What are patterns?

Linear Patterns, as the name suggests, are equations that have a consistent pattern when a number is plugged in. For example, 2x is an example of a linear equation since if you plug in any number, it will be accurate no matter what.

2, 4, 6, 8, 10, 12

2(1) = 2.  2(4) = 8. 2(6) = 12

The next kind of linear equation is something slightly different, if we add a constant to the equation, like 2x+1, we’ll get a different result.

3, 5, 7, 9, ,11, 13, 15, 17, 19, 21

2(1)+1 = 3. 2(4)+1 = 9. 2(6) = 13

It may not be immediate at first, but you can notice that it still goes up by 1. This type of equation is used for patterns that aren’t perfect multiples. Here’s another example, this time involving negatives.

29, 22, 15, 8, 1, -6, 13, -20

It’s clearly descending this time, so that always means the variable is negative, since no constant will stop it from going up, we can also tell that the pattern goes down by 7, so we continue from there.

-7x

Now that we know the variable, we can find out the constant as well by simply adding the variable to the first number in the pattern, in this case 29.

29+7=36

So our full equation is -7x +36, at this point we can plug in any number and it will work just fine

29, 22, 15, 8, 1, -6, 13, -20

-7(1) +36 = 29. -7(3) +36 = 15. -7(6)+36 = -6

 

Another method for finding the answers to a pattern equation is by using what’s called a T-chart, this chart allows you to look for the pattern if a number is missing. So let’s say our pattern is this:

We can put it into a T-chart, which is a simple chart that splits the amount of multiples and the values themselves like this:

The X value represents the multiples while the Y value represents the values of the numbers themselves, from here on out we can look for the equation by looking at clues. We can clearly tell that it goes up by 6 right off the bat, so we already have the variable 6x, then we can trial and error to look for something that matches our table:

Here we simply look for what matches, 1 to 3 is too low and 5 and above is too high, but 4 matches our chart perfectly, so now we have our equation – 6x+4.

 

Our next topic is graphing, in graphing you simply put the equation into a graph using dots, so if we had an equation like x+2 it would look like this on a graph

I’ll be covering how to accurately plot the dots on the line to make a correct graph. So here there are two intersecting lines, the Y axis and the X axis, and just like in the T-chart, the X axis controls the multiples while the Y represents the values themselves, the upper boxes represent the positive end and the bottom represents the negative end.

Let’s have a demonstration with 2x+4. The way you put an equation into a graph is like this, first you go up on the Y axis according to the constant which is +4

Then you go up by the variable while moving a tile to the right which is 2x, so you go up by 2 and move to the right

Keep doing this until you achieve the desired amount of dots on the graph.

Just remember to move the dots according to their signs as well. If the equation is something like -x-5 I would have to go down the Y axis like this:

Then go down by 5 and then move down and to the right by 1 like this:

A quick side note is that negative patterns always slant downhill, this is because even if the equation is something like -x+100, it WILL become negative at some point, and sometimes the Y axis is higher than the limit of the graph you have, like x+10

If you can’t go any further, simply move to the left, since this still counts since it’s now multiplying by negatives: x+10 = -1+10

And that’s what I learned in Grade 9 Patterns and Graphing