Identifying Proportional Relationships
Identifying Proportional Relationships is a Grade 7 math skill in Illustrative Mathematics, Chapter 2: Introducing Proportional Relationships. Students learn to determine whether two quantities are in a proportional relationship by checking if their ratios are equivalent or if the graph passes through the origin.
Key Concepts
Property A relationship between two quantities, $x$ and $y$, is proportional if their ratio $\frac{y}{x}$ is constant for all corresponding non zero values. This constant ratio is called the constant of proportionality, $k$. The relationship can be described by the equation: $$y = kx$$.
Examples The cost of buying apples at $3 per apple is a proportional relationship. 2 apples cost $6$: $\frac{6}{2} = 3$ 5 apples cost $15$: $\frac{15}{5} = 3$ The ratio is constant, so the relationship is proportional with the equation $y = 3x$. A car traveling at a constant speed of 50 miles per hour represents a proportional relationship between time ($x$) and distance ($y$). In 2 hours, the car travels 100 miles: $\frac{100}{2} = 50$ In 3.5 hours, the car travels 175 miles: $\frac{175}{3.5} = 50$ The ratio is constant, so the relationship is proportional with the equation $y = 50x$.
Explanation To determine if a relationship is proportional, check if the ratio of the dependent variable ($y$) to the independent variable ($x$) is the same for every pair of values. If this ratio is constant, the relationship is proportional. This constant value, often denoted by $k$, is the constant of proportionality. A key feature of proportional relationships is that when one quantity is zero, the other must also be zero.
Common Questions
What is a proportional relationship?
A proportional relationship is one where two quantities always have the same ratio. If y equals kx for some constant k, then x and y are proportional.
How do you identify a proportional relationship?
Check if the ratio y/x is constant for all value pairs. You can also check if the graph is a straight line passing through the origin (0,0).
What is a non-example of a proportional relationship?
If a taxi charges $2 per mile plus a $3 flat fee, the ratio of total cost to miles is not constant, so this is not proportional.
How do you identify a proportional relationship from a table?
Divide each y value by the corresponding x value. If all quotients are equal, the relationship is proportional.
What chapter covers proportional relationships in Illustrative Mathematics Grade 7?
Identifying proportional relationships is covered in Chapter 2: Introducing Proportional Relationships in Illustrative Mathematics Grade 7.