Learn on PengiBig Ideas Math, Course 2Chapter 5: Ratios and Proportions

Lesson 5: Slope

In this Grade 7 lesson from Big Ideas Math, Course 2 (Chapter 5: Ratios and Proportions), students learn how to find the slope of a line by calculating the ratio of vertical change to horizontal change between two points. The lesson also develops students' ability to interpret slope as a rate of change, using real-world contexts like comparing animal speeds graphed over time to connect steepness of a line to the concept of unit rate.

Section 1

Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

m=ΔyΔx=change in y-coordinatechange in x-coordinatem = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}
where Δx\Delta x is positive if we move right and negative if we move left, and Δy\Delta y is positive if we move up and negative if we move down.

Examples

  • A line passes through the points (3,5)(3, 5) and (7,13)(7, 13). Its slope is m=13−57−3=84=2m = \frac{13-5}{7-3} = \frac{8}{4} = 2.
  • For a line containing points (−2,9)(-2, 9) and (4,6)(4, 6), the slope is m=6−94−(−2)=−36=−12m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}.
  • The slope of a line passing through (100,50)(100, 50) and (120,40)(120, 40) is m=40−50120−100=−1020=−0.5m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5.

Section 2

Meaning of slope

Property

The slope of a line measures the rate of change of yy with respect to xx. In different situations, this rate might be interpreted as a rate of growth or a speed.

Examples

  • If a graph of distance (miles) vs. time (hours) has a slope of 5555, it represents an average speed of 55 miles per hour.
  • If a graph of cost (dollars) vs. weight (pounds) has a slope of 33, it means the price is 3 dollars per pound.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Slope of a Line

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

m=ΔyΔx=change in y-coordinatechange in x-coordinatem = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}
where Δx\Delta x is positive if we move right and negative if we move left, and Δy\Delta y is positive if we move up and negative if we move down.

Examples

  • A line passes through the points (3,5)(3, 5) and (7,13)(7, 13). Its slope is m=13−57−3=84=2m = \frac{13-5}{7-3} = \frac{8}{4} = 2.
  • For a line containing points (−2,9)(-2, 9) and (4,6)(4, 6), the slope is m=6−94−(−2)=−36=−12m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}.
  • The slope of a line passing through (100,50)(100, 50) and (120,40)(120, 40) is m=40−50120−100=−1020=−0.5m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5.

Section 2

Meaning of slope

Property

The slope of a line measures the rate of change of yy with respect to xx. In different situations, this rate might be interpreted as a rate of growth or a speed.

Examples

  • If a graph of distance (miles) vs. time (hours) has a slope of 5555, it represents an average speed of 55 miles per hour.
  • If a graph of cost (dollars) vs. weight (pounds) has a slope of 33, it means the price is 3 dollars per pound.