equal
groups of 75 hundredths,

00:02:15.490 --> 00:02:17.690
and that's our fourth equal group

00:02:17.690 --> 00:02:20.570
of 75 hundredths in the green again.

00:02:20.570 --> 00:02:24.170
So how many equal groups of 75 hundredths

00:02:24.170 --> 00:02:26.970
can you divide three wholes into?

00:02:26.970 --> 00:02:30.633
Well, clearly it is equal to four.


Introduction to division with partial quotients

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.480
- [Instructor] In this
video, we wanna compute

00:00:02.480 --> 00:00:07.480
what 833 divided by seven is.

00:00:08.220 --> 00:00:10.850
So I encourage you to pause this video

00:00:10.850 --> 00:00:13.580
and see if you can figure
that out on your own.

00:00:13.580 --> 00:00:14.900
All right, now let's
work through it together.

00:00:14.900 --> 00:00:16.410
And you might have appreciated,

00:00:16.410 --> 00:00:17.750
this is a little bit more difficult

00:00:17.750 --> 00:00:19.430
than things that we have done in the past.

00:00:19.430 --> 00:00:21.770
And in this video, I'm
going to show you a method

00:00:21.770 --> 00:00:24.350
that your parents have probably not seen,

00:00:24.350 --> 00:00:26.220
but you'll see that it's kind of fun,

00:00:26.220 --> 00:00:28.630
and it's called division
with partial quotients,

00:00:28.630 --> 00:00:32.070
which is a very fancy word,
but as I said, it'll be fun.

00:00:32.070 --> 00:00:35.142
So, the first thing I will
do is I will rewrite this

00:00:35.142 --> 00:00:40.142
as 833 divided by, divided by seven.

00:00:43.600 --> 00:00:46.350
So you can view these
as the same expression.

00:00:46.350 --> 00:00:48.780
The reason why we do it
this way is it formats it

00:00:48.780 --> 00:00:49.670
so it's a little bit easier

00:00:49.670 --> 00:00:52.380
to do our division with partial quotients.

00:00:52.380 --> 00:00:55.450
So the way that division
with partial quotients works,

00:00:55.450 --> 00:00:56.290
and once again, it's not the way

00:00:56.290 --> 00:00:58.310
that your parents probably
learned how to do it,

00:00:58.310 --> 00:00:59.610
is you just say, hey,

00:00:59.610 --> 00:01:02.490
how many times can seven go into 833?

00:01:02.490 --> 00:01:04.090
I don't have to get it exactly.

00:01:04.090 --> 00:01:06.750
I just wanna go under 833.

00:01:06.750 --> 00:01:09.230
And so my brain immediately thinks, well,

00:01:09.230 --> 00:01:12.760
seven hundreds is less than 833,

00:01:12.760 --> 00:01:16.220
so we're going to go into
833 at least 100 times.

00:01:16.220 --> 00:01:17.610
And so what we would do

00:01:17.610 --> 00:01:20.403
is we would write that hundred up here.

00:01:20.403 --> 00:01:23.020
We'd wanna be very careful
about our place value.

00:01:23.020 --> 00:01:25.110
You can view this column
as the hundreds column,

00:01:25.110 --> 00:01:27.550
this is the tens column,
this is the ones column,

00:01:27.550 --> 00:01:29.960
and then we wanna see, how
much do we have left over?

00:01:29.960 --> 00:01:33.860
How close did seven times 100 get us?

00:01:33.860 --> 00:01:38.363
So what we do is we multiply
100 times seven to get 700,

00:01:39.450 --> 00:01:42.980
and then we can subtract that
700 from 833 to figure out

00:01:42.980 --> 00:01:44.820
how much more we have left.

00:01:44.820 --> 00:01:49.017
And so 833 minus 700 is 133.

00:01:52.080 --> 00:01:54.150
And so we can then say, all right,

00:01:54.150 --> 00:01:56.920
we still have another 133 to go.

00:01:56.920 --> 00:02:00.730
So how many more times
can seven go into this?

00:02:00.730 --> 00:02:03.750
Well, seven goes into 133,

00:02:03.750 --> 00:02:05.790
once again, you don't
have to have it exactly.

00:02:05.790 --> 00:02:10.790
If you know seven times 10 is equal to 70,

00:02:10.930 --> 00:02:12.080
actually, let's go with that.

00:02:12.080 --> 00:02:13.880
We know we go at least 10 times.

00:02:13.880 --> 00:02:15.410
So let's write that up here.

00:02:15.410 --> 00:02:17.040
We're going at least 10 times,

00:02:17.040 --> 00:02:18.820
and to figure out how
much more we have left,

00:02:18.820 --> 00:02:22.560
let's multiply 10 times seven to get 70,

00:02:22.560 --> 00:02:27.560
and then we can subtract,
and we see that we have,

00:02:27.680 --> 00:02:29.670
let's see, three minus zero is three.

00:02:29.670 --> 00:02:32.980
13 tens minus seven tens is six tens.

00:02:32.980 --> 00:02:34.600
So we have 63 left.

00:02:34.600 --> 00:02:36.880
So seven definitely can go into 63.

00:02:36.880 --> 00:02:37.850
We're gonna keep doing this

00:02:37.850 --> 00:02:41.890
until we have a number
less than seven over here.

00:02:41.890 --> 00:02:44.610
So let's see, seven, how many
times does seven go into 63?

00:02:44.610 --> 00:02:46.810
You might know from your
multiplication tables

00:02:46.810 --> 00:02:49.290
that seven times nine is 63.

00:02:49.290 --> 00:02:50.870
So you could get it exactly.

00:02:50.870 --> 00:02:52.490
So you could just write that up here.

00:02:52.490 --> 00:02:55.080
We have nine more times
to go into the number,

00:02:55.080 --> 00:02:57.137
and then you would say
nine times seven is 63.

00:02:57.137 --> 00:02:59.940
You could say, hey, we got exactly there.

00:02:59.940 --> 00:03:01.560
We have nothing left over.

00:03:01.560 --> 00:03:04.570
And as long as this
number is less than seven,

00:03:04.570 --> 00:03:06.650
you know that you can't
divide seven anymore

00:03:06.650 --> 00:03:09.040
into our original number.

00:03:09.040 --> 00:03:10.800
And so, you're done.

00:03:10.800 --> 00:03:14.950
And so, how many times
does seven go into 833?

00:03:14.950 --> 00:03:16.880
Well, we said it went 100 times,

00:03:16.880 --> 00:03:18.690
and then we were able
to go another 10 times,

00:03:18.690 --> 00:03:20.900
and then we were able to
go another nine times.

00:03:20.900 --> 00:03:23.450
And so what we wanna do
is add these numbers.

00:03:23.450 --> 00:03:28.450
So you wanna add 100 plus 10 plus nine.

00:03:29.640 --> 00:03:32.540
When you add up all of
them, what do you get?

00:03:32.540 --> 00:03:35.300
You get nine ones, one 10, one hundred.

00:03:35.300 --> 00:03:38.760
You get 119, so this is equal to 119.

00:03:38.760 --> 00:03:41.580
All I did is I added these up.

00:03:41.580 --> 00:03:44.380
Now, I wanna be very clear
that you could do division

00:03:44.380 --> 00:03:46.910
with partial quotients and
not do it exactly like this.

00:03:46.910 --> 00:03:48.660
That's kind of why it's fun.

00:03:48.660 --> 00:03:50.550
So let's do it another way.

00:03:50.550 --> 00:03:52.870
So let's say we wanna figure out again

00:03:52.870 --> 00:03:57.243
how many times does
seven go into 833, 833?

00:04:01.410 --> 00:04:06.080
We could have said,
maybe it goes 150 times.

00:04:06.080 --> 00:04:08.280
So what you could have said
is you could have said,

00:04:08.280 --> 00:04:12.800
all right, my current guess
or estimate is 150 times,

00:04:12.800 --> 00:04:14.730
that I could multiply 150 times seven.

00:04:14.730 --> 00:04:15.563
How would I do that?

00:04:15.563 --> 00:04:17.990
Let's see, zero times seven is zero,

00:04:17.990 --> 00:04:21.440
five times seven is 35.

00:04:21.440 --> 00:04:25.023
You can carry the three, so to speak.

00:04:26.770 --> 00:04:31.770
One times seven is seven plus
that three is going to be 10.

00:04:31.990 --> 00:04:34.090
And so that gets us to 1050.

00:04:34.090 --> 00:04:36.490
Well, over here, we just
finished overshooting.

00:04:36.490 --> 00:04:38.050
It doesn't go 150 times.

