HighSchoolQ

➕ Algebra 1 · Linear Functions

Slope of a Line: notes and common mistakes

Study notes

Key ideas

  • Slope = rise ÷ run = vertical change ÷ horizontal change.
  • Read lines left to right: up is positive, down is negative.
  • Horizontal lines have slope 0. Vertical lines have undefined slope.
  • Slope is a rate of change: how much y changes for each 1 unit of x.

Steps

  1. Label your two points as (x1, y1) and (x2, y2).
  2. Subtract the y’s for the rise and the x’s for the run, in the same order.
  3. Divide rise by run and simplify.
  4. Sanity check the sign against the direction of the line.

Formulas

  • Slope formula: m = (y2 − y1) ÷ (x2 − x1) (when x2 ≠ x1)
  • Horizontal line: y = b, slope is 0
  • Vertical line: x = a, slope is undefined

Watch out

  • Keep the order of the points the same on top and bottom.
  • Be careful with negatives: 4 − (−2) = 6.
  • Rise is on top. If you get the reciprocal, you put the run on top.

Why it works

  • Why is the slope the same between any two points on a line? A line is straight, so every rise-and-run triangle is a scaled copy of the others (they are similar triangles).
  • Why is vertical slope undefined? The run is 0, and dividing by 0 has no answer.

Common mistakes

✗ MistakeSlope = run ÷ rise.
✓ InsteadSlope = rise ÷ run. Remember I before U: rise on top.
✗ MistakeA horizontal line has undefined slope.
✓ InsteadHorizontal lines have slope 0. Vertical lines are the undefined ones (HOY VUX).
✗ Mistake(y2 − y1) ÷ (x1 − x2) is fine.
✓ InsteadUse the same order on top and bottom, or the sign flips.

Teach it back

Explain to a friend how to find the slope between (2, 1) and (6, 9) and what the answer means.

Explaining it out loud shows what you really know. Did you…

  • I found the rise (9 − 1 = 8) and the run (6 − 2 = 4).
  • I divided to get slope 2.
  • I explained it as: up 2 for every 1 step to the right.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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