Learn the Meaning of Multiplying Decimals - [13] - Free Educational videos for Students in K-12

Learn the Meaning of Multiplying Decimals - [13] - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is called Multiplying decimals using models . This is part
00:06	one . Probably a better title would be understanding decimal
00:10	. Multiplication using pictures using models . I'm really excited
00:13	to teach you this because you'd be surprised how many
00:15	people , adults don't really understand what it means when
00:19	you multiply a number Times of decimal or a decimal
00:23	times of decimal . You know , we learn how
00:24	to do it in school but a lot of times
00:26	people don't really understand what it means to multiply by
00:30	.3 or multiply pie .4 or whatever . So here
00:34	what we're going to do is you will understand by
00:36	the end of this lesson exactly what multiplying by a
00:39	decimal does and what it means . And as we
00:41	build those skills in the future lessons will do a
00:43	lot of practice with decimal multiplication to get the hang
00:46	of it . So the first way I want to
00:48	talk about this is I want to talk about multiplying
00:50	something times 0.5 because it's the easiest thing to do
00:54	in the back of your mind as we talk about
00:56	this discussion . When you think of 0.5 of something
00:59	I want you to think of . 0.5 means the
01:01	same thing as half of something , decimals and fractions
01:04	go together like peanut butter and jelly . And I
01:06	want you to remember that 0.5 is exactly the same
01:10	thing as one half of something . So 0.5 is
01:12	the same as one half of something . So for
01:14	our first problem , we want to look at this
01:18	problem here . 0.4 times 0.5 . Now I'm gonna
01:21	teach you how to do this by hand by doing
01:23	the actual multiplication in a future lesson . This lesson
01:26	has nothing to do with that . This lesson is
01:27	for us to understand what does this actually mean ?
01:30	All right . So what we want to do first
01:32	is we want to understand that if I were taking
01:34	0.4 times let's pretend for a second that that we're
01:38	not actually multiplying by .5 . Let's pretend we're just
01:40	multiplying by one . What happens when you multiply something
01:43	by one ? It means you just keep the same
01:46	number right ? Two times one is 23 times ,
01:48	one is three and so on . So if it
01:51	were 0.4 times won , the answer would just be
01:54	0.4 . But we're not multiplying by one . C
01:57	. When you multiply by one , it keeps the
01:59	number unchanged , right ? But here we're multiplying by
02:02	0.5 , which is the same thing as half .
02:05	So what I want you to start thinking about is
02:07	when you multiply the thing that you're multiplying by when
02:11	it's a decimal , I think like that I want
02:13	you to think of That second thing you're multiplying by
02:15	is like a chopping function I want , I'm using
02:18	my hand here to think , I want you to
02:20	think of chopping so what you're doing is you're taking
02:22	something that's 0.4 and you're chopping it in half .
02:26	That's what if you read it out loud , you're
02:27	taking this , you're chopping up by this amount and
02:30	this decimal amount of 0.5 is the same thing as
02:33	cutting it in half . If we were chopping this
02:35	thing times one Then nothing would change , it would
02:38	be 0.4 times one is 0.4 , but we're multiplying
02:42	by instead of one , we're multiplying by a half
02:45	, so we're chopping it in half . So whatever
02:47	we start with , We multiply by .5 , we're
02:51	just cutting it in half and that's going to be
02:53	what the answer is . We want to now draw
02:55	pictures to show what this means . First of all
02:58	, we have some diagrams here . I know it
03:00	looks confusing . I want you to completely ignore this
03:02	on the right . This little diagram is going to
03:05	represent the .4 and this little diagram is going to
03:09	represent the .5 . Now , before you do anything
03:12	else , I'm going to draw kind of under here
03:15	, I'm gonna put that . This is one whole
03:19	and this thing right here is also one hole .
03:23	So this box here , it represents one and this
03:28	box , this entire box also represents one for now
03:31	completely ignore the answer . I don't want you to
03:33	look at the answer . All right . So if
03:35	this represents one then what is 0.4 ? Remember the
03:39	first decimal here , .4 is the 10th place .