00:04:38.050 --> 00:04:38.980
There's nothing left over.

00:04:38.980 --> 00:04:40.900
So 150 is too high.

00:04:40.900 --> 00:04:42.980
So we would wanna backtrack that.

00:04:42.980 --> 00:04:46.630
And then you would go, well
maybe I could go to 110.

00:04:46.630 --> 00:04:47.980
So let's try that out.

00:04:47.980 --> 00:04:51.810
So 110, and now let's multiply.

00:04:51.810 --> 00:04:53.960
Zero times seven is zero.

00:04:53.960 --> 00:04:55.940
One times seven is seven.

00:04:55.940 --> 00:04:57.820
One times seven is seven.

00:04:57.820 --> 00:05:01.210
So, 110 times seven is 770.

00:05:01.210 --> 00:05:02.043
So that works.

00:05:02.043 --> 00:05:04.700
It's less than 833, but
let's see what we have left.

00:05:04.700 --> 00:05:07.750
So we subtract, we get a three here,

00:05:07.750 --> 00:05:12.750
and then, let's see, 83 tens
minus 77 tens, that's six tens,

00:05:13.550 --> 00:05:15.770
and actually, that got
us there a lot faster.

00:05:15.770 --> 00:05:17.140
So then you could just know

00:05:17.140 --> 00:05:19.640
that hey, seven goes into 63 times.

00:05:19.640 --> 00:05:20.980
But let's say we didn't know that.

00:05:20.980 --> 00:05:21.880
We could say, all right,

00:05:21.880 --> 00:05:25.790
let's say I'm gonna estimate
it goes eight times.

00:05:25.790 --> 00:05:28.310
So you would put an eight up here.

00:05:28.310 --> 00:05:30.080
And then you say, how much
did we have left over?

00:05:30.080 --> 00:05:33.410
Eight times seven is 56.

00:05:33.410 --> 00:05:38.410
You subtract, and then 63
minus 56 is exactly seven.

00:05:40.130 --> 00:05:42.690
And you say, okay, look,
I can go one more time.

00:05:42.690 --> 00:05:44.390
So I'd write that up there.

00:05:44.390 --> 00:05:46.620
So one times seven is seven,

00:05:46.620 --> 00:05:48.940
and then you see we
have nothing left over.

00:05:48.940 --> 00:05:50.100
So we are done.

00:05:50.100 --> 00:05:51.660
So how many times did it go in?

00:05:51.660 --> 00:05:55.650
One plus eight is nine plus 110 is 119.

00:05:55.650 --> 00:05:57.630
So hopefully, you find that interesting,

00:05:57.630 --> 00:06:00.500
and I really want you to think
about why this is working.

00:06:00.500 --> 00:06:01.720
We're just trying to see how many times

00:06:01.720 --> 00:06:04.230
can we go in without overshooting it,

00:06:04.230 --> 00:06:05.630
and then what's left over?

00:06:05.630 --> 00:06:07.420
So how many more times can we go in,

00:06:07.420 --> 00:06:08.620
and then what's left over,

00:06:08.620 --> 00:06:10.210
and then how many more times can we go in,

00:06:10.210 --> 00:06:12.520
until what we have left
over is less than seven,

00:06:12.520 --> 00:06:15.523
so that we can't go
into it any more times.


Estimating division that results in non whole numbers

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.980
- [Instructor] So let's think
about something a little bit.

00:00:01.980 --> 00:00:06.953
What do you think 17 divided
by two is going to be?

00:00:08.010 --> 00:00:12.140
Well, you might immediately
realize that it's not obvious

00:00:12.140 --> 00:00:14.150
what you need to multiply two by

00:00:14.150 --> 00:00:17.870
in order to get to 17.

00:00:17.870 --> 00:00:19.990
There's no whole number
that I could put here

00:00:19.990 --> 00:00:21.890
that gets us to 17.

00:00:21.890 --> 00:00:25.720
We know that two times, let's see,

00:00:25.720 --> 00:00:28.970
two times eight is a
little bit less than that.

00:00:28.970 --> 00:00:31.740
Two times eight is equal to 16.

00:00:31.740 --> 00:00:35.600
And we know that two times nine is going

00:00:35.600 --> 00:00:39.640
to be a little bit more
than that, is equal to 18.

00:00:39.640 --> 00:00:41.520
And so this question mark,

00:00:41.520 --> 00:00:46.490
two times question mark is equal to 17.

00:00:46.490 --> 00:00:49.270
This helps us know that our question mark,

00:00:49.270 --> 00:00:51.470
our unknown value here is going

00:00:51.470 --> 00:00:54.700
to be between eight and nine.

00:00:54.700 --> 00:00:57.970
It is going to be eight point something.

00:00:57.970 --> 00:00:59.290
So let's do another example.

00:00:59.290 --> 00:01:00.710
If someone were to ask you,

00:01:00.710 --> 00:01:03.220
hey, let's think about 18 divided by four.

00:01:03.220 --> 00:01:04.550
And you don't have to figure out exactly

00:01:04.550 --> 00:01:07.680
what it is just yet, but
I want you to think about

00:01:07.680 --> 00:01:12.260
what two consecutive whole
numbers is 18 divided by four,

00:01:12.260 --> 00:01:14.563
whatever that is, what is that between?

00:01:15.830 --> 00:01:17.700
Well, we could do something similar.

00:01:17.700 --> 00:01:19.470
We could say, all right, let's see,

00:01:19.470 --> 00:01:22.530
four times four is equal to 16,

00:01:22.530 --> 00:01:24.670
so that doesn't quite get us to 18.

00:01:24.670 --> 00:01:29.590
We could say that four
times five is equal to 20,

00:01:29.590 --> 00:01:31.480
so that's more than 18.

00:01:31.480 --> 00:01:36.360
So four times question
mark is equal to 18.

00:01:36.360 --> 00:01:38.160
This helps us realize that

00:01:38.160 --> 00:01:40.410
because 18 is between 16 and 20,

00:01:40.410 --> 00:01:42.370
this question mark
right over here is going

00:01:42.370 --> 00:01:43.860
to be between four and five.

00:01:43.860 --> 00:01:48.700
So 18 divided by four is going
to be four point something.

00:01:48.700 --> 00:01:52.370
Now, another way of
thinking about or estimating

00:01:52.370 --> 00:01:56.380
what division might result
in is through an inequality.

00:01:56.380 --> 00:01:59.720
So if I were to say 87

00:01:59.720 --> 00:02:02.600
divided by nine,

00:02:02.600 --> 00:02:06.690
how would you compare
that to 10 over here?

00:02:06.690 --> 00:02:08.100
So there's a couple of
ways you could compare it.

00:02:08.100 --> 00:02:10.860
You could say, hey, maybe
this is greater than 10.

00:02:10.860 --> 00:02:12.850
Maybe it's equal to 10.

00:02:12.850 --> 00:02:15.150
Maybe it is less than 10.

00:02:15.150 --> 00:02:17.710
So pause this video and see
if you can figure that out.

00:02:17.710 --> 00:02:18.760
What would you put here?

00:02:18.760 --> 00:02:21.573
Is it less than, greater
than, or equal to 10?

00:02:22.800 --> 00:02:24.550
Well, we know it's not equal to 10.

00:02:24.550 --> 00:02:29.140
We know that 90 divided by nine

00:02:29.140 --> 00:02:31.290
is equal to 10.

00:02:31.290 --> 00:02:33.960
And let's see, if we go one
multiple of nine less than that,

00:02:33.960 --> 00:02:38.580
we know that 81 divided by nine

00:02:38.580 --> 00:02:41.230
is equal to nine.

00:02:41.230 --> 00:02:45.930
And so 87 divided by nine is
in between these two values.

00:02:45.930 --> 00:02:47.610
Let me do this in another color.

00:02:47.610 --> 00:02:51.920
So 87 divided by nine is going

00:02:51.920 --> 00:02:55.920
to be something in between nine and 10.

00:02:55.920 --> 00:03:00.920
And so it is going to be less than 10.

00:03:01.120 --> 00:03:03.370
So the whole point of
this video is to start

00:03:03.370 --> 00:03:05.270
to get us a little bit comfortable

00:03:05.270 --> 00:03:10.270
with this idea of dividing
where we might get an answer

00:03:10.490 --> 00:03:12.320
that is not a whole number.

00:03:12.320 --> 00:03:14.390
And in the future we'll
learn how to compute that,

00:03:14.390 --> 00:03:16.320
but for now we're just
learning to estimate

00:03:16.320 --> 00:03:18.660
and just get a sense of, hey, okay,

00:03:18.660 --> 00:03:21.120
this is going to be
between eight and nine.

