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## 7th grade (Eureka Math/EngageNY)

### Course: 7th grade (Eureka Math/EngageNY) > Unit 2

Lesson 2: Topic B: Multiplication and division of integers and rational numbers- Why a negative times a negative is a positive
- Why a negative times a negative makes sense
- Signs of expressions
- Multiplying positive & negative numbers
- Dividing positive and negative numbers
- Multiplying negative numbers
- Dividing negative numbers
- One-step equations with negatives (multiply & divide)
- Multiplying negative numbers review
- Dividing negative numbers review
- Rewriting decimals as fractions: 2.75
- Write decimals as fractions
- Rewriting decimals as fractions challenge
- Fraction to decimal: 11/25
- Worked example: Converting a fraction (7/8) to a decimal
- Fraction to decimal with rounding
- Converting fractions to decimals
- Multiplying positive and negative fractions
- Multiplying positive and negative fractions
- Dividing negative fractions
- Dividing positive and negative fractions
- Negative signs in fractions
- Negative signs in fractions
- Negative signs in fractions (with variables)
- Dividing mixed numbers
- Dividing mixed numbers with negatives
- Simplifying complex fractions
- Expressions with rational numbers
- Simplify complex fractions
- Equivalent expressions with negative numbers (multiplication and division)
- Equivalent expressions with negative numbers (multiplication and division)
- Why dividing by zero is undefined
- Dividing by zero

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# Equivalent expressions with negative numbers (multiplication and division)

In this video, we figure out whether or not some expressions are equivalent. To do this, we think about properties of multiplication and division.

## Want to join the conversation?

- I'm not sure I understand the different ways to break down the problems and do them differently. Can someone please help me?(10 votes)
- Hello kenbat2. You might find it helpful to go back and relearn/rewatch the video/s on the different properties, such as disttributive and commutative. There are always multiple ways of figuring out how to solve a problem. I hope this helps, and if anyone could point to these videos, please do so. Otherwise you will have to search here on KA.

Jennifer in AZ(0 votes)

- I have a question with an answer suggestions:

-x*(-y*x) = x^2*y. The solution says that it's the same as x*x*y, but what about the -x? Shouldn't it be -x^2?

I just have no clue why and how that could work.

Please help!(4 votes)- -yx is a single term, so you won't need to use the distributive property. The term, -yx has a negative sign, so does -x. Multiplying the two negative terms will make the negatives cancel out, so we are left with x^2y.

Hope this helps!(10 votes)

- looks like my name is different now

someone upvote or comment if you want same(7 votes) - I don't understand why -x*(-y*x) is not equal to -x*x*xy when using the distributive property? Please help! Thank You(4 votes)
- The distributive property is used to distribute something outside the parentheses to each of the terms inside the parentheses. Since you are multiplying, -y*x is a single term, and while what you are doing does not appear to follow any pattern, it appears you are trying to look at -yx as two terms, not one. If you had -x(-y+x) then you would get xy-x*x or xy-x^2.(6 votes)

- -12(-3/4) how do you solve this poblem(3 votes)
- What you do is you place -12 over 1 to become a fraction, then multiply the numerators and denominators as usual.(7 votes)

- How did he get a positive six when it was (-2)x3(5 votes)
- oh my god! I am in 6th grade and this is still super easy! I got a hundred on the quiz!!(4 votes)
- What do you mean why did you make another lesson on the ordering thing didn't we already do that in the lesson before that?(3 votes)
- This is like an in depth version of the other one(3 votes)

- why is the 2.2 in the equation 3.7 / 2.2 x (-9.4) turned into 1/2.2 when it is rewritten? (in the quiz)(2 votes)
- With your expression, to multiply fractions times whole numbers, you can divide whole number by 1, then you multiply the numerators and denominators, there are a variety of ways to write this. 3.7/2.2 * (-9.4) = -9.4/2.2 * 3.7 = (3.7 * -9.4)/2.2 = 1/2.2 (3.7*-9.4). Each way is equivalent to multiplying the numerators and denominators.(3 votes)

- What about fractions? How does -(x/-y)*(-z/-a) equal to x/a*(-(-z/y)?(3 votes)

