Math Antics - Integer Multiplication & Division - Free Educational videos for Students in K-12 | Lumos Learning

Math Antics - Integer Multiplication & Division - Free Educational videos for Students in k-12


Math Antics - Integer Multiplication & Division - By Mathantics



Transcript
00:03 Uh huh . Hi , welcome to Math Antics .
00:08 In our last video we learn how to add and
00:10 subtract integers . In this video , we're going to
00:13 learn how to multiply and divide integers . The good
00:16 news is that multiplying and dividing integers is easier than
00:20 adding and subtracting them because even though it involves both
00:24 positive and negative numbers , the multiplication and division still
00:28 works basically the same way as we learned in our
00:31 video about negative numbers . The negative numbers are like
00:35 a mirror image of the positive numbers on the number
00:37 line . Each number on the positive side has a
00:40 negative counterpart on the negative side . There's a two
00:44 on the positive side and there's a negative two on
00:46 the negative side . There's a five on the positive
00:49 side and there's a negative five on the negative side
00:52 . And so on realizing that can help us understand
00:56 that negative numbers are just like positive numbers but they
00:59 have a negative factor built into them . For example
01:03 , Think about the number 3 . 4 minutes .
01:05 Do you remember when we learned about factors , we
01:07 learned that one is always a factor of any number
01:11 and that's because multiplying by one doesn't change a number
01:14 three and one times three are the same value .
01:19 So if you have the number three you can factor
01:21 out one and you have one times three . On
01:24 the other hand , what if you have negative three
01:26 instead ? We could still factor out of one from
01:29 it and nothing would change . You'd have one times
01:31 negative three . But because the number is negative ,
01:34 we could also factor out a negative one . Doing
01:37 that would give us -1 times three . So one
01:41 way to think about negative numbers is to imagine that
01:44 they're just like positive numbers but they always have a
01:47 factor of negative one built into them . That means
01:50 if you want to change a positive number into a
01:52 negative number , all you have to do is multiply
01:55 it by a factor of negative one . five times
01:58 negative one is negative five , seven times negative one
02:01 is negative seven , 10 times negative one is negative
02:05 10 and so on . And one way to visualize
02:08 this is to see that multiplying by a factor of
02:11 negative one just switches a number from the positive side
02:15 of the number line to the negative side . Ah
02:18 But that raises an interesting question . What happens if
02:21 you multiply a number that's already on the negative side
02:24 by a factor of negative one Like -1 times -3
02:29 . Is that going to make it extra negative ,
02:32 nope . In fact it's going to do just the
02:34 opposite multiplying a negative number by another negative factor is
02:40 actually going to switch the answer back to the positive
02:42 side of the number line , multiplying by negative one
02:46 acts like a switch , no matter which side of
02:48 the number line you start on . If you start
02:51 with a positive then multiplying by negative one switches it
02:55 to negative . But if you start with a negative
02:57 multiplying by negative one switches it back to positive and
03:01 you can keep switching back and forth between the positive
03:04 and negative side of the number line by multiplying by
03:07 another negative one as many times as you want .
03:12 So multiplying by one negative gives us a negative multiplying
03:19 by two negatives gives us a positive multiplying by three
03:23 negatives gives us a negative multiplying by four negatives gives
03:27 us a positive multiplying by five negatives gives us a
03:30 negative Multiplying by six negatives gives us a positive and
03:34 so on . Did you see the pattern in a
03:37 multiplication problem . If you have an even number of
03:40 negative factors , they'll form pairs that will balance each
03:43 other out and we'll give you a positive answer .
03:46 That's because the pair negative one times negative one just
03:50 equals one , which has no effect on the answer
03:54 . But if you have an odd number of negative
03:56 factors that are being multiplied together after you balance out
04:00 all the pairs , there will always be one negative
04:03 factor leftover . That will give you a negative answer
04:06 in a minute . We'll see some examples of how
04:09 knowing this will help us when multiplying and dividing lots
04:12 of integers . But first , let's look at the
04:14 simple problem three times five . We know that the
04:18 answer is 15 . But now , thanks to negative
04:21 numbers , we know that there's three more variations of
04:24 this problem that we need to learn how to do
04:26 . This might seem a little complicated , but it's
04:29 really not . That's because we're going to get the
04:31 same number for the answer for all four problems .
04:34 It's just that the sign of the answer , whether
04:36 it's positive or negative will be different depending on how
04:39 many negative factors were multiplying . Basically when doing integer
04:43 multiplication and division , you can just pretend that the