00:03:21.120 --> 00:03:24.150
This is going to be between four and five.

00:03:24.150 --> 00:03:28.330
This is going to be between nine and 10,

00:03:28.330 --> 00:03:30.730
which just means it's going
to be nine point something,

00:03:30.730 --> 00:03:32.830
which means it's going to be less than 10.


Estimating quotients

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.700
- [Instructor] We are
told to use estimation

00:00:02.700 --> 00:00:06.310
to tell whether each estimated quotient

00:00:06.310 --> 00:00:09.530
is reasonable or unreasonable.

00:00:09.530 --> 00:00:12.910
So we have a bunch of
estimated quotients here.

00:00:12.910 --> 00:00:17.210
Someone is estimating that
2,419 divided by three

00:00:17.210 --> 00:00:20.480
is roughly equal to 7,500

00:00:21.542 --> 00:00:22.375
and we have to figure out

00:00:22.375 --> 00:00:23.208
is that reasonable or unreasonable?

00:00:23.208 --> 00:00:25.180
And then we have two more
quotients to figure out

00:00:25.180 --> 00:00:27.180
if it's unreasonable or reasonable.

00:00:27.180 --> 00:00:28.050
Pause this video

00:00:28.050 --> 00:00:29.570
and see if you can
figure it out on your own

00:00:29.570 --> 00:00:31.980
before we work through it together.

00:00:31.980 --> 00:00:33.350
All right, now let's
work through it together.

00:00:33.350 --> 00:00:35.100
The way that I'd like to tackle this

00:00:35.100 --> 00:00:38.470
is just try to do the estimation myself.

00:00:38.470 --> 00:00:40.674
I'll start with this first one.

00:00:40.674 --> 00:00:45.674
2,419 divided by three.

00:00:46.470 --> 00:00:47.920
One way to estimate

00:00:47.920 --> 00:00:51.120
is try to think about what
numbers are these close to

00:00:51.120 --> 00:00:53.440
that are a little bit
easier to compute with.

00:00:53.440 --> 00:00:55.260
Some people would say they're
a little bit friendlier

00:00:55.260 --> 00:00:56.600
so to speak.

00:00:56.600 --> 00:00:59.210
Well three is already
a pretty simple number

00:00:59.210 --> 00:01:00.170
a pretty friendly number.

00:01:00.170 --> 00:01:03.550
2,419, well it's a little
bit more complicated.

00:01:03.550 --> 00:01:05.140
But I could try to think of

00:01:05.140 --> 00:01:10.140
is there there a multiple of
three that's close to 2,419

00:01:10.300 --> 00:01:13.410
that is a fairly friendly number?

00:01:13.410 --> 00:01:17.010
I do know that three times eight is 24.

00:01:17.010 --> 00:01:19.520
So if I had something like 2,400

00:01:19.520 --> 00:01:21.360
which is the same thing as 2,400

00:01:21.360 --> 00:01:23.680
that'd be easy to divide by three.

00:01:23.680 --> 00:01:27.060
So I could say that
this is roughly equal to

00:01:27.060 --> 00:01:30.840
2,400 divided by three.

00:01:30.840 --> 00:01:35.840
All I did is I said, hey
2,419 for estimation purposes

00:01:36.350 --> 00:01:39.360
let's just say it's
roughly equal to 2,400.

00:01:39.360 --> 00:01:44.150
And then 2,400 is fairly
straightforward to divide by three.

00:01:44.150 --> 00:01:45.710
This over here

00:01:46.750 --> 00:01:49.300
24 divided by three is eight

00:01:49.300 --> 00:01:53.003
so 2,400 divided by three is 800.

00:01:54.190 --> 00:01:56.240
This is going to be 800.

00:01:56.240 --> 00:02:01.240
So 2,419 divided by three
should be roughly around 800.

00:02:02.040 --> 00:02:05.250
They're saying that it's roughly 7,500.

00:02:05.250 --> 00:02:06.870
That's a lot more than 800.

00:02:06.870 --> 00:02:08.610
If they said 750

00:02:08.610 --> 00:02:10.010
then I'd say, hey that's
pretty reasonable.

00:02:10.010 --> 00:02:12.240
But they're saying 7,500.

00:02:12.240 --> 00:02:14.830
That's almost 10 times more than 800

00:02:14.830 --> 00:02:18.453
so, this I would say is unreasonable.

00:02:19.420 --> 00:02:20.940
All right, let's think about this next one

00:02:20.940 --> 00:02:22.210
and again, pause the video

00:02:22.210 --> 00:02:24.750
and see if you can have a go at it.

00:02:24.750 --> 00:02:28.630
All right, I'm just going
to try to estimate again.

00:02:28.630 --> 00:02:32.940
So 435 divided by five.

00:02:32.940 --> 00:02:34.923
Let's say that's roughly equal to.

00:02:36.960 --> 00:02:40.740
An easy multiple of
five that's close to 435

00:02:40.740 --> 00:02:42.340
if I wanna think of a friendly number

00:02:42.340 --> 00:02:43.760
it might be 400.

00:02:43.760 --> 00:02:48.500
So I'll say that's roughly
equal to 400 divided by five.

00:02:49.620 --> 00:02:54.390
Now, we know that 40 divided by five is

00:02:54.390 --> 00:02:55.750
is eight, sorry.

00:02:55.750 --> 00:03:00.710
And so 40 10s divided
by five is eight 10s.

00:03:00.710 --> 00:03:04.490
So this is going to be equal to 80.

00:03:04.490 --> 00:03:05.580
And you could think
about it the other way.

00:03:05.580 --> 00:03:10.470
Five times eight 10s is going to be 40 10s

00:03:10.470 --> 00:03:12.633
which is the same thing as 400.

00:03:13.620 --> 00:03:16.640
This thing should be roughly equal to 80.

00:03:16.640 --> 00:03:18.470
Well, that's exactly what we estimated

00:03:18.470 --> 00:03:20.453
so that feels very reasonable.

00:03:22.040 --> 00:03:24.273
All right, this last one.

00:03:25.628 --> 00:03:28.230
Pause the video again, see
if you can have a go at that.

00:03:28.230 --> 00:03:33.230
All right, so we have
6,202 divided by two.

00:03:33.620 --> 00:03:35.650
We could say that's approximately equal to

00:03:35.650 --> 00:03:38.290
or roughly equal to.

00:03:38.290 --> 00:03:40.320
Well, if I just wanted
to do it very roughly

00:03:40.320 --> 00:03:42.680
I could say, hey 6,202

00:03:42.680 --> 00:03:43.870
let's just go to 6,000

00:03:43.870 --> 00:03:45.780
just to make things really easy.

00:03:45.780 --> 00:03:47.160
Divided by two.

00:03:47.160 --> 00:03:51.130
And so this is going to be
six 1000s divided by two

00:03:51.130 --> 00:03:54.500
would be three 1000s or 3,000.

00:03:54.500 --> 00:03:56.090
But here they have 300.

00:03:56.090 --> 00:03:58.540
That's one tenth of what I have over here.

00:03:58.540 --> 00:04:01.603
So this is unreasonable.

00:04:02.700 --> 00:04:03.533
And we're done.


Describing numerical relationships with polynomial identities

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.510
- [Instructor] What we're going
to do in this video is use

00:00:02.510 --> 00:00:05.620
what we know about polynomials
and how to manipulate them

00:00:05.620 --> 00:00:07.930
and what we've talked about
of whether two polynomials

00:00:07.930 --> 00:00:10.220
are equal to each other for all values

00:00:10.220 --> 00:00:12.220
of the variable that they're written in,

00:00:12.220 --> 00:00:14.080
so whether we're dealing
with a polynomial identity.

00:00:14.080 --> 00:00:16.940
And we're going to use
those skills in order

00:00:16.940 --> 00:00:21.730
to prove some properties of
relationships between numbers.

00:00:21.730 --> 00:00:24.420
So if I were to list out some integers,

00:00:24.420 --> 00:00:29.130
I could go zero, I could
go one, I could go two,

00:00:29.130 --> 00:00:32.080
three, four, five.

00:00:32.080 --> 00:00:34.000
And if I were to list
out the squares of these,

00:00:34.000 --> 00:00:36.350
if I were to create a
sequence of integer squares,

00:00:36.350 --> 00:00:38.970
well, zero squared would be zero,

00:00:38.970 --> 00:00:41.240
one squared would be one,

00:00:41.240 --> 00:00:43.760
two squared is four,

00:00:43.760 --> 00:00:46.040
three squared is nine,

00:00:46.040 --> 00:00:48.100
four squared is 16,

00:00:48.100 --> 00:00:50.490
five squared is 25.

