NAME THE TOOL

How steep is it?

Coach

Remember the rate, coins per day? Draw the bank on a graph: days go across, coins go up. A fast bank makes a steep line. A slow bank makes a gentle one.

Slope measures how steep. Go across 1 step. How far up do you go? That is the slope. It is the rate you can see.

Quick check: Two staircases. One climbs 3 for each 1 across. The other climbs 1 for each 1 across. Which is steeper?

THE SYMBOLS

Slope = rise ÷ run

slope = rise ÷ run = (y₂ + (−y₁)) ÷ (x₂ + (−x₁))

Rise is the change in y, the up-down change. Run is the change in x, the across change. Both are Δs: slope = Δy ÷ Δx.

Going down from left to right, the rise is negative, so the slope is negative. A flat line has slope 0.

In a rule like y = 2x + 5, the line climbs 2 for each 1 across, so the slope is 2, the number times x.

Quick check: A line goes from (0, 6) to (3, 0). Is the slope positive or negative?

WORKED EXAMPLE

Find the slope from (1, 3) to (3, 11)

Strategy: Rise over run, both as new + (−old)

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

Two points. We want how steep the line is.

Why? Slope is a rate, not a height.

2

Plan

rise = 11 + (−3); run = 3 + (−1). Then divide.

Why? Rise and run are both changes.

3

Solve

rise = 8, run = 2, slope = 8 ÷ 2 = 4.

Why? Rise goes on top.

4

Look back

From (1, 3), 2 across at slope 4 climbs 8, to y = 11. ✓

Why? Walking the line lands on the second point.

Stair way

Each 1 across, the line climbs 4: 3 → 7 → 11.

Symbol way

(11 + (−3)) ÷ (3 + (−1)) = 8 ÷ 2 = 4

Quick check: True or false: the slope here is 8, because the line climbs 8.

SEE IT

The slope triangle

From (2, 1) to (4, 7): run 2, rise 6, slope 3.

Trap: Flipping rise and run

2 ÷ 6 = 1/3 puts the run on top. Slope puts rise on top: 6 ÷ 2 = 3.

Trap: Dividing by an x-value

6 ÷ 4 uses the x of the point. The run is 4 + (−2) = 2.

Trap: Giving the height

The point (4, 7) is 7 high. That is a height, not a slope.

Quick check: On this triangle, which number is the run?

WHAT TO LOOK FOR

Clue words and easy mix-ups

slope, steep, how fast it climbsrise ÷ run.
how high, the value, the pointA height. Not a slope.
goes down, falls, slopes downNegative slope.
y = 2x + 5Slope 2. The + 5 is where the line starts on the y-axis.
coins vs days graphSlope = coins per day, the rate.

Quick check: In y = −2x + 5, what is the slope?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

Slope from (5, 2) to (1, 10). I take 2 + (−10) = −8 on top, and 1 + (−5) = −4 on the bottom. −8 ÷ −4 = 2. Uphill.

Coach

I started with a different point on top and on the bottom. Same point first both times: (2 + (−10)) ÷ (5 + (−1)) = −8 ÷ 4 = −2. The line falls.

Check habit: Pick one point as first. Use it first on top and first on the bottom.

Quick check: From (5, 2) to (1, 10), is the slope 2 or −2?

TRAP CHECK

Run this before every answer

Rise on top, run on the bottom.
Both as new + (−old), same point first.
Uphill positive, downhill negative, flat 0.
Answer the slope, not a height or the rise.