Slope

This graph shows the movement of a racer over time, and here's the question we're trying to answer.

How much does the distance increase as the time increases by one second?

Or if we think about the graph, how much does the y axis increase every time the x axis increases by one.

There's a special word for this relationship when we analyze the line of a graph. It's called the slope of the line.

The slope tells us how steep the line is.

To find slope, we need to look at two points on the line. That's because slope compares how much the line rises as it moves across the graph. We want to find precise points by locating where the line crosses an intersection along the graph. That way, we can clearly read both its x and y values.

We already know one extremely valuable point, the origin. Zero zero. That's where the racer starts. At zero seconds and a distance of zero meters.

Now let's look for other points on the line that land exactly on an intersection of the grid. This point right here located at seven along the x axis, and point four along the y axis.

And now, let's interpret what these points tell us. The values along the x axis increase by seven as the y axis increases by point four.

So here's our guiding question again. How much does the distance increase as the time increases by one second? Or how much does the y axis increase as the x axis increases by one? That's really what slope means.

We can think of this just like our earlier bar models. If this bar represents seven seconds and a distance of point four meters, then we can split that bar into seven equal chunks, one for each second.

If we are splitting the bar into seven equal sections, we also need to divide point four into seven equal groups. Point four divided by seven.

I can use a calculator for precision or round it off to the nearest thousandth place. And it rounds to point zero five seven.

That's the slope of the graph.

That number, point zero five seven, tells us how much the line of the graph is going up for every one unit it goes over.

But in more practical terms, it's telling us how much the meters increase for every one second of time. That's just like unit rate.

And that's because on a constant rate line graph, the slope and the unit rate are the same in value, but they describe two different things. The slope tells us what's happening on the graph, and the unit rate tells us what's happening in the real world situation.