Ninth Grade · Math · week 4

Slope: the rate in the line

Compute slope as rise over run and interpret slope and intercept in real contexts.

Slope is how much y changes for each single step of x — a rate, wearing graph clothes. Between two points, slope = (y₂ − y₁)/(x₂ − x₁): the rise over the run. Uphill left-to-right is positive, downhill is negative, flat is zero.

In y = mx + b, the m IS the slope and b is the y-intercept — the starting value when x = 0. In real life these have names: b is the flat fee, the starting height, the initial savings; m is the per-hour rate, the growth per week, the burn per mile. Reading m and b out of a situation is the week's superpower.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Find slope in the house: a staircase (rise/run of a step!), a ramp, a graph in the news.
  • Always ask the pair: "what's the START (b)? what's the RATE (m)?" for every scenario this week.
  • Compute one slope from two points, then confirm it against the equation's m — seeing them agree is the click.
  • Watch the subtraction order: pick which point is "point 1" and stay consistent top and bottom.
Talk about it:
  • Why does slope mean the same thing as a rate?
  • What real thing is the y-intercept in the kayak problem?
  • What does a negative slope look like walking left to right?

Mastery looks like: Given a scenario, they can write y = mx + b, and given two points, compute slope with signs intact.

Common stumbles: Flipping rise/run; inconsistent subtraction order between numerator and denominator; calling b the rate.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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