The Distributive Property In Arithmetic

Hi, I’m Rob. Welcome to Math Antics! In this video, we’re going to talk about really important math concept called “The Distributive Property”. Well… at least that’s what it’s was called sometimes. You may hear it referred to as “The Distributive Law” or even the “The Distributive Property of Multiplication over Addition” by people who want to sound really technical. But no matter what it’s called, the concept of the Distributive Property is the same. Before we actually dive into that concept, there are two quick things that will help make it easier to understand. The first is simply knowing what the word “Distribute” mean. To distribute something means to give it to each member of group. …like an old-fashioned paper boy delivering newspapers to each house in neighborhood. One for you… One for you… Gee! Sorry!! One for you… And one for you… The second thing you need to know is the order of operations rules of arithmetic, which we cover in another video, so you might want to watch that if you’re not familiar with those rules already. That’s because the Distributive Property is actually way of allowing us to change the order of operations we do in certain types of problems. To see what mean, have look at this simple arithmetic expression: three times the group 4 plus 6. We’re going to simplify this expression in two different ways. The first way will just use the basic order of operations rules that you already know. But the second way will use the Distributive Property. And if we do the arithmetic right, both ways will give us the same answer. So for the first way, our order of operations rules tell us that we need to do any operations inside of groups first. And since these parentheses form group, we first need to add the 4 and 6 which gives us 10. Next, we can multiply that by the 3 which gives us final answer of 30. Now let’s use the Distributive Property. The Distributive Property allows us to change this expression into different form. Instead of multiplying the 3 by the whole group at once, we can distribute that factor of 3 and multiply it by each member of the group individually. That means we’ll make copy of the ‘3 times’ for each member of the group… the 4 and the 6. So after applying the Distributive Property, our expression looks like this: 3 times 4 PLUS 3 times 6. Because we distributed the multiplication to each member of the group, the group isn’t needed anymore so the parentheses can go away. Now we can continue to simplify this new form using our order of operations rules. Those rules tell us to do multiplication before addition, so 3 times 4 is 12 and 3 times 6 is 18. The last step is to add those two results together. 12 + 18 = 30. Well look at that… we got the same answer in both cases, which means our original expressions are equivalent, even though they have different forms. In the first form, the factor ‘3’ is being multiplied by the entire group all at once, so we needed to do the addition inside the group first. But in the second form, we used the distributive property to rearrange the expression so that the factor of ‘3’ is multiplied by each member of the group individually, instead of the whole group all at once. Distributing that factor made the group go away, so we didn’t have to do the addition inside that group first. So the Distributive Property is basically way of getting rid of group that is being multiplied by factor. If you distribute the factor to each member of the group, you’ll get the same answer you would if you calculate what’s in the group first and then multiply. And it works no matter how many members are in the group. Like in this problem, we have to multiply 4 by the group (1 + 2 + 3). Again, let’s try simplifying this both ways. In the first way, we start by simplifying what’s in the group. 1 + 2 + 3 = 6 and then we multiply 4 times 6 which gives us 24. Now let's use the Distributive Property. We distribute factor of ‘4’ to each member of the group which makes the group go away and allows us to do those multiplications individually. 4 times 1 is 4 4 times 2 is 8 and 4 times 3 is 12 Finally, we add up those three individual answers: 4 plus 8 is 12 and 12 plus 12 is 24. See, the Distributive Property gave us another way to arrive at the same answer. It’s like the Distributive Property is an alternate path that you can take to arrive at the same point! Wa….What are you doing here!? I’m always here! Okay, great, we have two ways to get the same answer. But, why do we need two different ways to do the same calculation? And it seemed like the Distributive Property way was even more complicated that the regular way. Why would we ever want to use it? That’s good question. And it’s true! There are times where the Distributive Property way is harder …like in our first problem. It was easier to just go ahead and simplify the group first because it’s easy to multiply 3 times 10 mentally. BUT… there are also times when the Distributive Property way is easier, …like in this case: 8 times the group (50 + 3) If we decide to simplify the group first in this problem, we end up needing to multiply 8 times 53 which is not so easy to do mentally. But if we apply the Distributive Property instead, we can change the expression into 8 × 50 plus 8 × 3. And that’s easier to do mentally. 8 times 50 is 400 and 8 times 3 is 24. So the answer is 424. Realizing that the Distributive Property can make some calculations easier to do mentally can come in really handy for certain basic multi-digit multiplication problems. That’s because you can break up the multi-digit factor into “group sum” …ya know… like expanded form. and then distribute the other factor to the members of that group. …sound confusing? Here’s what mean… Let’s say you need to multiply 5 times 47. Well, you could just use the multi-digit multiplication procedure, OR, you could change this into problem where the Distributive Property will make it little easier to do. The key is to realize that you can replace the ’47’ with ’40 plus 7’. Then the problem becomes 5 times the group (40 + 7) and the Distributive Property lets us change that into 5 times 40 plus 5 times 7. Those two multiplications are easy to do! 5 times 40 is 200 and 5 times 7 is 35. So our answer is 200 plus 35 …or 235. Wanna see another example? Let’s apply that same idea to this multiplication problem: 3 times 127 But instead of 127, let’s change that into the group (100 + 20 + 7). We need to multiply that by ‘3’ and the Distributive Property lets us distribute that multiplication to each member of the group: 3 × 100 + 3 × 20 + 3 × 7 That helps because we can do those mentally: 3 times 100 is 300 3 times 20 is 60 and 3 times 7 is 21. All that’s left to do is add those three products up which is not too hard to do mentally either. 300 + 60 + 21 gives us 381 as our final answer. Now before we wrap up, there’s one more important thing you should know about the Distributive Property. You already know that the Distributive Property works when the members of group are being added, but it works the same way for members of group that are being subtracted. …like in this problem: 7 times the group (10 - 4) You could do this problem the typical way and simplify the group first. 10 minus 4 is 6 and then 7 times 6 gives us 42. OR… you can use the distributive property. You distribute the ‘7 times’ to both members of the group to get 7 times 10 MINUS 7 times 4. 7 times 10 equals 70 and 7 times 4 is 28. And 70 minus 28 equals 42. Again, both ways are equivalent! So the distributive Property works for groups of ANY size and it works the same for group members that are being added OR subtracted. …even if there’s mixture of addition and subtraction in the group. But The Distributive Property DOESN’T work when the members of group are being multiplied or divided. For example if you have 5 times the group (2 × 3) You CAN’T distribute copy of the factor ‘5’ to each member of the group without getting completely different answer. And the same goes for division. If the members of the group are being divided, like 4 times the group (6 ÷ 2) you will NOT get the right answer if you distribute the factor ‘4’ to each member. That’s why the 'technical' name is the Distributive Property of Multiplication over Addition. You’re distributing the multiplication over all the members of group that are being ADDED. And the reason that it also works for subtraction is that subtraction is really just negative form of addition, since subtraction and addition are inverse operations. Alright, so the Distributive Property is handy way to rearrange arithmetic expressions. It’s like tool that you can use in certain situations if you think it will make particular calculation easier to do. And even if you don’t end up using The Distributive Property whole lot for arithmetic problems, it’s still really important math concept that will be even more useful when you get to Algebra. Until then, be sure to practice what you’ve learned in this video by trying some of the exercise problems. Practice is the best way to make sure you really understand. As always, thanks for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com