Two pages of visual notes for students to keep: the key words with graphs, the four kinds of slope, four solved problems in the style of the Linear World Cup game, and where slope shows up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Slope and y = mx + b
Grade 8 · CCSS 8.EE.B.6 · Pairs with the Linear World Cup game
Name
The big idea: a straight line climbs at the same steepness the whole way, like a staircase where every step is the same. That steepness is the slope: how far it goes up (rise) for how far it goes across (run). In Linear World Cup, each pass is a line y = mx + b: it starts at the ball (0, b) and climbs m for every 1 across.
Key vocabulary
Linear equation
An equation whose graph is a straight line, like y = 1.5x + 1.
Rise / run
Rise: the up-or-down change. Run: the left-to-right change between two points.
Slope (m)
How steep a line is: m = rise ÷ run. A roof that rises 3 ft for every 4 ft across has slope 3/4.
y-intercept (b)
Where the line crosses the y-axis: the point (0, b). Here b = 2.
Slope-intercept form
y = mx + b
m is the slope; b is the y-intercept. In y = 2x − 3, m = 2 and b = −3.
Proportional line
A line through the origin (0, 0): y = mx, with b = 0.
The four kinds of slope
PositiveGoes up left to right
NegativeGoes down left to right
ZeroFlat: y = b
UndefinedStraight up and down (x = a number)
How to read slope from a graph: start at one point on the line, count up or down (rise; down is negative), then count right to the next point (run). Slope = rise ÷ run. Pick points where the line crosses grid corners.
✓ Check: from P, up 2 and right 1 lands on the line
2. A line going down
Find the slope through A(0, 4) and B(2, 0).
Rise: 0 − 4 = −4 (down 4)
Run: 2 − 0 = 2 (right 2)
Slope: m = −4 ÷ 2 = −2
✓ Check: it goes down left to right, so the slope is negative
3. Write the pass: y = mx + b
The ball is at (0, −2). The goal is at (3, 4).
b is where the line starts on the y-axis: b = −2
m = rise ÷ run = (4 − (−2)) ÷ 3 = 6 ÷ 3 = 2
The pass: y = 2x − 2
✓ Check: x = 3 gives y = 2(3) − 2 = 4, the goal
4. Real life: a taxi fare
The starting fee is the y-intercept: b = 4
The cost per mile is the slope: m = 1.50
Cost: y = 1.5x + 4. For 10 miles: 1.5(10) + 4 = $19
✓ Check: 0 miles costs $4, the starting fee
Watch out
Slope is rise over run (up-down change on top), not run over rise.
Subtract in the same order on top and bottom: (y2 − y1) ÷ (x2 − x1).
A line going down from left to right has a negative slope.
Where you see it in real life
Stairs: every step has the same rise and run.Roofs: builders call the slope the "pitch."Taxis: a starting fee (b) plus a cost per mile (m).Ramps: a wheelchair ramp's slope can't be too steep.Soccer: a straight pass follows a line.