Using associate property to simplify multiplication

In this video, we're going to think about how we can use our knowledge of multiplying single-digit numbers to multiply things that might involve two digits.

So, for example, let's start with what is 5 times 18. You can pause the video and see how you might try to approach this, and then we'll do it together.

All right, so if we're trying to tackle 5 times 18, one strategy could be to say, "Hey, can I re-express 18 as the product of two numbers?" The one that jumps out at me is that eighteen is the same thing as two times nine.

So I could rewrite five times eighteen; this is the same thing as 5 times instead of 18, I could write 2 times 9. Now, why does this help us? Well, instead of multiplying the 2 times 9 first to get the 18 and then multiplying that by 5, what we could do is we could multiply the 5 times the 2 first.

And you might be thinking, "Wait, wait, hold on a second! Before you did the 2 times 9 first, and now you're telling me that you're going to change the order? You're going to say, 'Hey, let's multiply the 5 times 2 first'? Is that ok?"

And the simple answer is yes, it is ok. If you are multiplying a string of numbers, you can do them in any order that you choose. So this is often known as the associative property of multiplication.

We can associate the two with the nine first; we can multiply those first, or we can have an association with the five and the two. We can multiply those two first.

Now, why is that helpful? Well, what is five times two? Well, that's pretty straightforward; that's going to be equal to ten. So this is going to be equal to ten, we're doing that same color: 10 times 9.

Now, 10 times 9 is a lot more straightforward for most of us than 5 times 18. 10 times 9 is equal to 90.

Let's do another example: Let's say we want to figure out what 3 times 21 is. Pause this video and see if you can work through that. There's multiple ways to do it, but see if you could do it the way we just approached this first example.

Well, as you could imagine, we want to re-express 21 as the product of smaller numbers. So we could rewrite 21 as 3 times 7 maybe. And so if we rewrite it as 3 times 7, and now we do the 3 times 3 first, so I'm going to put parentheses there, which we can do because of the associative property of multiplication—a fancy word for something that is hopefully a little bit intuitive.

Well then, this is going to be equal to what's 3 times 3? It is 9, and then times 7, which you may already know is equal to 63.

Let's do another example; this is kind of fun. Let's say we want to figure out what 14 times 5 is. Pause this video and see if you can figure that out.

Well, we could once again try to break up 14 into the product of smaller numbers. 14 is 2 times 7, so we can rewrite this as 2 times 7 or 7 times 2. I'm writing it as 7 times 2 because I want to associate the 2 with the 5 to get the 10 times 5, and then I could multiply the 2 times 5 first.

So this is going to give us 7 times 10, which is of course equal to 70.

One more example: let's say we want to calculate 15 times 3. How would you tackle that?

Well, we can break up 15 into 5 times 3. 5 times 3, and then we can multiply that, of course, by this 3, and then we can multiply the 3s together first, and then this amounts to 5 times 9.

5 times 9, you might already be familiar with this; this is going to be equal to 45.

And another way to get to 45, you could say, "Hey, 5 times 10 is 50, so 5 times 9 is going to be 5 less than that, which is also 45."