Multiplication as groups of objects
I have these three star patches, I guess you could call them, right over here. And so I could say, if I had one group of three star patches, how many star patches do I have? So I literally have one group of three star patches. Well, that means that I have three star patches. 1, 2, 3. This is my one group of three. Now let's make it a little bit more interesting. Let's say that I had two groups. Let's say that I had two groups of three. So that's one group, and then here's a second group. Here's two groups of three. So how many total star patches do I have now? Well, I have two groups of three. Or another way of thinking about it is this is 3 plus 3. This is equal to 3 plus 3, which is equal to 6. So we see 1 times 3-- one group of 3 is 3. Two groups of 3, which is literally two 3's, is 6. Let's make it even more interesting. Let's have three groups of 3. Now, what is this going to be equal to? Well, it's three groups of 3. So I could write this as three groups, 3 times 3. And how many of these star patches do I now have? Well, this is going to be 3 plus 3 plus 3. It's going to be 3 plus 3 plus 3. Notice I have three 3's. I have two 3's. I have one 3. So this is 3 plus 3 plus 3 is equal to 9. And you can count them. 1, 2, 3, 4, 5, 6, 7, 8, 9, or you could just count by 3's. 3, 6, 9. And I think you see where this is going. Let's keep incrementing it. Let's get four groups of 3. So let's think about what 4 times 3 is. 1, 2, 3, and 4. This right over here is four groups of 3. We could write this down as 4 times 3, which is the same thing as 3 plus 3 plus 3 plus 3. Notice I have four 3's. One 3, two 3's, three 3's, four 3's. One 3, two 3's, three 3's, four 3's. So we get 3, 6, 9, 12. So what I encourage you to do now, now that the video is almost over, is to keep going. I want you to figure out what 5 times 3 is, and 6 times 3, and 7 times 3, and 8 times 3, and 9 times 3, and 10 times 3. And I'll give you a little hint. You don't always have to draw the star patches, but it's nice to visualize it. We saw 4 times 3 is literally four 3's. Well, 5 times 3 is going to be five 3's. So 2, 3, 4, 5. Which is equal to 3, 6, 9, 12, 15. So I encourage you to think about what all of these are after this video is done.