00:00:50.490 --> 00:00:54.510
And we could, of course,
keep going in either case.

00:00:54.510 --> 00:00:55.930
But the first thing I
want you to think about,

00:00:55.930 --> 00:00:57.720
before you even write down a polynomial

00:00:57.720 --> 00:00:59.080
or try to construct one,

00:00:59.080 --> 00:01:02.460
is look at this sequence
of integer squares.

00:01:02.460 --> 00:01:04.280
And do you see any pattern

00:01:04.280 --> 00:01:07.500
in terms of the difference
between successive terms

00:01:07.500 --> 00:01:09.993
of this sequence of integer squares?

00:01:11.240 --> 00:01:14.020
All right, now let's
think about this together.

00:01:14.020 --> 00:01:18.180
So to go from zero to one, you add one.

00:01:18.180 --> 00:01:22.930
And to go from one to four, you add three.

00:01:22.930 --> 00:01:25.940
To go from four to nine,

00:01:25.940 --> 00:01:27.660
you add five.

00:01:27.660 --> 00:01:31.360
To go from nine to 16, you add seven.

00:01:31.360 --> 00:01:33.020
It seems like a pattern here.

00:01:33.020 --> 00:01:34.880
As we go to successive terms

00:01:34.880 --> 00:01:36.990
of this sequence of integer squares,

00:01:36.990 --> 00:01:40.340
we're adding increasing odd numbers.

00:01:40.340 --> 00:01:42.720
So I'm guessing that if I add nine here,

00:01:42.720 --> 00:01:44.880
which is the next odd
number, I'm gonna get to 25,

00:01:44.880 --> 00:01:46.890
and that indeed is the case.

00:01:46.890 --> 00:01:48.210
And you could test that out.

00:01:48.210 --> 00:01:49.670
Well, what, if I add 11,

00:01:49.670 --> 00:01:51.690
which would be the next odd
number, what do I get to?

00:01:51.690 --> 00:01:55.550
I get to 36, which is the square of six.

00:01:55.550 --> 00:01:58.650
But how can we feel good
that this always is true,

00:01:58.650 --> 00:02:01.150
that this never breaks down?

00:02:01.150 --> 00:02:02.800
Well, one way to do it is

00:02:02.800 --> 00:02:04.500
to think a little bit more generally,

00:02:04.500 --> 00:02:06.830
and that's where our
algebra and our knowledge

00:02:06.830 --> 00:02:08.760
of polynomials are going to be useful.

00:02:08.760 --> 00:02:10.360
So let's say we go all the way,

00:02:10.360 --> 00:02:11.700
and we're just speaking generally now.

00:02:11.700 --> 00:02:14.520
So we have the number n, and
then with the next number

00:02:14.520 --> 00:02:17.120
after that is going to be n plus one.

00:02:17.120 --> 00:02:20.390
And then if we think about
what the corresponding terms

00:02:20.390 --> 00:02:23.510
in the sequence of
integer squares would be,

00:02:23.510 --> 00:02:25.440
well, that would be, when we square it,

00:02:25.440 --> 00:02:27.600
when we get to n, we would get n squared.

00:02:27.600 --> 00:02:29.470
And when we get to n plus one,

00:02:29.470 --> 00:02:32.340
we would have n plus one squared.

00:02:32.340 --> 00:02:34.350
And let's see if we could
think about what the difference

00:02:34.350 --> 00:02:36.370
between these two things are.

00:02:36.370 --> 00:02:38.628
The difference between 25 and 16 is nine.

00:02:38.628 --> 00:02:40.780
Difference between 16 and nine is seven.

00:02:40.780 --> 00:02:41.970
So let's think about what the difference

00:02:41.970 --> 00:02:44.690
between n plus one
squared and n squared is.

00:02:44.690 --> 00:02:46.400
And how do we write that as a polynomial?

00:02:46.400 --> 00:02:49.490
Well, it'll just be n plus one squared

00:02:49.490 --> 00:02:52.150
minus n squared.

00:02:52.150 --> 00:02:54.590
And now let's see if we can rewrite this,

00:02:54.590 --> 00:02:56.480
algebraically manipulate
this so we can set up

00:02:56.480 --> 00:02:59.960
a polynomial identity that
describes this pattern

00:02:59.960 --> 00:03:01.470
that we just saw.

00:03:01.470 --> 00:03:04.399
So what I'll do is I'm
just going to expand out

00:03:04.399 --> 00:03:07.640
n plus one squared right over there.

00:03:07.640 --> 00:03:12.340
So that is going to be n
squared plus two n plus one.

00:03:12.340 --> 00:03:15.540
And then we have this
minus n squared here,

00:03:15.540 --> 00:03:17.440
so minus n squared.

00:03:17.440 --> 00:03:22.440
And so we see that n squared
minus n squared cancels out.

00:03:22.620 --> 00:03:25.750
And so we can rewrite
everything we have here as

00:03:25.750 --> 00:03:28.370
n plus one squared

00:03:28.370 --> 00:03:29.830
minus n squared.

00:03:29.830 --> 00:03:32.480
So this is really the difference
between successive terms

00:03:32.480 --> 00:03:34.620
in our sequence of integer squares

00:03:34.620 --> 00:03:38.760
is going to be equal to two n plus one

00:03:38.760 --> 00:03:40.870
for any integer n.

00:03:40.870 --> 00:03:44.970
Well, for any integer n, what
is two n plus one going to be?

00:03:44.970 --> 00:03:45.820
And especially here,

00:03:45.820 --> 00:03:49.330
we're dealing with the positive integers.

00:03:49.330 --> 00:03:53.560
Well, for any integer n, this
is going to be an odd integer.

00:03:53.560 --> 00:03:57.330
If you take any integer,
you multiply it by two,

00:03:57.330 --> 00:03:59.380
this part is going to be even.

00:03:59.380 --> 00:04:00.480
But then you add one to that,

00:04:00.480 --> 00:04:02.360
you're going to get an odd integer.

00:04:02.360 --> 00:04:03.920
And you can also see

00:04:03.920 --> 00:04:08.140
that this increases by two as n increases.

00:04:08.140 --> 00:04:09.770
So when you go from one odd integer,

00:04:09.770 --> 00:04:12.330
you go add two to the next odd integer.

00:04:12.330 --> 00:04:14.220
You add two to the next odd integer,

00:04:14.220 --> 00:04:17.270
which is exactly what is described there.

00:04:17.270 --> 00:04:19.000
So this is pretty neat.

00:04:19.000 --> 00:04:20.730
We've just used a little bit of algebra,

00:04:20.730 --> 00:04:23.270
a little bit of what we know
about polynomial identities

00:04:23.270 --> 00:04:26.720
to show that the difference
between successive terms

00:04:26.720 --> 00:04:30.010
in this sequence of integer
squares right over here

00:04:30.010 --> 00:04:32.983
is going to be increasing odd numbers.


Introduction to standard way of multiplying multidigit numbers

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:00.870
- [Instructor] What we're going to do

00:00:00.870 --> 00:00:02.010
in this video is think about how

00:00:02.010 --> 00:00:05.790
we might multiply 592 times seven.

00:00:05.790 --> 00:00:07.490
And in general we're gonna think about

00:00:07.490 --> 00:00:09.210
how we would approach
multiplying something

00:00:09.210 --> 00:00:11.610
that has multiple digits times something

00:00:11.610 --> 00:00:13.170
that has one digit.

00:00:13.170 --> 00:00:14.690
And the way we're going
to do it is the way

00:00:14.690 --> 00:00:15.890
that, if you were to ask your parents,

00:00:15.890 --> 00:00:17.690
it's probably the way that they do it.

00:00:17.690 --> 00:00:20.040
And so the typical approach is

00:00:20.040 --> 00:00:23.230
you would write the larger
multi digit number on top

00:00:23.230 --> 00:00:25.880
and then you would write the
smaller single digit number

00:00:25.880 --> 00:00:28.140
below that, and since
it's in the ones place,

00:00:28.140 --> 00:00:30.580
the seven, you would put it
in the ones place column.

00:00:30.580 --> 00:00:31.971
So you'd put it right below the ones place

00:00:31.971 --> 00:00:35.390
in the larger number,
so right below that two.

00:00:35.390 --> 00:00:38.070
And then you'd write the
multiplication symbol.

00:00:38.070 --> 00:00:39.620
And the way you'd think about it is,

00:00:39.620 --> 00:00:41.930
all right, I'm just gonna
take each of these places

00:00:41.930 --> 00:00:43.760
and multiply it by the seven.

