Math patterns: toothpicks

I want to make little townhouse shapes with toothpicks. So this would be my first townhouse. I've used 3 toothpicks so far-- 4, 5, and 6. So that is my first townhouse. Now, let me make a little table here keeping track of things. So I'll do that in white. So here's my table to keep track of things. So this is the number of houses, and then this is the toothpicks that I'm using to make that house. So this first house here, took me 6 toothpicks-- 1, 2, 3, 4, 5, 6. Now let's make our second house. And these are going to be townhouses. They're going to share common walls. So I'm going to add 1, 2, 3, 4, 5 toothpicks for my second house. Now, why did I only have to add 5 and not 6? Well, they shared a common wall here so I didn't have to add another toothpick here for this left-hand side wall. So starting with the first house, I really just had to add 5 toothpicks. I had to add 5 toothpicks to get to now 11 total toothpicks if I want two houses. I think you see the trend here. What about 3 of these? Well, this is going to be another 5-- 1, 2, 3, 4, 5 toothpicks. So we're going to add 5 again and get to 16. Let's do 4 just for good measure. So the fourth one, we're going to add another 5-- 1, 2, 3, 4, 5. So the fourth one, we're going to add another 5 gets us to 21. Now, I want to think about, can we, using this pattern, figure out how many toothpicks it would take for us to, say, make 50 of these townhouses or even 500 of these townhouses, or even 5,000 of them? Now we just have to look at this pattern here and see can we come up with an equation for each of these actual values? So, for example, we see a pattern that-- well, we already recognize that we started with 6, and we're adding 5 every time we add a house. So when you add the second house, you add 5 once. The third house, you start with 6, and you add 5 twice. The fourth house, you start with 6 and you add 5 three times. So let's actually write that down. So 21 is equal to-- you start with 6, you start with this 6 here, and then you add 5 three times, plus 5 times 3. When you had the 3 houses, once again, you started with 6 and you added 5 two times. Let me do that same color. And you added 5 two times. Plus 5 times 2. When you had 2 houses, you started with 6 again. This is equal to 6 and you added 5 once, so plus 5 times 1. And then when you had 1 house-- and it'll fit the same pattern-- you started with 6, and how many times did you add 5? Well, you didn't add 5. You could say that you added 5 zero times. So you might see a little pattern here. However many houses you needed, you take one less than that and multiply it by 5, add that to 6, and you get the number of toothpicks. And actually, let me rewrite this. So I could rewrite this as 6 plus 5 times 4 minus 1. I could write this as 6 plus 5 times 3 minus 1. You could write this as 6 plus 5 times 2 minus 1. You could rewrite this as 6 plus 5 times 1 minus 1. And maybe that makes a pattern a little bit clearer. This 4 is right over here. This 3 is right over here. This 2 is right over here. And then this 1 is right over here. So now, I think we are ready to think about what would happen if we wanted to make 50 houses. So let's try to do that. Let me do that in orange. This right over here is our 50th house. So this is the shared left wall it has. This is the 50th house right over here. So how many total toothpicks for 50 houses? So if we have 50 houses, well, we can use the pattern that we came up with. It's going to be equal to, starting with our 6, the first house requires 6. And then we're going to add 5 for each incremental house, so plus 5 for each incremental house. And how many incremental houses are there going to be? Well, there are going to be 50 minus 1 incremental houses. Why minus 1? Well, you already built one of them with the 6. Then for every extra one-- so there's going to be 49 extra houses-- you're going to add 5 toothpicks apiece. So this is going to be equal to 6 plus 5 times 49. And that is 245. So 6 plus 245 is equal to 251 sticks. And what's really neat about this pattern we just came up with is you could use it to figure out how many sticks you would need for a million of these little toothpick townhouses.