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

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.