Application: Comparing Rates by Comparing Slopes
Application: Comparing Rates by Comparing Slopes is a Grade 7 math skill in Big Ideas Math Advanced 2, Chapter 4: Graphing and Writing Linear Equations. Students learn to compare multiple proportional relationships on the same coordinate plane by analyzing their slopes — the steeper the line, the larger the slope and the faster the rate of change. This skill is applied to real-world contexts like speed comparisons.
Key Concepts
When multiple proportional relationships are graphed on the same coordinate plane, they can be compared by analyzing their slopes. The relationship with the larger slope (constant of proportionality) has a steeper line and represents a faster rate of change.
Common Questions
How do you compare rates using slopes of graphs?
Graph both proportional relationships on the same coordinate plane. The line with the steeper slope (larger value) represents the faster rate. For example, y = 5x is steeper and faster than y = 2x.
What does a steeper slope mean in a real-world context?
A steeper slope means a greater rate of change. For example, two cars with equations d = 60t and d = 45t: the first car travels faster because its slope (60) is larger.
How do you order proportional relationships from slowest to fastest?
Compare the slopes (coefficients of x). Order them from smallest to largest. For example, y = 0.5x, y = x, and y = 3x go from slowest to fastest.
Do all proportional relationships pass through the origin?
Yes. All proportional relationships have the form y = kx and pass through the origin (0, 0). Their slopes fan out from this common point at different angles.