00:00:43.760 --> 00:00:47.500
So, for example, if I'm
taking those two ones

00:00:47.500 --> 00:00:49.130
and I'm multiplying it times seven,

00:00:49.130 --> 00:00:51.060
well that's gonna be 14 ones.

00:00:51.060 --> 00:00:52.880
Well there's no digit for 14.

00:00:52.880 --> 00:00:55.150
I can only put four of
those ones over here.

00:00:55.150 --> 00:00:57.940
And then the other 10 ones I can express

00:00:57.940 --> 00:01:00.960
as one 10, and so I'd put it up there.

00:01:00.960 --> 00:01:02.150
Sometimes when people learn it,

00:01:02.150 --> 00:01:03.836
they say, hey, two times seven is 14.

00:01:03.836 --> 00:01:06.520
I write the four and I carry the one.

00:01:06.520 --> 00:01:07.620
But all you're doing is you're saying,

00:01:07.620 --> 00:01:10.910
hey, 14 is one 10 plus four ones.

00:01:10.910 --> 00:01:14.011
But then you move over to the tens place.

00:01:14.011 --> 00:01:17.400
You say, hey, what's
nine tens times seven?

00:01:17.400 --> 00:01:20.010
Well, nine times seven is 63,

00:01:20.010 --> 00:01:23.730
so nine tens times seven is 63 tens,

00:01:23.730 --> 00:01:26.280
plus another 10 is 64 tens.

00:01:26.280 --> 00:01:28.374
You can only put four
of those tens over here.

00:01:28.374 --> 00:01:33.374
So the other 60 tens you
can express as six hundreds,

00:01:33.470 --> 00:01:34.860
so you can stick that right over there.

00:01:34.860 --> 00:01:36.930
Now a lot of people would
explain that as saying,

00:01:36.930 --> 00:01:40.850
hey, nine times seven
is 63, plus one is 64.

00:01:40.850 --> 00:01:42.670
Write the four and carry the six.

00:01:42.670 --> 00:01:45.730
But hopefully you understand
what we mean by carrying.

00:01:45.730 --> 00:01:47.644
You're really trying to write 64 tens.

00:01:47.644 --> 00:01:50.600
Only four of those tens
can be expressed over here.

00:01:50.600 --> 00:01:52.200
Or that's maybe the cleanest way to do it.

00:01:52.200 --> 00:01:55.770
And then the other 60 tens you
can express as six hundreds.

00:01:55.770 --> 00:01:57.820
And then, last but not least,

00:01:57.820 --> 00:02:02.820
five hundreds times seven
is going to be 35 hundreds,

00:02:04.240 --> 00:02:08.690
and then you add six
hundreds, you get 41 hundreds.

00:02:08.690 --> 00:02:13.690
So 41 hundreds, so it's 4,144.

00:02:13.830 --> 00:02:16.174
Now I wanna reconcile this, or connect it,

00:02:16.174 --> 00:02:19.810
to with other ways that
you might have seen this.

00:02:19.810 --> 00:02:22.390
So let's say that, let's do this again.

00:02:22.390 --> 00:02:27.363
So if we were to write 592 times seven.

00:02:28.470 --> 00:02:31.300
So one way that we've
approached it in the past is

00:02:31.300 --> 00:02:33.290
we say, all right, what's two times seven?

00:02:33.290 --> 00:02:35.570
Well that's going to be 14.

00:02:35.570 --> 00:02:37.250
Notice that's the same 14.

00:02:37.250 --> 00:02:39.960
We're just representing it
a little bit differently.

00:02:39.960 --> 00:02:44.150
Then we might say, well
what is nine times seven?

00:02:44.150 --> 00:02:45.763
Gonna do the same color.

00:02:46.900 --> 00:02:48.960
And this really nine tens times seven.

00:02:48.960 --> 00:02:53.100
That's 63 tens, so you might
write it right over there,

00:02:53.100 --> 00:02:55.420
which is the same thing as 630.

00:02:55.420 --> 00:02:57.200
And then you could think about

00:02:57.200 --> 00:03:00.330
what is five hundreds times seven?

00:03:00.330 --> 00:03:03.780
Well that's 35 hundreds, so
you could write it like that.

00:03:03.780 --> 00:03:08.780
Same thing as 3500, and then
you would add everything up.

00:03:09.330 --> 00:03:11.530
So you have a total of four ones.

00:03:11.530 --> 00:03:14.060
You have a total of four tens.

00:03:14.060 --> 00:03:16.150
You have a total of 11 hundreds.

00:03:16.150 --> 00:03:17.700
You could right 100 there.

00:03:17.700 --> 00:03:20.720
And then regroup the other 10 hundreds

00:03:20.720 --> 00:03:23.047
into the thousands place as 1000.

00:03:23.047 --> 00:03:26.370
1000 plus three thousands is 4000.

00:03:26.370 --> 00:03:27.920
So we got the exact same answer because

00:03:27.920 --> 00:03:29.205
we essentially did the same thing.

00:03:29.205 --> 00:03:32.260
Over here, when we were carrying it,

00:03:32.260 --> 00:03:34.530
we were essentially
regrouping things from here

00:03:34.530 --> 00:03:37.290
and you could think about
it where we're condensing

00:03:37.290 --> 00:03:38.920
our writing versus what we did here.

00:03:38.920 --> 00:03:41.400
Here we just very systematically
said two times seven,

00:03:41.400 --> 00:03:42.709
nine times seven, five times seven,

00:03:42.709 --> 00:03:45.160
but we made sure to
keep track of the places

00:03:45.160 --> 00:03:46.195
to figure out what each of those,

00:03:46.195 --> 00:03:49.300
you could think of it as
partial products, would be,

00:03:49.300 --> 00:03:50.460
and then we added.

00:03:50.460 --> 00:03:52.370
While here we carried along the way,

00:03:52.370 --> 00:03:54.270
essentially regrouping
the values when we said,

00:03:54.270 --> 00:03:56.480
hey, two ones times seven ones,

00:03:56.480 --> 00:03:58.520
that's 14 ones, which is the same thing

00:03:58.520 --> 00:04:01.790
as four ones plus one 10.

00:04:01.790 --> 00:04:03.050
And so on, and so forth.

00:04:03.050 --> 00:04:05.740
So I encourage you, one, it's
good to learn this method.

00:04:05.740 --> 00:04:09.090
It's the most common
way that folks multiply.

00:04:09.090 --> 00:04:11.690
Once again, your parents
probably learned it this way,

00:04:11.690 --> 00:04:13.022
but it's really valuable to understand

00:04:13.022 --> 00:04:15.280
why these two things are the same thing,

00:04:15.280 --> 00:04:16.510
so really ponder that.

00:04:16.510 --> 00:04:17.470
Think about that.

00:04:17.470 --> 00:04:20.270
And see if you can, if it all makes sense

00:04:20.270 --> 00:04:21.103
what's going on there.

00:04:21.103 --> 00:04:23.943
You're not just blindly
memorizing the steps.


Strategies for multiplying decimals

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.023
- [Instructor] In this video,

00:00:01.023 --> 00:00:02.980
we're going to further build our intuition

00:00:02.980 --> 00:00:05.030
for multiplying decimals.

00:00:05.030 --> 00:00:07.100
So let's say that we wanted to figure out

00:00:07.100 --> 00:00:11.640
what eight times seven tenths is.

00:00:11.640 --> 00:00:13.550
Pause this video and see if you can figure

00:00:13.550 --> 00:00:15.243
this out on your own.

00:00:16.110 --> 00:00:18.130
All right, now there's several ways

00:00:18.130 --> 00:00:21.400
that we could approach what
eight times seven tenths is.

00:00:21.400 --> 00:00:23.830
We could view this as eight times,

00:00:23.830 --> 00:00:27.240
and we could write seven
tenths as a fraction.

00:00:27.240 --> 00:00:31.360
So we can re-express this as seven tenths,

00:00:31.360 --> 00:00:34.810
seven over 10 is the same thing as 0.7.

00:00:34.810 --> 00:00:37.590
And we already know how
to multiply fractions,

00:00:37.590 --> 00:00:41.140
you could view this as being equal to,

00:00:41.140 --> 00:00:45.070
eight is the same thing as
eight over one or eight wholes,

00:00:45.070 --> 00:00:48.400
I guess you could say,
times seven over 10,

00:00:48.400 --> 00:00:52.290
times seven tenths, which
is going to be equal to,

00:00:52.290 --> 00:00:56.400
if we multiply our numerator,
we're going to get 56.