03:43	That means that we take whatever we're talking about ,
03:45	we chop it into 10 pieces and 0.4 means four
03:49	out of 10 pieces because it's the 10th place .
03:52	So if we want to represent four out of 10
03:54	pieces , notice I've already chopped this into 1234 56789
03:59	10 pieces . This is 10 equal pieces . This
04:02	hole is cut into 10 equal pieces , so if
04:05	we want to represent what 0.4 really looks like they
04:08	will take the whole and we will only discuss or
04:11	talk about four of those 10 pieces . So there's
04:15	one out of 10 1/10 Here's two out of 10
04:18	to 10s , here's three out of 10 which is
04:21	three tens and here's four out of 10 which is
04:25	4/10 . So this shading of this much of the
04:28	whole represents 0.4 . How do we know that ?
04:32	Because 0.4 is 4/10 or four out of 10 pieces
04:37	. We've taken the whole thing , we've cut it
04:39	into 10 pieces and we only shade four of them
04:41	. So this box here , this shaded region here
04:44	represents 0.4 of something you can think of . This
04:47	being like a sandwich . And if you cut this
04:50	much of the sandwich away , then this is 0.4
04:52	of the sandwich . So that's the number we're multiplying
04:56	to begin with . Now , what are we going
04:58	to chop it by ? What are we going to
04:59	multiply by ? We said if we were multiplying by
05:03	instead of this one , then this whole thing would
05:06	be shaded with one and then the answer would be
05:09	exactly the same . 0.4 times one is 0.4 .
05:12	Nothing changes . But we're multiplying by 0.5 . Let's
05:16	write down what 0.5 would look like on here .
05:19	I have this box again into 23456789 10 pieces .
05:24	So what we want to do is we want to
05:27	represent that In terms of a picture . So 0.5
05:31	means five out of 10 pieces . So here is
05:34	one out of those 10 pieces . Here's two out
05:37	of those 10 pieces . Here's three out of those
05:40	10 pieces , 0.3 here , 0.4 because it's four
05:44	out of 10 pieces . And here 0.5 which is
05:47	five out of the 10 pieces . Notice that this
05:50	represents half . It's literally half of the entire object
05:54	. If you cut this entire thing in half ,
05:56	this is a half and this is the other half
05:59	. So I told you in the beginning that 0.5
06:01	is the same thing as one half . This is
06:03	why because if I take an object and cut it
06:06	into 10th like 10 pieces , 10th and I shade
06:10	five of the 10 pieces , I'm gonna have half
06:12	of it . So when you multiply by .5 you're
06:15	chopping it by this . So I'm gonna write here
06:18	chop right ? So the multiplication means chop you take
06:23	what you start with which is this much of a
06:25	sandwich . And this second picture is only telling you
06:28	how much you're chopping it and in this case we're
06:31	just chopping it in half . So if I know
06:33	that I have this much of the sandwich which is
06:35	less than half of it to begin with . And
06:37	this is telling me when I multiply that I'm going
06:39	to chop this in half , then I know without
06:42	doing any math that the final answer is just gonna
06:45	be half of what I started with because I'm multiplying
06:47	by 0.5 which is a half . So before I
06:49	do anything else over there , I know that the
06:52	answer is going to be about this much because it's
06:54	going to be half of what I started with .
06:56	So what I want you to do in your mind
06:59	is I want you to use this as a chop
07:02	. We can kind of extend this , you can
07:05	kind of like extend this over here and kind of
07:08	like do a little dot , dot , dot ,
07:09	I'll kind of like do it right here , I
07:10	guess a little dot dot dot , because we're gonna
07:11	chop this in half . This is using this amount
07:15	is used to chop this thing in half is to
07:18	chop chop this thing in half . So I can
07:21	kind of like kind of go through here like this
07:24	. And I know over here that I had four
07:26	columns and kind of like reproduce 1234 columns and kind
07:29	of go through here and you see what I'm doing
07:32	. I'm finding where these intersect which is all of
07:34	this right here . All of this right here .
