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CCSS.Math:

I want to make little townhouse shapes with toothpicks so this would be my first townhouse I've used three toothpicks so far four five and six so that is my first townhouse and 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 number of houses and then this is the toothpicks toothpicks that I'm using to make that house so this first toothpick or this is the first house here took me six toothpicks one two three four five six now let's make our second house our second house and these are going to be townhouses they're going to share common walls so I'm going to add one two three four five toothpicks for my second house now why did I only have to add five and not six 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 five to that to add five toothpicks to get to now 11 total toothpicks if I want two houses I think you see the trend here what about three what about three of these well this is going to be another five one two three four five toothpicks so we're going to add five again I'm going to add five again and get to 16 let's do four just for good measure so the fourth one we're going to add another five one two three four five so the fourth one we're going to add another five gets us two gets us to 21 now I want to think about can we using this pattern figure out how many it would how many toothpicks it would take for us to say make 50 of these townhouses or even 500 of these townhouses or even 5000 of them and 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 where we started with 6 and we're adding 5 every time every time we add a house so instance you so when you add the second house you add five once the third house you add [ __ ] you start with six and you add five twice the fourth house you start with six and you add five three times so let's actually write that down so 21 is equal to you start with six you start with this six here you start with six and then you add five three x plus 5 times three when you had the three houses once again you started with six you started with six and you added five two times you added five we do that same color and you added five two x plus five times two when you had when you had two houses you started with six again this is equal to six and you added five once so plus five plus five times one and then when you had one house when you one happen when you had one house and it'll fit the same pattern you start with six and how many times did you add five well you didn't add five you could say that you added five you added five zero times so you might see a little pattern here however many houses you needed you take one less than that multiply it by five add that to six 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 the pattern a little bit closer or clearer this 4 is right over here this 3 is right over here this 2 this 2 is right over here and then this one 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 so this right over here is do that in orange this right over here is our 50th house so this is this this is a shared left wall it has this is a 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 so it's going to be equal to starting with our six the first house requires six and then we're going to add five for each incremental house so plus five for each incremental house and how many incremental houses are there going to be well they're going to be 50 minus one incremental houses Y minus one well you already built one of them with the six then for every extra one so there's going to be 49 extra houses you're going to add five toothpicks apiece so this is going to be equal to six plus five times 49 and that is 245 so 6 plus 245 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 how many sticks you would need for a million of these little toothpick townhouses