00:00:56.400 --> 00:00:59.080
And if we multiply our
denominators, we get tenths.

00:00:59.080 --> 00:01:00.110
And that makes sense.

00:01:00.110 --> 00:01:04.870
If I have eight times seven
tenths, I end up with 56 tenths.

00:01:04.870 --> 00:01:08.600
Now 56 tenths can also be written as,

00:01:08.600 --> 00:01:13.560
this is the same thing
as 50, plus six over 10,

00:01:13.560 --> 00:01:18.560
which is the same thing as
50 over 10, plus six over 10.

00:01:18.670 --> 00:01:22.670
And so this is the same thing
as, this is five wholes,

00:01:22.670 --> 00:01:26.860
so five and six tenths,
five and six tenths,

00:01:26.860 --> 00:01:31.223
which we can write as five
and six tenths, or 5.6.

00:01:32.470 --> 00:01:35.070
And it's always good to do a
little bit of a reality check,

00:01:35.070 --> 00:01:38.290
whenever you get an answer when
you're multiplying decimals.

00:01:38.290 --> 00:01:41.970
Say, okay, seven tenths is
a little bit less than one.

00:01:41.970 --> 00:01:43.330
So we would expect this product,

00:01:43.330 --> 00:01:44.810
if we're multiplying eight times something

00:01:44.810 --> 00:01:46.470
a little bit less than one,

00:01:46.470 --> 00:01:47.650
we would expect the product to be

00:01:47.650 --> 00:01:49.220
a little bit less than eight.

00:01:49.220 --> 00:01:51.040
So 5.6 makes sense.

00:01:51.040 --> 00:01:52.440
If for some reason we got,

00:01:52.440 --> 00:01:55.480
the we you computed something
and you were to get 60,

00:01:55.480 --> 00:01:56.680
you say, well, that doesn't make sense,

00:01:56.680 --> 00:01:58.160
I should get a value less than eight.

00:01:58.160 --> 00:02:00.710
And similarly, if you somehow got a value

00:02:00.710 --> 00:02:02.780
or product of like one, you're like, well,

00:02:02.780 --> 00:02:05.030
that's a lot less than
eight, I should get something

00:02:05.030 --> 00:02:07.970
that is seven tenths of eight.

00:02:07.970 --> 00:02:10.180
Now, another way that
you could approach this

00:02:10.180 --> 00:02:14.300
is you could view this as the
same thing as eight times,

00:02:14.300 --> 00:02:15.690
and once again, I'm just gonna write this

00:02:15.690 --> 00:02:17.913
in a different way, eight times seven,

00:02:18.760 --> 00:02:20.830
eight times seven tenths.

00:02:20.830 --> 00:02:24.840
So, if you have eight
times seven of something,

00:02:24.840 --> 00:02:26.650
what is that going to be equal to?

00:02:26.650 --> 00:02:29.510
Well, eight times seven, that's 56.

00:02:29.510 --> 00:02:30.343
So you're going to be,

00:02:30.343 --> 00:02:34.112
this is going to be equal to 56 tenths,

00:02:34.112 --> 00:02:36.850
56 tenths.

00:02:36.850 --> 00:02:39.760
And one way to think about 56 tenths,

00:02:39.760 --> 00:02:44.760
56 tenths is the same thing as 50 tenths,

00:02:44.850 --> 00:02:46.930
50, let me color code that differently.

00:02:46.930 --> 00:02:51.690
So this is going to be the
same thing as 50 tenths,

00:02:51.690 --> 00:02:55.700
50 tenths, plus six tenths,

00:02:55.700 --> 00:03:00.700
plus the six tenths, get
right tenths, six tenths,

00:03:01.890 --> 00:03:06.190
and 50 tenths is the
same thing as five ones.

00:03:06.190 --> 00:03:09.710
So five ones, and six tenths,

00:03:09.710 --> 00:03:11.690
which is exactly what we have here,

00:03:11.690 --> 00:03:14.770
five ones, and six tenths.

00:03:14.770 --> 00:03:17.080
Let's do another example,

00:03:17.080 --> 00:03:19.560
that's a little bit more involved.

00:03:19.560 --> 00:03:24.560
So let's say that we want to figure out,

00:03:24.670 --> 00:03:27.860
we want to figure out what is

00:03:27.860 --> 00:03:31.773
three times 0.87.

00:03:33.890 --> 00:03:36.333
Pause this video and
try to figure that out.

00:03:37.440 --> 00:03:40.360
Well, once again, there's
many ways to approach it.

00:03:40.360 --> 00:03:43.310
But we could just start with
the way that we just looked at.

00:03:43.310 --> 00:03:47.210
We could say, hey, this is
the same thing as three times,

00:03:47.210 --> 00:03:49.510
and we can re-express this as,

00:03:49.510 --> 00:03:52.300
this is the same thing as 87 hundredths.

00:03:52.300 --> 00:03:54.423
87 hundredths,

00:03:56.380 --> 00:04:01.090
and so if I have three
times 87 of something,

00:04:01.090 --> 00:04:02.600
what am I going to be left with?

00:04:02.600 --> 00:04:03.650
Well, this is going to be equal

00:04:03.650 --> 00:04:05.620
to some number of hundredths,

00:04:05.620 --> 00:04:07.550
and to figure out that, we just figure out

00:04:07.550 --> 00:04:09.190
what's three times 87?

00:04:09.190 --> 00:04:12.270
So 87 times three,

00:04:12.270 --> 00:04:15.200
seven times three is 21,

00:04:15.200 --> 00:04:18.000
we regroup that two, becomes two 10s.

00:04:18.000 --> 00:04:21.450
And then eight times three is 24.

00:04:21.450 --> 00:04:24.140
And that's really 24 10s
plus those other two 10s,

00:04:24.140 --> 00:04:29.140
so we get 26 10s, which is
the same thing as 206 10s,

00:04:29.210 --> 00:04:30.540
but it's gonna be 261.

00:04:30.540 --> 00:04:34.910
So the three times 87 of
something is going to be 261

00:04:34.910 --> 00:04:38.600
of that something, and this
case something is hundredths.

00:04:38.600 --> 00:04:42.800
So this is 261 hundredths.

00:04:42.800 --> 00:04:45.600
So how do we express this as a decimal?

00:04:45.600 --> 00:04:47.820
Well, there's a couple of
ways that you can approach it.

00:04:47.820 --> 00:04:50.240
You can think about is
this is the ones place,

00:04:50.240 --> 00:04:54.200
this is the tenths place,
this is the hundredths place.

00:04:54.200 --> 00:04:57.920
And so very clearly, 100th here

00:04:57.920 --> 00:05:01.020
would be one in the hundredths place.

00:05:01.020 --> 00:05:06.020
If you have 60 hundredths, which
is what the six represents,

00:05:06.630 --> 00:05:11.080
60 hundredths is the
same thing as six tenths.

00:05:11.080 --> 00:05:15.843
And then last but not least,
if you have 200 hundredths,

00:05:16.830 --> 00:05:19.300
that's the same thing as two wholes.

00:05:19.300 --> 00:05:20.930
Another way to think about it is,

00:05:20.930 --> 00:05:22.520
you go to the hundredths place,

00:05:22.520 --> 00:05:26.040
and then you start from
there, but you write out 261,

00:05:26.040 --> 00:05:30.070
one, the 60 hundredths, and
then the 200 hundredths,

00:05:30.070 --> 00:05:33.080
and you get 2.61.

00:05:33.080 --> 00:05:35.250
Now another way that you
could have approached this,

00:05:35.250 --> 00:05:37.740
and we saw this in the last example,

00:05:37.740 --> 00:05:39.340
is you could say hey, this
is going to be the same thing

00:05:39.340 --> 00:05:44.340
as three times at 87 hundredths.

00:05:44.377 --> 00:05:45.530
(mumbles) These are all equivalent,

00:05:45.530 --> 00:05:46.510
but hopefully one of these,

00:05:46.510 --> 00:05:47.840
or more than one of these register

00:05:47.840 --> 00:05:49.830
with you of what's really going on.

00:05:49.830 --> 00:05:53.960
Well, this is going to be the
same thing as three wholes,

00:05:53.960 --> 00:05:57.880
times 87 hundredths, 87 hundredths.

00:05:57.880 --> 00:06:00.740
And so this is going to be
equal to, in the numerator,

00:06:00.740 --> 00:06:03.660
we have three times 87.

00:06:03.660 --> 00:06:08.660
Three times 87 hundredths,
one times 100 is 100.

