Mathematics
Grade 9
15 min
Identify proportional relationships
Identify proportional relationships
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Introduction & Learning Objectives
Learning Objectives
Identify a direct proportional relationship from a table of values by checking for a constant ratio.
Determine if a graph represents a proportional relationship by verifying it is a straight line passing through the origin.
Calculate the constant of proportionality, k, from a table, graph, or equation.
Write the equation for a direct proportional relationship in the form y = kx.
Distinguish between direct proportional relationships, inverse relationships, and other non-proportional linear relationships.
Solve real-world problems by identifying and using proportional relationships.
Ever wonder how a streaming service knows how much data you'll use for a 2-hour movie versus a 30-minute show? 📱 That's a proportional relationship in action!
This t...
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Key Concepts & Vocabulary
TermDefinitionExample
Proportional RelationshipA relationship between two quantities where their ratio is constant. As one quantity changes, the other changes by the same factor.If 1 ticket costs $5, then 3 tickets cost $15. The ratio of cost to tickets ($5/1 and $15/3) is always 5.
Constant of Proportionality (k)The constant value of the ratio between two proportional quantities. It's the 'k' in the equation y = kx.In the relationship between total cost (y) and number of apples (x) at $0.50 per apple, the constant of proportionality is 0.50, since y/x = 0.50.
Direct VariationA specific type of proportional relationship where one variable is a constant multiple of another. When one variable is zero, the other is also zero.The distance (d) you travel at a constant speed (s)...
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Core Formulas
The Constant Ratio Test (from a table)
k = \frac{y}{x}
For any (x, y) pair in a proportional relationship (where x is not zero), the ratio y/x will always be the same constant, k. Use this to test if a table of values represents a proportional relationship.
The Proportional Relationship Equation
y = kx
This is the standard form for a direct proportional relationship. 'y' varies directly with 'x', and 'k' is the constant of proportionality. Any relationship that can be written in this form is proportional.
The Graphical Test
A straight line through the origin (0,0)
A relationship is proportional if and only if its graph is a straight line that passes through the point (0,0). If it's a curve or if it's a straight line that doesn&...
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Challenging
If y varies directly with x, and y = 15 when x = 6, what is the value of x when y = 25?
A.10
B.62.5
C.2.5
D.16
Challenging
The perimeter P of a regular hexagon is directly proportional to its side length s. The area A of a regular hexagon is given by A = (3√3/2)s². Which statement is correct?
A.Both P and A have a proportional relationship with s.
B.Only P has a proportional relationship with s.
C.Only A has a proportional relationship with s.
D.Neither P nor A has a proportional relationship with s.
Challenging
A student analyzes the table below and claims the relationship is proportional. They state, 'It's proportional with k=2 because for the first point, x/y = 4/8 = 0.5, and for the second, x/y = 6/12 = 0.5.' What is the primary error in the student's reasoning?
A.The student should have added x and y.
B.The student correctly identified the relationship as proportional.
C.The student calculated x/y instead of y/x to find the constant.
D.The student did not use the last data point in the table.
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