Direct variation
Direct variation in Grade 8 Saxon Math Course 3 describes a proportional relationship between two variables where one is always a constant multiple of the other, expressed as y = kx, where k is the constant of variation. Students identify direct variation from tables, graphs (line through origin), and equations, and calculate the constant of variation. This concept is the foundation for linear functions and proportional reasoning.
Key Concepts
Property When two variables are proportional, one value is found by multiplying the other by a constant factor, $k$. This relationship is defined by the equation: $$y = kx$$.
Examples Your pay depends on hours worked: $Total Pay = 12 \text{ dollars} \times hours$. The perimeter of a square depends on its side length: $P = 4 \times s$.
Explanation Think of it like this: whatever the independent variable (x) does, the dependent variable (y) copies it, but scaled by the special number k. If x doubles, y doubles. If x is cut in half, so is y. Their ratio is always constant and predictable.
Common Questions
What is direct variation in Grade 8 math?
Direct variation is a relationship of the form y = kx, where k is the constant of variation. As x increases, y increases proportionally; as x decreases, y decreases.
How do you verify direct variation from a table?
Calculate y/x for each row. If this ratio is constant throughout the table, the relationship is direct variation, and that constant ratio is k.
What does the graph of a direct variation look like?
The graph of y = kx is a straight line passing through the origin (0, 0). The slope of the line equals the constant of variation k.
What is the constant of variation?
The constant of variation k is the value of y/x for any point on the line. It represents the unit rate, or how much y changes per unit change in x.
How does Saxon Math Course 3 teach direct variation?
Saxon Math Course 3 identifies direct variation from tables and graphs, computes the constant of variation, and uses it to solve problems involving proportional real-world relationships.