07:39	Because literally all I have done , All I've done
07:45	is I've taken what I've started with which is this
07:48	much 0.4 of a sandwich or whatever and have multiplied
07:51	by this , which is a chopping factor . This
07:54	tells me how much to chop , which means I
07:55	chop this much away and that is how much I
07:58	have left in my answer . So then in order
08:01	to figure out what the answer is going to be
08:05	, I need to count squares over here , right
08:07	? Because if I take this and I slice it
08:10	this way , I'm going to be dividing these into
08:12	more squares . So notice here , I had these
08:15	were sliced into 10 equal pieces and this is sliced
08:18	into 10 equal pieces . But once I do the
08:20	chopping now it's like the whole entire sandwiches chopped into
08:23	how many pieces ? 123456789 10 . That's 10 this
08:28	way and 10 this way 10 times 10 is 100
08:32	. So this answer here is represented out of 100
08:36	pieces of the sandwich . How many do I have
08:39	? 123456789 10 11 12 13 14 15 16 17
08:45	18 1920 20 hundreds . So the answer is going
08:50	to be 20 100 0.20 right ? Because when you
08:56	read decimals , remember you take a look at how
08:59	many spots you have . This is the 10th place
09:01	and this is the hundreds place . So if you
09:03	look at the whole number , it's 20 out of
09:05	100 because this position here Is the hundreds place .
09:09	So you read everything after the decimal 20 is what's
09:12	after the decimal , but it goes to the hundreds
09:14	place . So it's 20 out of 100 pieces ,
09:17	so 20 hundreds , so 200.0 point 20 is how
09:21	we represent that . Now , getting the answer in
09:24	0.2 is something I'm going to show you how to
09:27	do later , but you can kind of think about
09:29	it . It makes sense because if you start with
09:31	0.4 and you cut it in half , it makes
09:33	sense that the answer should be 0.2 because half of
09:36	four is too right , that's what makes sense .
09:39	We're gonna learn how to do it by hand in
09:42	a minute . But for now again just want you
09:44	to realize when you take something and cut it into
09:46	10 pieces and take 4/10 and then you multiply by
09:50	100.5 , which means you cut it in half ,
09:52	you chop it . This is the amount that I
09:54	have in my answer , but I'm gonna express the
09:57	answer in terms of hundreds because then I can line
10:01	up everything properly and count the squares here , so
10:03	I cut it into 100 pieces and I have 20
10:05	of them , so 20 hundreds . The answer is
10:08	that you get here , I don't really care that
10:11	much if you're getting the correct answers yet , because
10:13	I'm going to show you how to multiply by hand
10:15	right now , what I'm really trying to teach you
10:17	is to multiply decimals , you're taking the first number
10:20	. And the second thing the thing you're multiplying by
10:22	is like a chopping factor which just cuts the thing
10:25	in half and we're expressing the answer in terms of
10:28	hundreds because that way we can just fill the squares
10:31	out here and and write the answer down . So
10:35	that was our first problem . We have to do
10:37	a lot of talking for that first problem . Um
10:40	And um I think before I do the next problem
10:43	, I want to take a step back and just
10:46	kind of like talk a little bit more for a
10:47	second . Let's just say forget about pictures for a
10:50	second . What if I ask you , what if
10:53	I take the number eight ? Not 0.8 , just
10:55	eight . And I multiply it by 0.5 . Now
10:59	that you know what's happening here , what do you
11:01	think is gonna happen ? What you're telling me or
11:04	what I'm telling you is that if I take eight
11:06	uh I don't know , eight bananas and I multiplied
11:09	by 80.5 , I'm chopping it by 0.5 . We
11:12	know that 0.5 means I have . So when you
11:14	multiply , it means chop it means chop it by
11:17	that much . By that factor . If we take
11:19	eight bananas and we chop it in half . What
11:21	do we have left eight bananas when you cut it
11:23	in half ? How many do you have four ?
