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### Course: 8th grade (Illustrative Mathematics)>Unit 3

Lesson 8: Extra practice: Slope

# Intro to slope

Walk through a graphical explanation of how to find the slope from two points and what it means.
We can draw a line through any two points on the coordinate plane.
Let's take the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis as an example:
The slope of a line describes how steep a line is. Slope is the change in y values divided by the change in x values.
Let's find the slope of the line that goes through the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis:
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, 6, end color #e07d10, divided by, start color #1fab54, 2, end color #1fab54, end fraction, equals, 3
Use the graph below to find the slope of the line that goes through the points left parenthesis, 1, comma, 2, right parenthesis and left parenthesis, 6, comma, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals

Notice that both of the lines we've looked at so far have been increasing and have had positive slopes as a result. Now let's find the slope of a decreasing line.

## Negative slope

Let's find the slope of the line that goes through the points left parenthesis, 2, comma, 7, right parenthesis and left parenthesis, 5, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, minus, 6, end color #e07d10, divided by, start color #1fab54, 3, end color #1fab54, end fraction, equals, minus, 2
Wait a minute! Did you catch that? The change in y values is negative because we went from 7 down to 1. This led to a negative slope, which makes sense because the line is decreasing.
Use the graph below to find the slope of the line that goes through the points left parenthesis, 1, comma, 9, right parenthesis and left parenthesis, 4, comma, 0, right parenthesis.
start text, S, l, o, p, e, end text, equals

## Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in y) divided by the "run" (change in x).
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, start text, R, i, s, e, end text, end color #e07d10, divided by, start color #1fab54, start text, R, u, n, end text, end color #1fab54, end fraction

## Let's practice!

Heads up! All of the examples we've seen so far have been points in the first quadrant, but that won't always be the case in the practice problems.
1) Use the graph below to find the slope of the line that goes through the points left parenthesis, 7, comma, 4, right parenthesis and left parenthesis, 3, comma, 2, right parenthesis.
start text, S, l, o, p, e, end text, equals

2) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 6, comma, 9, right parenthesis and left parenthesis, 2, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals

3) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 8, comma, minus, 3, right parenthesis and left parenthesis, 4, comma, minus, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals

4) Use the graph below to find the slope of the line that goes through the points left parenthesis, 4, comma, 5, right parenthesis and left parenthesis, 9, comma, 5, right parenthesis.
start text, S, l, o, p, e, end text, equals

5) Use the graph below to choose the slope of the line that goes through the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 3, comma, 8, right parenthesis.
start text, S, l, o, p, e, end text, equals

## Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with a slope of 5 is steeper than a line with a slope of start fraction, 1, divided by, 2, end fraction

7) A line with a slope of minus, 5 is steeper than a line with a slope of minus, start fraction, 1, divided by, 2, end fraction

## Want to join the conversation?

• How can the slope value (1/2 or 5) be used in real life, and how can we use it in math?

Thanks!
• It could be used to simulate the steepness of a mountain/hill.
• why does math have to be so confusing?
• If you work hard then eventually math won't be as confusing!
• Can somebody tell me how to easily visualized. which slope is steeper?
• Try to think about it like this, imagine you are running up (positive slope) or down (negative slope) a flight of stairs.

If the slope is a larger number, than it is the same as taking several steps at once.
If the slope is a smaller number, it is as if you are taking less steps at once, not going up the flight of stairs as quickly.

Imagine a slope of 1, means for ever step you take with your feet you only go up one stair.

Imagine a slope of 5, this means for every step you take you go up 5 stairs. You get to the top and rise much quicker.

___________

Another visual example: Imagine you are going skiing. As you go down a slope, you expect the slope to be negative. You come from up high on the y axis and go down.

If now you go to a ski slope at -5 that means for every meter you glide forward on your skis towards to bottom of the hill (x-axis), you also go down 5 meters on the hills height (y-axis). This is very steep as you can imagine.

Now imagine you are going down a ski slope of only 1/2. This means for every 1 meter you glide forward on your ski you only get 1/2 meter further down the hills height.
• My dad left to get milk. He never came back :)
• My mom gets milk every week, and she always comes back! I confuse.
(1 vote)
• lmao what is y=mx+b gonna do for me in life
• Genuine question- when will i EVER use this IRL?
• no but you will for your math class so
• Wait, if it's based on absolute value, how come in the video Slope & direction of a line, it was that the lesser negative was greater?
• In the Slope and Direction video we were determining a slope's direction on the number line, its location, and using values of specific numbers, so we were using the number line, (and smaller negative numbers have a greater value, as they are closer to zero and positive values).

But…
★for Steepness we're no longer comparing direction or location, we're now comparing how vertical a line is to another.

By comparing the absolute values of the slopes we can find which line has a greater change in y than the relative change to x.

Find which line increases or decreases further and faster on the y-axis than another line.

So by finding which is furthest from zero, the one with the greater absolute value, we know which line is steeper and had a more extreme change in y than the other line.

(ㆁωㆁ) Hope this helps someone!
• Is there another way of finding a slope without a graph?
• It should give you points on the graph, use the formula y2-y1/x2-x1, to find the slope, (y2 a y-coordinate, of one of the point, you are not multiplying anything)
Hope this helps :)
• Does anyone know how to open khan academy
• I mean... you're clearly in already
• What does steeper mean, greater slope? In that case, shouldn't a slope of -1/2 be steeper than -5, as -1/2 is greater than -5?