## Video transcript

- [Voiceover] What I
hope to do in this video is a bunch of examples to show us that a lot of the properties
we've been dealing with in arithmetic, the distributive property, associative property,
commutative property, that these apply just as
well to negative numbers. But with that said, it is
good to actually see it used using negative numbers, or see these properties applied just to make sure that we
understand what's going on. So these exercises, these
are all from Kahn Academy. Oops. So this first one says, which of the following
expressions are equivalent to negative two times the
quantity five minus three? Now you could, of course, just figure out that five minus three is two and then multiply that times negative two and you would get negative four. And you could see which of
these is equal to negative four. And that would be fair, but
the whole point of this video is to understand that
look, maybe I could apply the distributive property here. So let's do that. So what I could do is I could
distribute this negative two. I could multiply it times five and then I could multiply it by, I could either do it as it's going to be negative two times five plus negative two times negative three. Or you could view it as
negative two times five minus negative two times positive three. Now let me write those two things down. So you could do this as negative two times five, negative two times five plus negative plus negative
two times negative three. You could view it that way. Or you could view it as negative two, negative two times five minus minus and if I'm putting a minus here then I'm gonna view this
as a positive three, that we're subtracting a positive three. So minus negative two times positive three. Notice I either wrote the negative here and wrote a positive here, or I wrote the negative here
and made this a positive three. But these are going to be equivalent. Either way I've distributed
this negative two. Notice I have a negative two, negative two. And what are these going to be equal to? Well, negative two times
five is negative 10 and then, and then negative two times negative three is positive six, or over here negative two
times three is negative six, but then we subtract it so
it's just gonna get positive, you're gonna get positive six either way. This right over here is positive six and this over here,
subtracting a negative six would give you positive six. So you get negative ten plus six. And that's this choice right over here. Which of course does
evaluate to negative four, which this expression does evaluate to. This one up here evaluates to negative 16. And of course I won't select
this because I found an answer. Let's do several more. Which of the following
expressions are equivalent to negative S times T times S? Select all that apply. And here we can't just substitute, we can't just evaluate it and see, oh what do these evaluate to. We need to do a little bit of manipulation of these variables. Well there's a couple of
ways to think about it. One, we could change the order in which we multiply these things. So we could view this as negative S, that's that, let me write it a little bit neater. We could view it as negative S times S, times S times T. Times, oops, let me do
that in a different color, times S times T. Times T. And do any of the choices look like that? Well almost. Instead of saying negative S times S, this says S times negative S. And because multiplication, once again, I'm not a big fan of using the word because it sounds complicated, but it's commutative. A times B is the same thing as B times A. So I can rewrite this as, I can rewrite this, I can swap these two and write this as S times, S times negative S times T, times T. Times T. All I did is I swapped these two. This negative S and this S. I just swapped them and I got exactly what I have right over here. Now let's just make sure
that this one does not apply and maybe the easiest way
is to try to simplify this. And the best way I could think about that is by distributing this S. So if I distribute this S, what I'm going to get, this is going to be equal to S times T, which is ST. Or I could even, I can write it like this. I could write it S times T, like that. And then I have minus S times S, so minus S times S. I could write it that was
or I could write minus S squared if I want to. That's the same thing as S times S. But this is very different. This is very different. Here I'm just taking the
product of three variables here, I have two different terms, taking the product of two variables here and then the product I guess you could say I'm taking S squared or
I'm taking S times S. So this is not, this is not the same thing. Which of the following
expressions are equal to negative X times,
and then in parentheses negative Y times X? And I forgot to mention
it, but like always, pause the video, try to
work them out by yourself before I do them. Alright, select all that apply. So let's just try to
manipulate this a little bit. So once again, multiplication,
it's associative. I could, so it's negative X times, times negative Y times X. So, the way it's written here, I could do these first, that's essentially what's written over here. Or it's associative. Instead, I could do these first. And the reason why I find this interesting is a negative times a negative is going to be a positive. So this is going to be the same thing, this thing over here is
going to be the same thing as positive X times positive Y. Negative times a negative is a positive. So you're gonna get positive X times Y and then you're multiplying by an X again. Multiplying by an X again. Now the other thing we
know about multiplication is it's commutative. We can change the order
in which we multiply. Because I don't see this quite, I don't see this going on over here yet. So let's see, if I change the order, if I put the Xs, if I
multiply the Xs first, I could write this as, I could write this as X times X, X times X times Y, times Y. All I did is I swapped these two. Once again, I can swap the
order when I'm multiplying. And I don't quite see
this yet, but X times X, that's the same thing as X squared. This thing right over
here, that's X squared. So this is going to be X squared, X squared times Y, which is exactly what we have over there. Now what does this one evaluate to? Now this one actually
evaluates to a number. Because regardless of what X you pick, X minus X, that's going to be zero. Zero times anything is going to be zero. So this thing is going
to be equal to zero. So this is different than
what's going on over here. So I definitely would not pick that. Let's do one more. Which of the following expressions are equivalent to A times
negative 10 plus 11? Well once again, you know the instinct when you see something like this well I'd love to distribute that A, it's just sitting there. So let's do that. A times negative 10 would
be negative 10 times A, or negative 10A. And then A times 11, so it's gonna be plus, A times 11 is the same
thing as 11 times A, which we could write as 11A. Now which of these choices are that? Negative 10A plus 11A. So this is negative 10A plus 11A, so this one looks right. Now what about this one? Well here they just swapped the order. If you put the 11A first,
you could write it this way. If you write the 11A first, we could write 11A and then we have, instead of saying it negative 10A, we just say minus 10A. Once again I just took, all I did is I took this thing and I put it out front. So these two things are
actually equivalent. So I would select that one as well.