11:25	Now ? I'm going to show you how to do
11:27	it by hand , but I want you to know
11:29	what it means , and eight times 80.5 is four
11:32	. What if I give you , you know uh
11:34	16 times 0.5 we're gonna take 16 you know ,
11:40	pumpkins , and we're gonna chop it in half ,
11:42	because their 160.5 means a half . And so what's
11:45	going to happen is you take this and cut it
11:47	in half . So you're going to basically write that
11:49	down and look at it . You can convince yourself
11:51	that that's equal to 88 times two is 16 .
11:53	So when we cut this in half , we know
11:55	the answer is eight . When we multiply by point
11:58	half 80.0.5 we cut things in half . When we
12:01	multiply by one , we don't change anything . When
12:05	we multiply by zero we get zero and anything in
12:08	between that we multiply by any decimal is just going
12:11	to chop it by that much . So for instance
12:13	if we're just thinking about it here , if we
12:17	have zero right here and we have one right here
12:21	, we're gonna right down chopping factors . We multiply
12:25	by 0.5 . Where does 0.5 live here ? Between
12:29	zero and one ? Exactly in the middle is 0.50
12:33	point five is half . So it's right in the
12:35	middle it's between zero and one . It's called one
12:37	half . What I'm trying to get you to see
12:39	is that if you take a number and multiplying by
12:41	a chopping factor . If we multiply by point half
12:44	0.0.5 we're multiplying by a half . So we cut
12:46	it in half . If we multiply this by one
12:49	, anywhere over here by by one , then we're
12:52	going to get more and more of the original thing
12:54	. So 16 times one would be 16 , right
12:58	? If we go the other direction here and multiply
13:00	by 0 , 16 times zero is zero . So
13:04	if we multiply by one , we get the original
13:06	number . If we multiply by zero , we just
13:09	get zero . If we multiply by a half a
13:12	00.5 we get half of what we started with .
13:14	If I multiply by some number over here , you
13:18	know like this number might be 0.25 or a number
13:22	over here . This number might be 0.8 . All
13:25	that's happening is you're chopping it by by more .
13:28	If I multiply by .8 then I'm going to get
13:31	something closer to 16 . If I multiply by one
13:35	then I'll exactly get 16 . If I multiply by
13:37	something less than .5 , I multiply by .25 or
13:41	.1 or point something even lower , then I'm chopping
13:45	and getting less and less and less . All I'm
13:47	trying to get you to see is when you multiply
13:49	by decimals , that second thing you're multiplying by is
13:52	a chopping factor . The closer your chopping factor is
13:55	to the number one , then the closer you get
13:58	everything that you started with , but the farther away
14:01	you're chopping factor is 20 The less you get what
14:04	you started with . And if you chop exactly by
14:06	200.5 of course you just cut the thing in half
14:09	. And so now that we have the idea ,
14:11	let's fill out the rest of these problems are going
14:12	to go very fast now , What if we start
14:15	with 0.2 ? And we multiply by the chopping factor
14:17	of .8 , so we're gonna start with this and
14:20	we're not gonna cut it in half , we're going
14:21	to get more than half of it because we're chopping
14:24	up by something bigger than a half , we're yeah
14:27	, we're chopping up by something bigger than a half
14:28	. So 0.2 would be two out of 10 slices
14:33	of something . So this thing is cut into 10
14:35	pieces , 100.2 is equal to this , much of
14:38	whatever you're starting with , and I'm multiplying by 0.8
14:42	and that means I'm multiplying by something , I'm multiplying
14:46	something . Buy something that is much larger than uh
14:51	than a half . I'm multiplying by , here's one
14:55	, Here's two out of 10 , here's three out
14:58	of 10 , here's four , Here's five , here's
15:03	6 , here's seven and here's 88 out of 10
15:09	. So this is my chopping factor , I'll just
15:11	write it on the top here to remind you chop
15:13	, this is what you're shopping by , so if
15:15	I put a little dash line through here and I
15:18	can carry that dash line over here , then I
15:22	can see what's going to happen if I start with
15:23	this much and this is a chopping factor . It's
15:26	telling me that I'm only keeping this much of it
15:28	, I'm throwing the rest away and I'm chopping ,
15:30	I'm keeping all of this . If I were to
15:33	chop with a factor of one , that would be
15:35	the entire thing and then I would keep everything because
15:38	something times one is itself . But you can see
15:40	here that by chopping by this much , I'm only
15:42	going to keep this amount , so I go over
15:44	here and I shade that amount . And so it
15:48	was the first two columns up until this point .