00:06:09.060 --> 00:06:11.000
Three times 87 hundredths,
well, we already know

00:06:11.000 --> 00:06:16.000
what three times 87 is, this
is equal to 261 hundredths.

00:06:16.730 --> 00:06:20.610
And you can see 100
goes into 261, two times

00:06:20.610 --> 00:06:23.850
and you're left with 61 hundredths.

00:06:23.850 --> 00:06:26.780
So these are all
equivalent representations.

00:06:26.780 --> 00:06:29.840
And just reminder, so it's
always good to estimate.

00:06:29.840 --> 00:06:32.240
And so what you have is you
have three times something

00:06:32.240 --> 00:06:34.200
that's a little bit less than one.

00:06:34.200 --> 00:06:36.710
So you would expect a value,
a little bit less than three.

00:06:36.710 --> 00:06:40.550
And so 2.61 also meets that sniff test,

00:06:40.550 --> 00:06:41.530
that this seems about right.

00:06:41.530 --> 00:06:44.620
If for some reason you got 26 or 261,

00:06:44.620 --> 00:06:48.750
that would be way off or
even if you got 0.261,

00:06:48.750 --> 00:06:50.900
that would also feel way off.

00:06:50.900 --> 00:06:52.483
So hopefully this is helpful.


Worked examples for standard algorithm exercise

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.760
- We're not going to do
a few examples questions

00:00:02.760 --> 00:00:04.470
from the Khan academy exercise

00:00:04.470 --> 00:00:06.680
on the standard algorithms.

00:00:06.680 --> 00:00:08.290
We're asked, which of the following

00:00:08.290 --> 00:00:12.080
correctly multiplies 74 times 8

00:00:12.080 --> 00:00:15.610
using the standard algorithm.

00:00:15.610 --> 00:00:18.140
So pause this video and
see if you can work on that

00:00:18.140 --> 00:00:20.410
before we do it together.

00:00:20.410 --> 00:00:22.290
All right, now let's just remind ourselves

00:00:22.290 --> 00:00:23.720
what the standard algorithm is.

00:00:23.720 --> 00:00:26.090
In fact, let's just remind
ourselves what an algorithm is.

00:00:26.090 --> 00:00:29.040
An algorithm is a series
of steps that you can do

00:00:29.040 --> 00:00:30.930
to do something, so
you'll often hear about

00:00:30.930 --> 00:00:32.300
a computer algorithm.

00:00:32.300 --> 00:00:34.070
But you can also have algorithms in math,

00:00:34.070 --> 00:00:36.240
just a method for doing something.

00:00:36.240 --> 00:00:37.590
And the standard algorithm,

00:00:39.540 --> 00:00:41.830
that's the typical, or the standard, way

00:00:41.830 --> 00:00:43.450
that a lot of people will tackle

00:00:43.450 --> 00:00:47.260
a multiplication question
or computation like this.

00:00:47.260 --> 00:00:49.850
But just as a reminder,
in the standard algorithm

00:00:49.850 --> 00:00:52.460
if we're multiplying 74 times 8

00:00:52.460 --> 00:00:54.340
we would write the eight in the ones place

00:00:54.340 --> 00:00:56.590
right below the four in the ones place,

00:00:56.590 --> 00:01:00.280
and then you multiply each of
these places times the eight.

00:01:00.280 --> 00:01:04.840
So you would start with
the four times the eight,

00:01:04.840 --> 00:01:06.740
you would get 32.

00:01:06.740 --> 00:01:11.270
32 you can express as
two ones and three tens.

00:01:11.270 --> 00:01:13.090
So we'll put that three up there.

00:01:13.090 --> 00:01:17.830
And then you would multiple
the seven times the eight.

00:01:17.830 --> 00:01:22.830
Seven times eight is 56, and
that's going to be 56 tens

00:01:24.490 --> 00:01:28.410
because it's seven 10s
times eight is 56 tens.

00:01:28.410 --> 00:01:30.780
Plus the three 10s you had before

00:01:30.780 --> 00:01:34.350
get you to 59 tens, and so
you would write over here.

00:01:34.350 --> 00:01:38.990
That's 59 tens, and so this would be 592.

00:01:38.990 --> 00:01:40.150
Now when you look at the choices,

00:01:40.150 --> 00:01:43.860
that's exactly what
happened here in choice C.

00:01:43.860 --> 00:01:46.240
Just for kicks we can see what went wrong

00:01:46.240 --> 00:01:47.073
in these other ones.

00:01:47.073 --> 00:01:48.360
Let's see, in this first one,

00:01:48.360 --> 00:01:50.700
when we multiplied the,

00:01:50.700 --> 00:01:55.060
when we multiplied the
four ones times eight ones,

00:01:55.060 --> 00:01:57.260
according to this, this person somehow got

00:01:57.260 --> 00:02:00.230
three ones and two tens, and 23.

00:02:00.230 --> 00:02:03.150
Four times eight is not 23,
you can rule that one out.

00:02:03.150 --> 00:02:04.730
And here when they multiplied

00:02:04.730 --> 00:02:07.670
the four ones times the eight ones,

00:02:07.670 --> 00:02:10.090
that would be the 32.

00:02:10.090 --> 00:02:14.470
So it's two ones and
then another three tens,

00:02:14.470 --> 00:02:16.580
so there should've been a three up here.

00:02:16.580 --> 00:02:18.260
And so that way when you multiplied

00:02:18.260 --> 00:02:19.390
the seven tens times eight,

00:02:19.390 --> 00:02:21.760
you get 56 but then you
add this other three tens

00:02:21.760 --> 00:02:24.375
so you would really
need to get to 59 tens.

00:02:24.375 --> 00:02:27.030
So that's why that one didn't work.

00:02:27.030 --> 00:02:29.090
Let's do another example here.

00:02:29.090 --> 00:02:32.500
And this is going to be with
a different type of question.

00:02:32.500 --> 00:02:35.360
So here we are told that,

00:02:35.360 --> 00:02:37.630
Don starts to use the standard algorithm

00:02:37.630 --> 00:02:39.800
to solve 418 times five.

00:02:39.800 --> 00:02:42.030
His work is shown below.

00:02:42.030 --> 00:02:45.160
What number should Don replace Y with?

00:02:45.160 --> 00:02:47.493
Pause this video and see
if you can figure it out.

00:02:48.450 --> 00:02:50.040
Okay, so the way to think about this,

00:02:50.040 --> 00:02:51.750
this might at first
confuse you a little bit,

00:02:51.750 --> 00:02:56.150
because Y, and why is there a
Y there, in the first place?

00:02:56.150 --> 00:02:57.610
But what they're really trying to get at

00:02:57.610 --> 00:03:02.310
is making sure that you, or we, understand

00:03:02.310 --> 00:03:03.760
what Don is trying to do when

00:03:03.760 --> 00:03:05.560
he's trying to do the standard algorithms.

00:03:05.560 --> 00:03:08.210
So as we just highlighted in last example,

00:03:08.210 --> 00:03:09.360
the way that we would tackle this

00:03:09.360 --> 00:03:10.420
with the standard algorithm,

00:03:10.420 --> 00:03:14.610
and actually let me write
it down, 418 times five.

00:03:14.610 --> 00:03:18.030
We would say eight ones times five ones

00:03:18.030 --> 00:03:20.850
is going to be 40 ones.

00:03:20.850 --> 00:03:25.720
40 ones we can write as
zero ones and four tens.

00:03:25.720 --> 00:03:28.230
And so that looks like
the place where they,

00:03:28.230 --> 00:03:30.010
where Don stopped computing.

00:03:30.010 --> 00:03:32.210
So he's on his way to
solving the whole thing

00:03:32.210 --> 00:03:34.680
he just partially computed it so far.

00:03:34.680 --> 00:03:39.210
But just by doing that, we
know what the Y should be.

00:03:39.210 --> 00:03:41.770
The Y should be four.

00:03:41.770 --> 00:03:44.240
So what number should
Don replace the Y with?

00:03:44.240 --> 00:03:45.960
He should replace it with a four,

00:03:45.960 --> 00:03:49.420
and it's representing
Y tens, or four tens.

00:03:49.420 --> 00:03:52.470
And of course you could keep
going with this computation.

00:03:52.470 --> 00:03:55.930
If Don were to say, okay,
I have one ten times five,

00:03:55.930 --> 00:03:58.470
that'd be five tens,
plus another four tens,

00:03:58.470 --> 00:04:00.080
that's nine tens.

00:04:00.080 --> 00:04:02.680
And then last but not least,

00:04:02.680 --> 00:04:04.930
if Don wanted to figure out, well,

00:04:04.930 --> 00:04:07.710
he's got four hundreds here times five

00:04:07.710 --> 00:04:09.220
is twenty hundreds,

00:04:09.220 --> 00:04:12.670
which we can express as zero
hundreds and two thousands.

