How to extend a growing pattern and fill in missing numbers
lets say you have the pattern 4, 7, __, 13 and you want to find that missing number. the way you would do that is by putting the information you have onto a T-chart.
like so:
Now to find that missing number your going to have to find the equation. To find the equation is pretty simple, first you need to see how much the pattern is changing by. As you can see the pattern above changes from 4 to 7 in one move and with basic math we can see its changing by 3. With that information we can start to make our equation by putting how much its changing by (3) multiply X and = to Y. We end up with 3x=y but with the equation 3x=y we end up getting the pattern 3, 6, 9, 12… which is not what we are looking for so to make the right pattern we need to see how much we are going over/under from our target value.
ex: x = 1 and y = 4 for x = 1 our target value is 4
with that in mind we are getting 3 instead of 4 with our equation (3x=y) so that means we need to add 1 to it making our final equation 3x+1=y and if we put 1, 2, 3, 4, 5, 6… into our X slot we will always end up with this pattern.
How to extend the pattern and fill in the missing numbers
to fill in the missing numbers its quite simple we get our rule and use it.
for the missing number in the pattern above (4, 7, __, 13)
we can see that the missing number is the 3rd in the pattern using our equation we replace X with 3 and find Y.
3x+1=y
3(3)+1=y
9+1=y
10=y
as we can see the missing number was 10 completing the pattern at 4, 7, 10, 13
now that pattern is pretty small but believe it or not it actually extends infinitely to the left and the right. To show you that I’m going to teach you how to grow your patterns.
To grow the pattern above (4, 7, 10, 13) you need the rule (3x+1=y)
then you need to extend your T-chart, fill in the Y’s and write down the next numbers in the pattern.
ex:
How to fill a T-chart when you only have the equation?
to do that it’s very simple first you get a blank T-chart. Then for X you fill 1, 2, 3, 4 for the X side and for the Y side you run the X number through the equation and what Y equals is Y.
ex:
2x-2=y
2(1)-2=y
2-2=y
y=0
so x = 1 means y = 0
and so on keep doing that until you fill in the first 4 numbers in your T-chart and you should have something like this.
now with this you also have the pattern which is: 0, 2, 4, 6
How to graph with patterns
this is a basic if you’ve never graphed before to plot a point on a graph you need the x axis and the y axis which would look something like this: (2, 4) which would make a point here:
now you can do the same thing with T-charts to make a visual pattern!!!
so the T chart:
Looks like this on a graph:
now this pattern can be easily extended just by looking at it.
You can see for each time it goes 1 to the right it goes 2 up so you can guess where the next few dots are going to pop up.
another way you could have come up with this line is by looking at the equation. By looking at the equation 2x-2=Y you can see that the equation is telling you to make a line that goes up by 2 each time and a the x axis being 1 y is going to be 0 and +2 on y each time you go up one on x.
to graph equations that have fraction in it might not be easy and that’s why for your 3 or 4 numbers your going to put in your T-chart need to be smartly picked. for the expression 
So what I recommend doing to help you with these are to pick 0 and two other numbers that its being divided by so for this case x is divided by 4 so i’m going to pick 4 and 8 for my other two numbers.
making this:
and being graphed like this:
Horizontal and vertical lines
horizontal lines are lines that have a set number on y.
ex: y = 4 would be a horizontal line because X could be anything and no mater what y is 4 same story for vertical lines but vertical lines use x = … instead of y
vertical line:
Horizontal line:
