Slope: One Key Mistake to Avoid

Show a photo of a ramp or roof. Ask: 'Does this slope go up or down? How would you measure that direction?'

• Slope measures both steepness and direction
• Direction is given by the sign (positive = rising, negative = falling)
• Today we’ll focus on the one step that decides the sign.

Present two points: A(–3, 5) and B(–5, –3). Ask: 'Is the slope positive or negative?' Many students guess wrong because they subtract 5 – (–3) instead of (–3) – 5.

• Common mistake: using y₁ – y₂ instead of y₂ – y₁
• The sign of the numerator reverses if you swap the order
• This error makes a rising line look like it falls.

Drag two points on a coordinate plane. Watch the formula m = (y₂ – y₁)/(x₂ – x₁) update live and see how swapping y₂ and y₁ flips the sign.

• The numerator (y₂ – y₁) controls the sign
• Positive numerator → rising line
• Negative numerator → falling line

Show two calculations side by side: correct: (–3) – 5 = –8 vs. error: 5 – (–3) = +8. Emphasize that the steepness is the same (8), but the direction flips because of the sign.

• The absolute value of the numerator gives steepness
• Only the sign tells direction
• Keep point order consistent: always second y minus first y.

Rule: Slope numerator = y₂ – y₁. Keep the points in the same order for both numerator and denominator. The sign of the numerator alone decides if the line rises or falls.

• Order: (second point's y) – (first point's y)
• Sign of numerator = slope direction
• Steepness = absolute value of numerator / run