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Video transcript

- What I hope to do in this video is several examples from the Khan Academy exercise on negative number addition and subtraction equivalent expressions. This is just to really make sure that we fully digested how we can compute with negative and positive numbers, and also the properties of arithmetic that we can apply when we compute with them. Let's get started. It says Raul tried to evaluate an expression step by step. All right, they start with the expression, they give us the steps. They say find Raul's mistake. We have Step 1, Step 2, Step 3, or maybe Raul did not make a mistake. Like always, I encourage you to pause the video and try to solve it on your own before I work through it. All right, so let's look at this first step. It says 3 minus negative 5 minus 7. It looks like he correctly recognized that subtracting a negative 5 is the same thing as adding a positive 5. He replaced the minus negative 5 with the plus 5 right over there. That seems right. All right. Step 1 seems all right. Then, what does he do next? He has two options. He could say 3 plus 5 is equal to 8, and then subtract 7 from that, an you would get 1. He could even say 5 minus 7 is negative 2, and then add that to 3, and get 1. It looks like Raul tried to go that second way, but he made a slight mistake. He said 5 minus 7 is equal to 2 not negative 2. 5 minus 7 should be negative 2, but he wrote a positive 2 right over here. His mistake is in Step 2. Then obviously, Step 3 becomes wrong, because Step 2. This is where he first makes the mistake. Let's do several more of these. Which of the following expressions are equivalent to 6 plus negative 4 minus 5. All right, so let's look at all the options here. This first one, let's see, has a negative 5. Well, the negative 5, this minus 5 if you put it out front, you could represent it as a negative 5. Then it has a negative 4 here. Here, we're adding a negative 4, or I guess you could say, we have minus 4 here, which is the same thing as adding a negative 4. That seems reasonable. Then, you have plus 6, and you just had a positive 6 right over here. This one actually seem equivalent. It just the order, they just switched the order. Instead of adding the 6 first, they add the positive 6 last. Instead of having the minus 5 at the end, they put a negative 5 in the beginning. Instead of saying adding negative 4, they just said minus 4. These are all equivalent. This one is going to be an equivalent expression. Now, let's see, this one has ... it has the 6 and the minus 4, they just changed the order. Now, it has positive 5. That's not going to work, because we're subtracting 5 out here. If we want to put the 5 out front, it needs to be a negative 5. This isn't going to be right. I'm not going to select none of the above, because I found a choice that is equivalent. Let's keep going, it's a lot of fun. Which of the following expressions are equivalent to negative 4 plus 4 plus 5 in parenthesis. If I were to compute this, I would say negative 4 plus, and I would add the 4 and the 5 first. I would get 9. It would be negative 4 plus 9 which is equal to 5. Let's see. Zero plus 5, well that's going to be equal to 5. That's equivalent. I mean, I could just do it again. Negative 4 plus 9 is 5, which is the same thing as 9 minus 4. This right here is 5. Now, this is negative 4 minus 4, which would be negative 8. Then, plus 5 would be negative 3. That's not equivalent. In order to be equivalent, this should evaluate out to be 5. I'm not going to select that, and I'm not going to select none of the above, because I all ready found a choice that I like. Let's do a few more of these. One again, which are equivalent? Negative 5 plus 7 minus negative 3. Here you have negative 5 plus 7 plus 3. Well, that make sense. If you subtract a negative 3, that's the same thing as adding a 3. They didn't change the negative 5 plus 7. They didn't change the negative 5 plus 7. I definitely select this one. Definitely select that. Now, let's look at this one. Negative 5, here they're subtracting a 5, so that seems reasonable. Adding 7, they're adding 7. Here, they're subtracting a negative 3, which is the same thing as adding a positive 3. Here, they just have a negative 3 here. If this was a positive 3, then we would be in business. This is a negative 3, so this isn't going to work. I will not select that. I won't select none of the above, because I found a choice. Let's do one more of these. Which of the following expressions are equivalent to X plus Y, and since there's parenthesis around it I would do that first, and then I would subtract Z. Select all that apply. All right. Here, they have Z plus X plus Y. They have the X plus Y part, but then they have a Z here, while over here they have minus Z. If there were a negative Z here, this and this would of been equivalent, but they don't have a negative Z here. I would not select that one. Now, this one you have X plus Y minus Z. Well this, they just changed the parenthesis. Instead of doing the X plus Y first, they're dong the Y minus Z first. This goes back to addition and subtraction. You have the associative property. You could associate these. You could start with adding these two first, and then you do the subtracting, or you could do the subtraction first. Then, you add this one. If you don't believe me, you could try it out with some numbers, and I encourage you to try it out with some negative numbers. It will indeed all work out. All they do is they change where the parenthesis are. They did the Y minus Z first, and then they do the Y minus Z, and then they add it to the X, as opposed to doing the X plus Y first, then subtracting the Z. I would go with that one.