04:46 negatives aren't there while you multiply or divide and then
04:50 you count up how many negative factors you have to
04:53 figure out the sign of the answer . If there
04:55 is an even number of negative factors , the answer
04:58 will be positive . But if there's an odd number
05:00 of negative factors , then the answer will be negative
05:03 . So in the first problem we don't have any
05:06 negative factors . So our answer is just going to
05:09 be positive . 15 , which we already knew In
05:12 the second problem , we have only one negative factor
05:15 being multiplied , which means our answer will be negative
05:18 15 . In the third problem , we also have
05:21 just one negative factor , which means that her answer
05:24 will also be negative 15 . And in the fourth
05:27 problem we have to negative factors being multiplied . That's
05:30 an even number of negative factors . So they'll balance
05:33 each other out . They'll switch and switch back and
05:35 they'll give us an answer of positive 15 . That's
05:39 pretty easy . Right ? And the really great news
05:42 is that because multiplication and division are so closely related
05:46 their inverse operations , the rules about negative factors are
05:49 exactly the same for division problems . For example ,
05:53 if you have the problem eight divided by two ,
05:56 shown here , infraction for him , the answer is
05:58 positive for But if you have eight divided by negative
06:02 two , then there's one negative . So the answer
06:05 is negative for likewise , if you have negative eight
06:09 divided by two , then there's still just one negative
06:12 . So the answer will also be negative for But
06:16 if both of the numbers are dividing our -8 divided
06:20 by -2 , then there's two negatives in the division
06:23 problem . So the answer will be positive for And
06:26 one way to see that a negative divided by a
06:28 negative gives you a positive is to realize that if
06:31 you factor out the negative one on the top and
06:34 you factor out the negative one on the bottom ,
06:36 then you have a pair of common factors that you
06:38 can cancel just like you would if you were simplifying
06:41 a fraction . Okay , are you ready for some
06:44 more complicated examples in these problems ? We're going to
06:47 be combining both multiplication and division . And the good
06:51 news is that it doesn't matter whether a negative factor
06:54 is being multiplied or divided . You still get to
06:57 count it when figuring out if you have an even
06:59 or an odd number of factors . How about this
07:02 one ? Negative three times eight over negative too .
07:06 Well on the top negative three times eight is going
07:09 to give us negative 24 since three times eight is
07:13 24 there's only one negative factor And then -24 divided
07:19 by -2 is going to be positive 12 . Since
07:22 24 divided by two is 12 and we have two
07:26 negatives , which means our answer will be positive .
07:29 Here's another good example negative one times negative eight times
07:34 negative six over the quantity negative three times negative four
07:39 . All five of the numbers are negative . And
07:41 since that's an odd number of negative factors , we
07:44 know that the answer is going to be negative watch
07:47 and see on the top negative one times negative eight
07:51 is positive eight Positive eight times -6 is -48 .
07:56 And on the bottom we have negative three times negative
07:59 four which is positive 12 . And then as the
08:03 final step -48 divided by 12 is -4 . So
08:08 it worked . Our final answer is negative because we
08:11 had an odd number of negative factors . But what
08:15 if we made one slight change to that same problem
08:18 ? What if we made one of the numbers on
08:19 top positive instead of negative ? Then we only have
08:23 four negative numbers which is even so the answer should
08:27 be positive for let's see if it is On the
08:30 top negative one times positive eight is negative , eight
08:35 -8 times negative six is positive 48 . And on
08:40 the bottom we have negative three times negative four which
08:44 is positive 12 . And then as the final step
08:48 , 48 divided by 12 is positive for just as
08:52 I suspected . So do you see what I mean
08:55 ? All the multiplication and division works the same .
08:58 It's just that you have to figure out whether the
09:00 answer is going to be positive or negative and to
09:03 do that , all you have to do is figure
09:05 out if you have an even or an odd number
09:08 of negative factors in the multiplication or division , if
09:12 there's an odd number of negative factors , then the
09:14 answer will be negative . And if there is an
09:16 even number of negative factors , the answer will be
09:19 positive . And remember this process only works for integer
09:24 multiplication and division . If you have a problem that
09:27 also contains integer addition and subtraction , you need to
09:31 do those operations using the rules we learned in the
09:33 last video as always , the way to get good
09:37 at Math is to practice So be sure to work
09:39 some integer multiplication and division problems to make sure you've
09:42 got it . Thanks for watching Math Antics And I'll
09:45 see you next time . Learn more at Math Antics
09:49 dot com
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