Alissa Fong

**MA, Stanford University**

Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

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One of the most important things to understand about lines is the **definition of slope**. Slope is the 'steepness' of the line, also commonly known as rise over run. We can calculate slope by dividing the change in the x-value between two points over the change in the y-value. In order to understand the importance of the definition of slope, one should understand how to interpret graphs and how to write an equation.

One of the most important ideas you're

going to see in your whole study

of algebra is the idea of slope.

And slope just refers to

how steep a line is.

And that's kind of the basic definition.

That's something that I help -- or helps

me remember what slope means.

There's a whole lot of fancy notation

that goes along with it.

And we'll get into that in just a second.

But if you can just remember in the back

of your head that slope means how steep

the line is, it will

help you a lot.

So one thing you see is that anytime I

draw a straight line, the slope ratio

stays the same for

that entire line.

And here's what I mean.

Let's say those are my X and Y axes

and I have this line right here.

The line is equally steep

the entire time.

It has the same slope ratio.

And we're going to get in a second

what I mean by ratio.

And here's what I'm talking about.

When we talk about slope or steepness,

the way it's defined is by change of

Y over change of X. Like

in a fraction.

Change in Y on top of change in X.

That's why we call it a ratio.

Remember ratio is like a fraction.

So if I were to draw a little triangle

here that represents how steep my line

is, this would be my change in Y

piece because Y is up and down.

This would be my change in X piece,

because X is horizontal.

And whatever those numbers were on the

graph, I would write as a fraction.

That's one thing you want

to keep in mind.

Change in Y on top of change in X.

Sometimes we write it using this little

triangle.

This triangle is the Greek letter

delta, which is tricky.

Not only do you have to learn math

but now you have to learn Greek.

This means change in Y on top of change

in X. That delta just represents

the word change.

And a third way we write this is using

the letter M. M stands for slope,

and if I had two points,

I'll use them up here.

Let's say I had this point I'm going to

call it X from my first point and then

Y from my first point.

Here's my second point.

X from my second point I'm going to use

that little 2 to show it's my second

point. Y for my second point.

Then there's a formula I could use using

those X and Y numbers to find M, or

the slope.

And the way I would write that is Y to

take away Y1 on top of X2 take away X1.

This is the same thing just

written in a different way.

I'm finding out how much did my Y values

change and putting that on top of how

much did my X values change

in a fraction.

So this formula is really important anytime

you have two points like this.

These little numbers down here are tricky.

It doesn't mean take your Y value and

multiply it by 2 or take your Y value

and multiply it by 1. What it means

is we're just notating using

what's called a subscript.

That's my second Y value.

I'm subtracting my first Y value.

And we'll get into that more later as you

guys start your homework problems.

A couple other things I want you to keep

in mind with slope, and that is

sometimes slopes are positive numbers

and sometimes slopes are negative

numbers. And here's how you can tell.

Anytime you're given a graph, there's

my axis and here's my line.

Think about if you were like skateboarding

and you were coming along, and you

hit this ramp.

This is an uphill ramp.

We call that a positive slope.

As opposed to if you hit this line here,

here you come along skateboarding,

I don't know why that's a skateboarding noise.

I just made that up. You're skateboarding.

You hit that thing, you're going down.

That's why we call it a negative slope.

Anytime a line increases from left

to right it's called a positive.

Decreases from left to right,

it's called a negative.

That's a whole lot to keep in mind when

you guys approach your problems.

This is all mathematical notation that

you're going to see over and over and

over again.

So you'll get used to it.

The one last thing I want to leave you

with is thinking about word problems,

because a lot of times graphs, as we

know, represent the real world.

Like this graph might be growth per day,

or how much something changes over

time.

So slope becomes really important when

you're looking at word problems.

It can be growth per day.

Like let's say my slope was 5. I would

change like five inches per day

or something like that.

It's really important when you're looking

at word problems that you keep in

mind the units that go with your X

and Y values so you can relate the

slope back to the real world.

You guys are going to do all

kinds of practice with this.

Again, slope is a really, really important

idea that's going to show up over

and over, but I think you'll be successful

if you can just remember that

slope means how steep

the line is.