Algebra

Math patterns example 2 | Applying mathematical reasoning | Pre-Algebra | Khan Academy

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I want to make small models of town houses with the help of toothpicks. This will be my first house. So far I have used 3 toothpicks, 4, 5 and 6. This is my first house. Let me make a small table here to write things down. I will do it in white. Here is the table in which I will write things down. This is the number of houses, and these are the toothpicks, which I use to make this house. In this first house here I used 6 toothpicks – 1, 2, 3, 4, 5, 6. Now let’s make our second house. These will be town houses. They will share common walls. So I will add 1, 2, 3, 4, 5 toothpicks for my second house. Why did I have to add only 5 and not 6? They share a common wall here, so I don’t have to add another toothpick. for this wall on the left. I start with her first house you just need to add 5 toothpicks. I need to add 5 toothpicks to get a total of 11 toothpicks if I want two houses. I think you see the trend here. What happens to 3 of them? There will be 5 – 1, 2, 3, 4, 5 toothpicks. We will add 5 again and get 16. Let’s do 5 just for easier calculation. Fourth, we will add another 5 – 1, 2, 3, 4, 5. In the fourth we will add another 5, which will give 21. Now I want to think about whether we can, using this model, to find how many toothpicks we will need, to make 50 of these townhouses, or 500 of these townhouses, or 5,000? We just have to follow this pattern and see if we can come up with an equation for each one of these specific values? For example, we see a model in which – well, we have already accepted, that we start with 6 and we add 5 every time we add a house. So when you add a second house, you add 5 once. In the third house you start with 6 and add 5 twice. The fourth house – you start with 6 and add 5 three times. Let’s write this down below. 21 is equal to – you start with 6, you start with this 6 here and then add 5 three times, plus 5 by 3. Let’s repeat, when there were 3 houses, it started with 6 and add 5 twice. Let me do it in the same color. Add 5 twice. Plus 5 by 2. When you have 2 houses, you start again with 6. This is equal to 6 and add 5 once, plus 5 over 1. And then when you have 1 house – and it will uses the same model – started with 6, and how many times did you add 5? Okay, don’t add 5. You can say add 5 zero times. Maybe you see some pattern here. No matter how many houses you need, you get one less than that and multiply it by 5, add that to 6, and you get the number of toothpicks. In fact, let me rewrite this. I can write it again, like 6 plus 5, 4 minus 1. I can write this as 6 plus 5, 3 minus 1. You can write this as 6 plus 5, 2 minus 1 You can write this again as 6 plus 5, 1 minus 1. And maybe that makes the model a little clearer. This 4 is here. This 3 is here. This 2 is here. And then – this 1 is here. Now we are ready to think what if we want to make 50 houses. Let’s try to do that. Let me do it in orange. Here is our fiftieth house. This is the common left wall that has. This is the fiftieth house here. How many toothpicks in total for 50 houses? If we have 50 houses, fine, we can use the model we have reached. This is going to be equal to, we start with 6, the first house requires 6. And then we add 5 for each subsequent house, so plus 5 for each subsequent house. And how many next houses will there be here? There will be 50 minus 1 next house. Why minus 1? He has already built one of them with the six sticks. Then for each subsequent – will there are 49 more houses – you will add 5 toothpicks for the house. So that’s going to be equal to 6 plus 5 over 49. And that’s 245. 6 plus 245 equals 251 sticks. This model we just came up with, can be used to find how many sticks you will need for a million of these small town houses made of toothpicks.

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