[0:01]Hey everyone, it's Justin again. In this video, we're going to practice multiplying with positive and negative integers. Let's get into it. In today's lesson, you'll learn the rules for multiplying with positive and negative integers, learn why those rules are true, and review how to multiply numbers with more than one digit. Oh, hold on. It looks like Mia has a reminder for us. Oh, that's right. The word product means the answer we get when we multiply. Thanks, Mia. Let's find some products. Let's start with something we know. We'll find the product of a positive number and another positive number. Let's say we have 4 * 2. Now, what is this really saying? Remember that multiplication is just a shortcut for repeated addition. What we're really saying is 4 added together two times. So we have 4 + 4. How many times are we adding the number four together? Two times. And two fours give us the answer positive 8. Now, as we know, we can multiply terms together in any order. So, what if we had written it as 2 * 4? That would mean two added together four times, or 2 + 2 + 2 + 2, which would give us positive 8 as well. Since we know that repeatedly adding positive numbers always gives us a positive result, we can also see that any positive number times any positive number will always give us a positive answer. Here are some quick practice problems. If you haven't yet memorized your times tables, there's a printed times table with your PDF. However, it will save you a lot of time if you can memorize your basic multiplication facts, which will then make it easier for you to solve more complicated problems later. Pause the video and complete these problems on your PDF.
[2:20]Okay, let's see how you did. Did you get 56, 45, and 60? Great. In each case here, we multiplied a positive number times another positive number. But what do you think will happen if we multiply some negative numbers? Let's start by multiplying a negative number times a positive number. This time, we have -4 * +2. Now, what is this really saying? Right. It's saying add -4 two times. So I add -4 + -4 and get -8. So, this seems to give us a rule that a negative number times a positive number equals a negative number. But let's double check it with some new numbers. Let's say I have -6 * +3. What is this really saying? Right. It's saying add -6 three times. So I have -6 + -6 + -6, and if I add these together, what do I get? I get -18. We know that repeatedly combining negative numbers will always give us another negative number. So, we know that a negative number times a positive number does result in a negative number. But what if we switch it around and multiply a positive times a negative? How do you think that will change things? Now we have +4 * -2. What does that really mean? Can we add four together -2 times? Actually, this -2 is telling us to subtract 4 two times. Oh, so this should actually look like I'm subtracting +4 two times over. This looks a little complicated, but recall from our previous lessons that subtracting a positive is the same thing as adding a negative, which means that we can rewrite this as an addition problem. So now it looks like our previous problem, adding two negative fours, which gives us negative eight. It doesn't matter if our negative sign is on the first number or on the second number. Whenever we multiply a positive number times a negative number, we will always get a negative number. Here are a few more practice problems. Please pause the video and complete these ones on your own in your PDF.
[5:22]All right, how'd it go? You should have gotten -28, -27, and -35.
[5:35]Our final combination is multiplying a negative number times another negative number. What do you think will happen? Here we have -4 * -2. What is this really saying? It looks like it's saying add -4 together -2 times. But how do we do that? What did that mean again? Oh, right. It means we're subtracting -4 two times. But you know that subtracting a negative is the same as adding a positive. So we can rewrite this as +4 + +4, and we can simplify that by simply writing it as 4 + 4, which we know is 8. This example helps us see that a negative number times a negative number equals a positive number. Let's practice these. Pause the video and complete these problems in your PDF.
[6:50]Okay. Since each problem had us multiplying a negative number times a negative number, I bet you got a positive answer for all of these. 42, 24, and 36. Nice work. We've seen four different kinds of problems so far and we found a different pattern in each one. Let's summarize these patterns we've seen so far into some rules for multiplying positive and negative integers. Here are the four situations we looked at today. But remember, it doesn't matter which order we multiply. If we multiply one positive and one negative, we always get a negative result. And remember that if we multiplied two positives or two negatives, we got positive answers. So, let's simplify this a bit so that you only have two rules to remember. If you multiply two of the same type of number, the result is positive. If it's two different types of numbers, the result is negative. Now that you know the rules for multiplying with negative numbers and why the rules are true, let's review how to multiply numbers with more than one digit. Okay. We are going to find the product of -132 and 24. Remember that a product is the result of multiplication. You probably don't know this answer yet, but you know if it's going to be negative or positive, don't you? Right. The product will be negative. So, one way to solve this is to just ignore the negative sign at first. As long as we remember to add it back in later so that our answer is negative. Ready? Here we go. Let's get into it. How do we multiply -132 * 24? Remember, we know our answer will be negative, so we're going to ignore the negative sign for now. It will not affect how we solve the problem. We start by lining the numbers up vertically, aligning their ones and tens places above each other. Remember, this 24 is really 20 + 4, so we're actually going to multiply 132 * 4 and then multiply the 132 * 20.
[11:10]Then we will add the results together. Remember, when we're adding positive integers, we start at the right, adding each place value together. Remember to regroup by carrying one to the next place value. And then we have our answer. Except that we need to remember that we multiplied one negative number and one positive number. So we actually know that our answer is -3168. Oh, nice work. Okay, mathematicians, let's check back in with our objectives. How did you do? Do you feel ready to multiply positive and negative integers? Do you understand why the rules are true? Can you multiply numbers with more than one digit? Great. Well, you've now completed your video and guided practice. Your next step is to complete the activities in your PDF. After that, it'll be time to earn some medals in your practice game. Great multiplying, and I'll see you next time.