00:04:12.670 --> 00:04:14.940
Or you just do this as 20 hundreds.

00:04:14.940 --> 00:04:17.550
So they're not asking us to
do the entire computation.

00:04:17.550 --> 00:04:18.420
We're just trying to figure out

00:04:18.420 --> 00:04:20.690
what Don did essentially
in this first step.

00:04:20.690 --> 00:04:23.740
What number should he have
written here instead of a Y?

00:04:23.740 --> 00:04:25.790
So Y could be replaced with a four.

00:04:25.790 --> 00:04:27.420
It's representing the four,

00:04:27.420 --> 00:04:29.820
which is really in the
tens place, so four tens.


Dividing by a two digit number

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.480
- [Instructor] In this
video we're going to get

00:00:01.480 --> 00:00:05.100
a little bit of practice
divided by a two digit number.

00:00:05.100 --> 00:00:07.670
So let's say that we have 4,781

00:00:10.700 --> 00:00:14.150
divided by 32.

00:00:14.150 --> 00:00:16.090
Pause this video and see
if you can figure out

00:00:16.090 --> 00:00:17.020
what that's going to be.

00:00:17.020 --> 00:00:17.970
And if there is a remainder,

00:00:17.970 --> 00:00:20.420
figure out what that
remainder is going to be.

00:00:20.420 --> 00:00:22.460
All right, now let's
work on this together.

00:00:22.460 --> 00:00:27.460
So first let me rewrite 4,781

00:00:27.980 --> 00:00:32.960
and this is going to be divided by 32.

00:00:32.960 --> 00:00:35.110
Now whenever we divide by anything

00:00:35.110 --> 00:00:36.420
that has more than one digit,

00:00:36.420 --> 00:00:37.670
so it's a little bit of an art.

00:00:37.670 --> 00:00:39.180
The way we're going to tackle it is

00:00:39.180 --> 00:00:42.610
a little trial and error using estimation.

00:00:42.610 --> 00:00:44.457
So we'll start by saying "Hey,

00:00:44.457 --> 00:00:46.860
"how many times does 32 go into four?"

00:00:46.860 --> 00:00:49.860
Well, 32 doesn't go into four
at all, so then we'll move on.

00:00:49.860 --> 00:00:54.200
How many times does 32 go into 47?

00:00:54.200 --> 00:00:58.840
Well it's pretty clear that
32 goes one time into 47.

00:00:58.840 --> 00:01:02.890
So I'll put the one right over
there, above the seven in 47.

00:01:02.890 --> 00:01:04.850
And if you're saying, "Hey,
how did Sal know that?"

00:01:04.850 --> 00:01:07.400
Well, just remember, two
times 32 would be 64.

00:01:07.400 --> 00:01:09.110
That's more than 47.

00:01:09.110 --> 00:01:14.110
So 32 goes into 47 one time,
and we multiply one times 32.

00:01:14.450 --> 00:01:17.210
One times two is two,
one times three is three.

00:01:17.210 --> 00:01:19.720
You know that one times 32 is 32.

00:01:19.720 --> 00:01:23.250
And then we subtract 32 from 47.

00:01:23.250 --> 00:01:26.170
Let's do that, and we get

00:01:26.170 --> 00:01:28.890
seven minus two is five

00:01:28.890 --> 00:01:31.700
and four minus three is one.

00:01:31.700 --> 00:01:34.280
And the way you make sure
that you did that step right

00:01:34.280 --> 00:01:38.210
is whatever we get over
here should be less than 32.

00:01:38.210 --> 00:01:40.170
If whatever we have here is 32 or greater,

00:01:40.170 --> 00:01:43.010
that means we could of had
a larger number up here.

00:01:43.010 --> 00:01:47.260
But then our next step is
we bring down the eight

00:01:47.260 --> 00:01:51.940
and we say, "How many
times does 32 go into 158?"

00:01:51.940 --> 00:01:52.970
Now this is a little bit tricky.

00:01:52.970 --> 00:01:54.670
If you were estimating how many times

00:01:54.670 --> 00:01:58.610
does 30 go into 150,

00:01:58.610 --> 00:01:59.610
you might say five.

00:01:59.610 --> 00:02:01.820
And actually this is for
kicks, let's just try that out.

00:02:01.820 --> 00:02:03.640
Let's see if five works.

00:02:03.640 --> 00:02:07.830
So let's see, five times two is 10.

00:02:07.830 --> 00:02:09.620
We'll put zero ones and then

00:02:09.620 --> 00:02:12.250
we will have one 10 there, carry the one.

00:02:12.250 --> 00:02:15.730
Fives times three is 15, plus one is 16.

00:02:15.730 --> 00:02:16.960
So that didn't work out.

00:02:16.960 --> 00:02:19.210
It almost worked out, but not quite.

00:02:19.210 --> 00:02:23.320
160 is larger 158 so
five is too much there.

00:02:23.320 --> 00:02:27.960
So we need to go down to four.

00:02:27.960 --> 00:02:30.480
So four times two is eight.

00:02:30.480 --> 00:02:33.340
Four times three is 12.

00:02:33.340 --> 00:02:38.073
And now we subtract and 158 minus 128,

00:02:38.073 --> 00:02:40.190
eight mins eight is zero,

00:02:40.190 --> 00:02:42.500
five tens minus two tens is three tens,

00:02:42.500 --> 00:02:43.650
and then we have no hundreds.

00:02:43.650 --> 00:02:46.640
And then this is right,
'cause 30 is less than 32.

00:02:46.640 --> 00:02:49.320
And then we can bring down,

00:02:49.320 --> 00:02:51.800
we can bring down that one.

00:02:51.800 --> 00:02:55.060
Now how many times does 32 go into 301?

00:02:55.060 --> 00:02:56.157
Well we might be tempted to say,

00:02:56.157 --> 00:02:57.287
"Well, this is close to 30,

00:02:57.287 --> 00:03:00.000
"this is close to 300, so maybe it's 10."

00:03:00.000 --> 00:03:03.680
But 10 times 32 would be
320, so that'd be too much.

00:03:03.680 --> 00:03:07.340
So I feel good about nine as my estimate.

00:03:07.340 --> 00:03:11.190
Nine times two is 18, carry the one.

00:03:11.190 --> 00:03:16.130
Nine times three is 27, plus one is 28.

00:03:16.130 --> 00:03:18.650
And then we can subtract.

00:03:18.650 --> 00:03:20.860
We're actually going to have
to do a little regrouping here.

00:03:20.860 --> 00:03:22.350
You might be able to do it in your head.

00:03:22.350 --> 00:03:24.970
To go from 288 to 301, let's see,

00:03:24.970 --> 00:03:28.080
you would add 12 to get to
300 and then one more, 13.

00:03:28.080 --> 00:03:29.620
So you might be able to do it in your head

00:03:29.620 --> 00:03:32.180
that this is going to be
13, or you could regroup.

00:03:32.180 --> 00:03:33.247
You could say, "All right, let's see,

00:03:33.247 --> 00:03:36.047
"one is less than eight, can
I regroup from the 10s place?

00:03:36.047 --> 00:03:37.337
"No, I have nothing there
so I have to regroup

00:03:37.337 --> 00:03:38.817
"from the hundreds place.

00:03:38.817 --> 00:03:40.257
"So I'm gonna take one of those hundreds,

00:03:40.257 --> 00:03:41.467
"I'm left with two hundreds left.

00:03:41.467 --> 00:03:43.187
"And then I'm going to have 10 10s.

00:03:43.187 --> 00:03:44.777
"And then I could take one of those 10s,

00:03:44.777 --> 00:03:47.157
"so I have nine 10s left,
and give it to the ones place

00:03:47.157 --> 00:03:48.687
"so now I have 11 ones.

00:03:48.687 --> 00:03:52.307
"And 11 minus eight is 3,
nine 10s minus eight 10s

00:03:52.307 --> 00:03:54.377
"is one 10, and then hundreds,

00:03:54.377 --> 00:03:56.700
"200 minus 200 is just a zero."

00:03:56.700 --> 00:04:00.210
And so there we have it,
this is less than 32.

00:04:00.210 --> 00:04:02.160
We have nothing left to bring down

00:04:02.160 --> 00:04:05.300
and so we're left with
the remainder of 13.

00:04:05.300 --> 00:04:08.210
So this is going to be equal to 149

00:04:08.210 --> 00:04:11.993
with a remainder of 13, and we're done.