15:51	So I can kind of fill these out like this
15:54	. And then I can , instead of expressing intense
15:57	, I'll express these squares in hundreds . How many
15:59	hundreds do I have ? 123456789 10 , 11 ,
16:05	12 , 13 , 14 , 15 , 16 hundreds
16:08	. 16 hundreds , mean 0.16 This is the 10th
16:14	place . This is the hundreds place . So when
16:17	you read it it's 1616 squares out of 100 .
16:21	Is how much I keep whenever I start with this
16:23	much and I chop it by this much . All
16:26	right . Just like here , we had 20 squares
16:29	. So it was 0.20 . So .2 times .8
16:33	is more than half of what I started with .
16:35	And it comes out to be 0.16 . Again ,
16:38	we're going to be doing these by hand in a
16:40	few lessons , showing how to do the multiplication here
16:42	, we're just trying to understand what's going on .
16:45	All right , let's take a look at the next
16:48	two problems . We only have two more in this
16:50	lesson here . We start out with 0.7 and we're
16:54	not gonna cut it in half . We're gonna cut
16:56	it less than half . 0.5 would be cutting in
16:59	half . We're cutting it in less than that .
17:01	We start out with 0.7 . So here's one out
17:04	of 10 , here's two out of 10 , here's
17:07	three , here's four out of 10 or 100.4 ,
17:11	here's 0.5 , here's 0.6 and here's finally 0.7 .
17:16	So this purple shaded region represents how much we start
17:19	with . Now we're chopping by something less than half
17:22	. So we'll go over here and say three out
17:24	of 10 were chopping by here's one out of 10.1
17:28	, here's two out of 10 or 100.2 , here's
17:31	three out of 10 or 100.3 . This is what
17:33	we're gonna use to chop what we have over here
17:36	. So I'll just draw a little dotted line through
17:38	here . This is how much we're going to keep
17:40	, that's what we chop . So we will just
17:42	extend this over here and then you can see of
17:44	course it went 2.7 , so 1234567 columns . It
17:49	kind of went to right here and then you can
17:51	see if we just kind of like copy it here
17:55	, we're gonna shade all of this right here ,
17:59	notice it's exactly this amount . This amount right here
18:06	. This amount of stuff is what we're shading right
18:09	here , we're just chopping it into 100 pieces .
18:11	So we can express it better so we can express
18:14	the amount better . How many squares out of 100
18:17	do we have ? 123456789 10 11 12 13 14
18:22	15 16 17 18 1920 21 21 squares out of
18:28	121 out of 121 100 0.21 tense place , hundreds
18:36	place . Read it together 2121 out of 100 .
18:40	So 1000.7 times 0.3 . When we do by hand
18:44	later , we'll find out that it comes out to
18:46	exactly be 0.21 and that's what this is representing is
18:50	we take this thing and we chop it by something
18:53	just a little bit less than half . And we
18:55	get A number of 0.21 , which represents about less
18:59	than half . A little bit less than half of
19:01	what we started with . All right , here's our
19:04	last problem for this lesson . We're going to take
19:07	0.8 and we're gonna multiply by 0.4 . So we're
19:10	going to start with this amount and we're going to
19:12	chop it pretty close to to halfway , but not
19:16	quite here . We were chopping by .3 here ,
19:18	we're going to chop a little bit more , but
19:20	it's still not quite halfway . It's a little bit
19:23	less than half . 0.5 would be chopping it in
19:26	half . So let's represent 8/10 . Here's 1/10 0.1
19:32	here , 0.2 here , 0.3 here , 0.40 point
19:38	50.60 point seven and 08 So , this entire purple
19:47	shaded area means 8/10 of a whole . It's less
19:51	than a whole , it's 8/10 of a whole .
19:54	Now this guy over here , we're chopping it by
19:56	just a little bit less than half . So let's
19:58	see out of 10 here , 0.1 here , 0.2
20:03	here , 0.3 or 3/10 . And here 0.4 ,
20:08	which is 4/10 . Notice that going one more up
20:11	would be exactly half . This is just a little
20:13	bit less than half . So then we will take
20:16	and we will cut and literally chop this thing by
20:19	that amount . And this is what the answer will
20:21	be this much stuff . We start with this ,
20:24	we use this to chop it and this is all
20:26	that we have left . It should be a little
20:28	bit less than half of what we started . So
20:31	we go over here , we can extend this line
20:33	, see if I can make it line up .
20:35	Exactly . Yeah . And then we need to go
20:37	over to here as well and then we can start
20:39	shading . So here is make sure I don't overshoot
20:42	here . Yeah , it goes to this far uh
20:45	like this , make sure we catch everything over here
20:48	. And we were just basically shading what the answer
20:50	is , which we already have , but now we're
20:52	representing it in hundreds because uh we can represent the
20:57	exact answer . Notice this is the answer . After
20:59	we chopped it , we're just taking that and cutting
21:01	the whole thing into 100 pieces to see how many
21:04	we have . 12345678 9 , 10 11 12 13
21:10	14 15 16 17 18 1920 21 22 23 24
21:15	25 26 27 28 29 30 31 32 32 squares
21:20	out of 100 is 0.32 This is the tense ,
21:26	this is the hundreds . Read it together , it's
21:27	32 out of 100 . So in this lesson we
21:32	have learned , believe it or not , something really
21:34	important . I do not care if you know that
21:37	0.8 times 0.4 is 0.32 point 32 I do not
21:41	care if you know that 320.7 times 0.3 point 21
21:45	I don't care about that . I will show you
21:47	how to write it down and calculate it in a
21:49	few lessons . All I care about is you understand
21:52	what's happening ? The first number is just something less
21:56	than one . It's represented by a shaded region .
21:58	The second number you multiply by is a chopping factor
22:02	that tells you how much of the first thing we're
22:05	basically going to chop and throw away . So if
22:07	we start out with this much of something and we
22:09	chop it by exactly one half . This represents one
22:12	half . This represents the chopping factor , Then literally
22:15	this is the answer , we take that answer and
22:18	represented in hundreds . In this case we had 20
22:20	squares out of a total of 100 so the answer
22:23	was 0.20 . If we take something and multiply by
22:27	chopping factor larger than 0.5 , then we're chopping a
22:31	larger amount and we're going to keep more than half
22:34	of what we started with because we're chopping it by
22:36	something bigger than one half . We count the squares
22:39	get 10.16 squares out of 100 If we chop by
22:43	something less than five , both of these cases here
22:45	was we're chopping by three Here we're chopping by .4
22:48	. In both cases , we used it to just
22:51	cut what we started with and then our answer is
22:54	what's going on here , so when you're multiplying by
22:56	decimals you're chopping , it's like you have an axe
22:59	and you're cutting the tree down the chopping factor .
23:02	The thing you're multiplying by is just telling you how
23:05	much to chop away . That's it . As we
23:07	go through the next lesson , we'll get a little
23:08	more practice with these kinds of problems and then as
23:12	we move beyond that We will , we will learn
23:16	how to write these down and calculate the answer ,
23:18	but when you get the answer you'll know what it
23:20	means . And again , as we said before ,
23:22	if you take something and multiply by .5 , you're
23:26	cutting it exactly in half . If you multiply by
23:28	something more and more and more than .5 , then
23:31	you're going to keep more and more and more of
23:33	what you started with Until you get to the point
23:35	where you just multiply by one and then you keep
23:37	everything 16 times one would be 16 . If you
23:40	multiply by less than .5 , you get closer and
23:42	closer to zero so that eventually if you multiply by
23:46	zero you'll just get zero . So that's why I
23:48	say we're multiplying by a chopping factor . Watch this
23:51	a second time . I really think it's important .
23:53	Move on to the next lesson . We'll do more
23:56	practice with these pictures and then we'll start multiplying decimals
23:59	by hand .

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